TWIRPX.COM
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Section 1.1. The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Section 1.2. Models and derivatives markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Section 1.3. Using derivatives the right way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Section 1.4. Nineteen steps to using derivatives the right way . . . . . . . . . . . . . . . . . . . . 19
Literature Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 1.1. Payoff of derivative which pays the 10m times the excess of the square of the
decimal interest rate over 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 1.2. Hedging with forward contract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Panel A. Income to unhedged exporter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Panel B. Forward contract payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Panel C. Hedged firm income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 1.3. Payoff of share and call option strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 1.3.A. Payoff of buying one share of Amazon.com at $75 . . . . . . . . . . . 26
Figure 1.3.B. Payoff of buying a call option on one share of Amazon.com with
exercise price of $75 for a premium of $10. . . . . . . . . . . . . . . . . . . . . . . 26
Chapter 1: Introduction
August 5, 1999
© René M. Stulz 1997, 1999
Throughout history, the weather has determined the fate of nations, businesses, and
individuals. Nations have gone to war to take over lands with a better climate. Individuals have
starved because their crops were made worthless by poor weather. Businesses faltered because the
goods they produced were not in demand as a result of unexpected weather developments. Avoiding
losses due to inclement weather was the dream of poets and the stuff of science fiction novels - until
it became the work of financial engineers, the individuals who devise new financial instruments and
strategies to enable firms and individuals to better pursue their financial goals. Over the last few years,
financial products that can be used by individuals and firms to protect themselves against the financial
consequences of inclement weather have been developed and marketed. While there will always be
sunny and rainy days, businesses and individuals can now protect themselves against the financial
consequences of unexpectedly bad weather through the use of financial instruments. The introduction
of financial instruments that help firms and individuals to deal with weather risks is just one example
of the incredible growth in the availability of financial instruments for managing risks. Never in the
course of history have firms and individuals been able to mitigate the financial impact of risks as
effectively through the use of financial instruments as they can now.
There used to be stocks and bonds and not much else. Knowing about stocks and bonds was
enough to master the intricacies of financial markets and to choose how to invest one’s wealth. A
manager had to know about the stock market and the bond market to address the problems of his
firm. Over the last thirty years, the financial instruments available to managers have become too
numerous to count. Not only can managers now protect their firms against the financial consequences
of bad weather, there is hardly a risk that they cannot protect their firm against if they are willing to
Chapter 2, page 1
pay the appropriate price or a gamble that they cannot take through financial instruments. Knowing
stocks and bonds is therefore not as useful as it used to be. Attempting to know all existing financial
instruments is no longer feasible. Rather than knowing something about a large number of financial
instruments, it has become critical for managers to have tools that enable them to evaluate which
financial instruments - existing or to be invented - best suit their objectives. As a result of this
evolution, managers and investors are becoming financial engineers.
Beyond stocks and bonds, there is now a vast universe of financial instruments called
derivatives. In chemistry, a derivative is a compound “derived from another and containing essential
elements of the parent substance.”1 Derivatives in finance work on the same principle as in chemistry.
They are financial instruments whose payoffs are derived from something else, often but not
necessarily another financial instrument. It used to be easier to define the world of derivatives. Firms
would finance themselves by issuing debt and equity. Derivatives would then be financial instruments
whose payoffs would be derived from debt and equity. For instance, a call option on a firm’s stock
gives its owner the right to buy the stock at a given price, the exercise price. The call option payoff
is therefore derived from the firm’s stock. Unfortunately, defining the world of derivatives is no
longer as simple. Non-financial firms now sell derivatives to finance their activities. There are also
derivatives whose value is not derived from the value of financial instruments directly. Consider a
financial instrument of the type discussed in chapter 18 that promises its holder a payment equal to
ten million dollars times the excess of the square of the decimal interest rate over 0.01 in 90 days
where the interest rate is the London Interbank Offer Rate (LIBOR) on that day as reported by the
1
The American Heritage Dictionary of the English Language, Third Edition (1982),
Houghton Mifflin Company, Boston.
Chapter 2, page 2
British Bankers Association. Figure 1.1. plots the payoff of that financial instrument. If the interest
rate is 10%, this instrument therefore pays nothing, but if the interest rate is 20%, this instrument pays
$300,000 (i.e., 0.20 squared minus 0.01 times ten million). Such an instrument does not have a value
that depends directly on a primitive asset such as a stock or a bond.
Given the expansion of the derivatives markets, it is hard to come up with a concise definition
of derivatives that is more precise than the one given in the previous paragraph. A derivative is a
financial instrument with contractually specified payoffs whose value is uncertain when the contract
is initiated and which depend explicitly on verifiable prices and/or quantities. A stock option is a
derivative because the payoff is explicitly specified as the right to receive the stock in exchange of the
exercise price. With this definition of a derivative, the explicit dependence of payoffs on prices or
quantities is key. It distinguishes derivatives from common stock. The payoffs of a common stock are
the dividend payments. Dividends depend on all sorts of things, but this dependence is not explicit.
There is no formula for a common stock that specifies the size of the dividend at one point in time.
The formula cannot depend on subjective quantities or forecasts of prices: The payoff of a derivative
has to be such that it can be determined mechanically by anybody who has a copy of the contract.
Hence, a third party has to be able to verify that the prices and/or quantities used to compute the
payoffs are correct. For a financial instrument to be a derivative, its payoffs have to be determined
in such a way that all parties to the contract could agree to have them defined by a mathematical
equation that could be enforced in the courts because its arguments are observable and verifiable.
With a stock option, the payoff for the option holder is receiving the stock in exchange of paying the
exercise price. The financial instrument that pays $300,000 if the interest rate is 20% is a derivative
by this definition since the contractually specified payoff in ninety days is given by a formula that
Chapter 2, page 3
depends on a rate that is observable. A judge could determine the value of the contract’s payoff by
determining the appropriate interest rate and computing the payoff according to the formula. A share
of IBM is not a derivative and neither is a plain vanilla bond issued by IBM. With this definition, the
variables that determine the payoffs of a derivative can be anything that the contracting parties find
useful including stock prices, bond prices, interest rates, number of houses destroyed by hurricanes,
gold prices, egg prices, exchange rates, and the number of individual bankruptcies within a calendar
year in the U.S.
Our definition of a derivative can be used for a weather derivative. Consider a financial
contract that specifies that the purchaser will receive a payment of one hundred million dollars if the
temperature at La Guardia at noon exceeds 80 degrees Fahrenheit one hundred days or more during
a calendar year. If one thinks of derivatives only as financial instruments whose value is derived from
financial assets, such a contract would not be called a derivative. Such a contract is a derivative with
our definition because it specifies a payoff, one hundred million dollars, that is a function of an
observable variable, the number of days the temperature exceeds 80 degrees at La Guardia at noon
during a calendar year. It is easy to see how such a weather derivative would be used by a firm for
risk management. Consider a firm whose business falls dramatically at high temperatures. Such a firm
could hedge itself against weather that is too hot by purchasing such a derivative. Perhaps the
counterparty to the firm would be an ice cream manufacturer whose business suffers when
temperatures are low. But it might also be a speculator who wants to make a bet on the weather.
At this point, there are too many derivatives for them to be counted and it is beyond
anybody’s stamina to know them all. In the good old days of derivatives - late 1970s and early 1980s
- a responsible corporate manager involved in financial matters could reasonably have detailed
Chapter 2, page 4
knowledge of all economically relevant derivatives. This is no longer possible. At this point, the key
to success is being able to figure out which derivatives are appropriate and how to use them given
one’s objectives rather than knowing a few of the many derivatives available. Recent history shows
that this is not a trivial task. Many firms and individuals have faced serious problems using derivatives
because they were not well equipped to evaluate their risks and uses. Managers that engaged in
poorly thought out derivatives transactions have lost their jobs, but not engaging in derivatives
transactions is not an acceptable solution. While derivatives used to be the province of finance
specialists, they are now intrinsic to the success of many businesses and businesses that do not use
them could generally increase their shareholder’s wealth by using them. A finance executive who
refuses to use derivatives because of these difficulties is like a surgeon who does not use a new
lifesaving instrument because some other surgeon made a mistake using it.
The remainder of this chapter is organized as follows. We discuss next some basic ideas
concerning derivatives and risk management. After explaining the role of models in the analysis of
derivatives and risk management, we discuss the steps one has to take to use derivatives correctly.
We then turn to an overview of the book.
Section 1.1. The basics.
Forward contracts and options are often called plain vanilla derivatives because they are
the simplest derivatives. A forward contract is a contract where no money changes hands when the
contract is entered into but the buyer promises to purchase an asset or a commodity at a future date,
the maturity date, at a price fixed at origination of the contract, the forward price, and where the
seller promises to deliver this asset or commodity at maturity in exchange of the agreed upon price.
Chapter 2, page 5
An option gives its holder a right to buy an asset or a commodity if it is a call option or to sell an
asset or a commodity if it is a put option at a price agreed upon when the option contract is written,
the exercise price. Predating stocks and bonds, forward contracts and options have been around a
long time. We will talk about them throughout the book. Let’s look at them briefly to get a sense of
how derivatives work and of how powerful they can be.
Consider an exporter who sells in Europe. She will receive one million euros in ninety days.
The dollar value of the payment in ninety days will be one million times the dollar price of the euro.
As the euro becomes more valuable, the exporter receives more dollars. The exporter is long in the
euro, meaning that she benefits from an increase in the price of the euro. Whenever cash flow or
wealth depend on a variable - price or quantity - that can change unexpectedly for reasons not under
our control, we call such a variable a risk factor. Here, the dollar price of a euro is a risk factor. In
risk management, it is always critical to know what the risk factors are and how their changes affect
us. The sensitivity of cash flow or wealth to a risk factor is called the exposure to that risk factor.
The change in cash flow resulting from a change in a risk factor is equal to the exposure times the
change in the risk factor. Here, the risk factor is the dollar price of the euro and the cash flow impact
of a change in the dollar price of the euro is one million times the change in the price of the euro. The
exposure to the euro is therefore one million euros. We will see that measuring exposure is often
difficult. Here, however, it is not.
In ninety days, the exporter will want to convert the euros into dollars to pay her suppliers.
Let’s assume that the suppliers are due to receive $950,000. As long as the price of the euro in ninety
days is at least 95 cents, everything will be fine. The exporter will get at least $950,000 and therefore
can pay the suppliers. However, if the price of the euro is 90 cents, the exporter receives $900,000
Chapter 2, page 6
dollars and is short $50,000 to pay her suppliers. In the absence of capital, she will not be able to pay
the suppliers and will have to default, perhaps ending the business altogether. A forward contract
offers a solution for the exporter that eliminates the risk of default. By entering a forward contract
with a maturity of 90 days for one million euros, the exporter promises to deliver one million euros
to the counterparty who will in exchange pay the forward price per euro times one million. For
instance, if the forward price per euro is 99 cents, the exporter will receive $990,000 in ninety days
irrespective of the price of the euro in ninety days. With the forward contract, the exporter makes
sure that she will be able to pay her suppliers.
Panel A of figure 1.2. shows the payoff of the position of the exporter in the cash market, i.e.,
the dollars the exporter gets for the euros it sells on the cash market if she decides to use the cash
market to get dollars in 90 days. Panel B of figure 1.2. shows the payoff of a short position in the
forward contract. A short position in a forward contract benefits from a fall in the price of the
underlying. The underlying in a forward contract is the commodity or asset one exchanges at
maturity of the forward contract. In our example, the underlying is the euro. The payoff of a short
position is the receipt of the forward price per unit times the number of units sold minus the value of
the price of the underlying at maturity times the number of units delivered. By selling euros through
a forward contract, our exporter makes a bigger profit from the forward contract if the dollar price
of the euro falls more. If the euro is at $1.1 at maturity, our exporter agreed to deliver euros worth
$1.1 per unit at the price of $0.99, so that she loses $0.11 per unit or $110,000 on the contract. In
contrast, if the euro is at $0.90 at maturity, the exporter gets $0.99 per unit for something worth $0.9
per unit, thereby gaining $90,000 on the forward contract. With the forward contract, the long -, i.e.,
the individual who benefits from an increase in the price of the underlying - receives the underlying
Chapter 2, page 7
at maturity and pays the forward price. His profit is therefore the cash value of the underlying at
maturity minus the forward price times the size of the forward position. The third panel of Figure
1.1., panel C, shows how the payoff from the forward contract added to the payoff of the cash
position of the exporter creates a risk-free position. A financial hedge is a financial position that
decreases the risk resulting from the exposure to a risk factor. Here, the cash position is one million
euros, the hedge is the forward contract. In this example, the hedge is perfect - it eliminates all the
risk so that the hedged position, defined as the cash position plus the hedge, has no exposure to the
risk factor.
Our example has three important lessons. First, through a financial transaction, our exporter
can eliminate all her risk without spending any cash to do so. This makes forward contracts
spectacularly useful. Unfortunately, life is more complicated. Finding the best hedge is often difficult
and often the best hedge is not a perfect hedge.
Second, to eliminate the risk of the hedged position, one has to be willing to make losses on
derivatives positions. Our exporter takes a forward position such that her hedged cash flow has no
uncertainty - it is fixed. When the euro turns out to be worth more than the forward price, the
forward contract makes a loss. This is the case when the price of the euro is $1.1. This loss exactly
offsets the gain made on the cash market. It therefore makes no sense whatsoever to consider
separately the gains and losses of derivatives positions from the rest of the firm when firms use
derivatives to hedge. What matters are the total gain and loss of the firm.
Third, when the exporter enters the forward contract, she agrees to sell euros at the forward
price. The counterparty in the forward contract therefore agrees to buy euros at the forward price.
No money changes hands except for the agreed upon exchange of euros for dollars. Since the
Chapter 2, page 8
counterparty gets the mirror image of what the exporter gets, if the forward contract has value at
inception for the exporter in that she could sell the forward contract and make money, it has to have
negative value for the counterparty. In this case, the counterparty would be better off not to enter the
contract. Consequently, for the forward contract to exist, it has to be that it has no value when
entered into. The forward price must therefore be the price that insures that the forward contract has
no value at inception.
Like the forward contract, many derivatives have no value at inception. As a result, a firm can
often enter a derivatives contract without leaving any traces in its accounting statements because no
cash is used and nothing of value is acquired. To deal with derivatives, a firm has to supplement its
conventional accounting practices with an accounting system that takes into account the risks of
derivatives contracts. We will see how this can be done.
Let’s now consider options. The best-known options are on common stock. Consider a call
option on Amazon.com. Suppose the current stock price is $75 and the price of a call option with
exercise price of $75 is $10. Such an option gives the right to its holder to buy a fixed number of
shares of Amazon.com stock at $75. Options differ as to when the right can be exercised. With
European options, the right can only be exercised at maturity. In contrast, American options are
options that can be exercised at maturity and before. Consider now an investor who believes that
Amazon.com is undervalued by the market. This individual could buy Amazon.com shares and could
even borrow to buy such shares. However, this individual might be concerned that there is some
chance he will turn out to be wrong and that something bad might happen to Amazon.com. In this
case, our investor would lose part of his capital. If the investor wants to limit how much of his capital
he can lose, he can buy a call option on Amazon.com stock instead of buying Amazon.com shares.
Chapter 2, page 9
In this case, the biggest loss our investor would make would be to lose the premium he paid to
acquire the call option. Figure 1.3. compares the payoff for our investor of holding Amazon.com
shares and of buying a call option instead at maturity of the option assuming it is a European call
option. If the share price falls to $20, the investor who bought the shares loses $55 per share bought,
but the investor who bought the call option loses only the premium paid for the call of $10 per share
since he is smart enough not to exercise a call option that requires him to pay $75 for shares worth
$20. If the share price increases to $105, the investor who bought the shares gains $30 per share, but
the investor who bought options gains only $20 since he gains $30 per share when he exercises the
call option but had paid a premium of $10 per share. Our investor could have used a different
strategy. He could have bought Amazon.com shares and protected himself against losses through the
purchase of put options. A put option on a stock gives the right to sell shares at a fixed price. Again,
a put option can be a European or an American option.
With our example of a call option on Amazon.com, the investor has to have cash of $75 per
share bought. He might borrow some of that cash, but then his ability to invest depends on his credit.
To buy a call, the investor has to have cash of $10. Irrespective of which strategy the investor uses,
he gets one dollar for each dollar that Amazon.com increases above $75. If the share price falls below
$75, the option holder loses all of the premium paid but the investor in shares loses less as long as the
stock price does not fall by $10 or more. Consider an investor who is limited in his ability to raise
cash and can only invest $10. This investor can get the same gain per dollar increase in the stock price
as an investor who buys a share if he buys the call. If this investor uses the $10 to buy a fraction of
a share, he gets only $0.13 per dollar increase in the share price. To get a gain of one dollar from a
one dollar increase in the share price, our investor with $10 would have to borrow $65 to buy one
Chapter 2, page 10
share. In other words, he would have to borrow $6.5 for each dollar of capital. Option strategies
therefore enable the investor to lever up his resources without borrowing explicitly. The same is true
for many derivatives strategies. This implicit leverage can make the payoff of derivatives strategies
extremely volatile. The option strategy here is more complicated than a strategy of borrowing $65
to buy one share. This is because the downside risk is different between the borrowing strategy and
the option strategy. If the stock price falls to $20, the loss from the call strategy is $10 but the loss
from the borrowing strategy is $55. The option payoff is nonlinear. The gain for a one dollar increase
in the share price from $75 is not equal to minus the loss for a one dollar decrease in the share price
from $75. This nonlinearity is typical of derivatives. It complicates the analysis of the pricing of these
financial instruments as well as of their risk.
Call and put options give their holder a right. Anybody who has the right but not the
obligation to do something will choose to exercise the right to make himself better off. Consequently,
a call option is never exercised if the stock price is below the exercise price and a put option is never
exercised if the stock price is above the exercise price. Whoever sells an option at initiation of the
contract is called the option writer. The call option writer promises to deliver shares for the exercise
price and the put option writer promises to receive shares in exchange of the exercise price. When
an option is exercised, the option writer must always deliver something that is worthwhile. For the
option writer to be willing to deliver something worthwhile upon exercise, she must receive cash
when she agrees to enter the option contract. The problem is then to figure out how much the option
writer should receive to enter the contract.
Chapter 2, page 11
Section 1.2. Models and derivatives markets.
To figure out the price of an option, one has to have a model. To figure out whether it is
worthwhile to buy an option to hedge a risk, one has to be able to evaluate whether the economic
benefits from hedging the risk outweigh the cost from purchasing the option. This requires a model
that allows us to quantify the benefits of hedging. Models therefore play a crucial role in derivatives
and risk management. Models are simplified representations of reality that attempt to capture what
is essential. One way to think of models is that they are machines that allow us to see the forest rather
than only trees. No model is ever completely right because every model always abstracts from some
aspects of the real world. Since there is no way for anybody to take into account all the details of the
real world, models are always required to guide our thinking. It is easy to make two mistakes with
models. The first mistake is to think that a model is unrealistic because it misses some aspect of the
real world. Models do so by necessity. The key issue is not whether models miss things, but rather
whether they take enough things into account that they are useful. The second mistake is to believe
that if we have a model, we know the truth. This is never so. With a good model, one knows more
than with a bad model. Good models are therefore essential. Things can still go wrong with good
models because no model is perfect.
Stock options were traded in the last century and much of this century without a satisfactory
model that allowed investors and traders to figure out a price for these options. Markets do not have
to have a model to price something. To obtain a price, an equilibrium for a product where demand
equals supply is all that is required. Operating without a model is like flying a plane without
instruments. The plane can fly, but one may not get where one wants to go. With options, without
a model, one cannot quantify anything. One can neither evaluate a market price nor quantify the risk
Chapter 2, page 12
of a position. Lack of a model to price options was therefore a tremendous impediment to the growth
of the option market. The lack of a model was not the result of a lack of trying. Even Nobel
prizewinners had tried their hand at the problem. People had come up with models, but they were just
not very useful because to use them, one had to figure out things that were not observable. This lasted
until the early 1970s. At that time, two financial economists in Boston developed a formula that
revolutionized the field of options and changed markets for derivatives forever. One, Fischer Black,
was a consultant. The other one, Myron Scholes, was an assistant professor at MIT who had just
earned a Ph.D. in finance from the University of Chicago. These men realized that there was a trading
strategy that would yield the same payoff as an option but did not use options. By investing in stocks
and bonds, one could obtain the same outcome as if one had invested in options. With this insight and
the help of a third academic, Robert Merton, they derived a formula that was instantly famous except with the editors of academic journals who, amazingly, did not feel initially that it was
sufficiently useful to be publishable. This formula is now called the Black-Scholes formula for the
pricing of options. With this formula, one could compute option prices using only observable
quantities. This formula made it possible to assess the risk of options as well as the value of portfolios
of options.
There are few achievements in social sciences that rival the Black-Scholes formula. This
formula is tremendously elegant and represents a mathematical tour-de-force. At the same time, and
more importantly, it is so useful that it has spawned a huge industry. Shortly after the option pricing
formula was discovered, the Chicago Board of Trade started an options exchange. Business on this
exchange grew quickly because of the option pricing formula. Traders on the exchange would have
calculators with the formula programmed in them to conduct business. When Fischer Black or Myron
Chapter 2, page 13
Scholes would show up at the exchange, they would receive standing ovations because everybody
knew that without the Black-Scholes formula, business would not be what it was.
The world is risky. As a result, there are many opportunities for trades to take place where
one party shifts risks to another party through derivatives. These trades must be mutually beneficial
or otherwise they would not take place. The purchaser of a call option wants to benefit from stock
price increases but avoid losses. He therefore pays the option writer to provide a hedge against
potential losses. The option writer does so for appropriate compensation. Through derivatives,
individuals and firms can trade risks and benefit from these trades. Early in the 1970s, this trading of
risks took place through stock options and forward transactions. However, this changed quickly. It
was discovered that the Black-Scholes formula was useful not only to price stock options, but to
price any kind of financial contract that promises a payoff that depends on a price or a quantity.
Having mastered the Black-Scholes formula, one could price options on anything and everything.
This meant that one could invent new instruments and find their value. One could price exotic
derivatives that had little resemblance to traditional options. Exotic derivatives are all the derivatives
that are not plain vanilla derivatives or cannot be created as a portfolio of plain vanilla derivatives.
The intellectual achievements involved in the pricing of derivatives made possible a huge industry.
Thirty years ago, the derivatives industry had no economic importance. We could produce countless
statistics on its current importance. Measuring the size of the derivatives industry is a difficult
undertaking. However, the best indicator of the growth and economic relevance of this industry is
that observers debate whether the derivatives markets are bigger and more influential than the
markets for stocks and bonds and often conclude that they are.
Because of the discovery of the Black-Scholes formula, we are now in a situation where any
Chapter 2, page 14
type of financial payoff can be obtained at a price. If a corporation would be better off receiving a
large payment in the unlikely event that Citibank, Chase, and Morgan default in the same month, it
can go to an investment bank and arrange to enter the appropriate derivatives contract. If another
corporation wants to receive a payment which is a function of the square of the yen/dollar exchange
rate if the volatility of the S&P500 exceeds 35% during a month, it can do so. There are no limits to
the type of financial contracts that can be written. However, anybody remembers what happened in
their youth when suddenly their parents were not watching over their shoulders. Without limits, one
can do good things and one can do bad things. One can create worthwhile businesses and one can
destroy worthwhile businesses. It is therefore of crucial importance to know how to use derivatives
the right way.
Section 1.3. Using derivatives the right way.
A corporate finance specialist will see new opportunities to take positions in derivatives all
the time. He will easily think of himself as a master of the universe, knowing which instruments are
too cheap, which are too expensive. As it is easy and cheap to take positions in derivatives, this
specialist can make dramatic changes in the firm’s positions in instants. With no models to measure
risks, he can quickly take positions that can destroy the firm if things go wrong. Not surprisingly,
therefore, some firms have made large losses on derivatives and some firms have even disappeared
because of derivatives positions that developed large losses.
The first thing to remember, therefore, is that there are few masters of the universe. For every
corporate finance specialist who thinks that a currency is overvalued, there is another one who thinks
with the same amount of conviction that currency is undervalued. It may well be that a corporate
Chapter 2, page 15
finance specialist is unusually good at forecasting exchange rates, but typically, that will not be the
case. To beat the market, one has to be better than the investors who have the best information - one
has to be as good as George Soros at the top of his game. This immediately disqualifies most of us.
If mutual fund managers whose job it is to beat the market do not do so on average, why could a
corporate finance specialist or an individual investor think that they have a good enough chance of
doing so that this should direct their choice of derivatives positions? Sometimes, we know something
that has value and should trade on it. More often, though, we do not.
A firm or an individual that take no risks hold T-bills and earn the T-bill rate. Rockfeller had
it right when he said that one cannot get rich by saving. If one is to become rich, one has to take risks
to exploit valuable opportunities. Valuable opportunities are those where we have a comparative
advantage in that they are not as valuable to others. Unfortunately, the ability to bear risk for
individuals or firms is limited by lack of capital. An individual who has lost all his wealth cannot go
back to the roulette table. A firm that is almost bankrupt cannot generally take advantage of the same
opportunities as a firm that is healthy. This forces individuals and firms to avoid risks that are not
profitable so that they can take on more risks that are advantageous. Without derivatives, this is often
impossible. Derivatives enable individuals and firms to shed risks and take on risks cheaply.
To shed risks that are not profitable and take on the ones that are profitable, it is crucial to
understand the risks one is exposed to and to evaluate their costs and benefits. Risks cannot be
analyzed without statistics. One has to be able to quantify risks so that one can understand their costs
and so that one can figure out whether transactions decrease or increase risk and by how much. When
one deals with risks, it is easy to let one’s biases take charge of ones decisions. Individuals are just
not very good at thinking about risks without quantitative tools. They will overstate the importance
Chapter 2, page 16
of some risks and understate the importance of others. For instance, individuals put too much weight
on recent past experience. If a stock has done well in the recent past, they will think that it will do
unusually well in the future so that it has little risk. Yet, a quick look at the data will show them that
this is not so. They will also be reluctant to realize losses even though quantitative analysis will show
that it would be in their best interest to do so. Psychologists have found many tendencies that people
have in dealing with risk that lead to behavior that cannot be justified on quantitative grounds. These
tendencies have even led to a new branch of finance called behavioral finance. This branch of finance
attempts to identify how the biases of individuals influence their portfolio decisions and asset returns.
To figure out which risks to bear and which risks to shed, one therefore must have models
that allow us to figure out the economic value of taking risks and shedding risks. Hence, to use
derivatives in the right way, one has to be able to make simple statements like the following: If I keep
my exposure to weather risk, the value of my firm is X; if I shed my exposure to weather risk, the
value of my firm after purchasing the appropriate financial instruments is Y; if Y is greater than X,
I shed the weather risk. For individuals, it has to be that the measure of their welfare they focus on
is affected by a risk and they can establish whether shedding the risk makes them better off than
bearing it. To figure out the economic value of taking risks and shedding risks, one has to be able to
quantify risks. This requires statistics. One has to be able to trace out the impact of risks on firm value
or individual welfare. This requires economic analysis. Finally, one must know how a derivative
position will affect the risks the firm is exposed to. This requires understanding the derivatives and
their pricing. A derivative could eliminate all of a risk, but it may be priced so that one is worse off
without the risk than with. A derivatives salesperson could argue that a derivative is the right one to
eliminate a risk we are concerned about, but a more thorough analysis might reveal that the derivative
Chapter 2, page 17
actually increases our exposure to other risks so that we would be worse off purchasing it.
To use derivatives the right way, one has to define an objective function. For a firm, the
objective function is generally to maximize shareholder wealth. For an investor, there will be some
measure of welfare that she focuses on. Objective functions are of little use unless we can measure
the impact of choices on our objectives. We therefore have to be able to quantify how various risks
affect our objective function. Doing so, we will find some risks that make us worse off and, possibly,
others that make us better off. Having figured out which risks are costly, we need to investigate
whether there are derivatives strategies that can be used to improve our situation. This requires us
to be able to figure out the impact of these strategies on our objective function. The world is not
static. Our exposures to risks change all the time. Consequently, derivatives positions that were
appropriate yesterday may not be today. This means that we have to be able to monitor these
positions and monitor our risk exposures to be able to make changes when it is appropriate to do so.
This means that we must have systems in place that make it possible to monitor our risk exposures.
Using derivatives the right way means that we look ahead and figure out which risks we
should bear and how. Once we have decided which risks we should bear, nature has to run its course.
In our example of the exporter to Europe, after she entered the forward contract, the euro ended up
either above or below the forward price of $0.99. If it ended up above, the exporter actually lost
money on the contract. The temptation would be to say that she made a poor use of derivatives since
she lost money on a derivative. This is simply not the way to think about derivatives use. When the
decision was made to use the derivative, the exporter figured out that she was better off hedging the
currency risk. She had no information that allowed her to figure out that the price of the euro was
going to appreciate and hence could not act on such information. At that time, it was as likely that
Chapter 2, page 18
the exchange rate would fall and that she would have gained from her forward position. If a derivative
is bought to insure against losses, it is reasonable to think that about half the time, the losses one
insures against will not take place and the derivative will therefore not produce a gain to offset losses.
The outcome of a derivatives transactions does not tell us whether we were right or wrong in entering
the transaction any more than whether our house burns down or not tells us whether we were right
or wrong to buy fire insurance. Houses almost never burn down, so that we almost always make a
loss on fire insurance. We buy the insurance because we know ex ante that we are better off shedding
the financial risk of having to replace the house.
Section 1.4. Nineteen steps to using derivatives the right way.
This book has nineteen chapters. Each chapter will help you to understand better how to use
derivatives the right way. Because of the biases in decision making in the absence of quantitative
evaluations, risk has to be evaluated using statistical tools that are not subject to the hidden biases
of the human mind. We therefore have to understand how to measure risk. In chapters 2 through 4,
we investigate how to measure risk and how risk affects firm value and the welfare of individuals. A
crucial issue in risk measurement is that lower tail risks - risks that things can go wrong in a serious
way - affect firm value and individual welfare in ways that are quite different from other risks. Small
cash flow fluctuations around their mean generally have little impact on firm value. Extreme outcomes
can mean default and bankruptcy. It is therefore essential to have quantitative measures of these lower
tail risks. We therefore introduce such measures in chapter 4. These measures enable us to assess the
impact of derivatives strategies on risk and firm value. As we consider different types of derivatives
throughout the book, we will have to make sure that we are able to use our risk measures to evaluate
Chapter 2, page 19
these derivatives.
After chapter 4, we will have the main framework of risk management and derivatives use in
place. We will know how to quantify risks, how to evaluate their costs and benefits, and how to make
decisions when risk matters. This framework is then used throughout the rest of the book to guide
us in figuring out how to use derivatives to manage risk. As we progress, however, we learn about
derivatives uses but also learn more about how to quantify risk. We start by considering the uses and
pricing of plain vanilla derivatives. In chapter 5, we therefore discuss the pricing of forward contracts
and of futures contracts. Futures contracts are similar to forward contracts but are traded on
organized exchanges. Chapters 6 through 9 discuss extensively how to use forward and futures
contracts to manage risk. We show how to set up hedges with forwards and futures. Chapter 8
addresses many of the issues that arise in estimating foreign exchange rate exposures and hedging
them. Chapter 9 focuses on interest rate risks.
After having seen how to use forwards and futures, we turn our attention on options. Chapter
10 shows why options play an essential role in risk management. We analyze the pricing of options
in chapters 11 and 12. Chapter 12 is completely devoted to the Black-Scholes formula. Unfortunately,
options complicate risk measurement. The world is full of options, so that one cannot pretend that
they do not exist to avoid the risk measurement problem. Chapter 13 therefore extends our risk
measurement apparatus to handle options and more complex derivatives. Chapter 14 covers fixed
income options. After chapter 14, we will have studied plain vanilla derivatives extensively and will
know how to use them in risk management. We then move beyond plain vanilla derivatives. In chapter
15, we address the tradeoffs that arise when using derivatives that are not plain vanilla derivatives.
We then turn to swaps in chapter 16. Swaps are exchanges of cash flows: one party pays well-defined
Chapter 2, page 20
cash flows to the other party in exchange for receiving well-defined cash flows from that party. The
simplest swap is one where one party promises to pay cash flows corresponding to the interest
payments of fixed rate debt on a given amount to a party that promises to pay cash flows
corresponding to the payments of floating rate debt on the same amount. We will see that there are
lots of different types of swaps. In chapter 17, we discuss the pricing and uses of exotic options.
Credit risks are important by themselves because they are a critical source of risk for firms. At the
same time, one of the most recent growth areas in derivative markets involves credit derivatives,
namely derivatives that can be used to lay off credit risks. In chapter 18, we analyze credit risks and
show how they can be eliminated through the uses of credit derivatives.
After developing the Black-Scholes formula, eventually Fischer Black, Robert Merton, and
Myron Scholes all played major roles in the business world. Fischer Black became a partner at
Goldman Sachs, dying prematurely in 1996. Robert Merton and Myron Scholes became partners in
a high-flying hedge fund company named Long-term Capital Management that made extensive use
of derivatives. In 1997, Robert Merton and Myron Scholes received the Nobel Memorial Prize in
Economics for their contribution to the pricing of options. In their addresses accepting the prize, both
scholars focused on how derivatives can enable individuals and firms to manage risks. In September
1998, the Federal Reserve Bank of New York arranged for a group of major banks to lend billions
of dollars to that hedge fund. The Fed intervened because the fund had lost more than four billion
dollars of its capital, so it now had less than half a billion dollars to support a balance sheet of more
than $100 billion. Additional losses would have forced the long-term capital fund to unwind positions
in a hurry, leading regulators to worry that this would endanger the financial system of the Western
world. The press was full of articles about how the best and the brightest had failed. This led to much
Chapter 2, page 21
chest-beating about our ability to manage risk. Some thought that if Nobel prizewinners could not
get it right, there was little hope for the rest of us. James Galbraith in an article in the Texas Observer
even went so far as to characterize their legacy as follows: “They will be remembered for having tried
and destroyed, completely, utterly and beyond any redemption, their own theories.”
In the last chapter, we will consider the lessons of this book in the light of the LTCM
experience. Not surprisingly, we will discover that the experience of LTCM does not change the main
lessons of this book and that, despite the statements of James Galbraith, the legacy of Merton and
Scholes will not be the dramatic losses of LTCM but their contribution to our understanding of
derivatives. This book shows that risks can be successfully managed with derivatives. For those who
hear about physicists and mathematicians flocking to Wall Street to make fortunes in derivatives, you
will be surprised to discover that this book is not about rocket science. If there was a lesson from
LTCM, it is that derivatives are too important to be left to rocket scientists. What makes financial
markets different from the experiments that physicists focus on in their labs, it is that financial history
does not repeat itself. Markets are new every day. They surprise us all the time. There is no
mathematical formula that guarantees success every time.
After studying this book, all you will know is how, through careful use of derivatives, you can
increase shareholder wealth and improve the welfare of individuals. Derivatives are like finely tuned
racing cars. One would not think of letting an untutored driver join the Indianapolis 500 at the wheel
of a race car. However, if the untutored driver joins the race at the wheel of a Ford Escort, he has
no chance of ever winning. The same is true with derivatives. Untutored users can crash and burn.
Nonusers are unlikely to win the race.
Chapter 2, page 22
Literature Note
Bernstein (1992) provides a historical account of the interaction between research and practice in the
history of derivatives markets. The spectacular growth in financial innovation is discussed in Miller
(1992). Finnerty (1992) provides a list of new financial instruments developed since the discovery of
the Black-Scholes formula. Black () provides an account of the discovery of the Black-Scholes
formula. Allen and Gale (), Merton (1992), and Ross (1989) provide an analysis of the determinants
of financial innovation and Merton (1992).
Chapter 2, page 23
Figure 1.1. Payoff of derivative which pays the 10m times the excess of the square of the
decimal interest rate over 0.01.
Million dollars
0.8
0.6
0.4
0.2
5
10
15
20
25
30
Interest rate in percent
Chapter 2, page 24
Figure 1.2. Hedging with forward contract. The firm’s income is in dollars and the exchange rate
is the dollar price of one euro.
Unhedged income
Income to firm
999,000
900,000
Exchange rate
0.90
0.99
Panel A. Income to unhedged exporter. The exporter receives euro 1m in 90 days, so that the
dollar income is the dollar price of the euro times 1m if the exporter does not hedge.
Unhedged income
Income to firm
Income to firm
99,000
Forward
loss
999,000
Forward
gain
Forward
loss
0.9
Hedged income
Forward
gain
Exchange rate
Exchange rate
0.99
0.99
Panel B. Forward contract payoff. The
forward price for the euro is $0.99. If the spot
exchange rate is $0.9, the gain from the
forward contract is the gain from selling euro
1m at $0.99 rather than $0.9.
Panel C. Hedged firm income. The firm sells
its euro income forward at a price of $0.99 per
euro. It therefore gets a dollar income of
$990,000 for sure, which is equal to the
unhedged firm income plus the forward
contract payoff.
Chapter 2, page 25
Figure 1.3. Payoff of share and call option strategies.
Payoff
+30
Amazon.com price
-55
20
75
105
Figure 1.3.A. Payoff of buying one share of Amazon.com at $75.
Payoff
20
0
Amazon.com price
-10
20
75
105
Figure 1.3.B. Payoff of buying a call option on one share of Amazon.com with exercise price
of $75 for a premium of $10.
Chapter 2, page 26
Chapter 2, page 27
Chapter 2: Investors, Derivatives, and Risk Management
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section 2.1. Evaluating the risk and the return of individual securities and portfolios. . . . 3
Section 2.1.1. The risk of investing in IBM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Section 2.1.2. Evaluating the expected return and the risk of a portfolio. . . . . . 11
Section 2.2. The benefits from diversification and their implications for expected returns.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Section 2.2.1. The risk-free asset and the capital asset pricing model . . . . . . . . . 21
Section 2.2.2. The risk premium for a security. . . . . . . . . . . . . . . . . . . . . . . . . . 24
Section 2.3. Diversification and risk management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Section 2.3.1. Risk management and shareholder wealth. . . . . . . . . . . . . . . . . . 36
Section 2.3.2. Risk management and shareholder clienteles . . . . . . . . . . . . . . . . 41
Section 2.3.3. The risk management irrelevance proposition. . . . . . . . . . . . . . . 46
1) Diversifiable risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2) Systematic risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3) Risks valued by investors differently than predicted by the CAPM . . . 46
Hedging irrelevance proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Section 2.4. Risk management by investors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Section 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Literature Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Questions and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 2.1. Cumulative probability function for IBM and for a stock with same return and
twice the volatility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Figure 2.2. Normal density function for IBM assuming an expected return of 13% and a
volatility of 30% and of a stock with the same expected return but twice the
volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 2.3. Efficient frontier without a riskless asset . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 2.4. The benefits from diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 2.5. Efficient frontier without a riskless asset. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Figure 2.6. Efficient frontier with a risk-free asset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 2.7. The CAPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Box: T-bills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Box: The CAPM in practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
SUMMARY OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 2: Investors, Derivatives, and Risk Management
December 1, 1999
© René M. Stulz 1997, 1999
Chapter objectives
1. Review expected return and volatility for a security and a portfolio.
2. Use the normal distribution to make statements about the distribution of returns of a portfolio.
3. Evaluate the risk of a security in a portfolio.
4. Show how the capital asset pricing model is used to obtain the expected return of a security.
4. Demonstrate how hedging affects firm value in perfect financial markets.
5. Show how investors evaluate risk management policies of firms in perfect financial markets.
6. Show how investors can use risk management and derivatives in perfect financial markets to make
themselves better off.
1
During the 1980s and part of the 1990s, two gold mining companies differed dramatically in
their risk management policies. One company, Homestake, had a policy of not managing its gold price
risk at all. Another company, American Barrick, had a policy of eliminating most of its gold price risk
using derivatives. In this chapter we investigate whether investors prefer one policy to the other and
why. More broadly, we consider how investors evaluate the risk management policies of the firms
in which they invest. In particular, we answer the following question: When does an investor want
a firm in which she holds shares to spend money to reduce the volatility of its stock price?
To find out how investors evaluate firm risk management policies, we have to study how
investors decide to invest their money and how they evaluate the riskiness of their investments. We
therefore examine first the problem of an investor that can invest in two stocks, one of them IBM and
the other a fictional one that we call XYZ. This examination allows us to review concepts of risk and
return and to see how one can use a probability distribution to estimate the risk of losing specific
amounts of capital invested in risky securities. Throughout this book, it will be of crucial importance
for us to be able to answer questions such as: How likely is it that the value of a portfolio of securities
will fall by more than 10% over the next year? In this chapter, we show how to answer this question
when the portfolio does not include derivatives.
Investors have two powerful risk management tools at their disposal that enable them to
invest their wealth with a level of risk that is optimal for them. The first tool is asset allocation. An
investor’s asset allocation specifies how her wealth is allocated across types of securities or asset
classes. For instance, for an investor who invests in equities and the risk-free asset, the asset
allocation decision involves choosing the fraction of his wealth invested in equities. By investing less
in equities and more in the risk-free asset, the investor reduces the risk of her invested wealth. The
2
second tool is diversification. Once the investor has decided how much to invest in an asset class,
she has to decide which securities to hold and in which proportion. A portfolio’s diversification is the
extent to which the funds invested are distributed across securities to reduce the dependence of the
portfolio’s return on the return of individual securities. If a portfolio has only one security, it is not
diversified and the investor always loses if that security performs poorly. A diversified portfolio can
have a positive return even though some of its securities make losses because the gains from the other
securities can offset these losses. Diversification therefore reduces the risk of funds invested in an
asset class. We will show that these risk management tools imply that investors do not need the help
of individual firms to achieve their optimal risk-return takeoff. Because of the availability of these risk
management tools, investors only benefit from a firm’s risk management policy if that policy
increases the present value of the cash flows the firm is expected to generate. In the next chapter, we
demonstrate when and how risk management by firms can make investors better off.
After having seen the conditions that must be met for a firm’s risk management policy to
increase firm value, we turn to the question of when investors have to use derivatives as additional
risk management tools. We will see that derivatives enable investors to purchase insurance, to hedge,
and to take advantage of their views more efficiently.
Section 2.1. Evaluating the risk and the return of individual securities and portfolios.
Consider the situation of an investor with wealth of $100,000 that she wants to invest for one
year. Her broker recommends two common stocks, IBM and XYZ. The investor knows about IBM,
but has never heard of XYZ. She therefore decides that first she wants to understand what her wealth
will amount to after holding IBM shares for one year. Her wealth at the end of the year will be her
3
initial wealth times one plus the rate of return of the stock over that period. The rate of return of the
stock is given by the price appreciation plus the dividend payments divided by the stock price at the
beginning of the year. So, if the stock price is $100 at the beginning of the year, the dividend is $5,
and the stock price appreciates by $20 during the year, the rate of return is (20 + 5)/100 or 25%.
Throughout the analysis in this chapter, we assume that the frictions which affect financial
markets are unimportant. More specifically, we assume that there are no taxes, no transaction costs,
no costs to writing and enforcing contracts, no restrictions on investments in securities, no differences
in information across investors, and investors take prices as given because they are too small to affect
prices. Financial economists call markets that satisfy the assumptions we just listed perfect financial
markets. The assumption of the absence of frictions stretches belief, but it allows us to zero in on
first-order effects and to make the point that in the presence of perfect financial markets risk
management cannot increase the value of a firm. In chapter 3, we relax the assumption of perfect
financial markets and show how departures from this assumption make it possible for risk
management to create value for firms. For instance, taxes make it advantageous for firms and
individuals to have more income when their tax rate is low and less when their tax rate is high. Risk
management with derivatives can help firms and individuals achieve this objective.
Section 2.1.1. The risk of investing in IBM.
Since stock returns are uncertain, the investor has to figure out which outcomes are likely and
which are not. To do this, she has to be able to measure the likelihood of possible returns. The
statistical tool used to measure the likelihood of various returns for a stock is called the stock’s
return probability distribution. A probability distribution provides a quantitative measure of the
4
likelihood of the possible outcomes or realizations for a random variable. Consider an urn full of balls
with numbers on them. There are multiple balls with the same number on them. We can think of a
random variable as the outcome of drawing a ball from the urn. The urn has lots of different balls, so
that we do not know which number will come up. The probability distribution specifies how likely
it is that we will draw a ball with a given number by assigning a probability for each number that can
take values between zero and one. If many balls have the same number, it is more likely that we will
draw a ball with that number so that the probability of drawing this number is higher than the
probability of drawing a number which is on fewer balls. Since a ball has to be drawn, the sum of the
probabilities for the various balls or distinct outcomes has to sum to one. If we could draw from the
urn a large number of times putting the balls back in the urn after having drawn them, the average
number drawn would be the expected value. More precisely, the expected value is a probability
weighted “average” of the possible distinct outcomes of the random variable. For returns, the
expected value of the return is the return that the investor expects to receive. For instance, if a stock
can have only one of two returns, 10% with probability 0.4 and 15% with probability 0.6, its expected
return is 0.4*10% + 0.6*15% or 13%.
The expected value of the return of IBM, in short IBM’s expected return, gives us the
average return our investor would earn if next year was repeated over and over, each time yielding
a different return drawn from the distribution of the return of IBM. Everything else equal, the investor
is better off the greater the expected return of IBM. We will see later in this chapter that a reasonable
estimate of the expected return of IBM is about 13% per year. However, over the next year, the
return on IBM could be very different from 13% because the return is random. For instance, we will
find out that using a probability distribution for the return of IBM allows us to say that there is a 5%
5
chance of a return greater than 50% over a year for IBM. The most common probability distribution
used for stock returns is the normal distribution. There is substantial empirical evidence that this
distribution provides a good approximation of the true, unknown, distribution of stock returns.
Though we use the normal distribution in this chapter, it will be important later on for us to explore
how good this approximation is and whether the limitations of this approximation matter for risk
management.
The investor will also want to know something about the “risk” of the stock. The variance
of a random variable is a quantitative measure of how the numbers drawn from the urn are spread
around their expected value and hence provides a measure of risk. More precisely, it is a probability
weighted average of the square of the differences between the distinct outcomes of a random variable
and its expected value. Using our example of a return of 10% with probability 0.6 and a return of
15% with probability 0.4, the decimal variance of the return is 0.4*(0.10 - 0.13)2 + 0.6*(0.15 - 0.13)2
or 0.0006. For returns, the variance is in units of the square of return differences from their expected
value. The square root of the variance is expressed in the same units as the returns. As a result, the
square root of the return variance is in the same units as the returns. The square root of the variance
is called the standard deviation. In finance, the standard deviation of returns is generally called the
volatility of returns. For our example, the square root of 0.0006 is 0.0245. Since the volatility is in
the same units as the returns, we can use a volatility in percent or 2.45%. As returns are spread
farther from the expected return, volatility increases. For instance, if instead of having returns of 10%
and 15% in our example, we have returns of 2.5% and 20%, the expected return is unaffected but the
volatility becomes 8.57% instead of 2.45%.
If IBM’s return volatility is low, the absolute value of the difference between IBM’s return
6
and its expected value is likely to be small so that a return substantially larger or smaller than the
expected return would be surprising. In contrast, if IBM’s return volatility is high, a large positive or
negative return would not be as surprising. As volatility increases, therefore, the investor becomes
less likely to get a return close to the expected return. In particular, she becomes more likely to have
low wealth at the end of the year, which she would view adversely, or really high wealth, which she
would like. Investors are generally risk-averse, meaning that the adverse effect of an increase in
volatility is more important for them than the positive effect, so that on net they are worse off when
volatility increases for a given expected return.
The cumulative distribution function of a random variable x specifies, for any number X,
the probability that the realization of the random variable will be no greater than X. We denote the
probability that the random variable x has a realization no greater than X as prob(x # X). For our urn
example, the cumulative distribution function specifies the probability that we will draw a ball with
a number no greater than X. If the urn has balls with numbers from one to ten, the probability
distribution function could specify that the probability of drawing a ball with a number no greater than
6 is 0.4. When a random variable is normally distributed, its cumulative distribution function depends
only on its expected value and on its volatility. A reasonable estimate of the volatility of the IBM
stock return is 30%. With an expected return of 13% and a volatility of 30%, we can draw the
cumulative distribution function for the return of IBM. The cumulative distribution function for IBM
is plotted in Figure 2.1. It is plotted with returns on the horizontal axis and probabilities on the
vertical axis. For a given return, the function specifies the probability that the return of IBM will not
exceed that return. To use the cumulative distribution function, we choose a value on the horizontal
axis, say 0%. The corresponding value on the vertical axis tells us the probability that IBM will earn
7
less than 0%. This probability is 0.32. In other words, there is a 32% chance that over one year, IBM
will have a negative return. Such probability numbers are easy to compute for the normal distribution
using the NORMDIST function of Excel. Suppose we want to know how likely it is that IBM will
earn less than 10% over one year. To get the probability that the return will be less than 10%, we
choose x = 0.10. The mean is 0.13 and the standard deviation is 0.30. We finally write TRUE in the
last line to obtain the cumulative distribution function. The result is 0.46. This number means that
there is a 46% chance that the return of IBM will be less than 10% over a year.
Our investor is likely to be worried about making losses. Using the normal distribution, we
can tell her the probability of losing more than some specific amount. If our investor would like to
know how likely it is that she will have less than $100,000 at the end of the year if she invests in IBM,
we can compute the probability of a loss using the NORMDIST function by noticing that a loss means
a return of less than 0%. We therefore use x = 0 in our above example instead of x = 0.1. We find that
there is a 33% chance that the investor will lose money. This probability depends on the expected
return. As the expected return of IBM increases, the probability of making a loss falls.
Another concern our investor might have is how likely it is that her wealth will be low enough
that she will not be able to pay for living expenses. For instance, the investor might decide that she
needs to have $50,000 to live on at the end of the year. She understands that by putting all her wealth
in a stock, she takes the risk that she will have less than that amount at the end of the year and will
be bankrupt. However, she wants that risk to be less than 5%. Using the NORMDIST function, the
probability of a 50% loss for IBM is 0.018. Our investor can therefore invest in IBM given her
objective of making sure that there is a 95% chance that she will have $50,000 at the end
of the year.
8
The probability density function of a random variable tells us the change in prob(x # X)
as X goes to its next possible value. If the random variable takes discrete values, the probability
density function tells us the probability of x taking the next higher value. In our example of the urn,
the probability density function tells us the probability of drawing a given number from the urn. We
used the example where the urn has balls with numbers from one through ten and the probability of
drawing a ball with a number no greater than six is 0.4. Suppose that the probability of drawing a ball
with a number no greater than seven is 0.6. In this case, 0.6 - 0.4 is the probability density function
evaluated at seven and it tells us that the probability of drawing the number seven is 0.2. If the
random variable is continuous, the next higher value than X is infinitesimally close to X.
Consequently, the probability density function tells us the increase in prob(x # X) as X increases by
an infinitesimal amount, say ,. This corresponds to the probability of x being between X and X + ,.
If we wanted to obtain the probability of x being in an interval between X and X + 2,, we would add
the probability of x being in an interval between X and X + , and then the probability of x being in
an interval between X + , to X + 2,. More generally, we can also obtain the probability that x will
be in an interval from X to X’ by “adding up”the probability density function from X to X’, so that
the probability that x will take a value in an interval corresponds to the area under the probability
density function from X to X’.
In the case of IBM, we see that the cumulative distribution function first increases slowly, then
more sharply, and finally again slowly. This explains that the probability density function of IBM
shown in Figure 2.2. first has a value close to zero, increases to reach a peak, and then falls again.
This bell-shaped probability density function is characteristic of the normal distribution. Note that this
bell-shaped function is symmetric around the expected value of the distribution. This means that the
9
cumulative distribution function increases to the same extent when evaluated at two returns that have
the same distance from the mean on the horizontal axis. For comparison, the figure shows the
distribution of the return of a security that has twice the volatility of IBM but the same expected
return. The distribution of the more volatile security has more weight in the tails and less around the
mean than IBM, implying that outcomes substantially away from the mean are more likely. The
distribution of the more volatile security shows a limitation of the normal distribution: It does not
exclude returns worse than -100%. In general, this is not a serious problem, but we will discuss this
problem in more detail in chapter 7.
When interpreting probabilities such as the 0.18 probability of losing 50% of an investment
in IBM, it is common to state that if our investor invests in IBM for 100 years, she can expect to lose
more than 50% of her beginning of year investment slightly less than two years out of 100. Such a
statement requires that returns of IBM are independent across years. Two random variables a and
b are independent if knowing the realization of random variable a tells us nothing about the realization
of random variable b. The returns to IBM in years i and j are independent if knowing the return of
IBM in year i tells us nothing about the return of IBM in year j. Another way to put this is that,
irrespective of what IBM earned in the previous year (e.g., +100% or - 50%), our best estimate of
the mean return for IBM is 13%.
There are two good reasons for why it makes sense to consider stock returns to be
independent across years. First, this seems to be generally the case statistically. Second, if this was
not roughly the case, there would be money on the table for investors. To see this, suppose that if
IBM earns 100% in one year it is likely to have a negative return the following year. Investors who
know that would sell IBM since they would not want to hold a stock whose value is expected to fall.
10
By their actions, investors would bring pressure on IBM’s share price. Eventually, the stock price will
be low enough that it is not expected to fall and that investing in IBM is a reasonable investment. The
lesson from this is that whenever security prices do not incorporate past information about the history
of the stock price, investors take actions that make the security price incorporate that information.
The result is that markets are generally weak-form efficient. The market for a security is weak-form
efficient if all past information about the past history of that security is incorporated in its price. With
a weak-form efficient market, technical analysis which attempts to forecast returns based on
information about past returns is useless. In general, public information gets incorporated in security
prices quickly. A market where public information is immediately incorporated in prices is called a
semi-strong form efficient market. In such a market, no money can be made by trading on
information published in the Wall Street Journal because that information is already incorporated
in security prices. A strong form efficient market is one where all economically relevant information
is incorporated in prices, public or not. In the following, we will call a market to be efficient when
it is semi-strong form efficient, so that all public information is incorporated in prices.
Section 2.1.2. Evaluating the expected return and the risk of a portfolio.
To be thorough, the investor wants to consider XYZ. She first wants to know if she would
be better off investing $100,000 in XYZ rather than in IBM. She finds out that the expected return
of XYZ is 26% and the standard deviation of the return is 60%. In other words, XYZ has twice the
expected return and twice the standard deviation of IBM. This means that, using volatility as a
summary risk measure, XYZ is riskier than IBM. Figure 2.2. shows the probability density function
of a return distribution that has twice the volatility of IBM. Since XYZ has twice the expected return,
11
its probability density function would be that distribution moved to the right so that its mean is 26%.
It turns out that the probability that the price of XYZ will fall by 50% is 10.2%. Consequently, our
investor cannot invest all her wealth in XYZ because the probability of losing $50,000 would exceed
5%.
We now consider the volatility and expected return of a portfolio that includes both IBM and
XYZ shares. At the end of the year, the investor’s portfolio will be $100,000 times one plus the
return of her portfolio. The return of a portfolio is the sum of the return on each security in the
portfolio times the fraction of the portfolio invested in the security. The fraction of the portfolio
invested in a security is generally called the portfolio share of that security. Using wi for the portfolio
share of the i-th security in a portfolio with N securities and Ri for the return on the i-th security, we
have the following formula for the portfolio return:
N
∑ w R = Portfolio Return
i
i
(2.1.)
i =1
Suppose the investor puts $75,000 in IBM and $25,000 in XYZ. The portfolio share of IBM is
$75,000/$100,000 or 0.75. If, for illustration, the return of IBM is 20% and the return on XYZ is 10%, applying formula (2.1.) gives us a portfolio return in decimal form of:
0.75*0.20 + 0.25*(-0.10) = 0.125
In this case, the wealth of the investor at the end of the year is 100,000(1+0.125) or $125,000.
At the start of the year, the investor has to make a decision based on what she expects the
12
distribution of returns to be. She therefore wants to compute the expected return of the portfolio and
the return volatility of the portfolio. Denote by E(x) the expected return of random variable x, for any
x. To compute the portfolio’s expected return, it is useful to use two properties of expectations. First,
the expected value of a random variable multiplied by a constant is the constant times the expected
value of the random variable. Suppose a can take value a1 with probability p and a2 with probability
1-p. If k is a constant, we have that E(k*a) = pka1 + (1-p)k a2 = k[pa1 + (1-p)a2] = kE(a).
Consequently, the expected value of the return of a security times its portfolio share, E(wiRi) is equal
to wiE(Ri). Second, the expected value of a sum of random variables is the sum of the expected values
of the random variables. Consider the case of random variables which have only two outcomes, so
that a1 and b1 have respectively outcomes a and b with probability p and a2 and b2 with probability
(1-p). With this notation, we have E(a + b) = p(a1 + b1) + (1-p)(a2 + b2) = pa1 + (1-p)a2+ pb1 +(1p)b2 = E(a) + E(b). This second property implies that if the portfolio has only securities 1 and 2, so
that we want to compute E(w1R1 + w2R2), this is equal to E(w1R1) + E(w2R2), which is equal to
w1E(R1) + w2E(R2) because of the first property. With these properties of expectations, the expected
return of a portfolio is therefore the portfolio share weighted average of the securities in the portfolio:
N
∑ w E(R ) = Portfolio Expected Return
i
i
(2.2.)
i= 1
Applying this formula to our problem, we find that the expected return of the investor’s portfolio in
decimal form is:
0.75*0.13 + 0.25*0.26 = 0.1625
Our investor therefore expects her wealth to be 100,000*(1 + 0.1625) or $116,250 at the end of the
13
year.
Our investor naturally wants to be able to compare the risk of her portfolio to the risk of
investing all her wealth in IBM or XYZ. To do that, she has to compute the volatility of the portfolio
return. The volatility is the square root of the variance. The variance of a portfolio’s return is the
expected value of the square of the difference between the portfolio’s return and its expected return,
E[Rp - E(Rp)]2. To get the volatility of the portfolio return, it is best to first compute the variance and
then take the square root of the variance. To compute the variance of the portfolio return, we first
need to review two properties of the variance. Denote by Var(x) the variance of random variable x,
for any x. The first property is that the variance of a constant times random variable a is the constant
squared times the variance of a. For instance, the variance of 10 times a is 100 times the variance of
a. This follows from the definition of the variance of a as E[a - E(a)]2. If we compute the variance
of ka, we have E[ka - E(ka)]2. Since k is not a random variable, we can remove it from the
expectation to get the variance of ka as k2E[a - E(a)]2. This implies that Var(wiRi) = wi2Var(Ri).
To obtain the variance of a + b, we have to compute E[a+b - E(a + b)]2. Remember that the square
of a sum of two terms is the sum of each term squared plus two times the cross-product of the two
numbers (the square of 5 + 4 is 52 + 42 + 2*5*4, or 81). Consequently:
Var(a + b) = E[a + b - E(a + b)]2
= E[a - E(a) + b - E(b)]2
= E[a - E(a)]2 + E[b - E(b)]2 + 2E[a - E(a)][b - E(b)]
= Var(a) + Var(b) + 2Cov(a,b)
14
The bold term is the covariance between a and b, denoted by Cov(a,b). The covariance is a measure
of how a and b move together. It can take negative as well as positive values. Its value increases as
a and b are more likely to exceed their expected values simultaneously. If the covariance is zero, the
fact that a exceeds its expected value provides no information about whether b exceeds its expected
value also. The covariance is closely related to the correlation coefficient. The correlation coefficient
takes values between minus one and plus one. If a and b have a correlation coefficient of one, they
move in lockstep in the same direction. If the correlation coefficient is -1, they move in lockstep in
the opposite direction. Finally, if the correlation coefficient is zero, a and b are independent. Denote
by Vol(x) the volatility of x, for any x, and by Corr(x,y) the correlation between x and y, for any x
and y. If one knows the correlation coefficient, one can obtain the covariance by using the following
formula:
Cov(a,b) = Corr(a,b)*Vol(a)*Vol(b)
The variance of a and b increases with the covariance of a and b since an increase in the covariance
makes it less likely that an unexpected low value of a is offset by an unexpected high value of b. It
therefore follows that the Var(a + b) increases with the correlation between a and b. In the special
case where a and b have the same volatility, a + b has no risk if the correlation coefficient between
a and b is minus one because a high value of one of the variables is always exactly offset by a low
value of the other, insuring that the sum of the realizations of the random variables is always equal
to the sum of their expected values. To see this, suppose that both random variables have a volatility
of 0.2 and a correlation coefficient of minus one. Applying our formula for the covariance, we have
15
that the covariance between a and b is equal to -1*0.2*0.2, which is -0.04. The variance of each
random variable is the square of 0.2, or 0.04. Applying our formula, we have that Var(a + b) is equal
to 0.04 + 0.04 - 2*0.04 = 0. Note that if a and b are the same random variables, they have a
correlation coefficient of plus one, so that Cov(a,b) is Cov(a,a) = 1*Vol(a)Vol(a), which is Var(a).
Hence, the covariance of a random variable with itself is its variance.
Consider the variance of the return of a portfolio with securities 1 and 2, Var(w1R1 + w2R2).
Using the formula for the variance of a sum, we have that Var(w1R1 + w2R2) is equal to Var(w1R1)
+ Var(w2R2) +2Cov(w1R1,w2R2). Using the result that the variance of ka is k2Var(a), we have that
w12Var(R1) + w22Var(R2) +2w1w2Cov(R1,R2). More generally, therefore, the formula for the variance
of the return of a portfolio is:
N
N
N
∑ w Var(R ) + ∑ ∑ w w Cov(R , R ) = Variance of Portfolio Return (2.3.)
i =1
2
i
i
i = 1 j≠ i
i
j
i
j
Applying the formula to our portfolio of IBM and XYZ, we need to know the covariance between
the return of the two securities. Let’s assume that the correlation coefficient between the two
securities is 0.5. In this case, the covariance is 0.5*0.30*0.60 or 0.09. This gives us the following
variance:
0.752*0.32 + 0.252*0.62 +2*0.25*0.75*0.5*0.3*0.6 = 0.11
The volatility of the portfolio is the square root of 0.11, which is 0.33. Our investor therefore
discovers that by investing less in IBM and investing some of her wealth in a stock that has twice the
volatility of IBM, she can increase her expected return from 13% to 16.25%, but in doing so she
16
increases the volatility of her portfolio from 30% to 32.70%. We cannot determine a priori which
of the three possible investments (investing in IBM, XYZ, or the portfolio) the investor prefers. This
is because the portfolio has a higher expected return than IBM but has higher volatility. We know that
the investor would prefer the portfolio if it had a higher expected return than IBM and less volatility,
but this is not the case. An investor who is risk-averse is willing to give up some expected return in
exchange for less risk. If the investor dislikes risk sufficiently, she prefers IBM to the portfolio
because IBM has less risk even though it has less expected return. By altering portfolio shares, the
investor can create many possible portfolios. In the next section, we study how the investor can
choose among these portfolios.
Section 2.2. The benefits from diversification and their implications for expected returns.
We now consider the case where the return correlation coefficient between IBM and XYZ
is zero. In this case, the decimal variance of the portfolio is 0.07 and the volatility is 26%. As can be
seen from the formula for the expected return of a portfolio (equation (2.1.)), the expected return of
a portfolio does not depend on the covariance of the securities that compose the portfolio.
Consequently, as the correlation coefficient between IBM and XYZ changes, the expected return of
the portfolio is unchanged. However, as a result of selling some of the low volatility stock and buying
some of the high volatility stock, the volatility of the portfolio falls from 30% to 26% for a constant
expected return. This means that when IBM and XYZ are independent, an investor who has all her
wealth invested in IBM can become unambiguously better off by selling some IBM shares and buying
shares in a company whose stock return has twice the volatility of IBM.
That our investor wants to invest in XYZ despite its high volatility is made clear in figure 2.3.
17
In that figure, we draw all the combinations of expected return and volatility that can be obtained by
investing in IBM and XYZ. We do not restrict portfolio shares to be positive. This means that we
allow investors to sell shares of one company short as long as all their wealth is fully invested so that
the portfolio shares sum to one. With a short-sale, an investor borrows shares from a third party and
sells them. When the investor wants to close the position, she must buy shares and deliver them to
the lender. If the share price increases, the investor loses because she has to pay more for the shares
she delivers than she received for the shares she sold. In contrast, a short-sale position benefits from
decreases in the share price. With a short-sale, the investor has a negative portfolio share in a security
because she has to spend an amount of money at maturity to unwind the short-sale of one share equal
to the price of the share at the beginning of the period plus its return. Consequently, if a share is sold
short, its return has to be paid rather than received. The upward-sloping part of the curve drawn in
figure 2.3. is called the efficient frontier. Our investor wants to choose portfolios on the efficient
frontier because, for each volatility, there is a portfolio on the efficient frontier that has a higher
expected return than any other portfolio with the same volatility. In the case of the volatility of IBM,
there is a portfolio on the frontier that has the same volatility but a higher expected return, so that
IBM is not on the efficient frontier. That portfolio, portfolio y in the figure, has an expected return
of 18.2%. The investor would always prefer that portfolio to holding only shares of IBM.
The investor prefers the portfolio to holding only shares of IBM because she benefits from
diversification. The benefit of spreading a portfolio’s holdings across different securities is the
volatility reduction that naturally occurs when one invests in securities whose returns are not perfectly
correlated: the poor outcomes of some securities are offset by the good outcomes of other securities.
In our example, XYZ could do well when IBM does poorly. This cannot happen when both securities
18
are perfectly correlated because then they always do well or poorly together. As the securities become
less correlated, it becomes more likely that one security does well and the other poorly at the same
time. In the extreme case where the correlation coefficient is minus one, IBM always does well when
XYZ does poorly, so that one can create a portfolio of IBM and XYZ that has no risk. To create such
a portfolio, choose the portfolio shares of XYZ and IBM that sum to one and that set the variance
of the portfolio equal to zero. The portfolio share of XYZ is 0.285 and the portfolio share of IBM
is 0.715. This offsetting effect due to diversification means that the outcomes of a portfolio are less
dispersed than the outcomes of many and sometimes all of the individual securities that comprise the
portfolio.
To show that it is possible for diversification to make the volatility of a portfolio smaller than
the volatility of any security in the portfolio, it is useful to consider the following example. Suppose
that an investor can choose to invest among many uncorrelated securities that all have the same
volatility and the same expected return as IBM. In this case, putting the entire portfolio in one
security yields a volatility of 30% and an expected return of 13%. Dividing one’s wealth among N
such uncorrelated securities has no impact on the expected return because all securities have the same
expected return. However, using our formula, we find that the volatility of the portfolio is:
 N

Volatility of portfolio =  ∑ (1 / N) 2 * 0.32 
 i= 1

= ( N *(1 / N) 2 *0.32 )
0.5
0.5
= ((1 / N) *0.32 ) 0.5
Applying this result, we find that for N = 10, the volatility is 9%, for N = 100 it is 3%, and for N =
1000 it is less than 1%. As N is increased further, the volatility becomes infinitesimal. In other words,
19
by holding uncorrelated securities, one can eliminate portfolio volatility if one holds sufficiently many
of these securities! Risk that disappears in a well-diversified portfolio is called diversifiable risk. In
our example, all of the risk of each security becomes diversifiable as N increases.
In the real world, though, securities tend to be positively correlated because changes in
aggregate economic activity affect most firms. For instance, news of the onset of a recession is
generally bad news for almost all firms. As a result, one cannot eliminate risk through diversification
but one can reduce it. Figure 2.4. shows how investors can substantially reduce risk by diversifying
using common stocks available throughout the world. The figure shows how, on average, the
variance of an equally-weighted portfolio of randomly chosen securities is related to the number of
securities in the portfolio. As in our simple example, the variance of a portfolio falls as the number
of securities is increased. This is because, as the number of securities increases, it becomes more likely
that some bad event that affects one security is offset by some good event that affects another
security. Interestingly, however, most of the benefit from diversification takes place when one goes
from one security to ten. Going from 50 securities in a portfolio to 100 does not bring much in terms
of variance reduction. Another important point from the figure is that the variance falls more if one
selects securities randomly in the global universe of securities rather than just within the U.S. The
lower variance of the portfolios consisting of both U.S. and foreign securities reflects the benefits
from international diversification.
Irrespective of the universe from which one chooses securities to invest in, there is always an
efficient frontier that has the same form as the one drawn in figure 2.3. With two securities, they are
both on the frontier. With more securities, this is no longer the case. In fact, with many securities,
individual securities are generally inside the frontier so that holding a portfolio dominates holding a
20
single security because of the benefits of diversification. Figure 2.5. shows the efficient frontier
estimated at a point in time. Because of the availability of international diversification, a welldiversified portfolio of U.S. stocks is inside the efficient frontier and is dominated by internationally
diversified portfolios. In other words, an investor holding a well-diversified portfolio of U.S. stocks
would be able to reduce risk by diversifying internationally without sacrificing expected return.
Section 2.2.1. The risk-free asset and the capital asset pricing model.
So far, we assumed that the investor could form her portfolio holding shares of IBM and
XYZ. Let’s now assume that there is a third asset, an asset which has no risk over the investment
horizon of the investor. An example of such an asset would be a Treasury Bill, which we abbreviate
as T-bill. T-bills are discount bonds. Discount bonds are bonds where the interest payment comes
in the form of the capital appreciation of the bond. Hence, the bond pays par at maturity and sells for
less than par before maturity. Since they are obligations of the Federal government, T-bills have no
default risk. Consequently, they have a sure return if held to maturity since the gain to the holder is
par minus the price she paid for the T-bill. Consider the case where the one-year T-bill has a yield of
5%. This means that our investor can earn 5% for sure by holding the T-bill. The box on T-bills
shows how they are quoted and how one can use a quote to obtain a yield.
By having an asset allocation where she invests some of her money in the T-bill, the investor
can decrease the volatility of her end-of-year wealth. For instance, our investor could put half of her
money in T-bills and the other half in the portfolio on the frontier with the smallest volatility. This
minimum-volatility portfolio has an expected return of 15.6% and a standard deviation of 26.83%.
As a result (using our formulas for the expected return and the volatility of a portfolio), her portfolio
21
would have a volatility of 13.42% and an expected return of 12.8%. All combinations of the
minimum-volatility portfolio and the risk-free asset lie on a straight line that intersects the efficient
frontier at the minimum-volatility portfolio. Portfolios on that straight line to the left of the minimumvolatility portfolio have positive investments in the risk-free asset. To the right of the minimumvolatility portfolio, the investor borrows to invest in stocks. Figure 2.6. shows this straight line.
Figure 2.6. suggests that the investor could do better by combining the risk-free asset with a portfolio
more to the right on the efficient frontier than the minimum-volatility portfolio because then all
possible combinations would have a higher return. However, the investor cannot do better than
combining the risk-free asset with portfolio m. This is because in that case the straight line is tangent
to the efficient frontier at m. There is no straight line starting at the risk-free rate that touches the
efficient frontier at least at one point and has a steeper slope than the line tangent to m. Hence, the
investor could not invest on a line with a steeper slope because she could not find a portfolio of
stocks to create that line!
There is a tangency portfolio irrespective of the universe of securities one uses to form the
efficient frontier as long as one can invest in a risk-free asset. We already saw, however, that investors
benefit from forming portfolios using the largest possible universe of securities. As long as investors
agree on the expected returns, volatilities, and covariances of securities, they end up looking at the
same efficient frontier if they behave optimally. In this case, our reasoning implies that they all want
to invest in portfolio m. This can only be possible if portfolio m is the market portfolio. The market
portfolio is a portfolio of all securities available with each portfolio share the fraction of the market
value of that security in the total capitalization of all securities or aggregate wealth. IBM is a much
smaller fraction of the wealth of investors invested in U.S. securities (on June 30, 1999, it was 2.09%
22
of the S&P500 index). Consequently, if the portfolio share of IBM is 10%, there is too much demand
for IBM given its expected return. This means that its expected return has to decrease so that
investors want to hold less of IBM. This process continues until the expected return of IBM is such
that investors want to hold the existing shares of IBM at the current price. Consequently, the demand
and supply of shares are equal for each firm when portfolio m is such that the portfolio of each
security in the portfolio corresponds to its market value divided by the market value of all securities.
If all investors have the same views on expected returns, volatilities, and covariances of
securities, all of them hold the same portfolio of risky securities, portfolio m, the market portfolio.
To achieve the right volatility for their invested wealth, they allocate their wealth to the market
portfolio and to the risk-free asset. Investors who have little aversion to risk borrow to invest more
than their wealth in the market portfolio. In contrast, the most risk-averse investors put most or all
of their wealth in the risk-free asset. Note that if investors have different views on expected returns,
volatilities and covariances of securities, on average, they still must hold portfolio m because the
market portfolio has to be held. Further, for each investor, there is always a tangency portfolio and
she always allocates her wealth between the tangency portfolio and the risk-free asset if she trades
off risk and return. In the case of investors who invest in the market portfolio, we know exactly their
reward for bearing volatility risk since the expected return they earn in excess of the risk-free rate is
given by the slope of the tangency line. These investors therefore earn (E(Rm) - Rf)/Fm per unit of
volatility, where Rm is the return of portfolio m, Fm is its volatility, and Rf is the risk-free rate. The
excess of the expected return of a security or of a portfolio over the risk-free rate is called the risk
premium of that security or portfolio. A risk premium on a security or a portfolio is the reward the
investor expects to receive for bearing the risk associated with that security or portfolio. E(Rm) - Rf
23
is therefore the risk premium on the market portfolio.
Section 2.2.2. The risk premium for a security.
We now consider the determinants of the risk premium of an individual security. To start this
discussion, it is useful to go back to our example where we had N securities with uncorrelated
returns. As N gets large, a portfolio of these securities has almost no volatility. An investor who
invests in such a portfolio should therefore earn the risk-free rate. Otherwise, there would be an
opportunity to make money for sure if N is large enough. A strategy which makes money for sure
with a net investment of zero is called an arbitrage strategy. Suppose that the portfolio earns 10%
and the risk-free rate is 5%. Investing in the portfolio and borrowing at the risk-free rate earns five
cents per dollar invested in the portfolio for sure without requiring any capital. Such an arbitrage
strategy cannot persist because investors make it disappear by taking advantage of it. The only way
it cannot exist is if each security in the portfolio earns the risk-free rate. In this case, the risk of each
security in the portfolio is completely diversifiable and consequently no security earns a risk premium.
Now, we consider the case where a portfolio has some risk that is not diversifiable. This is
the risk left when the investor holds a diversified portfolio and is the only risk the investor cares
about. Because it is not diversifiable, such risk is common to many securities and is generally called
systematic risk. In the aggregate, there must be risk that cannot be diversified away because most
firms benefit as economic activity unexpectedly improves. Consequently, the risk of the market
portfolio cannot be diversified because it captures the risk associated with aggregate economic
fluctuations. However, securities that belong to the market portfolio can have both systematic risk
and risk that is diversifiable.
24
The market portfolio has to be held in the aggregate. Consequently, since all investors who
invest in risky securities do so by investing in the market portfolio, expected returns of securities must
be such that our investor is content with holding the market portfolio as her portfolio of risky
securities. For the investor to hold a security that belongs to the market portfolio, it has to be that the
risk premium on that security is just sufficient to induce her not to change her portfolio. This means
that the change in the portfolio’s expected return resulting from a very small increase in the investor’s
holdings of the security must just compensate for the change in the portfolio’s risk. If this is not the
case, the investor will change her portfolio shares and will no longer hold the market portfolio.
To understand how the risk premium on a security is determined, it is useful to note that the
formula for the variance of the return of an arbitrary portfolio, Rp, can be written as the portfolio
share weighted sum of the return covariances of the securities in the portfolio with the portfolio. To
understand why this is so, remember that the return covariance of a security with itself is the return
variance of that security. Consequently, the variance of the return of a portfolio can be written as the
return covariance of the portfolio with itself. The return of the portfolio is a portfolio share weighted
sum of the returns of the securities. Therefore, the variance of the return of the portfolio is the
covariance of the portfolio share weighted sum of returns of the securities in the portfolio with the
return of the portfolio. Since the covariance of a sum of random variables with Rp is equal to the sum
of the covariances of the random variables with Rp, it follows that the variance of the portfolio returns
is equal to a portfolio share weighted sum of the return covariances of the securities with the portfolio
return:
N
N
i= 1
i=1
Cov(R p , R p ) = Cov( ∑ w i R i , R p ) = ∑ w i Cov(R i , R p )
25
(2.4.)
Consequently, the variance of the return of a portfolio is a portfolio share weighted average of the
covariances of the returns of the securities in the portfolio with the return of the portfolio. Equation
(2.4.) makes clear that a portfolio is risky to the extent that the returns of its securities covary with
the return of the portfolio.
Let’s now consider the following experiment. Suppose that security z in the market portfolio has
a zero return covariance with the market portfolio, so that Cov(Rz,Rm) = 0. Now, let’s decrease
slightly the portfolio share of security z and increase by the same decimal amount the portfolio share
of security i. Changing portfolio shares has a feedback effect: Since it changes the distribution of the
return of the portfolio, the return covariance of all securities with the portfolio is altered. If the
change in portfolio shares is small enough, the feedback effect becomes a second-order effect and can
be ignored. Consequently, as we change the portfolio shares of securities i and z, none of the
covariances change in equation (2.4.). By assumption, the other portfolio shares are kept constant.
Letting the change in portfolio share of security i be ), so that the portfolio share after the change
is wi + ), the impact on the volatility of the portfolio of the change in portfolio shares is therefore
equal to )cov(Ri,Rp) - )cov(Rz,Rp). Since Cov(Rz,Rp) is equal to zero, increasing the portfolio share
of security i by ) and decreasing the portfolio share of security z by ) therefore changes the volatility
of the portfolio by )cov(Ri,Rp). Security i can have a zero, positive, or negative return covariance
with the portfolio. Let’s consider each case in turn:
1) Security i has a zero return covariance with the market portfolio. In this case, the
volatility of the portfolio return does not change if we increase the holding of security i at the expense
of the holding of security z. Since the risk of the portfolio is unaffected by our change, the expected
26
return of the portfolio should be unaffected, so that securities z and i must have the same expected
return. If the two securities have a different expected return, the investor would want to change the
portfolio shares. For instance, if security z has a higher expected return than security i, the investor
would want to invest more in security z and less in security i. In this case, she would not hold the
market portfolio anymore, but every other investor would want to make the same change so that
nobody would hold the market portfolio.
2) Security i has a positive covariance with the market portfolio. Since security i has a
positive return covariance with the portfolio, it is more likely to have a good (bad) return when the
portfolio has a good (bad) return. This means holding more of security i will tend to increase the
good returns of the portfolio and worsen the bad returns of the portfolio, thereby increasing its
volatility. This can be verified by looking at the formula for the volatility of the portfolio return: By
increasing the portfolio share of security i and decreasing the portfolio share of security z, we increase
the weight of a security with a positive return covariance with the market portfolio and thereby
increase the weighted sum of the covariances. If security z is expected to earn the same as security
i, the investor would want to sell security i to purchase more of security z since doing so would
decrease the risk of the portfolio without affecting its expected return. Consequently, for the investor
to be satisfied with its portfolio, security i must be expected to earn more than security z.
3) Security i has a negative return covariance with the market portfolio. In this case, the
reasoning is the opposite from the previous case. Adding more of security i to the portfolio reduces
the weighted sum of the covariances, so that security i has to have a lower expected return than
security z.
27
In equilibrium it must the case that no investor wants to change her security holdings. The
reasoning that we used leads to three important results that must hold in equilibrium:
Result I: A security’s expected return should depend only on its return covariance with
the market portfolio. Suppose that securities i and j have the same return covariance with the
market portfolio but different expected returns. A slight increase in the holding of the security that
has the higher expected return and decrease in the holding of the other security would create a
portfolio with a higher expected return than the market portfolio but with the same volatility. In that
case, no investor would want to hold the market portfolio. Consequently, securities l and j cannot
have different expected returns in equilibrium.
Result II: A security that has a zero return covariance with the market portfolio should
have an expected return equal to the return of the risk-free security. Suppose that this is not the
case. The investor can then increase the expected return of the portfolio without changing its volatility
by investing slightly more in the security with the higher expected return and slightly less in the
security with the lower expected return.
Result III: A security’s risk premium is proportional to its return covariance with the
market portfolio. Suppose that securities i and j have positive but different return covariances with
the market portfolio. If security i has k times the return covariance of security j with the market
portfolio, its risk premium must be k times the risk premium of security j. To see why this is true, note
that we can always combine security i and security z in a portfolio. Let h be the weight of security i
28
in that portfolio and (1-h) the weight of security z. We can choose h to be such that the portfolio of
securities i and z has the same return covariance with the market portfolio as security j:
hCov(R i , R m ) + (1 − h)Cov(R z , R m ) = Cov(R j , R m )
(2.5.)
If h is chosen so that equation (2.5.) holds, the portfolio should have the same expected excess return
as security j since the portfolio and the security have the same return covariance with the market
portfolio. Otherwise, the investor could increase the expected return on her invested wealth without
changing its return volatility by investing in the security and shorting the portfolio with holdings of
securities i and z if the security has a higher expected return than the portfolio or taking the opposite
position if the security has a lower expected return than the portfolio. To find h, remember that the
return covariance of security z with the market portfolio is zero and that the return covariance of
security i with the market portfolio is k times the return covariance of security j with the market
portfolio. Consequently, we can rewrite equation (2.5.) as:
hCov(R i , R m ) + (1 − h)Cov(R z , R m ) = h * k * Cov(R j , R m )
(2.6.)
Therefore, by choosing h equal to 1/k so that h*k is equal to one, we choose h so that the
return covariance of the portfolio containing securities z and i with the market portfolio is the same
as the return covariance with the market portfolio of security j. We already know from the Second
Result that security z must earn the risk-free rate. From the First Result, it must therefore be that the
portfolio with portfolio share h in security i and (1-h) in security z has the same expected return as
29
security j:
hE(R i ) + (1 − h)R f = E(R j )
(2.7.)
If we subtract the risk-free rate from both sides of this equation, we have h times the risk premium
of security j on the left-hand side and the risk premium of security j on the right-hand side. Dividing
both sides of the resulting equation by h and remembering that h is equal to 1/k, we obtain:
[
E(R i ) − R f = k E(R j ) − R f
]
(2.8.)
Consequently, the risk premium on security i is k times the risk premium on security j as
predicted. By definition, k is the return covariance of security i with the market portfolio divided by
the return covariance of security j with the market portfolio. Replacing k by its definition in the
equation and rearranging, we have:
E(R i ) − R f =
[
Cov(R i , R m )
E(R j ) − R f
Cov(R j , R m )
]
(2.9.)
This equation has to apply if security j is the market portfolio. If it does not, one can create a security
that has the same return covariance with the market portfolio as the market portfolio but a different
expected return. As a result, the investor would increase her holding of that security and decrease her
holding of the market portfolio if that security has a higher expected return than the market portfolio.
The return covariance of the market portfolio with itself is simply the variance of the return of the
market portfolio. Let’s choose security j to be the market portfolio in our equation. Remember that
30
in our analysis, security i has a return covariance with the market portfolio that is k times the return
covariance of security j with the market portfolio. If security j is the market portfolio, its return
covariance with the market portfolio is the return variance of the market portfolio. When security j
is the market portfolio, k is therefore equal to Covariance(Ri,Rm)/Variance(Rm), which is called
security i’s beta, or $i. In this case, our equation becomes:
Capital asset pricing model
E(R i ) - R f = βi [ E(R m ) - R f ]
(2.10.)
The capital asset pricing model (CAPM) tells us how the expected return of a security is
determined. The CAPM states that the expected excess return of a risky security is equal to the
systematic risk of that security measured by its beta times the market’s risk premium. Importantly,
with the CAPM diversifiable risk has no impact on a security’s expected excess return. The relation
between expected return and beta that results from the CAPM is shown in figure 2.7.
Our reasoning for why the CAPM must hold with our assumptions implies that the CAPM
must hold for portfolios. Since a portfolio’s return is a portfolio share weighted average of the return
of the securities in the portfolio, the only way that the CAPM does not apply to a portfolio is if it does
not apply to one or more of its constituent securities. We can therefore multiply equation (2.10.) by
the portfolio share and add up across securities to obtain the expected excess return of the portfolio:
N
[
]
[
E(R pf ) − R f = ∑ w i E(R i ) − R f = ∑ w i β i E(R m ) − R f
i=1
]
The portfolio share weighted sum of the securities in the portfolio is equal to the beta of the portfolio
31
obtained by dividing the return covariance of the portfolio with the market with the return variance
of the market. This is because the return covariance of the portfolio with the market portfolio is the
portfolio share weighted average of the return covariances of the securities in the portfolio:
N
N
N
Cov(R i , R m )
Cov(w i R i , R m )
=
∑ w i Var(R ) = ∑ Var(R
i=1
i=1
m
m)
Cov( ∑ w i R i , R m )
i =1
Var(R m )
=
Cov(R p , R m )
Var(R m )
= βp
To see how the CAPM model works, suppose that the risk-free rate is 5%, the market risk premium
is 6% and the $ of IBM is 1.33. In this case, the expected return of IBM is:
Expected return of IBM = 5% + 1.33*[6%] = 13%
This is the value we used earlier for IBM. The box on the CAPM in practice shows how
implementing the CAPM for IBM leads to the above numbers.
Consider a portfolio worth one dollar that has an investment of $1.33 in the market portfolio
and -$0.33 in the risk-free asset. The value of the portfolio is $1, so that the portfolio share of the
investment in the market is (1.33/1) and the portfolio share of the investment in the risk-free asset is
(-0.33/1). The beta of the market is one and the beta of the risk-free asset is zero. Consequently, this
portfolio has a beta equal to the portfolio share weighted average of the beta of its securities, or
(1.33/*1)*1 + (- 0.33/1)*0 =1.33, which is the beta of IBM. If the investor holds this portfolio, she
has the same systematic risk and therefore ought to receive the same expected return as if she had
invested a dollar in IBM. The actual return on IBM will be the return of the portfolio plus
32
diversifiable risk that does not affect the expected return. The return of the portfolio is RF + 1.33(Rm
- RF). The return of IBM is RF + 1.33(Rm - RF ) plus diversifiable risk whose expected value is zero.
If IBM is expected to earn more than the portfolio, then the investor can make an economic profit
by holding IBM and shorting the portfolio. By doing this, she invests no money of her own and bears
no systematic risk, yet earns a positive expected return corresponding to the difference between the
expected return of the portfolio and of IBM.
Section 2.3. Diversification and risk management.
We now discuss how an investor values a firm when the CAPM holds. Once we understand
this, we can find out when risk management increases firm value. For simplicity, let’s start with a firm
that lives one year only and is an all-equity firm. At the end of the year, the firm has a cash flow of
C and nothing else. The cash flow is the cash generated by the firm that can be paid out to
shareholders. The firm then pays that cash flow to equity as a liquidating dividend. Viewed from
today, the cash flow is random. The value of the firm today is the present value of receiving the cash
flow in one year. We denote this value by V. To be specific, suppose that the firm is a gold mining
firm. It will produce 1M ounces of gold this year, but after that it cannot produce gold any longer and
liquidates. For simplicity, the firm has no costs and taxes are ignored. Markets are assumed to be
perfect. At the end of the year, the firm makes a payment to its shareholders corresponding to the
market value of 1M ounces of gold.
If the firm is riskless, its value V is the cash flow discounted at the risk-free rate, C/(1+Rf).
For instance, if the gold price is fixed at $350 an ounce and the risk-free rate is 5%, the value of the
firm is $350M/(1+0.05) or $333.33M. The reason for this is that one can invest $333.33M in the risk33
free asset and obtain $350M at the end of the year. Consequently, if firm value differs from
$333.33M, there is an arbitrage opportunity. Suppose that firm value is $300m and the firm has one
million shares. Consider an investor who buys a share and finances the purchase by borrowing. At
maturity, she has to repay $300(1+0.05), or $315 and she receives $350, making a sure profit of $35
per share. If firm value is more than the present value of the cash flow, investors make money for sure
by selling shares short and investing the proceeds in the risk-free asset.
Let’s now consider the case where the cash flow is random. In our example, this is because
the price of gold is random. The return of a share is the random liquidating cash flow C divided by
the firm value at the beginning of the year, V, minus one. Since C is equal to a quantity of gold times
the price of gold, the return is perfectly correlated with the return on an ounce of gold so that the firm
must have the same beta as gold. We know from the CAPM that any financial asset must have the
same expected return as the expected return of a portfolio invested in the market portfolio and in the
risk-free asset that has the same beta as the financial asset. Suppose, for the sake of illustration, that
the beta of gold is 0.5. The expected return on a share of the firm has to be the expected return on
a portfolio with a beta of 0.5. Such a portfolio can be constructed by investing an amount equal to
half the value of the share in the market portfolio and the same amount in the risk-free asset. If the
firm is expected to earn more than this portfolio, investors earn an economic profit by investing in the
firm and financing the investment in the firm by shorting the portfolio since they expect to earn an
economic profit without taking on any systematic risk. This strategy has risk, but that risk is
diversifiable and hence does not matter for investors with diversified portfolios.
We now use this approach to value the firm. Shareholders receive the cash flow in one year
for an investment today equal to the value of the firm. This means that the cash flow is equal to the
34
value of the firm times one plus the rate of return of the firm. We know that the expected return of
the firm has to be given by the CAPM equation, so that it is the risk-free rate plus the firm’s beta
times the risk premium on the market. Consequently, firm value must be such that:
(
[
E(C) = V 1 + R f + β E(R m ) − R f
])
(2.12.)
If we know the distribution of the cash flow C, the risk-free rate Rf, the $ of the firm, and the risk
premium on the market E(RM) - Rf, we can compute V because it is the only variable in the equation
that we do not know. Solving for V, we get:
E(C)
[
1 + R f + β E(R M ) − R F
]
= V
(2.13.)
Using this formula, we can value our gold mining firm. Let’s say that the expected gold price is $350.
In this case, the expected payoff to shareholders is $350M, which is one million ounces times the
expected price of one ounce. As before, we use a risk-free rate of 5% and a risk premium on the
market portfolio of 6%. Consequently:
E(C)
$350M
=
= $324.074M
1 + R f + β[E(R M ) − R f ] 1 + 0.05 + 0.5(0.06)
We therefore obtain the value of the firm by discounting the cash flow to equity at the expected return
required by the CAPM. Our approach extends naturally to firms expected to live more than one year.
The value of such a firm for its shareholders is the present value of the cash flows to equity. Nothing
else affects the value of the firm for its shareholders - they only care about the present value of cash
35
the firm generates over time for them. We can therefore value equity in general by computing the sum
of the present values of all future cash flows to equity using the approach we used above to value one
such future cash flow. For a levered firm, one will often consider the value of the firm to be the sum
of debt and equity. This simply means that the value of the firm is the present value of the cash flows
to the debt and equity holders.
Cash flow to equity is computed as net income plus depreciation and other non-cash charges
minus investment. To get cash flow to the debt and equity holders, one adds to cash flow to equity
the payments made to debt holders. Note that the cash flow to equity does not necessarily correspond
each year to the payouts to equity. In particular, firms smooth dividends. For instance, a firm could
have a positive cash flow to equity in excess of its planned dividend. It would then keep the excess
cash flow in liquid assets and pay it to shareholders later. However, all cash generated by the firm
after debt payments belongs to equity and hence contributes to firm value whether it is paid out in a
year or not.
Section 2.3.1. Risk management and shareholder wealth.
Consider now the question we set out to answer in this chapter: Would shareholders want
a firm to spend cash to decrease the volatility of its cash flow when the only benefit of risk
management is to decrease share return volatility? Let’s assume that the shareholders of the firm are
investors who care only about the expected return and the volatility of their wealth invested in
securities, so that they are optimally diversified and have chosen the optimal level of risk for their
portfolios. We saw in the previous section that the volatility of the return of a share can be
decomposed into systematic risk, which is not diversifiable, and other risk, unsystematic risk, which
36
is diversifiable. We consider separately a risk management policy that decreases unsystematic risk and
one that decreases systematic risk. A firm can reduce risk through financial transactions or through
changes in its operations. We first consider the case where the firm uses financial risk management
and then discuss how our reasoning extends to changes in the firm’s operations to reduce risk.
1) Financial risk management policy that decreases the firm’s unsystematic risk.
Consider the following situation. A firm has a market value of $1 billion. Suppose that its
management can sell the unsystematic risk of the firm’s shares to an investment bank by paying
$50M. We can think of such a transaction as a hedge offered by the investment bank which exactly
offsets the firm’s unsystematic risk. Would shareholders ever want the firm to make such a payment
when the only benefit to them of the payment is to eliminate the unsystematic risk of their shares? We
already know that firm value does not depend on unsystematic risk when expected cash flow is given.
Consider then a risk management policy eliminating unsystematic risk that decreases expected cash
flow by its cost but has no other impact on expected cash flow. Since the value of the firm is the
expected cash flow discounted at the rate determined by the systematic risk of the firm, this risk
management policy does not affect the rate at which cash flow is discounted. In terms of our
valuation equation, this policy decreases the numerator of the valuation equation without a change
in the denominator, so that firm value decreases.
Shareholders are diversified and do not care about diversifiable risks. Therefore, they are not
willing to discount expected cash flow at a lower rate if the firm makes cash flow less risky by
eliminating unsystematic risk. This means that if shareholders could vote on a proposal to implement
risk management to decrease the firm’s diversifiable risk at a cost, they would vote no and refuse to
incur the cost as long as the only effect of risk management on expected cash flow is to decrease
37
expected cash flow by the cost of risk management. Managers will therefore never be rewarded by
shareholders for decreasing the firm’s diversifiable risk at a cost because shareholders can eliminate
the firm’s diversifiable risk through diversification at zero cost. For shareholders to value a decrease
in unsystematic risk, it has to increase their wealth and hence the share price.
2) Financial risk management policy that decreases the firm’s systematic risk. We now
evaluate whether it is worthwhile for management to incur costs to decrease the firm’s systematic risk
through financial transactions. Consider a firm that decides to reduce its beta. Its only motivation for
this action is that it believes that it will make its shares more attractive to investors. The firm can
easily reduce its beta by taking a short position in the market since such a position has a negative beta.
The proceeds of the short position can be invested in the risk-free asset. This investment has a beta
of zero. In our discussion of IBM, we saw that a portfolio of $1.33 invested in the market and of
$0.33 borrowed in the risk-free asset has the same beta as an investment of one dollar invested in the
market. Consequently, if IBM was an all-equity firm, the management of IBM could make IBM a
zero beta firm by selling short $1.33 of the market and investing the proceeds in the risk-free asset
for each dollar of market value. Would investors be willing to pay for IBM management to do this?
The answer is no because the action of IBM’s management creates no value for its shareholders. In
perfect financial markets, a shareholder could create a zero beta portfolio long in IBM shares, short
in the market, and long in the risk-free asset. Hence, the investor would not be willing to pay for the
management of IBM to do something for her that she could do at zero cost on her own.
Remember that investors are assumed to be optimally diversified. Each investor chooses her
optimal asset allocation consisting of an investment in the risk-free asset and an investment in the
market portfolio. Consequently, if IBM were to reduce its beta, this means that the portfolio that
38
investors hold will have less risk. However, investors wanted this level of risk because of the reward
they expected to obtain in the form of a higher expected return. As IBM reduces its beta, therefore,
investors no longer hold the portfolio that they had chosen. They will therefore want to get back to
that portfolio by changing their asset allocation. In perfect financial markets, investors can costlessly
get back to their initial asset allocation by increasing their holdings of the risky portfolio and
decreasing their holdings of the safe asset. They would therefore object if IBM expended real
resources to decrease its beta. It follows from this discussion that investors choose the level of risk
of their portfolio through their own asset allocation. They do not need the help of firms whose shares
they own for that purpose.
Comparing our discussion of the reduction in unsystematic risk with our discussion of the
reduction in systematic risk creates a paradox. When we discussed the reduction in unsystematic risk,
we argued that shareholders cannot gain from a costly reduction in unsystematic risk undertaken for
the sole purpose of decreasing the return volatility because such a reduction decreases the numerator
of the present value formula without affecting the denominator. Yet, the reduction in systematic risk
obviously decreases the denominator of the present value formula for shares since it decreases the
discount rate. Why is it then that decreasing systematic risk does not increase the value of the shares?
The reason is that decreasing systematic risk has a cost, in that it decreases expected cash flow. To
get rid of its systematic risk, IBM has to sell short the market. Selling short the market earns a
negative risk premium since holding the market long has a positive risk premium. Hence, the expected
cash flow of IBM has to fall by the risk premium of the short sale. The impact of the short sale on
firm value is therefore the sum of two effects. The first effect is the decrease in expected cash flow
and the second is the decrease in the discount rate. The two effects cancel out. They have to for a
39
simple reason. Going short the market is equivalent to getting perfect insurance against market
fluctuations. In perfect markets, insurance is priced at its fair value. This means that the risk premium
IBM would earn by not changing its systematic risk has to be paid to whoever will now bear this
systematic risk. Hence, financial risk management in this case just determines who bears the
systematic risk, but IBM’s shareholders charge the same price for market risk as anybody else since
that price is determined by the CAPM. Consequently, IBM management cannot create value by
selling market risk to other investors at the price that shareholders would require to bear that risk.
We have focused on financial risk management in our reasoning. A firm could change its
systematic risk or its unsystematic risk by changing its operations. Our reasoning applies in this case
also, but with a twist. Let’s first look at unsystematic risk. Decreasing unsystematic risk does not
make shareholders better off if the only benefit of doing so is to reduce share return volatility. It does
not matter therefore whether the decrease in share volatility is due to financial transactions or to
operating changes. In contrast, if the firm can change its operations costlessly to reduce its beta
without changing its expected cash flow, firm value increases because expected cash flow is
discounted at a lower rate. Hence, decreasing cash flow beta through operating changes is worth it
if firm value increases as a result. On financial markets, every investor charges the same for systematic
risk. This means that nobody can make money from selling systematic risk to one group of investors
instead of another. The ability to change an investment’s beta through operating changes depends on
technology and strategy. A firm can become more flexible so that it has less fixed costs in cyclical
downturns. This greater flexibility translates into a lower beta. If flexibility has low cost but a large
impact on beta, the firm’s shareholders are better off if the firm increases its flexibility. If greater
flexibility has a high cost, shareholders will not want it because having it will decrease share value.
40
Hedging that only reduces systematic or idiosyncratic risk for shareholders has no impact on
the value of the shares in our analysis. This is because when the firm reduces risk, it is not doing
anything that shareholders could not do on their own equally well with the same consequences.
Shareholders can diversify to eliminate unsystematic risk and they can change their asset allocation
between the risk-free investment and the investment in the market to get the systematic risk they want
to bear. With our assumption of perfect markets, risk management just redistributes risk across
investors who charge the same price for bearing it. For risk management to create value for the firm,
the firm has to transfer risks to investors for whom bearing these risks is less expensive than it is for
the firm. With the assumptions of this section, this cannot happen. Consequently, for this to happen,
there must exist financial market imperfections.
Section 2.3.2. Risk management and shareholder clienteles.
In the introduction, we mentioned that one gold firm, Homestake, had for a long time a policy
of not hedging at all. Homestake justified its policy as follows in its 1990 annual report (p. 12):
“So that its shareholders might capture the full benefit of increases in the price of gold,
Homestake does not hedge its gold production. As a result of this policy, Homestake’s
earnings are more volatile than those of many other gold producers. The Company believes
that its shareholders will achieve maximum benefit from such a policy over the long-term.”
The reasoning in this statement is that some investors want to benefit from gold price movements and
that therefore giving them this benefit increases firm value. These investors form a clientele the firm
41
caters to. Our analysis so far does not account for the possible existence of clienteles such as
investors wanting to bear gold price risks. With the CAPM, investors care only about their portfolio’s
expected return and volatility. They do not care about their portfolio’s sensitivity to other variables,
such as gold prices. However, the CAPM has limitations in explaining the returns of securities. For
instance, small firms earn more on average than predicted by the CAPM. It is possible that investors
require a risk premium to bear some risks other than the CAPM’s systematic risk. For instance, they
might want a risk premium to bear inflation risk. The existence of such risk premia could explain why
small firms earn more on average than predicted by the CAPM. It could be the case, then, that
investors value gold price risk. To see the impact of additional risk premia besides the market risk
premium on our reasoning about the benefits of hedging, let’s suppose that Homestake is right and
that investors want exposures to specific prices and see what that implies for our analysis of the
implications of hedging for firm value.
For our analysis, let’s consider explicitly the case where the firm hedges its gold price risk
with a forward contract on gold. We denote by Q the firm’s production and by G the price of gold
in one year. There is no uncertainty about Q.
Let’s start with the case where there is no clientele effect to have a benchmark. In this case,
the CAPM applies. Empirically, gold has a beta close to zero. Suppose now for the sake of argument
that the gold beta is actually zero. In this case, all the risk of the gold producing firm is diversifiable.
This means that hedging does not affect the gold firm’s value if our analysis is right. Let’s make sure
that this is the case. The firm eliminates gold price risk by selling gold forward at a price F per unit,
the forward price. In this case, the value of the cash flow to shareholders when the gold is sold is FQ,
which is known today. Firm value today is FQ discounted at the risk-free rate. If the firm does not
42
hedge, the expected cash flow to shareholders is E(G)Q. In this case, firm value is obtained by
discounting E(G)Q at the risk-free rate since there is no systematic risk. The difference between the
hedged value of the firm and its unhedged value is [F - E(G)]Q/(1 + Rf). With our assumptions, the
hedged firm is worth more than the unhedged firm if the forward price exceeds the expected spot
price, which is if F - E(G) is positive. Remember that with a short forward position we receive F for
delivering gold worth G per ounce. F - E(G) is therefore equal to the expected payoff from selling
one ounce of gold forward at the price F. If this expected payoff is positive, it means that one expects
to make a profit on a short forward position without making an investment since opening a forward
contract requires no cash. The only way we can expect to make money without investing any of our
own in the absence of arbitrage opportunities is if the expected payoff is a reward for bearing risk.
However, we assumed that the risk associated with the gold price is diversifiable, so that F - G
represents diversifiable risk. The expected value of F - G has to be zero, since diversifiable risk does
not earn a risk premium. Consequently, FQ = E(G)Q and hedging does not affect the firm’s value.
Consider now the case where gold has a positive beta. By taking a long forward position in
gold which pays G - F, one takes on systematic risk. The only way investors would enter such a
position is if they are rewarded with a risk premium, which means that they expect to make money
out of the long forward position. Hence, if gold has systematic risk, it must be that E(G) > F, so that
the expected payout to shareholders is lower if the firm hedges than if it does not. If the firm is
hedged, the cash flow has no systematic risk and the expected cash flow is discounted at the risk-free
rate. If the firm is not hedged, the cash flow has systematic risk, so that the higher expected cash flow
of the unhedged firm is discounted at a higher discount rate than the lower expected cash flow of the
hedged firm. The lower discount rate used for the unhedged firm just offsets the fact that expected
43
cash flow is lower for the hedged firm, so that the present value of expected cash flow is the same
whether the firm hedges or not. An investor today must be indifferent between paying FQ in one year
or receiving GQ at that time. Otherwise, the forward contract has a value different from zero since
the payoff to a long forward position to buy Q units of gold is GQ - FQ, which cannot be the case.
Hence, the present value of FQ and of GQ must be the same.
The argument extends naturally to the case where some investors value exposure to gold for
its own sake. Since investors value exposure to gold, they are willing to pay for it and hence the risk
premium attached to gold price risk is lower than predicted by the CAPM. This means that exposure
to gold lowers the discount rate of an asset relative to what it would be if exposure to gold was not
valuable to some investors. Consequently, investors who want exposure to gold bid up forward prices
since forward contracts enable these investors to acquire exposure to gold. If the firm does not hedge,
its share price reflects the benefit from exposure to gold that the market values because its discount
rate is lower. In contrast, if the firm hedges, its shareholders are no longer exposed to gold. However,
to hedge, the firm sells exposure to gold to investors by enabling them to take long positions in
forward contracts. Consequently, the firm earns a premium for selling exposure to gold that increases
its expected cash flow. Hence, shareholders can earn the premium for gold exposure either by having
exposure to gold and thereby lowering the discount rate that applies to the firm’s cash flow or by
selling exposure to gold and earning the risk premium that accrues to short forward positions. The
firm has a natural exposure to gold and it is just a matter of which investors bear it. In our reasoning,
it does not matter whether the firm’s shareholders themselves are shareholders who value gold
exposure or not. If the firm gets rid of its gold exposure, the firm’s shareholders can buy it on their
own. Irrespective of how investors who value gold exposure get this exposure, they will have to pay
44
the same price for it since otherwise the same good - gold exposure - would have different prices on
the capital markets, making it possible for investors to put on arbitrage trades that profit from these
price differences. As a result, if the firm’s shareholders value gold exposure, they will get it one way
or another, but they will always pay the same price for it which will make them indifferent as to where
they get it.
An important lesson from this analysis is that the value of the firm is the same whether the firm
hedges or not irrespective of how the forward price is determined. This lesson holds because there
are arbitrage opportunities if hedging creates value with our assumptions. Suppose that firm value
for the hedged firm is higher than for the unhedged firm. In this case, an investor can create a share
of the hedged firm on his own by buying a share of the unhedged firm and selling one ounce of gold
forward at the price F. His cash cost today is the price of a share since the forward position has no
cash cost today. Having created a share of the hedged firm through homemade hedging, the investor
can then sell the share hedged through homemade hedging at the price of the share of the hedged
firm. There is no difference between the two shares and hence they should sell for the same price.
Through this transaction, the investor makes a profit equal to the difference between the share price
of the hedged firm and the share price of the unhedged firm. This profit has no risk attached to it and
is, consequently, an arbitrage profit. Consequently, firm value must be the same whether the firm
hedges or not with our assumptions.
It follows from our analysis that the clientele argument of Homestake is not correct. With
perfect financial markets, anybody can get exposure to gold without the help of Homestake. For
instance, if Homestake can take a long forward position, so can investors. If Homestake increases its
value by hedging, then investors can buy the unhedged Homestake, create a hedged Homestake on
45
their own account with a forward transaction and sell it on the equity markets. On net, they make
money! This cannot be. Whenever investors can do on their own what the firm does, in other words,
whenever homemade hedging is possible, the firm cannot possibly contribute value through hedging.
Section 2.3.3. The risk management irrelevance proposition.
Throughout this section, we saw that the firm cannot create value by hedging risks when the
price of bearing these risks within the firm is the same as the price of bearing them outside of the firm.
In this chapter, the only cost of bearing risks within the firm is the risk premium attached to these
risks by the capital markets when they value the firm. The same risk premium is required by the
capital markets for bearing these risks outside the firm. Consequently, shareholders can alter the
firm’s risk on their own through homemade hedging at the same terms as the firm and the firm has
nothing to contribute to the shareholders’ welfare through risk management. Let’s make sure that this
is the case:
1) Diversifiable risk. It does not affect the share price and investors do not care about it
because it gets diversified within their portfolios. Hence, eliminating it does not affect firm value.
2) Systematic risk. Shareholders require the same risk premium for it as all investors. Hence,
eliminating it for the shareholders just means having other investors bear it at the same cost. Again,
this cannot create value.
3) Risks valued by investors differently than predicted by the CAPM. Again,
shareholders and other investors charge the same price for bearing such risks.
The bottom line from this can be summarized in the hedging irrelevance proposition:
46
Hedging irrelevance proposition. Hedging a risk does not increase firm value when the cost
of bearing the risk is the same whether the risk is born within the firm or outside the firm by the
capital markets.
Section 2.4. Risk management by investors.
We saw so far that firms have no reason to adopt risk management policies to help investors
manage their portfolio risks when investors are well-diversified. We now consider whether investors
have reasons to do more than simply diversify and allocate their wealth between the market portfolio
and the risk-free asset optimally. Let’s go back to the investor considered in the first section. Suppose
that investor does not want to take the risk of losing more than $50,000. For such an investor, the
only solution given the approach developed so far would be to invest in the risk-free asset at least the
present value of $50,000 invested at the risk-free rate. In one year, this investment would then
amount to $50,000. At the risk-free rate of 7%, this amounts to $46,729. She can then invest the rest
in the market, so her investment in the market is $52,271. With this strategy, the most the investor
can gain for each 1% return in the market is $522.71. The investor might want more exposure to the
market.
To get a different exposure to the market, the investor could create a levered position in
stocks. However, as soon as she borrows to invest more in the market, she takes the risk of ending
up with a loss on her levered position so that she has less than $50,000 at the end of the year. A static
investment strategy is one that involves no trades between the purchase of a portfolio and the time
the portfolio is liquidated. The only static investment strategy which guarantees that the investor will
have $50,000 at the end of the year but benefits more if the market increases than an investment in
47
the market of $52,271 is a strategy that involves buying call options.
Consider the following strategy. The investor invests $100,000 in the market portfolio minus
the cost of buying a put with an exercise price at $50,000. With this strategy, the investor has an
exposure to the market which is much larger than $50,000. The exact amount of the exposure
depends on the cost of the put. However, a put on an investment in the market at $100,000 with an
exercise price of $50,000 will be extremely cheap - less than one hundred dollars because the
probability of the market losing 50% is very small. Consequently, the investor can essentially invest
$100,000 in the market and eliminate the risk of losing more than $50,000. She could not achieve this
payoff with a static investment strategy without using derivatives. Using derivatives, the investor
could achieve lots of different payoffs. For instance, she could buy a call option on a $100,000
investment in the market portfolio with an exercise price of $100,000 and invest the remainder in the
risk-free asset. As long as the volatility of the market portfolio is not too high, this strategy
guarantees wealth in excess of $100,000 at the end of the year and participation in the market
increases from the current level equal to an investment in the market of $100,000. Such a strategy
is called a portfolio insurance strategy in that it guarantees an amount of wealth at the end of the
investment period at least equal to the current wealth. In chapter 13, we will see exactly how to
implement portfolio insurance.
Derivatives enable investors to achieve payoffs that they could not achieve otherwise. They
can also allow them to take advantage of information in ways that they could not otherwise. The
concept of market efficiency means that all public information is incorporated in prices. However, it
does not preclude that an investor might disagree with the market or have information that the market
does not have. Such an investor might then use derivatives to take advantage of her views.
48
Suppose, for instance, that our investor believes that it is highly likely that IBM will exceed $120 in
one year when it is $100 currently. If she feels that this outcome is more likely than the market thinks
it is, she can capitalize on her view by buying a call option with exercise price at $120. Even more
effectively, she might buy an exotic derivative such as a digital cash or nothing call with exercise price
at $120. This derivative pays off a fixed amount if and only if the price of the underlying exceeds the
exercise price. Suppose that this digital pays $100,000 if IBM exceeds $120. The digital pays all of
its promised payment if IBM sells for $120.01. In contrast, a call on a share of IBM with exercise
price of $120 pays only one cent if IBM is at $120.01. Consequently, the digital call is a more
effective option to take advantage of the view that there is an extremely high probability that IBM
will be above $120 in one year when one does not know more than that.
An important use of risk management by investors occurs when investors have information
that leads them to want to hold more of some securities than the market portfolio. For instance, our
investor might believe that IBM is undervalued, which makes its purchase attractive. If our investor
puts all her wealth in IBM to take advantage of her knowledge, she ends up with a portfolio that is
not at all diversified. To reduce the risk of her IBM position, she might decide to hedge the risks of
IBM that she can hedge. An obvious risk she can hedge is the market risk. She can therefore use a
derivative, such as an index futures contract, to hedge this risk. Another case where hedging is often
used is in international investment. An investor might believe that investing in a foreign country is
advantageous, but she feels that there is no reward to bearing foreign exchange risk. She can then use
derivatives to eliminate exchange rate risk.
This discussion shows that risk management can make individual investors better off. It can
allow investors to choose more appropriate payoff patterns from their investments, to hedge the risks
49
that they do not want to bear because they feel that the reward is too low, and to allow them to
capitalize more effectively on their views.
Section 2.5. Conclusion.
In this chapter, we first examined how investors evaluate the risk of securities. We saw how,
using a distribution function, one can evaluate the probability of various outcomes for the return of
a security. This ability we developed to specify the probability that a security will experience a return
smaller than some number will be of crucial importance throughout this book. We then saw that
investors can diversify and that this ability to diversify effects how they evaluate the riskiness of a
security. When holding a portfolio, an investor measures the riskiness of a security by its contribution
to the risk of the portfolio. Under some conditions, the best portfolio of risky securities an investor
can hold is the market portfolio. A security’s contribution to the risk of the market portfolio is its
systematic risk. A security’s unsystematic risk does not affect the riskiness of the portfolio. This
fundamental result allowed us to present the capital asset pricing model, which states that a security’s
risk premium is given by its beta times the risk premium on the market portfolio. The CAPM allowed
us to compute the value of future cash flows. We saw that only the systematic risk of cash flows
affects the rate at which expected future cash flows are discounted.
We then showed that in perfect financial markets, hedging does not affect firm value. We
showed that this is true for hedging through financial instruments systematic as well as unsystematic
risks. Further, we demonstrated that if investors have preferences for some types of risks, like gold
price risks, it is still the case that hedging is irrelevant in perfect financial markets. The reasoning was
straightforward and led to the hedging irrelevance proposition: if it costs the same for a firm to bear
50
a risk as it does for the firm to pay somebody else to bear it, hedging cannot increase firm value.
In the last section, we discussed when investors can be made better off using derivatives. This
is the case when diversification and asset allocation are not sufficient risk management tools. Such
a situation can arise, for instance, when the investor wants to construct a portfolio that requires no
trading after being put together, that guarantees some amount of wealth at some future date and yet
enables her to gain substantially from increases in the stock market. Such a portfolio is an insured
portfolio and it requires positions in options to be established. Though we saw how investors can be
made better off by using derivatives, we did not see how firms could increase their value through the
use of derivatives. The next chapter address this issue.
51
Literature Note
Much research examines the appropriateness of the assumption of the normal distribution for
security returns. Fama (1965) argues that stock returns are well-described by the normal distribution
and that these returns are independent across time. Later in this book, we will discuss some of the
issues raised in that research.
Fama (1970) presents the definitions of market efficiency and reviews the evidence. He
updated his review in Fama (1991). Over the recent past, several papers have questioned the
assumption that stock returns are independent across time. These papers examine the serial
correlation of stock returns. By serial correlation, we mean the correlation of a random variable
between two contiguous periods of equal length. Lo and McKinlay (1988) provide some evidence
of negative serial correlation for daily returns. Jegadeesh and Titman (1993) show that six month
returns have positive serial correlation. Finally, Fama and French (1988) find that returns measured
over periods of several years have negative serial correlation. This literature suggests that there is
some evidence that the distribution of returns depends on their recent history. Such evidence is
consistent with market efficiency if the recent history of stock returns is helpful to forecast the risk
of stocks, so that changes in expected returns reflect changes in risk. Later in this book, we will
discuss how to account for dependence in returns across time. None of the results of this chapter are
fundamentally altered. In particular, these results did not require the returns of IBM to be independent
across years. All they required was the distribution of the next year’s return to be normal.
The fundamental results on diversification and the CAPM were discovered respectively by
Markovitz (1952) and Sharpe (1964). Each of these authors was awarded a share of the Nobel
Memorial Prize in Economics in 1990. Textbooks on investments cover this material in much greater
52
details than we did here. Elton and Gruber provide an extensive presentation of portfolio theory. The
valuation theory using the CAPM is discussed in corporate finance textbooks or textbooks specialized
in valuation. For corporate finance textbooks, see Brealey and Myers (1999) or Jordan, Ross and
Westerfield. A book devoted to valuation is Copeland, Koller, and Murrin (1996).
The hedging irrelevance result is discussed in Smith and Stulz (1985). This result is a natural
extension of the leverage irrelevance result of Modigliani and Miller. They argue that in perfect
markets leverage cannot increase firm value. Their result led to the award of a Nobel Memorial Prize
in Economics in 1990 for Miller. Modigliani received such a Prize earlier for a different contribution.
Risk management in the gold industry has been the object of several papers and cases. The
firm American Barrick discussed in the introduction is the object of a Harvard Business School case.
Tufano (1996) compares the risk management practices of gold mining firms. His work is discussed
more in chapter 4.
One statement of the argument that foreign exchange hedging is advantageous for diversified
portfolios is Perold and Schulman (1988).
53
Key concepts
Cumulative distribution function, normal distribution, expected return, variance, covariance,
diversification, asset allocation, capital asset pricing model, beta, risk premium, homemade hedging,
risk management
54
Review questions
1. Consider a stock return that follows the normal distribution. What do you need to know to
compute the probability that the stock’s return will be less than 10% over the coming year?
2. What are the three types of financial market efficiency?
3. What does diversification of a portfolio do to the distribution of the portfolio’s return?
4. What is beta?
5. When does beta measure risk?
6. For a given expected cash flow, how does the beta of the cash flow affects its current value?
7. How does hedging affect firm value if financial markets are perfect?
8. Why can hedging affect a firm’s expected cash flow when it does not affect it’s value?
9. Why is it that the fact that some investors have a preference for gold exposure does not mean that
a gold producing firm should not hedge its gold exposure.
10. What is the “risk management irrelevance proposition”?
55
11. How come risk management cannot be used to change firm value but can be used to change
investor welfare when financial markets are perfect?
56
Questions and exercises
1. The typical level of the monthly volatility of the S&P500 index is about 4%. Using a risk premium
of 0.5% and a risk-free rate of 5% p.a., what is the probability that a portfolio of $100,000 invested
in the S&P500 will lose $5,000 or more during the next month? How would your answer change if
you used current interest rates from T-bills?
2. During 1997, the monthly volatility on S&P500 increased to about 4.5% from its typical value of
4%. Using the current risk-free rate, construct a portfolio worth $100,000 invested in the S&P500
and the risk-free asset that has the same probability of losing $5,000 or more in a month when the
S&P500 volatility is 4.5% as $100,000 invested in the S&P500 when the S&P500 volatility is 4%.
3. Compute the expected return and the volatility of return of a portfolio that has a portfolio share
of 0.9 in the S&P500 and 0.1 in an emerging market index. The S&P500 has a volatility of return of
15% and an expected return of 12%. The emerging market has a volatility of return of 30% and an
expected return of 10%. The correlation between the emerging market index return and the S&P500
is 0.1.
4. If the S&P500 is a good proxy for the market portfolio in the CAPM and the CAPM applies to the
emerging market index, use the information in question 3 to compute the beta and risk premium for
the emerging market index.
57
5. Compute the beta of the portfolio described in question 4 with respect to the S&P500.
6. A firm has an expected cash flow of $500m in one year. The beta of the common stock of the firm
is 0.8 and this cash flow has the same risk as the firm as a whole. Using a risk-free rate of 6% and a
risk premium on the market portfolio of 6%, what is the present value of the cash flow. If the beta
of the firm doubles, what happens to the present value of the cash flow?
7. With the data used in the previous question, consider the impact on the firm of hedging the cash
flow against systematic risk. If management wants to eliminate the systematic risk of the cash flow
completely, how could it do so? How much would the firm have to pay investors to bear the
systematic risk of the cash flow?
8. Consider the situation you analyzed in question 6. To hedge the firm’s systematic risk, management
has to pay investors to bear this risk. Why is it that the value of the firm for shareholders does not fall
when the firm pays other investors to bear the systematic risk of the cash flow?
9. The management of a gold producing firm agrees with hedging irrelevance result and has
concluded that it applies to the firm. However, the CEO wants to hedge because the price of gold has
fallen over the last month. He asks for your advice. What do you tell him?
10. Consider again an investment in the emerging market portfolio discussed earlier. Now, you
consider investing $100,000 in that portfolio because you think it is a good investment. You decide
58
that you are going to ignore the benefits from diversification, in that all your wealth will be invested
in that portfolio. Your broker nevertheless presents you with an investment in a default-free bond in
the currency of the emerging country which matures in one year. The regression beta in a regression
of the dollar return of the portfolio on the return of the foreign currency bond is 1. The expected
return on the foreign currency bond is 5% in dollars and the volatility is 10%. Compute the expected
return of a portfolio with $100,000 in the emerging market portfolio, -$50,000 in the foreign currency
bond, and $50,000 in the domestic risk-free asset which earns 5% per year. How does this portfolio
differ from the portfolio that has only an investment in the emerging market portfolio? Which one
would you choose and why? Could you create a portfolio with investments in the emerging market
portfolio, in the emerging market currency risk-free bond and in the risk-free asset which is has the
same mean but a lower volatility?
59
Figure 2.1. Cumulative probability function for IBM and for a stock with same return and
twice the volatility.
Probability
IBM
Stock with twice the volatility
Return in decimal form
The expected return of IBM is 13% and its volatility is 30%. The horizontal line corresponds to a
probability of 0.05. The cumulative probability function of IBM crosses that line at a return almost
twice as high as the cumulative probability function of the riskier stock. There is a 5% chance that
IBM will have a lower return than the one corresponding to the intersection of the IBM cumulative
distribution function and the horizontal line, which is a return of -36%. There is a 5% change that the
stock with twice the volatility of IBM will have a return lower than -0.66%.
60
Figure 2.2. Normal density function for IBM assuming an expected return of 13% and a
volatility of 30% and of a stock with the same expected return but twice the volatility.
Probability density
Probability density function of IBM
Probability density function
of the more volatile stock
Decimal return
This figure shows the probability density function of the one-year return of IBM assuming an
expected return of 13% and a volatility of 30%. It also shows the probability density function of the
one-year return of a stock that has the same expected return but twice the volatility of return of IBM.
61
Figure 2.3. Efficient frontier without a riskless asset. The function represented in the figure gives
all the combinations of expected return and volatility that can be obtained with investments in IBM
and XYZ. The point where the volatility is the smallest has an expected return of 15.6% and a
standard deviation of 26.83%. The portfolio on the upward-sloping part of the efficient frontier that
has the same volatility as a portfolio wholly invested in IBM has an expected return of 18.2%.
62
Figure 2.4. The benefits from diversification. This figure shows how total variance falls as more
securities are added to a portfolio. Consequently, 100% represents the variance of a typical U.S.
stock. As randomly chosen securities are added to the portfolio, its variance falls, but more so if the
stocks are chosen among both U.S. and non-U.S. stocks than only among U.S. stocks.
63
Figure 2.5. The efficient frontier using national portfolios. This figure shows the efficient frontier
estimated using the dollar monthly returns on country indices from 1980 to 1990.
64
Expected returns
Portfolio m
Volatility
Figure 2.6. Efficient frontier with a risk-free asset.
The function giving all the expected return and the volatility of all combinations of holdings in IBM
and XYZ is reproduced here. The risk-free asset has a return of 5%. By combining the risk-free asset
and a portfolio on the frontier, the investor can obtain all the expected return and volatility
combinations on the straight line that meets the frontier at the portfolio of risky assets chosen to form
these combinations. The figure shows two such lines. The line with the steeper slope is tangent to the
efficient frontier at portfolio m. The investor cannot form combinations of the risk-free asset and a
risky portfolio that dominate combinations formed from the risk-free asset and portfolio m.
65
Expected
return
The security
market line
E(Rm)
R
Beta
1
Figure 2.7. The CAPM. The straight line titled the security market line gives the expected return
of a security for a given beta. This line intersects the vertical axis at the risk-free rate and has a value
equal to the expected return on the market portfolio for a beta of one.
66
Box: T-bills.
T-bills are securities issued by the U.S. government that mature in one year or less. They pay
no coupon, so that the investor’s dollar return is the difference between the price paid on sale or at
maturity and the price paid on purchase. Suppose that T-bills maturing in one year sell for $95 per
$100 of face value. This means that the holding period return computed annually is 5.26% (100*5/95)
because each investment of $95 returns $5 of interest after one year.
Exhibit 2.2.1. shows the quotes for T-bills provided by the Wall Street Journal. T-bills are
quoted on a bank discount basis. The price of a T-bill is quoted as a discount equal to:
D(t,t+n/365) = (360/n)(100-P(t,t+n/365))
where D(t,t+n/365) is the discount for a T-bill that matures in n days and P(t,t+n/365) is the price of
the same T-bill. The bank discount method uses 360 days for the year. Suppose that the price of a
90-day T-bill, P(t,t+90/365), is $98.5. In this case, the T-bill would be quoted at a discount of 6.00.
From the discount rate, one can recover the price using the formula:
P(t,t+n/365) = 100 - (n/360)D(t,t+365/n)
For our example, we have 100 - (90/360)6.00 = 98.5.
In the Wall Street Journal, two discounts are quoted for each bill, however. This is because
the dealers have to be compensated for their services. Let’s look at how this works for the bill
maturing in 92 days. For that bill, there is a bid discount of 5.07 corresponding to a price of 98.7043
(100-(92/360)5.07) and asked discount of 5.05 corresponding to a price 98.7096. The dealer buys
at the lower price and sells at the higher price. However, when the dealer quotes discounts, the higher
discount corresponds to the buy price and the lower discount corresponds to the price at which the
bidder sells. This is because to get the price one has to subtract the discount multiplied by the fraction
of 360 days the bill has left before it matures.
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Box: The CAPM in practice.
The CAPM provides a formula for the expected return on a security required by capital
markets in equilibrium. To implement the CAPM to obtain the expected return on a security, we need
to:
Step 1. Identify a proxy for the market portfolio.
Step 2. Identify the appropriate risk-free rate.
Step 3. Estimate the risk premium on the market portfolio.
Step 4. Estimate the $ of the security.
If we are trying to find the expected return of a security over the next month, the next year, or longer
in the future, all steps involve forecasts except for the first two steps. Using discount bonds of the
appropriate maturity, we can always find the risk-free rate of return for the next month, the next year,
or longer in the future. However, irrespective of which proxy for the market portfolio one uses, one
has to forecast its risk premium and one has to forecast the $ of the security.
What is the appropriate proxy for the market portfolio? Remember that the market portfolio
represents how the wealth of investors is invested when the assumptions of the CAPM hold. We
cannot observe the market portfolio directly and therefore we have to use a proxy for it. Most
applications of the CAPM in the U.S. involve the use of some broad U.S. index, such as the S&P500,
as a proxy for the market portfolio. As capital markets become more global, however, this solution
loses its appeal. Investors can put their wealth in securities all over the world. Hence, instead of
investing in the U.S. market portfolio, one would expect investors to invest in the world market
portfolio. Proxies for the U.S. market portfolio are generally highly correlated with proxies for the
world market portfolio, so that using the U.S. market portfolio in U.S. applications of the CAPM is
unlikely to lead to significant mistakes. However, in smaller countries, the market portfolio cannot
be the market portfolio of these countries. For instance, it would not make sense to apply the CAPM
in Switzerland using a Swiss index as a proxy for the market portfolio. Instead, one should use a
world market index, such as the Financial Times-Goldman Sachs World Index or the Morgan StanleyCapital International Word Index.1
Having chosen a proxy for the market portfolio, one has to estimate its risk premium.
Typically, applications of the CAPM use the past history of returns on the proxy chosen to estimate
the risk premium. The problem with doing so is that the resulting estimate of the risk premium
depends on the period of time one looks at. The table shows average returns for the U.S. market
portfolio in excess of the risk-free rate over various periods of time as well as average returns for the
world market portfolio in dollars in excess of the risk-free rate. Estimates of the risk premium used
in practice have decreased substantially over recent years. Many practitioners now use an estimate
of 6%.
How do we get $? Consider a security that has traded for a number of years. Suppose that
the relation between the return of that security and the return on the proxy for the market portfolio
is expected to be the same in the future as it was in the past. In this case, one can estimate $ over the
past and apply it to the future. To estimate $, one uses linear regression. To do so, one defines a
1
For a detailed analysis of the difference in the cost of capital between using a local and a global
index, see Stulz (1995).
68
sample period over which one wants to estimate $. Typically, one uses five or six years of monthly
returns. Having defined the sample period, one estimates the following equation over the sample
period using regression analysis:
RC(t) = c + bRM(t) + e(t)
In this equation, e(t) is residual risk. It has mean zero and is uncorrelated with the return on the
market portfolio. Hence, it corresponds to idiosyncratic risk. The estimate for b will then be used as
the $ of the stock.
Let’ look at an example using data for IBM. We have data from January 1992 to the end of
September 1997, sixty-nine observations in total. We use as the market portfolio the S&P500 index.
Using Excel, we can get the beta estimate for IBM using the regression program in data analysis
under tools (it has to be loaded first if it has not been done already). We use the return of IBM in
decimal form as the dependent or Y variable and the return on the S&P500 as the independent or X
variable. We use a constant in the regression. The Excel output is reproduced below. The estimates
are:
Return on IBM = -0.00123 + 1.371152*Return on S&P500
The standard error associated with the beta estimate is 0.33592. The difference between the beta
estimate and the unknown true beta is a normally distributed random variable with zero mean. Using
our knowledge about probabilities and the normal distribution, we can find that there is a 95% chance
that the true beta of IBM is between 0.7053 and 2.037004. The t-statistic is the ratio of the estimate
to its standard error. It provides a test here of the hypothesis that the beta of IBM is greater than
zero. A t-statistic greater than 1.65 means that we can reject this hypothesis in that there is only a 5%
chance or less that zero is in the confidence interval constructed around the estimate of beta. Here,
the t-statistic is 4.110271. This means that zero is 4.110271 standard errors from 1.371152. The
probability that the true beta would be that many standard errors below the mean is the p-value
0.00011. As the standard error falls, the confidence interval around the coefficient estimates becomes
more narrow. Consequently, we can make stronger statements about the true beta. The R-square
coefficient of 0.201376 means that the return of the S&P500 explains a fraction 0.201376 of the
volatility of the IBM return. As this R-square increases, the independent variable explains more of
the variation in the dependent variable. As one adds independent variables in a regression, the Rsquare increases. The adjusted R-square takes this effect into account and hence is a more useful
guide of the explanatory power of the independent variables when comparing across regressions
which have different numbers of independent variables.
69
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.44875
R Square
0.201376
Adjusted R 0.189457
Square
S t a n d a r d 0.076619
Error
Observations 69
Intercept
X Variable
Coefficients Standard t-Stat
Error
-0.00123
0.010118 -0.12157
1.371152
0.333592 4.110271
P-value
0.903604
0.00011
70
Chapter 3: Creating value with risk management
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section 3.1. Bankruptcy costs and costs of financial distress. . . . . . . . . . . . . . . . . . . . . . 4
Section 3.1.1. Bankruptcy costs and our present value equation. . . . . . . . . . . . . 6
Section 3.1.2. Bankruptcy costs, financial distress costs, and the costs of risk
management programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Section 3.1.3. Bankruptcy costs and Homestake. . . . . . . . . . . . . . . . . . . . . . . . 10
Section 3.2. Taxes and risk management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Section 3.2.1. The tax argument for risk management. . . . . . . . . . . . . . . . . . . 16
1. Carry-backs and carry-forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. Tax shields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3. Personal taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Section 3.2.2. The tax benefits of risk management and Homestake . . . . . . . . . 18
Section 3.3. Optimal capital structure, risk management, bankruptcy costs and taxes. . . 19
Section 3.3.1. Does Homestake have too little debt? . . . . . . . . . . . . . . . . . . . . . 26
Section 3.4. Poorly diversified stakeholders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Section 3.4.1. Risk and the incentives of managers. . . . . . . . . . . . . . . . . . . . . . . 29
Section 3.4.2. Managerial incentives and Homestake. . . . . . . . . . . . . . . . . . . . . 33
Section 3.4.3. Stakeholders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Section 3.4.4. Are stakeholders important for Homestake? . . . . . . . . . . . . . . . . 34
Section 3.5. Risk management, financial distress and investment. . . . . . . . . . . . . . . . . . 35
Section 3.5.1. Debt overhang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Section 3.5.2. Information asymmetries and agency costs of managerial discretion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Section 3.5.3. The cost of external funding and Homestake. . . . . . . . . . . . . . . . 41
Section 3.6. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
BOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 3.1. Cash flow to shareholders and operating cash flow. . . . . . . . . . . . . . . . . . . 49
Figure 3.2. Creating the unhedged firm out of the hedged firm. . . . . . . . . . . . . . . . . . . 50
Figure 3.3. Cash flow to shareholders and bankruptcy costs . . . . . . . . . . . . . . . . . . . . . 51
Figure 3.4. Expected bankruptcy cost as a function of volatility . . . . . . . . . . . . . . . . . . 52
Figure 3.5. Taxes and cash flow to shareholders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 3.6. Firm after-tax cash flow and debt issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Chapter 3: Creating value with risk management
December 1, 1999
© René M. Stulz 1997, 1999
Chapter objectives
1. To understand when risk management creates value for firms.
2. To show how taxes, financial distress and bankruptcy costs, contracting costs, information
asymmetries, managerial incentives, stakeholder interests, large shareholders are reasons why risk
management creates value.
3. To determine what sorts of risk have to be hedged to create value.
Chapter 3, page 1
Chapter 2 shows that a risk management program cannot increase firm value when the cost
of bearing a risk is the same whether the risk is born within or outside the firm. This result is called
the risk management irrelevance proposition. The irrelevance proposition holds when financial
markets are perfect. If the proposition holds, any risk management program undertaken by the firm
can be implemented with the same outcome by an investor through homemade risk management. The
usefulness of the risk management irrelevance proposition is that it allows us to find out when
homemade risk management is not equivalent to risk management by the firm. This is the case
whenever risk management by the firm affects firm value in a way that cannot be mimicked by
investors on their own. In this chapter, we identify situations where there is a wedge between the cost
of bearing a risk within the firm and the cost of bearing it outside the firm. Such a wedge requires the
existence of financial markets imperfections.
To show why the risk management irrelevance proposition is right, chapter 2 uses the example
of a gold producing firm. We continue using this example in this chapter. The firm is exposed to gold
price risk. The risk can be born within the firm. In this case, the firm has low income if the price of
gold is unexpectedly low and high income if it is unexpectedly high. If the irrelevance proposition
holds, the only ex ante cost of bearing this risk within the firm is that shares are worth less if gold
price risk is systematic risk because shareholders require a risk premium for gold price risk. The only
cost to the firm of having the gold price risk born outside the firm is that the firm has to pay a risk
premium to induce capital markets to take that risk. The risk premium capital markets require is the
same as the one shareholders require. Consequently, it makes no difference for firm value whether
the gold price risk is born by shareholders or by the capital markets.
The risk management irrelevance proposition makes clear that for risk management to
Chapter 3, page 2
increase firm value it has to be more expensive to take a risk within the firm than to pay capital
markets to take it. In terms of our gold producing firm example, risk management creates value if an
unexpectedly low gold price has costs for the firm that it would not have for the capital markets. An
example is a situation where, if the gold price is unexpectedly low, the firm does not have funds to
invest and hence has to give up valuable projects. For this situation to occur, it has to be that the firm
finds it costly to raise funds on capital markets if the gold price is unexpectedly low. In this case, the
unexpected low gold price implies that shareholders lose income now, but in addition they lose future
income because the firm cannot take advantage of investment opportunities. The firm does not incur
this additional cost resulting from a low gold price if the gold price risk is born by the capital markets.
To take the gold price risk, the capital markets require a risk premium. The cost of having the capital
markets take the gold price risk is less than the cost the firm has to pay if it bears the risk within the
firm. If the firm bears the risk within the firm, the cost is the risk premium required by the firm’s
claimholders (the shareholders in an all-equity firm; the shareholders and debtholders otherwise) plus
the loss of profits due to the inability to invest optimally if the gold price is low.
In this chapter, we investigate how risk management can be used to increase firm value. We
discuss the reasons why a firm might find it more expensive to bear a risk within the firm than pay the
capital markets to bear that risk. This chapter will therefore show where the benefits of risk
management come from. More precisely, we show how risk management can decrease the present
value of bankruptcy costs and costs of financial distress, how it can decrease the present value of
taxes paid, how it can enable firms to make it more likely that they can take advantage of the positive
net present value projects available to them, and how it enables firms to provide better incentives for
managers. In the next chapter, we integrate these various sources of gain from risk management into
Chapter 3, page 3
an integrated theory of risk management.
Section 3.1. Bankruptcy costs and costs of financial distress.
In our analysis of when risk management does not create value in chapter 2, we take the
distribution of the firm’s cash flow before hedging, the cash flow from operations, as given. We
assume that the firm sells 100m ounces of gold at the end of the year and then liquidates. The firm
has no debt. The gold price is assumed to be normally distributed with a mean of $350 per ounce.
There are no operating costs for simplicity. All of the cash flow accrues to the firm’s shareholders.
This situation is represented by the straight line in figure 3.1. In that graph, we have cash flow to the
firm on the horizontal axis and cash flow to the holders of financial claims against the firm on the
vertical axis. Here, the only claimholders are the shareholders. In perfect financial markets, all cash
flows to the firm accrue to the firm’s claimholders, so that there is no gain from risk management.
In our analysis of chapter 2, the firm has a cash flow at the end of the year. It distributes the
cash flow to its owners, the shareholders, and liquidates. If the firm hedges, it sells its production at
the forward price so that the firm’s owners get the proceeds from selling the firm’s gold production
at the forward price. If the owners receive the proceeds from the hedged firm, they can recreate the
unhedged gold firm by taking a long forward position in gold on personal account as shown in figure
3.2. Suppose the forward price is $350. If the gold price turns out to be $450, for example, the
unhedged firm receives $350 an ounce by delivering on the forward contract while the unhedged firm
would receive $450 an ounce. An investor who owns the hedged firm and took a long forward
contract on personal account receives $350 per ounce of gold from the hedged firm plus $450 - $350
per ounce from the forward contract for a total payoff of $450 per ounce which is the payoff per
Chapter 3, page 4
ounce of the unhedged firm. Hence, even though the firm is hedged, the investor can create for
himself the payoff of the unhedged firm.
Consider now the case where the firm is a levered firm. We still assume that markets are
perfect and that the distribution of the cash flow from operations is given. With our assumptions,
there are no taxes. At the end of the year, the cash flow to the firm is used to pay off the debtholders
and shareholders receive what is left over. In this case, the firm’s claimholders still receive all of the
firm’s cash flow and the firm’s cash flow is not changed by leverage. However, now the claimholders
are the debtholders and the shareholders. Leverage does not affect firm value because the firm’s cash
flow is unaffected by leverage. Hence, leverage just specifies how the pie - the firm’s operating cash
flow - is divided among claimants - the debtholders and the shareholders. Since the cash flow to
claimholders is the firm’s cash flow, risk management does not affect firm value.
Default on debt forces the firm to file for bankruptcy or to renegotiate its debt. With perfect
financial markets, claims on the firm can be renegotiated at no cost instantaneously. If financial
markets are not perfect, this creates costs for the firm. For instance, the firm might have to hire
lawyers. Costs incurred as a result of a bankruptcy filing are called bankruptcy costs. We now
assume that financial markets are imperfect because of the existence of bankruptcy costs. These costs
arise because the firm has debt that it cannot service, so that it has to file for bankruptcy.
Consequently, the value of the unlevered firm exceeds the value of the levered firm by the present
value of the bankruptcy costs. Later, we will find that there are benefits to leverage, but for the
moment we ignore them. In figure 3.3., these bankruptcy costs create a wedge between cash flow to
the firm and cash flow to the firm’s claimholders. This wedge corresponds to the loss of income for
the owners when the firm is bankrupt.
Chapter 3, page 5
The importance of the bankruptcy costs depends on their level and on the probability that the
firm will have to file for bankruptcy. The probability that the firm will be bankrupt is the probability
that its cash flow will be too small to repay the debt. We know how to compute such a probability
from chapter 2 for a normally distributed cash flow. Figure 3.4 shows how the distribution of cash
flow from operations affects the probability of bankruptcy. If the firm hedges its risk completely, it
reduces its volatility to zero since the claimholders receive the present value of gold sold at the
forward price. Hence, in this case, the probability of bankruptcy is zero and the present value of
bankruptcy costs is zero also. As firm volatility increases, the present value of bankruptcy costs
increases because bankruptcy becomes more likely. This means that the present value of cash flow
to the firm’s claimholders falls as cash flow volatility increases. Therefore, by hedging, the firm
increases firm value because it does not have to pay bankruptcy costs and hence the firm’s
claimholders get all of the firm’s cash flow. In this case, homemade risk management by the firm’s
claimholders is not a substitute for risk management by the firm. If the firm does not reduce its risk,
its value is lower by the present value of bankruptcy costs. Consequently, all homemade risk
management can do is eliminate the risk associated with the firm value that is net of bankruptcy costs.
Section 3.1.1. Bankruptcy costs and our present value equation.
We now use our present value equation to show that risk management increases firm value
when the only financial market imperfection is the existence of bankruptcy costs. Remember that in
the absence of bankruptcy costs, the firm’s claimholders receive the cash flow at the end of the year
and the firm is liquidated. With our new assumptions, the claimholders receive the cash flow if the
firm is not bankrupt. Denote this cash flow by C. If the firm is bankrupt, the claimholders receive C
Chapter 3, page 6
minus the bankruptcy costs. Consequently, the value of the firm is now:
Value of firm = PV(C - Bankruptcy costs)
We know from the previous chapter that the present value of a sum of cash flows is the sum of the
present values of the cash flows. Consequently, the value of the firm is equal to:
Value of firm = PV(C) - PV(Bankruptcy costs)
= Value of firm without bankruptcy costs - Present value of bankruptcy costs
Let’s now consider the impact of risk management on firm value. If the hedge eliminates all risk, then
the firm does not incur the bankruptcy costs. Hence, the cash flow to the firm’s owner is what the
cash flow would be in the absence of bankruptcy costs, which is C. This means that with such a hedge
the claimholders get the present value of C rather than the present value of C minus the present value
of bankruptcy costs. Consequently, the gain from risk management is:
Gain from risk management = Value of firm hedged - Value of firm unhedged
= PV(Bankruptcy costs)
Let’s look at a simple example. We assume that the interest rate is 5% and gold price risk is
unsystematic risk. The forward price is $350. Because gold price risk is unsystematic risk, the
forward price is equal to the expected gold price from our analysis of chapter 2. As before, the firm
Chapter 3, page 7
produces 1m ounces of gold. Consequently, PV(C) is equal to $350M/1.05 or $333.33M. The
present value of the hedged firm is the same (this is because E(C) is equal to 1M times the expected
gold price which is the forward price). The present value of the bankruptcy costs requires us to
specify the debt payment and the distribution of the cash flow. Let’s say that the bankruptcy costs are
$20m, the face value of debt is $250m, and the volatility of the gold price is 20%. The firm is
bankrupt if the gold price falls below $250. The probability that the gold price falls below $250 is
0.077 using the approach developed in chapter 2. Consequently, the expected bankruptcy costs are
0.077*20M, or $1.54M. By the use of risk management, the firm insures that it is never bankrupt and
increases its value by the present value of $1.54M. Since gold price risk is assumed to be
unsystematic risk, we discount at the risk-free rate of 5% to get $1.47M (which is $1.54M/1.05).
In the presence of bankruptcy costs, the risk management irrelevance theorem no longer
holds. Because we assume that gold price risk is diversifiable, the cost of having the capital markets
bear this risk is zero. The risk is diversified in the capital markets and hence there is no risk premium
attached to it. In contrast, the cost of bearing the risk in the firm is $1.47M. The capital markets
therefore have a comparative advantage over the firm in bearing gold price risk. Note that if gold
price risk is systematic risk, then capital markets charge a risk premium for bearing gold price risk.
However, this risk premium is the same as the one shareholders charge in the absence of bankruptcy
costs. Hence, the capital markets still have a comparative advantage for bearing risk measured by the
bankruptcy costs saved by having the capital markets bear the risk. There is nothing that shareholders
can do on their own to avoid the impact of bankruptcy costs on firm value, so that homemade risk
management cannot eliminate these costs.
Chapter 3, page 8
Section 3.1.2. Bankruptcy costs, financial distress costs, and the costs of risk management
programs.
So far, we have described the bankruptcy costs as lawyer costs. There is more to bankruptcy
costs than lawyer costs. There are court costs and other administrative costs. An academic literature
exists that attempts to evaluate the importance of these costs. In a recent study of bankruptcy for 31
firms over the period from 1980 to 1986, Weiss (1990) finds that the average ratio of direct
bankruptcy costs to total assets is 2.8%. In his sample, the highest percentage of direct bankruptcy
costs as a function of total assets is 7%. Other studies find similar estimates of direct bankruptcy
costs. In addition, bankruptcy can have large indirect costs. For instance, management has to spend
a lot of time dealing with the firm’s bankruptcy proceedings instead of managing the firm’s
operations. Also, when the firm is in bankruptcy, management loses control of some decisions. For
instance, it might not be allowed to undertake costly new projects.
Many of the costs we have described often start taking place as soon as the firm’s financial
situation becomes unhealthy. Hence, they can occur even if the firm never files for bankruptcy or
never defaults. If management has to worry about default, it has to spend time dealing with the firm’s
financial situation. Management will have to find ways to conserve cash to pay off debtholders. In
doing so, it may have to cut investment which means the loss of future profits. Potential customers
might become reluctant to deal with the firm, leading to losses in sales. All these costs are often called
costs of financial distress. Specifically, costs of financial distress are all the costs that the firm incurs
because its financial situation has become tenuous. All our analysis of the benefit of risk management
in reducing bankruptcy costs holds for costs of financial distress also. Costs of financial distress
amount to a diversion of cash flow away from the firm’s claimholders and hence reduce firm value.
Chapter 3, page 9
Reducing firm risk to decrease the present value of these costs naturally increases firm value.
Reducing costs of financial distress is the most important benefit of risk management. Consequently,
we study in more detail how risk management can be used to reduce specific costs of financial distress
throughout the remainder of this chapter.
In our example, the gold mining firm eliminates all of the bankruptcy costs through risk
management. If there were costs of financial distress that occur when the firm’s cash flow is low, it
obviously could eliminate them as well through risk management. This is not the case in general
because there may be risks that are too expensive to reduce through risk management. We assumed
that there are no costs to reducing risk with risk management. Without risk management costs,
eliminating all bankruptcy and distress risks is optimal.
Transaction costs of risk management can be incorporated in our analysis in a straightforward
way. Suppose that there are transaction costs of taking positions in forward contracts. As transaction
costs increase, risk management becomes less attractive. If the firm bears a risk internally, it does not
pay these transaction costs. The transaction costs of risk management therefore simply increase the
cost of paying the capital markets to take the risk.
Section 3.1.3. Bankruptcy costs and Homestake.
In the previous chapter, we introduced Homestake as a gold mining firm which had a policy
of not hedging gold price exposure. As we saw, management based their policy on the belief that
Homestake’s shareholders value gold price exposure. We showed that this belief is wrong because
investors can get gold price exposure without Homestake on terms at least as good as those
Homestake offers and most likely on better terms. This raises the question of whether, by not
Chapter 3, page 10
hedging, Homestake’s value was lower than it would have been with hedging. Throughout this
chapter, for each source of value of hedging that we document, we investigate whether this source
of value applies to Homestake.
At the end of the 1990 fiscal year, Homestake had cash balances of more than $300 million.
In contrast, its long-term debt was $72 million and it had unused credit lines amounting to $245
million. Homestake could therefore repay all its long-term debt and still have large cash balances.
Bankruptcy is not remotely likely at that point for Homestake. However, suppose that Homestake
had a lot more long-term debt. Would bankruptcy and financial distress costs have been a serious
issue? Homestake’s assets are its mines and its mining equipment. These assets do not lose value if
Homestake defaults on its debt. If it makes sense to exploit the mines, the mines will be exploited
irrespective of who owns them. It follows from this that neither bankruptcy costs nor financial distress
costs provide an important reason for Homestake to practice risk management. Though the reduction
of financial distress costs is the most important benefit of risk management in general, there are firms
for which this benefit is not important and Homestake is an example of such a firm. In the next
chapter, we will consider a type of firm that is at the opposite end of the spectrum from Homestake,
namely financial institutions. For many financial institutions, even the appearance of a possibility of
financial distress is enough to force the firm to close. For instance, in a bank, such an appearance can
lead to a bank run where depositors remove their money from the bank.
Section 3.2. Taxes and risk management.
Risk management creates value when it is more expensive to take a risk within the firm than
to pay the capital markets to bear that risk. Taxes can increase the cost of taking risks within the firm.
Chapter 3, page 11
The idea that the present value of taxes paid can be affected by moving income into future years is
well-established. If a dollar of taxes has to be paid, paying it later is better. Derivatives are sometimes
used to create strategies that move income to later years and we will see such uses of derivatives in
later chapters. However, in this section, we focus on how managing risk, as opposed to the timing
of income, can reduce the present value of taxes. To understand the argument, it is useful to think
about one important tax planning practice. If an individual knows that in some future year, her tax
rate will be less, that individual should try to recognize income in that year rather than in years with
a higher tax rate. Hence, given the opportunity to defer taxation on current income through a pension
plan, the investor would want to do so if the tax rate when taxes are paid is lower than the tax rate
that would apply now to the income that is deferred. With risk management, rather than altering when
income is recognized to insure that it is recognized when one’s tax rate is low, one alters the risks one
takes to decrease income in a given tax year in states of the world where the tax rate is high and
increase income in the same tax year in states of the world in which the tax rate is low. By doing so,
one reduces the average tax rate one pays and hence reduces the present value of taxes paid.
To understand the role of risk management in reducing the present value of taxes, let’s
consider our gold firm. Generally, firm pay taxes only if their income exceeds some level. To reflect
this, let’s assume that the gold firm pays taxes at the rate of 50% if its cash flow exceeds $300M and
does not pay taxes if its cash flow is below $300M. For simplicity, we assume it is an all-equity firm,
so that there are no bankruptcy costs. Figure 3.5. graphs the after-tax cash flow of the firm as a
function of the pretax cash flow. Note that now there is a difference between the firm’s operating
cash flow and what its shareholders receive which is due to taxes. To simplify further, we assume that
there is a 50% chance the gold price will be $250 per ounce and a 50% chance it will be $450. With
Chapter 3, page 12
this, the expected gold price is $350. Assuming that gold price risk is unsystematic risk, the forward
price for gold is $350. As before, the interest rate is 5%. In the absence of taxes, therefore, the value
of the gold mining firm is the present value of the expected cash flow, $350M discounted at 5%, or
$333.33M. Now, with taxes, the present value of the firm for its shareholders is lower since the firm
pays taxes when the gold price is $450. In this case, the firm pays taxes of 0.5*(450-300)*1M, or
$75M. With taxes, the value of the firm’s equity is:
Value of firm with taxes
= PV(Gold sales - Taxes)
= PV(Gold sales) - PV(Taxes)
= PV(Firm without taxes) - PV(Taxes)
= $333.33m - $35.71M
= $297.62M.
Let’s figure out the cost to shareholders of having the firm bear gold price risk. To do that,
we have to compare firm value if the gold price is random with firm value if the gold price has no
volatility and is fixed at its expected value. Remember that the gold price can be either $250 or $450.
If the gold price is $250, the shareholders get $250 per ounce. Alternatively, if the gold price is $450,
they get $375 per ounce ($450 minus taxes at the rate of 50% on $150). The expected cash flow to
the shareholders is therefore 0.5*250 + 0.5*375 or $312.5 per ounce, which is $12.50 less than $325.
In this case, expected taxes are $37.5 per ounce. If the gold price is fixed at its expected value instead
so that cash flow is not volatile, shareholders receive $325 per ounce since they must pay taxes at the
rate of 50% on $50. In this case, expected taxes are $25 per ounce. Taking present values, the equity
Chapter 3, page 13
value is $309.52 per ounce in the absence of cash flow volatility and $297.62 if cash flow is volatile.
Hence, the cost to the shareholders of having the firm bear the gold price risk is $11.90 per ounce
or $11.90M for the firm as a whole.
Let’s look at the cost of having the capital markets bear the gold price risk. If gold price risk
is unsystematic risk, there is no risk premium. The capital markets charge a cost for bearing gold price
risk with our assumptions only if taxes affect the pricing of risky assets. Tax status differs across
investors. For instance, some investors pay no taxes because they are institutional investors; other
investors pay taxes as individuals in the top tax bracket. Let’s suppose that capital markets are
dominated by investors who pay no taxes and that therefore securities are priced ignoring taxes. In
this case, the CAPM applies to pretax returns and the cost of bearing gold price risk for the capital
markets is zero. Hence, the capital markets have a comparative advantage in bearing gold price risk.
If the firm has the capital markets bear gold price risk, it sells gold forward and gets $350m.
Shareholders get less because the firm has to pay taxes. They get $325m or $309.52m in present
value. This is the value of the firm if gold is not random. Hence, capital markets charge nothing for
bearing the risk, but if the firm bears the risk itself, the cost to the firm is $11.90m.
The reason the firm saves taxes through risk management is straightforward. If the firm’s
income is low, the firm pays no taxes. In contrast, if the firm’s income is high, it pays taxes. Shifting
a dollar from when the income is high to when the income is low saves the taxes that would be paid
on that dollar when the income is high. In our example, shifting income of a dollar from when income
is high to when income is low saves $0.50 with probability 0.5.
It should be clear why homemade risk management cannot work in this case. If the firm does
not use risk management to eliminate its cash flow volatility, its expected taxes are higher by $12.5M.
Chapter 3, page 14
This is money that leaves the firm and does not accrue to shareholders. If the firm does not hedge,
shareholders can eliminate the risk from holding the shares. However, through homemade risk
management, shareholders can only guarantee an expected payoff of $312.5M. Let’s figure out how
shareholders would practice homemade risk management. Note that there are only two possible
outcomes: $375 per share or $250 per share with equal probability. Hence, shareholders must take
a forward position per ounce that insures a payoff of $312.5. Let h be the short forward position per
ounce. We then need:
Gold price is $250:
$250 - h(350 - 250) = 312.5
Gold price is $450:
$375 - h(350 - 450) = 312.5
Solving for h, we get h to be equal to 0.6125. Hence, the investor must sell short 0.6125 ounces to
insure receipt of $312.5 per ounce at the end of the year. If the gold price is $250 an ounce, the
shareholders get $250 per share from the firm and 0.6125*(350 - 250), or $75, from the forward
position. This amounts to $325. The investor is unambiguously better off if the firm hedges directly.
The marginal investor is the one that would not invest in an asset if its expected return was
slightly less. If gold price risk is diversifiable for this investor, this means that the investor does not
expect to make a profit from taking gold price risk net of taxes. So far, we assumed that the marginal
investor pays no taxes. Suppose that, instead, the marginal investor has a positive tax rate. It turns
out that making this different assumption does not change our analysis as long as the marginal
investor’s tax rate is not too different when the gold price is high and when it is low. Since gold price
risk is diversifiable, one would not expect an unexpectedly high or low gold price to change the
Chapter 3, page 15
marginal investor’s tax bracket. Let’s see that this works. Suppose that the tax rate of the marginal
investor is 40% and that this tax rate is the same for the high and the low gold price. Remember that,
since the gold price risk is unsystematic risk for the marginal investor, the investor’s after-tax
expected profit from a forward contract should be zero. In this case, if the investor takes a long
forward position, she expects to receive after tax:
0.5*(1-0.4)*(450 - F) + 0.5*(1-0.4)*(250 - F) = 0
The forward price that makes the expected payoff of a long forward position equal to zero is $350
in this case. Hence, the tax rate does not affect the forward price. This means that when the marginal
investor has a constant positive tax rate, the capital markets still charge nothing to bear unsystematic
gold price risk.
Section 3.2.1. The tax argument for risk management.
The tax argument for risk management is straightforward: By taking a dollar away from a
possible outcome (often called a state of the world in finance) in which it incurs a high tax rate and
shifting it to a possible outcome where it incurs a low tax rate, the taxpayer - a firm or an investor reduce the present value of taxes to be paid. The tax rationale for risk management is one that is
extremely broad-based: it applies whenever income is taxed differently at different levels. The tax
code introduces complications in the analysis. Some of these complications decrease the value of
hedging, whereas others increase it. Here are some of these complications:
1. Carry-backs and carry-forwards. If a firm has negative taxable income, it can offset
Chapter 3, page 16
future or past taxable income with a loss this tax year. There are limitations to the ability to carry
losses back or forward. One set of limitations is that losses can be carried back or forward only for
a limited number of years. In addition, no allowance is made for the time-value of money. To see the
importance of the time-value of money, consider a firm which makes a gain of $100,000 this year and
a loss of $100,000 in three years. It has no other income. The tax rate is 30%. Three years from now,
the firm can offset the $100,000 gain of this year with the loss. However, it must pay $30,000 in taxes
this year and only gets back $30,000 in three years, so that it loses the use of the money for three
years.
2. Tax shields. Firms have a wide variety of tax shields. One of these tax shields is the tax
shield on interest paid. Another is the tax shield on depreciation. Firms also have tax credits. As a
result of the complexity of the tax code, the marginal tax rate of firms can be quite variable. Further,
tax laws change so that at various times firms and investors know that taxes will increase or fall. In
such cases, the optimal risk management program is one that increases cash flows when taxes are low
and decreases them when they are high.
3. Personal taxes. In our discussion, we assumed that the tax rate of the marginal investor
on capital markets is uncorrelated with firm cash flow. Even if this is not the case, a general result
holds: As long as the tax rate of investors differs from the tax rate of the corporation, it pays for the
corporation to reallocate cash flow from possible outcomes where the combined tax rate of investors
and the corporation is high to other possible outcomes where it is low.
It is difficult to incorporate all these real life complications in an analytical model to evaluate the
importance of the tax benefits of risk management. To cope with this problem, Graham and Smith
Chapter 3, page 17
(1999) use a simulation approach instead. Their paper does not take into account personal taxes, but
otherwise it incorporates all the relevant features of the tax code. They simulate a firm’s income and
then evaluate the tax benefit of hedging. They find that for about half the firms, there is a tax benefit
from hedging, and for these firms the typical benefit is such that a 1% decrease in the volatility of
taxable income for a given year decreases the present value of taxes by 1%.
Section 3.2.2. The tax benefits of risk management and Homestake
In 1990, Homestake paid taxes of $5.827 million dollars. It made a loss on continuing
operations because it wrote down its investment in North American Metals Corporation. Taxation
in extracting industries is noticeably complicated and hence makes it difficult to understand the extent
to which Homestake could reduce the present value of its taxes with risk management. However, the
annual report is useful in showing why the tax rate of Homestake differs from the statutory tax rate
of 34%. In thousand dollars, the taxes of Homestake differ from the statutory tax as follows:
Homestake made a loss of 13,500. 34% of that loss would yield taxes of
(4,600)
Depletion allowance
(8,398)
State income taxes, net of Federal benefit
Nondeductible foreign losses
(224)
18,191
Other-net
858
Total
$ 5,827
It is striking that Homestake paid taxes even though it lost money. The exact details of the
nondeductible foreign losses are not available from the annual report. It seems, however, that part
Chapter 3, page 18
of the problem is due to a non-recurring write-down which may well have been unpredictable.
Although risk management to smooth taxes is valuable, it is unlikely that Homestake could have
devised a risk management program that would have offset this write-down with foreign taxable
gains. At the same time, variations in the price of gold could easily lead to a situation where
Homestake would make losses. Avoiding these losses would smooth out taxes over time and hence
would increase firm value. Based on the information in the annual report, we cannot quantify this
benefit. Petersen and Thiagarajan (1998) compare American Barrick and Homestake in great detail.
In their paper, they find that Homestake has a tendency to time the recognition of expenses when gold
prices are high to smooth income, which may decrease the value of using derivatives to smooth
income.
Section 3.3. Optimal capital structure, risk management, bankruptcy costs and taxes.
Generally, interest paid is deductible from income. A levered firm therefore pays less in taxes
than an unlevered firm for the same operating cash flow. This tax benefit of debt increases the value
of the levered firm relative to the value of the unlevered firm. If a firm’s pretax income is random,
however, there will be circumstances where it cannot take full advantage of the tax benefit of debt
because its operating cash flow is too low. Hence, decreasing the volatility of cash flow can increase
firm value by making it less likely that the operating cash flow is too low to enable the firm to take
advantage of the tax shield of debt.
Let’s consider the impact on the tax shield from debt of a risk management program that
eliminates cash flow volatility. We use the same example as we did in the previous section, in that the
firm pays taxes if its cash flow exceeds $300M. However, now the firm issues debt and pays out the
Chapter 3, page 19
proceeds of debt to the shareholders. Suppose that the shareholders decide to issue debt with face
value of $200M. Since the risk of the firm is diversifiable, the debtholders expect to receive a return
of 5%. Since the firm is never bankrupt with an interest rate of 5%, the debt interest payment will be
5% of $200M, or $10M. At the end of the year, the firm has to make a debt payment of $210M. If
the firm’s cash flow at the end of the year is $250M because the gold price is $250 an ounce, the firm
pays no taxes. Hence, it does not get a tax benefit from having to pay $10M in interest. In contrast,
if the firm’s cash flow is $450M, the firm gets to offset $10M against taxable income of $150M and
hence pays taxes only on $140M. This means that the value of the firm is:
Value of levered firm = PV(Cash flow - taxes paid)
= PV(Cash flow) - PV(Taxes paid)
= 333.33M - 0.5*0.5*(450M - 300M - 10M)/1.05
= 333.33M - 33.33M
= 300M
The value of the firm is higher than in the previous section because the firm benefits from a tax shield
of $10m if the gold price is $450 per ounce. The expected value of that tax shield is 0.5*0.5*10M,
or $2.50M. The present value of that amount is $2.38M using the risk-free rate of 5% as the discount
rate since gold price risk is unsystematic risk. In the previous section, the value of the firm in the
absence of risk management was $297.62M, which is exactly $2.38M less than it is now. The
difference between $300M and $297.62M represents the benefit of the tax shield of debt. However,
when the cash flow is risky, the firm gets the benefit only when the price of gold is $450 an ounce.
Chapter 3, page 20
Consider the situation where the firm eliminates the volatility of cash flow. We know that in
this case the firm insures a cash flow pre-tax of $350M. Its taxable income with debt is $350M $10M - $300M, or $40M, so that it pays taxes of $20M for sure. Firm value is therefore the present
value of $350M - $20M, or $314.286M. In the absence of debt, hedged firm value is instead
$309.524M. Shareholders therefore gain $4.762M by issuing debt and hedging. The benefit from
hedging in this example comes solely from the increase in the tax shield of debt. Since the firm issues
risk-free debt, the interest rate on the debt does not depend on whether the firm hedges or not in our
example.
In our analysis so far, we took the firm’s debt as given. With a promised payment to
debtholders of $210M, the firm is never bankrupt yet benefits from the tax shield of debt. Let’s now
extend the analysis so that the tax shield of debt is maximized. If this involves increasing the firm’s
debt, we assume that the funds raised are paid out to shareholders in the form of a dividend so that
the firm’s investment policy is unaffected.1 Let’s start with the case where the firm eliminates gold
price risk and can guarantee to the debtholders that it will do so. Since in that case the firm has no
risk, it either never defaults or always defaults. One would expect the IRS to disallow a deduction
for debt that always defaults, so that the total debt and tax payments cannot exceed $350M for the
firm to benefit from a tax shield of debt. Since the firm never defaults with hedging, it can sell the new
debt so that it earns the risk-free rate of 5% since it is risk-free. To maximize firm value, we maximize
the expected cash flow minus taxes paid:
1
If the funds were kept within the firm, they would have to be invested which would
complicate the analysis because taxes would have to be paid on the investment income.
Chapter 3, page 21
Firm cash flow = 350M - 0.5Max(50M - 0.05F,0)
where F is the amount borrowed. Since the firm has no volatility, it is never bankrupt. Figure 3.6.
plots firm value imposing the constraint that total debt and tax payments cannot exceed $350M. Since
the tax shield increases with debt payments, the firm wants to issue as much debt as it can without
being bankrupt. This means that taxes plus debt payments have to equal $350M. Solving for F, we
get $317.073M. To see that this works, note that the firm has to pay taxes on 50M - 10M 0.05*117.073M, corresponding to taxes of $17.073M. The debt payments amount to $210M plus
1.05*$117.073, or $332.927M. The sum of debt payments and taxes is therefore exactly $350M. The
shareholders get nothing at maturity, so that the firm is an all-debt firm after the additional debt issue
(one could always have the shareholders hold a small amount of equity so that the firm is not an alldebt firm). They benefit from the increase in leverage because they receive a dividend worth
$317.073M when the debt is issued. If the firm issues only debt with principal of $200M, the
shareholders own a firm worth $314.286M in the form of a dividend of $200M and shares worth
$114.286. Hence, by issuing more debt, shareholders increased their wealth by a further $2.787M.
To check that the shareholders have chosen the right amount of debt, let’s see what happens
if they issue an additional $1M of debt that they pay out as a dividend. This would lead to an
additional debt payment of $1.05M. Taxes would fall by $0.025M because of the addition $0.05M
of interest payment, so that the firm would have to pay net an additional $1.025M. As a result of this
additional promised payment, the firm would always bankrupt. Suppose next that the firm issues $1M
less of debt. In this case, the dividend falls by $1M and the firm has $1.05M less of debt payments
at the end of the year. This has the effect of increasing taxes by $0.025M, so that shareholders get
Chapter 3, page 22
$1.025M at the end of the year. The present value of $1.025M is less than $1M, so that shareholder
equity falls as a results of decreasing debt.
Suppose now that the firm wants to maximize its value without using risk management. In this
case, if the optimum amount of debt is such that the debt payment exceeds $250M, the firm is
bankrupt with probability 0.5. Since, in this case, increasing the principal amount of debt further does
not affect bankruptcy costs, the firm’s best strategy would be to issue a sufficient amount of debt so
that it minimizes taxes paid when the gold price is $450 an ounce since it pays no taxes when the gold
price is $250 an ounce. Consequently, the firm chooses debt so that the debt payment plus taxes paid
are equal to the before-tax cash flow of $450M. If the firm does that, it will be bankrupt if the gold
price is $250 an ounce. If bankruptcy takes place, the firm incurs a bankruptcy cost which we assume
to be $20M as before so that bondholders get $230M in that case. Since the firm defaults when the
gold price is $250 an ounce, the yield of the debt has to exceed the risk-free rate so that the expected
return of the bondholders is the risk-free rate. The bondholders require an expected return equal to
the risk-free rate because we assumed that gold price risk is nonsystematic risk. Solving for the
promised interest rate and the amount borrowed, we find that the firm borrows a total amount of
316.29M promising to pay interest at the rate of 37.25%.2 The value of the firm to its shareholders
2
Let x be the promised interest rate on debt and F be the amount borrowed in millions. To
solve for x and F, we have to solve two equations:
0.5 * 230 + 0.5(1 + x )F
=F
1.05
450 - 0.5(150 - xF) - (1 + x)F = 0
The first equation states that the present value of the debt payoffs has to be equal to the principal.
The second equation states that the cash flow if the gold price is $450 an ounce has to be equal to
the debt payment and taxes.
Chapter 3, page 23
corresponds to the amount borrowed of 316.29M, so that the value of the firm is lower in the absence
of risk management.
The difference between firm value with risk management and firm value without risk
management depends crucially on the bankruptcy cost. As the bankruptcy cost increases, the risk
management solution becomes more advantageous. If the bankruptcy cost is $100M, the value of the
firm without risk management falls to $290.323. In this case, the firm has a higher value with no debt.
If there is no bankruptcy cost, the solution that involves no risk management has higher value than
the solution with risk management because it eliminates more taxes when the firm is profitable. In this
case, firm value is $322.581M. The problem with the solution without risk management is that it may
well not be possible to implement it: It involves the firm going bankrupt half the time in this example,
so that the IRS would probably disallow the deduction of the interest payment from taxable income,
viewing the debt as disguised equity.
It is important to note that if the firm engages in risk management but debtholders do not
believe that it will do so, then the debt is sold at a price that reflects the absence of risk management.
In this case, the yield on debt where there is a probability of default is higher to reflect the losses
bondholders make if the firm defaults. If shareholders are paying the bondholders for the risk of
bankruptcy, it may not make sense to remove that risk through risk management because doing so
might benefit mostly bondholders. If the firm issues debt so that there is no significant risk of
bankruptcy, it matters little whether bondholders believe that the firm will hedge or not. However,
many firms could increase their leverage without significantly affecting their probability of
bankruptcy. For such firms, risk management would be valuable because it implies that firms capture
Chapter 3, page 24
more of the tax shield of debt.
In the example discussed here, firm value depends on leverage because of the existence of
corporate taxes. If there is a tax shield of debt, more leverage decreases taxes to be paid. However,
with bankruptcy costs, there is an offsetting effect to the tax benefit of debt. As leverage increases,
it becomes more likely that the firm will be bankrupt and hence that bankruptcy costs will have to be
paid. This reasoning leads to an optimal capital structure for a firm. Without bankruptcy costs but a
tax shield of debt, the optimal capital structure would be for the firm to be all debt since this would
maximize the tax shield of debt. With bankruptcy costs but without a tax shield of debt, the firm
would not want any debt since there is no benefit to debt but only a cost. With bankruptcy costs and
with a tax shield of debt, the firm chooses its capital structure so that the tax benefit of an additional
dollar of debt equals the increase in bankruptcy costs. When bankruptcy costs are nontrivial, the firm
finds it worthwhile to decrease risk through risk management so that it can increase leverage. In
general, the firm cannot eliminate all risk through risk management so that there is still some
possibility of bankruptcy. As a result, the optimal solution is generally far from having only debt.
Further, as we will see later, there are other reasons for firms not to have too much debt.
All our tax discussion has taken place assuming a very simple tax code. One complication we
have ignored is that investors pay taxes too. Miller (1978) has emphasized that this complication can
change the analysis. Suppose investors pay taxes on coupon income but not on capital gains. In this
case, the required expected return on debt will be higher than on equity to account for the fact that
debt is taxed more highly than equity at the investor level. In this case, the tax benefit at the
corporation level of debt could be decreased because of the higher yield of debt. At this point,
though, the consensus among financial economists is that personal taxes limit the corporate benefits
Chapter 3, page 25
from debt but do not eliminate them. In any case, if the firm has tax shields, whether there are
personal taxes or not, the corporation will want to maximize their value.
Section 3.3.1. Does Homestake have too little debt?
The answer to this question is undoubtedly yes. Homestake is paying taxes every year. Most
years, its tax rate is close to the statutory rate of 34%. In 1990, as we saw, Homestake pays taxes at
a rate that even exceeds the statutory rate. Yet, we also saw that Homestake has almost no debt and
that its long-term debt is dwarfed by its cash balances. By increasing its debt, Homestake would
increase its tax shield of debt and reduce its taxes. Doing so, however, it would increase the
importance of risk management to avoid wasting the tax shield of debt when the price of gold moves
unfavorably.
Section 3.4. Poorly diversified stakeholders.
So far, we assumed that risk management involves no transaction costs. Suppose, however,
that homemade risk management is more expensive than risk management by the firm. In this case,
investors might care about the risks that the firm bears. However, investors who own large diversified
portfolios are relatively unaffected by the choices of an individual firm. On average, the risks balance
out except for systematic risks that have to be born by the economy as a whole and can easily be
controlled by an investor through her asset allocation. There are, however, investors for whom these
risks do not balance out because they have a large position in a firm relative to their wealth. These
investors might be large investors who value a control position. They might also be management who
has a large stake in the firm for control reasons or because of a compensation plan. Investors who
Chapter 3, page 26
cannot diversify away firm-specific risk want the firm to decrease it unless they can reduce risk more
cheaply through homemade risk management.
Consider the case of our gold producing firm. Now, this firm has one big shareholder whose
only investment is her holding of 10% of the shares of the firm. The shareholder is not diversified and
consequently cares about the diversifiable risk of the gold mining firm. Unless she strongly expects
the price of gold to be high at the end of the year, she wants to reduce the risk of her investment. To
do that, she could sell her stake and invest in a diversified portfolio and the risk-free asset. Second,
she could keep her stake but use homemade hedging. Third, she could try to convince the firm to
hedge.
Let’s assume that the investor wants to keep her stake intact. There are circumstances such
that the investor might prefer the firm to hedge instead of having to do it herself. This will be the case
when there are large setup costs for trading in financial instruments used to hedge and the firm has
already paid these costs. For instance, suppose that a banking relationship is required to set up the
appropriate hedge. The firm has such a relationship and the investor does not. In such a situation, the
investor would want the firm to hedge because homemade hedging is not possible. It is therefore
possible for the firm to have a comparative advantage in hedging. It is not clear, though, why the firm
would expend resources to hedge to please that large investor. If the only benefit of hedging is that
this large investor does not have to hedge on her own, the firm uses resources to hedge without
increasing firm value. If the large shareholder cannot hedge and sells her shares, she is likely to be
replaced by well-diversified shareholders who would not want the firm to pay to hedge and thereby
decrease its value. There is no clear benefit for the firm from having the large shareholder in our
example. If the firm gains from having the large shareholder, then it can make sense to hedge to make
Chapter 3, page 27
it possible for the large shareholder to keep her investment in the firm.
Having large shareholders can increase firm value. Small, highly diversified, shareholders have
little reason to pay much attention to what the firm is doing. Such shareholders hold the market
portfolio and cannot acquire easily information that allows them to beat the market. Evaluating the
actions of management takes time. Suppose that by spending one month of her time a shareholder has
a 10% chance of finding a way to increase the value of a firm by 5% and that, if this happens,
spending a month to convince management has a 20% chance of success. With these odds, a
shareholder whose time is worth $10,000 a month would need an investment in the firm of at least
$500,000 to justify starting the process of studying the firm. One might argue about the odds we give
the shareholder of finding something useful and of convincing management that she did. Most likely,
however, the true odds are much worse for the shareholder. Further, most diversified shareholders
have a smaller stake in a firm than $500,000. Hence, most diversified shareholders get no benefit
from evaluating carefully the actions of managers. A shareholder who has a stake of $10M in a firm
will follow the actions of management carefully even if the odds against her finding something that
would increase the value of the firm are worse than we assumed. The action of evaluating
management and trying to improve what it does is called monitoring management. Large
shareholders get greater financial benefits from monitoring management than small ones.
There are two reasons why monitoring by shareholders can increase firm value. First, an
investor might become a large shareholder because she has some ability in evaluating the actions of
management in a particular firm. Such an investor has knowledge and skills that are valuable to the
firm. If management chooses to maximize firm value, management welcomes such an investor and
listens to her carefully. Second, management does not necessarily maximize firm value. Managers
Chapter 3, page 28
maximize their welfare like all economic agents. Doing so sometimes involves maximizing firm value,
but other times it does not. What a manager does depends on his incentives. A manager whose only
income is a fixed salary from the firm wants to make sure that the firm can pay his salary. If an action
increases firm value but has much risk, that manager may decide against it because a firm that is
bankrupt cannot pay his salary. A large shareholder can make it more likely that management
maximizes firm value by monitoring management. For instance, a large shareholder might find that
management failed to take an action that maximizes firm value and might draw the attention of other
shareholders to her discovery. In some cases, a large shareholder might even convince another firm
to try to make a takeover attempt to remove management and take actions that maximize firm value.
A firm’s risk generally makes it unattractive for a shareholder to have a stake large enough that it is
worthwhile for him to monitor a firm. By hedging, a firm can make it more attractive for a
shareholder that has some advantage in monitoring management to take a large stake. As the large
shareholder takes such a stake, all other shareholders benefit from his monitoring.
Section 3.4.1. Risk and the incentives of managers.
Since managers, like all other individuals associated with the firm, pursue their own interests,
it is important for shareholders to find ways to insure that managers’ interests are to maximize the
value of the shares. One device at the disposal of shareholders for this purpose is the managerial
compensation contract. By choosing a managerial contract which gives managers a stake in how well
the firm does, shareholders help insure that managers are made better off by making shareholders
richer. If managers earn more when the firm does better, this induces them to work harder since they
benefit more directly from their work. However, managerial compensation related to the stock price
Chapter 3, page 29
has adverse implications for managers also. It forces them to bear risks that have nothing to do with
their performance. For instance, a firm may have large stocks of raw materials that are required for
production. In the absence of a risk management program, the value of these raw materials fluctuates
over time. Random changes in the value of raw materials may be the main contributors to the
volatility of a firm’s stock price, yet management has no impact on the price of raw materials. Making
managerial compensation depend strongly on the stock price in this case forces management to bear
risks, but provides no incentive effects and does not align management’s incentives with those of
shareholders. In fact, making managerial compensation depend strongly on the part of the stock
return which is not under control of management could be counterproductive. For instance, if the
value of raw materials in stock strongly affects the riskiness of managerial compensation, managers
might be tempted to have too little raw materials in stock.
If it is easy to know how variables that managers do not control affect firm value, then it
would be possible to have a management compensation contract that depends only on the part of the
stock return that is directly affected by managerial effort. Generally, however, it will be difficult for
outsiders to find out exactly which variables affect firm value. Hence, in general it will make sense
to tie managerial compensation to some measure of value created without trying to figure out what
was and what was not under management’s control. Management knows what is under its control.
It could reduce risk through homemade hedging, but it might be more cost effective to have the firm
hedge rather than having managers trying to do it on their own. If the firm can reduce its risk through
hedging, firm value depends on variables that management controls, so that having compensation
closely related to firm value does not force management to bear too much risk and does not induce
management to take decisions that are not in the interest of shareholders to eliminate this risk. This
Chapter 3, page 30
makes it possible to have managerial compensation closely tied to firm value, which means that when
managers work hard to increase their compensation, they also work hard to increase shareholder
wealth.
Having management own shares in the firm they manage ties their welfare more closely to the
welfare of the shareholders. Again, if management owns shares, they bear risk. Since managers are
not diversified shareholders, they care about the firm’s total risk. This may lead them to be
conservative in their actions. If the firm reduces risk through risk management, the total risk of the
firm falls. Consequently, managers become more willing to take risks. This means that firm-wide
hedging makes managerial stock ownership a more effective device to induce management to
maximize firm value.
Another argument can be made to let management implement a risk management program
within the firm. Suppose that instead of having compensation depend directly on firm value, it
depends on firm value indirectly in the sense that management’s employment opportunities depend
on the performance of the firm ‘s stock. In this case, in the absence of firm-made risk management,
firm value fluctuates for reasons unrelated to managerial performance. As a result, the market’s
perception of managers can fall even when managers perform well. This is a risk that managers cannot
diversify. Having a risk management program eliminates sources of fluctuation of the firm’s market
value that are due to forces that are not under control of management. This reduces the risk attached
to management’s human capital. Again, management is willing to have a lower expected
compensation if the risk attached to its human capital is lower. Hence, allowing management to have
a risk management program has a benefit in the form of expected compensation saved which increases
firm value. Possibly the greater benefit from allowing management to have a risk management
Chapter 3, page 31
program is that this makes it less likely that management undertakes risk reducing activities that
decrease firm value but also decrease the risks that management bears.
Not every form of compensation that depends on firm value induces management to try to
reduce firm risk. Suppose that management receives a large payment if firm value exceeds some
threshold. For instance, in our gold producing firm, suppose that management receives a $20m bonus
if firm value before management compensation exceeds $400m at the end of the year. In this case,
management receives no bonus if the firm hedges and has a 50% chance of getting a bonus if the firm
does not hedge. Obviously, management will not hedge. Management compensation contracts of this
types make management’s compensation a nonlinear function of firm value. If management owns call
options on the firm’s stock, it has incentives to take risks. Call options pay off only when the stock
price exceeds the exercise price. They pay more for large gains in the stock price. An option pays
nothing if the stock price is below the exercise price, no matter how low the stock price is. Hence,
managerial compensation in the form of options encourages management to take risks so that the
stock price increases sufficiently for the options to have value. In fact, management’s incentives might
be to take more risk than is required to maximize firm value.
To see how options might induce management to not hedge when hedging would maximize
firm value, let’s consider our gold firm example. Suppose that management owns a call option on
1,000 shares with exercise price of $350 a share. For simplicity, let’s assume that management
received these options in the past and that exercise of the options does not affect firm value. In our
gold firm example, suppose that there is a tax advantage to hedging as discussed in section 2. In this
case, firm value before managerial compensation is maximized if the firm hedges. Hedging locks in
a firm value before managerial compensation of $309.52. In this case, the options management has
Chapter 3, page 32
are worthless. If the firm does not hedge, there is a 50% chance that the shares will be worth $375
and hence a 50% chance that the options will pay off.
Section 3.4.2. Managerial incentives and Homestake.
The Homestake proxy statement for 1990 shows that the directors own 1.1% of the shares.
As mentioned earlier, two of the directors are executives of a company which owns 8.2% of
Homestake. The CEO of Homestake, Mr. Harry Conger, owns 137,004 shares directly and has the
right to acquire 243,542 shares through an option plan. The shares in the option plan have an average
exercise price of $14.43, but the share price in 1990 has a high of 23.6 and a low of 15.3.
Management holds few shares directly and much less than is typical for a firm of that size. Most of
management’s ownership is in the form of options. There is not much incentive for management to
protect its stake in the firm through hedging and management might benefit some from volatility
through its option holdings.
Section 3.4.3. Stakeholders.
So far in this section, we have considered individuals - large shareholders and managers - for
whom it was costly to be exposed to firm risk. We saw that the firm could benefit by reducing the
firm risk that these individuals are exposed to. The reason these individuals were exposed to firm risk
was that they could not diversify this risk away in their portfolio or use homemade hedging
effectively. There are other individuals associated with corporations that are in a similar situation. All
individuals or firms whose welfare depends on how well the firm is doing and who cannot diversify
the impact of firm risks on their welfare are in that situation. Such individuals and firms are often
Chapter 3, page 33
called stakeholders. It can be advantageous for a firm to reduce the risks that its stakeholders bear.
Often, it is important for the firm to have its stakeholders make long-term firm-specific investments.
For instance, the firm might want workers to learn skills that are worthless outside the firm. Another
example might be a situation where a firm wants a supplier to devote R&D to design parts that only
that firm will use. A final example is one where customers have to buy a product whose value
depends strongly on a warranty issued by the firm. In all these cases, the stakeholders will be reluctant
to make the investments if they doubt that the firm will be financially healthy. If the firm gets in
financial trouble, it may not be able to live up to its part of the bargain with the stakeholders. This
bargain is that the stakeholders invest in exchange for benefits from the firm over the long-term.
Hedging makes it easier for the firm to honor its bargain with the stakeholders.
If the firm does not reduce its risk, it may only be able to get the stakeholders to make the
requisite investments by bribing them to do so. This would mean paying workers more so that they
will learn the requisite skills, paying the suppliers directly to invest in R&D, and selling the products
more cheaply to compensate for the risks associated with the warranty. Such economic incentives can
be extremely costly. If hedging has low costs, it obviously makes more economic sense for the firm
to hedge rather than to use monetary incentives with its stakeholders.
Section 3.4.4. Are stakeholders important for Homestake?
No. The firm is financially healthy so that there is no risk of bankruptcy. The firm could suffer
large losses before bankruptcy would become an issue. Hence, those having relationships with the
firm have no good reason to worry about getting paid.
Homestake has one large shareholder, Case, Pomeroy and Co. This company owns 8.1% of
Chapter 3, page 34
the shares. Two executives of that company are represented on the board of directors. Case has been
decreasing its stake in Homestake and has a standstill agreement with Homestake that prevents it
from buying more shares and gives Homestake rights of first refusal when Case sells shares. This
large shareholder can hedge on its own if it chooses to do so.
Section 3.5. Risk management, financial distress and investment.
So far, we have paid little attention to the fact that firms have opportunities to invest in
valuable projects. For instance, suppose that the gold firm we focused on does not liquidate next
period but instead keeps producing gold. Now, this firm has an opportunity to open a new mine a
year from now. This mine will be profitable, but to open it a large investment has to be made. If the
firm does not have sufficient internal resources, it has to borrow or sell equity to finance the opening
of the mine. There are circumstances where a firm’s lack of internal resources makes it impossible
for the firm to take advantage of projects that it would invest in if it had more internal resources
because the costs of external financing are too high. In other words, it could be that the gold mining
firm in our example might not be able to open the mine because of a lack of internal resources. In this
case, the shareholders would lose the profits that would have accrued to them if the mine had been
opened. Firm value would have been higher had the firm managed its affairs so that it would not have
gotten itself into a situation where it cannot invest in profitable projects. In the remainder of this
section, we investigate the main reasons why firms might not be able to invest in profitable projects
and show how risk management can help firms avoid such situations.
Chapter 3, page 35
Section 3.5.1. Debt overhang.
Consider a firm with debt obligations that are sufficiently high that if the firm had to pay off
the debt today it could not do so and normal growth will not allow it to do so when the debt matures.
Such a situation creates a conflict between the firm’s creditors and the firm’s shareholders.
Shareholders want to maximize the value of their shares, but doing so can be inconsistent with
maximizing firm value and can reduce the value of the firm’s debt. First, shareholders may want to
take risks that are not beneficial to the firm as a whole. Second, they may be unwilling to raise funds
to invest in valuable projects. We look at these problems in turn.
We assumed that the firm cannot pay off its debt today. If the firm does not receive good
news, when the debt has to be paid off, shareholders will receive nothing and the creditors will own
the firm. Consequently, shareholders want to make it more likely that the firm will receive good news.
To see the difficulty this creates, consider the extreme case where the firm has no risk with its current
investment policies. In this case, unless shareholders do something, their equity is worthless. Suppose
however that they liquidate existing investments and go to Las Vegas with the cash they have
generated. They bet the firm at a game of chance. If they lose, the cash is gone. The loss is the
creditors’ loss since equityholders were not going to get anything anyway. If they win, the firm can
pay off the creditors and something is left for the equityholders. This strategy has a positive expected
profit for the shareholders so they want to undertake it - if they can. In contrast, this strategy has a
negative expected profit for the firm. This is because betting in Las Vegas is not a fair gamble - the
house has to make money on average. Excess leverage resulting from adverse shocks to firm value
therefore leads shareholders to do things that hurt firm value but help them. The possibility that there
is some chance that the firm might be in such a situation therefore reduces its value today. If low cost
Chapter 3, page 36
risk management can decrease the probability that the firm might ever find itself in such a situation,
it necessarily increases firm value. Unfortunately, if the firm reduces risk through risk management
but nevertheless finds itself in a situation where shareholders believe that gambling would benefit
them, they may choose to give up on risk management. Since they would give up on risk management
precisely when bondholders would value it the most -- because any loss is a loss for the bondholders
-- this possibility can make it harder to convince bondholders that the firm will consistently reduce
risk.
Consider again a firm with large amounts of debt that it could not repay if it had to do so
today. In contrast to the previous paragraph, we now assume that the firm cannot sell its assets. This
could be because bond covenants prohibit it from doing so or because the market for the firm’s assets
is too illiquid. Now, suppose that this firm has an investment opportunity. By investing $10m., the
firm acquires a project that has a positive net present value of $5m. This project is small enough that
the firm still could not repay its debt if it took the project and had to repay the debt today. The firm
does not have the cash. The only way it can invest is by raising funds. In this situation, shareholders
may choose not to raise the funds.
In our scenario, the firm could raise funds in two ways: It could borrow or issue equity. Since
the firm cannot repay its existing debt in its current circumstances, it cannot raise debt. Any debt
raised would be junior to the existing debt and would not be repaid unless the value of the firm
increases unexpectedly. Consequently, the firm would have to sell equity. Consider the impact of
having an investor invest one dollar in a new share. Since the firm is expected to default, that dollar
will most likely end up in the pockets of the creditors. If something good happens to the firm so that
equity has value, there will be more shareholders and the existing shareholders will have to share the
Chapter 3, page 37
payoff to equity with the new shareholders. This is most likely a no-win situation for the existing
shareholders: They do not benefit from the new equity if the firm is in default but have to share with
the new equity if the firm is not in default. Hence, even though the project would increase firm value,
existing shareholders will not want the firm to take it because it will not benefit them. The only way
the firm would take the project is if shareholders can renegotiate with creditors so that they get more
of the payoff of the project. If such a renegotiation is possible, it is often difficult and costly.
Sometimes, however, no such renegotiation succeeds. When the firm is valued by the capital markets,
its value is discounted because of the probability that it might not take valuable projects because its
financial health might be poor. Hence, reducing this probability through risk management increases
firm value as long as risk management is cheap.
Both situations we discussed in this section arise because the firm has too much debt. Not
surprisingly, in such cases the firm is said to have a debt overhang. A debt overhang induces
shareholders to increase risk and to avoid investing in valuable projects. The probability that the firm
might end up having a debt overhang in the future reduces its value today. Consequently, risk
management that reduces this probability increases firm value today. The costs associated with a debt
overhang are costs of financial distress. As with the costs of financial distress, the present value of
these costs could be reduced by having less debt. We saw, however, that debt has value and that it
is generally not an optimal strategy for a firm to be an all-equity firm. As long as a firm has some debt
and some risk, there is some possibility that it might end up having a debt overhang.
Section 3.5.2. Information asymmetries and agency costs of managerial discretion.
Most of our analysis derives benefits from risk management that depend on the existence of
Chapter 3, page 38
debt. Without debt of some sort, there are no bankruptcy costs and no tax benefits of debt to protect.
Generally, one thinks of costs of financial distress as costs resulting from difficulties the firm has to
cope with its debt service. In section 3.4., we saw that the existence of valuable stakeholders could
lead a firm to want to reduce risk. That argument for risk management does not depend on the
existence of debt. Let’s consider here another situation where risk management creates value even
for a firm that has no debt. Firm value fluctuates randomly. Hence, sometimes it will fall
unexpectedly. The problem with a low firm value is that it limits the firm’s ability to invest. Firms with
limited net worth can sometimes raise extremely large amounts of money, but typically they cannot.
Hence, if such firms want to invest massively, they may have trouble to do so.
The key problem management faces when trying to raise funds is that it knows more about
the firm’s projects than the outsiders it is dealing with. A situation where one party to a deal knows
more than the other is called a situation with an information asymmetry. Suppose that the firm’s
net worth with its existing projects is $10M. Management knows that by investing $100M the firm
can double its net worth. All management has to do then is find investors who will put up $100M.
If you are such an investor, you have to figure out the distribution of the return on your investment
based on the information provided to you by management. Generally, management has much to gain
by investing in the project. For instance, management compensation and perquisites increase with firm
size. Small firms do not have planes for managers; large firms do. As a result, management will be
enthusiastic about the project. This may lead to biases in its assessment and a tendency to ignore
problems. Even if management is completely unbiased and reveals all of the information it has to
potential investors, the potential investors cannot easily assess that management is behaving this way.
Potential investors know that often, management has enough to gain from undertaking the project
Chapter 3, page 39
that it might want to do so even if the chance of success is low enough that the project is a negative
net present value project. The costs associated with management’s opportunity to undertake projects
that have a negative net present value when it is advantageous for it to do so are called the agency
costs of managerial discretion. In the absence of managerial discretion, management would not have
this opportunity. Managerial discretion has benefits - without it, management cannot manage. It has
costs also, however. With managerial discretion, management pursues its own objectives, which
creates agency costs - the agent’s interests, management, are not aligned with the interests of those
who hire management, the principals, namely the shareholders.
Agency costs of managerial discretion make it harder for the firm to raise funds and increase
the costs of funds. If outsiders are not sure that the project is as likely to pay off as management
claims, they require a higher expected compensation for providing the funds. Clearly, if management
was truthful about the project, having to pay a higher expected compensation reduces the profits from
the project. This higher expected compensation could lead to a situation where the project is not
profitable because the cost of capital for the firm is too high because of costs of managerial discretion.
The firm could use a number of approaches to try to reduce the costs of managerial discretion
and hence reduce the costs of the funds raised. For instance, it could entice a large shareholder to
come on board. This shareholder would see the company from the inside and would be better able
to assess whether the project is valuable. However, if the firm could have invested in the project in
normal times more easily but has a low value now because of adverse developments unrelated to the
value of the project, a risk management strategy might have succeeded in avoiding the low firm value
and hence might have enabled the firm to take the project. For instance, if the firm’s value is much
larger, it might be able to borrow against existing assets rather than having to try to borrow against
Chapter 3, page 40
the project.
A risk management strategy that avoids bad outcomes for firm value might help the firm to
finance the project for another reason. Investors who look at the evolution of firm value have to
figure out what a loss in firm value implies. There could be many possible explanations for a loss in
firm value. For instance, firm value could fall because its stock of raw materials fell in value, because
the economy is in a recession, because a plant burned down, or because management is incompetent.
In general, it will be difficult for outsiders to figure out exactly what is going on. They will therefore
always worry that the true explanation for the losses is that management lacks competence.
Obviously, the possibility that management is incompetent makes it more difficult for management
to raise funds if is competent but outsiders cannot be sure. By reducing risk through risk
management, the firm becomes less likely to be in a situation where outsiders doubt the ability of
management.
Section 3.5.3. The cost of external funding and Homestake.
Our analysis shows that external funding can be more expensive than predicted by the CAPM
because of agency costs and information asymmetries. Agency costs and information asymmetries
create a wedge between the costs of internal funds and the costs of external funds. This wedge can
sometimes be extremely large. The box on “Warren Buffet and Catastrophe Insurance” provides an
example where taking on diversifiable risk is extremely rewarding. For Homestake, however, this is
not an issue. It turns out that Homestake could repay all its debt with its cash reserves, so that debt
overhang is not an issue. The firm also has enough cash that it could finance large investments out
of internal resources.
Chapter 3, page 41
Section 3.6. Summary.
In this chapter, we have investigated four ways in which firms without risk management can
leave money on the table:
1. These firms bear more direct bankruptcy costs than they should.
2. These firms pay more taxes than they should.
3. These firms pay more to stakeholders than they should.
4. These firms sometimes are unable to invest in valuable projects.
In this chapter, we have identified benefits from risk management that can increase firm value. In the
next chapter, we move on to the question of whether and how these benefits can provide the basis
for the design of a risk management program.
Chapter 3, page 42
Key concepts
Bankruptcy costs, financial distress costs, tax shield of debt, optimal capital structure, stakeholders,
debt overhang, agency costs of managerial discretion, costs of external funding.
Chapter 3, page 43
Review questions
1. How does risk management affects the present value of bankruptcy costs?
2. Why do the tax benefits of risk management depend on the firm having a tax rate that depends on
cash flow?
3. How do carry-back and carry-forwards affect the tax benefits of risk management?
4. How does risk management affect the tax shield of debt?
5. Does risk management affect the optimal capital structure of a firm? Why?
6. When does it pay to reduce firm risk because a large shareholder wants the firm to do it?
7. How does the impact of risk management on managerial incentives depend on the nature of
management’s compensation contract?
8. Is risk management profitable for the shareholders of a firm that has a debt overhang?
9. How do costs of external funding affect the benefits of risk management?
Chapter 3, page 44
Literature note
Smith and Stulz (1985) provide an analysis of the determinants of hedging policies that covers the
issues of bankruptcy costs, costs of financial distress, stakeholders, and managerial compensation.
Stulz (1983) examines optimal hedging policies in a continuous-time model when management is riskaverse. Diamond (1981) shows how hedging makes it possible for investors to evaluate managerial
performance more effectively. DeMarzo and Duffie (1991) and Breeden and Viswanathan (1998)
develop models where hedging is valuable because of information asymmetries between managers and
investors. Froot, Scharfstein and Stein (1993) construct a model that enables them to derive explicit
hedging policies when firms would have to invest suboptimally in the absence of hedging because of
difficulties in securing funds to finance investment. Stulz (1990,1996) discusses how hedging can
enable firms to have higher leverage. Stulz (1990) focuses on the agency costs of managerial
discretion. In that paper, hedging makes it less likely that the firm will not be able to invest in valuable
projects, so that the firm can support higher leverage. In that paper, debt is valuable because it
prevents managers from making bad investments. Tufano (1998) makes the point that reducing the
need of firms to go to the external market also enables managers to avoid the scrutiny of the capital
markets. This will be the case if greater hedging is not accompanied by greater leverage.
Bessembinder (1991) and Mayers and Smith (1987) also analyze how hedging can reduce the
underinvestment problem. Leland (1998) provides a continuous-time model where hedging increases
firm value because (a) it increases the tax benefits from debt and (b) it reduces the probability of
default and the probability of incurring distress costs. Ross (1997) also models the tax benefits of
hedging in a continuous-time setting. Petersen and Thiagarajan (1998) provide a detailed comparison
of how hedging theories apply to Homestake and American Barrick.
Chapter 3, page 45
BOX
Warren Buffet and the Catastrophe Insurance3
Let’s look at an interesting example where the costs of external finance can be computed directly and
turn out to be much larger than predicted by the CAPM. There exists a market for catastrophe
insurance. In this market, insurers provide insurance contracts that pay off in the event of events such
as earthquakes, tornadoes, and so on. Insurance companies hedge some of their exposure to
catastrophes by insuring themselves with re-insurers. A typical re-insurance contract promises to
reimburse an insurance company for claims due to a catastrophe within some range. For instance, an
insurance company could be reimbursed for up to $1 billion of Californian earthquake claims in excess
of $2 billion. Catastrophe insurance risks are diversifiable risks, so that bearing these risks should not
earn a risk premium. This means that the price of insurance should be the expected losses discounted
at the risk-free rate. Yet, in practice, the pricing of re-insurance does not work this way.
Let’s look at an example. In the Fall of 1996, Berkshire Hattaway, Warren Buffet’s company,
sold re-insurance to the California Earthquake Authority. The contract was for a tranche of $1.05
billion insured for four years. The annual premium was 10.75% of the annual limit, or $113M. The
probability that the reinsurance would be triggered was estimated at 1.7% at inception by EQE
International, a catastrophe risk modeling firm. Ignoring discounting, the annual premium was
therefore 530% the expected loss (530% is (0.1075/0.017) - 1 in percent). If the capital asset pricing
3
The source is Kenneth Froot, The limited financing of catastrophe risk: An overview, in
“The Financing of Property Casualty risks”, University of Chicago Press, 1997.
Chapter 3, page 46
model had been used to price the reinsurance contract, the premium would have been $17.85M in the
absence of discounting and somewhat less with discounting.
How can we make sense of this huge difference between the actual premium and the premium
predicted by the capital asset pricing model? A re-insurance contract is useless if there is credit risk.
Consequently, the re-insurer essentially has to have liquid assets that enable it to pay the claims. The
problem is that holding liquid assets creates agency problems. It is difficult to make sure that the reinsurer will indeed have the money when needed. Once the catastrophe has occurred, the underinvestment problem would prevent the re-insurer from raising the funds because the benefit from
raising the funds would accrue to the policy holders rather than to the investors. The re-insurer
therefore has to raise funds when the policy is agreed upon. Hence, in the case of this example, the
re-insurer would need - if it did not have the capital - to raise $1.05 billion minus the premium. The
investors would have to be convinced that the re-insurer will not take the money and run or take the
money and invests it in risky securities. Yet, because of the asset substitution problem, the re-insurer
has strong incentives to take risks unless its reputational capital is extremely valuable. In the absence
of valuable reputational capital, the re-insurer can gamble with the investors’ money. If the re-insurer
wins, it makes an additional profit. If it loses, the investors or the insurer’s clients lose.
There is another problem with re-insurance which is due to the information asymmetries and
agency costs in the investment industry. The re-insurer has to raise money from investors, but the
funds provided would be lost if a catastrophe occurs. Most investment takes place through money
managers that act as agents for individual investors. In the case of funds raised by re-insurance
companies, the money managers is in a difficult position. Suppose that he decides that investing with
a re-insurance firm is a superb investment. How can the individual investors who hire the money
Chapter 3, page 47
manager know that he acted in their interest if a catastrophe occurs? They will have a difficult time
deciding whether the money manager was right and they were unlucky or the money manager was
wrong. The possibility of such a problem will lead the money manager to require a much larger
compensation for investing with the re-insurance firm.
Berkshire Hataway has the reputational capital that makes it unprofitable to gamble with
investors’ money. Consequently, it does not have to write complicated contract to insure that there
will not be credit risk. Since it has already large reserves, it does not have to deal with the problems
of raising large amounts of funds for re-insurance purpose. Could these advantages be worth as much
as they appeared to be worth in the case of our example? Maybe not. However, there is no evidence
that there were credible re-insurers willing to enter cheaper contracts. With perfect markets, such reinsurers would have been too numerous to count.
Chapter 3, page 48
Figure 3.1. Cash flow to shareholders and operating cash flow.
The firm sells 1M ounces of gold at the end of the year and liquidates. There are no costs. The
expected gold price is $350.
Chapter 3, page 49
Figure 3.2. Creating the unhedged firm out of the hedged firm.
The firm produces 100M ounces of gold. It can hedge by selling 100M ounces of gold forward. The
expected gold price and the forward price are $350 per ounce. If the firm hedges and shareholders
do not want the firm to hedge, they can recreate the unhedged firm by taking a long position forward
in100M ounces of gold.
Chapter 3, page 50
Figure 3.3. Cash flow to shareholders and bankruptcy costs. The firm sells 1m ounces of gold
at the end of the year and liquidates. There are no costs. The expected gold price is $350.Bankruptcy
costs are $20M if cash flow to the firm is $250M. Expected cash flow to shareholders for unhedged
firm is 0.5 times cash flow if gold price is $250 plus 0.5 times cash flow if gold price is $450.
Chapter 3, page 51
Figure 3.4. Expected bankruptcy cost as a function of volatility.
The firm produces 100M ounces of gold and then liquidates. It is bankrupt if the price of gold is
below $250 per ounce. The bankruptcy costs are $20 per ounce. The gold price is distributed
normally with expected value of $350. The volatility is in dollars per ounce.
Chapter 3, page 52
Figure 3.5. Taxes and cash flow to shareholders.
The firm pays taxes at the rate of 50% on cash flow in excess of $300 per ounce. For simplicity, the
price of gold is either $250 or $450 with equal probability. The forward price is $350.
Chapter 3, page 53
Figure 3.6. Firm after-tax cash flow and debt issue.
The firm has an expected pre-tax cash flow of $350M. The tax rate is 0.5 and the risk-free rate is 5%.
The figure shows the impact on after-tax cash flow of issuing more debt, assuming that the IRS
disallows a deduction for interest of debt when the firm is highly likely to default.
After tax cash flow of hedged firm
330
325
320
315
310
305
Principal amount of debt
100
200
300
400
Optimal amount of
debt, $317.073M
Chapter 3, page 54
Chapter 4: An integrated approach to risk management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Section 4.1. Measuring risk for corporations: VaR, CaR, or what? . . . . . . . . . . . . . . . . . 4
Section 4.1.1. The choice of a risk measure at a financial institution . . . . . . . . . . 4
Section 4.1.2. The choice of a risk measure for a nonfinancial institution . . . . . . 16
Section 4.1.3. CaR or VaR for nonfinancial firms? . . . . . . . . . . . . . . . . . . . . . . 19
Section 4.2. CaR, VaR, and firm value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Section 4.2.1. The impact of projects on VaR . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Section 4.2.2. CaR and project evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Section 4.2.3. Allocating the cost of CaR or VaR to existing activities . . . . . . . 29
Section 4.3. Managing firm risk measured by VaR or CaR . . . . . . . . . . . . . . . . . . . . . . 36
Section 4.3.1. Increase capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Section 4.3.2. Project choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Section 4.3.1.C. Derivatives and financing transactions . . . . . . . . . . . . . 41
Section 4.3.3. The limits of risk management . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1. It is not rocket science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2. Credit risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3. Moral hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Section 4.4. An integrated approach to risk management . . . . . . . . . . . . . . . . . . . . . . . 46
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 4.1. Using the cumulative distribution function . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 4.2. Frequency distribution of two portfolios over one year horizon . . . . . . . . . . 54
Figure 4.3. VaR as a function of trade size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Box 4.1. VaR, banks, and regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Box 4.2. RAROC at Bank of America . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Technical Box 4.1. Impact of trade on volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 4: An integrated approach to risk management
December 13, 1999
© René M. Stulz 1998, 1999
Chapter 4, page 1
Chapter objectives
1. Introduce risk measures appropriate to financial and nonfinancial firms.
2. Define value at risk (VaR) and cash flow at risk (CaR).
3. Show how to use VaR and CaR to make investment decisions.
4. Demonstrate the use of VaR and CaR to evaluate the profitability of existing activities of a firm.
5. Investigate the strategies available to the firm to reduce the cost of risk.
6. Assess the limits of risk management.
Chapter 4, page 2
Chapter 3 showed that there are five major reasons why risk management can increase
shareholder wealth. These reasons are:
1. Risk management can decrease the present value of bankruptcy and financial distress costs.
2. It can make it more likely that the firm will be able to take advantage of valuable
investment opportunities.
3. It can decrease the present value of taxes paid by the corporation.
4. It can increase the firm’s debt capacity and allow it to make better use of the tax shield of
debt.
5. By reducing risk, risk management makes it possible for the firm to have large shareholders
that monitor management and to align management’s incentives better with the objectives
of shareholders.
Chapter 3 provided us with a catalog of benefits from risk management. Though such a
catalog is an important starting point in understanding risk management, it is only the first step in the
process of finding out how these benefits can be captured by a corporation. One might be tempted
to say that to capture these benefits, risk should be managed to maximize the value of the
corporation. However, such a way to look at risk management is too general to be useful. For risk
management to be used to maximize firm value, one must concretely define how risk is measured and
how it is managed. This chapter presents a framework that makes it possible to do that.
In the first part of this chapter, we show how the benefits of risk management presented in
Chapter 3 lead to the choice of a risk measure. This risk measure depends on the characteristics of
Chapter 4, page 3
the firm. Armed with a risk measure, a firm can evaluate the impact of new projects on its risk and
take into account the contribution of existing activities to risk when it evaluates their profitability.
Having specified the choice of a risk measure, we then discuss the tools available for risk
management. Firms can manage their cost of total risk through equity, through their choice of
projects or through transactions in financial markets. We discuss the costs and benefits of these
various tools to manage the cost of risks and argue that derivatives are generally the most cost
effective approach for a corporation to manage its total risk.
Section 4.1. Measuring risk for corporations: VaR, CaR, or what?
In this section, we analyze the considerations that affect a firm’s choice of a risk measure.
To understand these considerations, we focus on two important concrete examples that allow us to
examine the tradeoffs involved in choosing risk measures. The first part of the section examines
financial institutions. The second part of the section analyzes the choice of a risk measure for nonfinancial corporations.
Section 4.1.1. The choice of a risk measure at a financial institution.
The analysis of chapter 3 allows us to understand how risk management makes it possible to
increase the value of a financial institution. Financial institutions generally have customers who are
also creditors. The owner of a checking account lends his money to the bank. The firm which enters
into a derivatives contract with an investment bank is a creditor of the bank if the firm expects to
receive a payment from the bank at maturity. The buyer of a life insurance policy owns a claim against
the insurance company from which she bought the policy. In general, the customers of a financial
Chapter 4, page 4
institution are extremely sensitive to its credit risk because they cannot diversify this risk.
Government guarantees for some types of financial institutions and some types of claims against
financial institutions can reduce the concerns about credit risk for some customers of financial
institutions. For instance, deposits of less than $100,000 are federally insured in the U.S. Government
guarantees shift concerns about credit risks partly from customers to regulatory agencies that have
to make up shortfalls.
An investor who buys bonds issued by a financial institution is in general willing to trade off
expected return and risk for these bonds. If the financial institution has high credit risk, the investor
holds the bonds as long as the expected return is high enough to compensate her for the credit risk.
Risky bonds are generally held as part of a diversified portfolio, so that the required risk premium
depends on the contribution of the bonds to the risk of the portfolio. Bondholders are not customers
of the financial institution.
Customers of financial institutions have a dramatically different attitude toward credit risk
than bondholders. An individual who is willing to hold risky bonds as part of a diversified portfolio
generally wants her checking account to have no risk. The reason for this is straightforward. If the
checking account has significant credit risk, a check of $100 is not sufficient to buy an object with
a price of $100. This is because the store that sells the object wants to be compensated for the credit
risk incurred between the moment the object is sold and the moment the check clears. Consequently,
each transaction made using a check drawn on the checking account would require a negotiation to
determine the appropriate compensation for credit risk paid to the person or firm that receives the
check. This would destroy the usefulness of the checking account, which is that checks are substitutes
for cash. A firm that enters a forward contract with a financial institution wants the forward contract
Chapter 4, page 5
to serve as an effective hedge. If the financial institution has substantial credit risk, it might not deliver
on the forward contract. This possibility would sharply decrease the usefulness of the forward
contract as a hedging device. Finally, consider the case of an insurance contract. No customer of an
insurance company would be willing to buy a life-insurance contract that costs half as much as other
insurance contracts but has a 0.5 probability of paying off because of credit risk.
It follows from this discussion that a small probability of default risk can have a dramatic
impact on the business of a financial institution. As default risk becomes significant, customers
withdraw their checking accounts, the derivatives business dries up, and the life insurance business
disappears. The value of a financial institution for its shareholders depends crucially on its ability to
retain old customers and acquire new ones. Shareholder wealth in a financial institution is fragile,
probably more so than at any other type of corporation. Because customers are creditors in financial
institutions, financial institutions are highly levered in comparison to other firms. The more business
a financial institution has for a given amount of capital, the greater its leverage. An adverse shock to
a financial institution can therefore make its equity disappear quickly as customers react to the shock
by withdrawing their business. Costs of financial distress are very large for financial institutions
because generally the mere hint of the possibility of financial distress can create a run on a financial
institution that eliminates its value as an ongoing business.
The fact that a financial institution’s ability to do business depends so critically on its
creditworthiness means that it must control its risks with extreme care. It must make sure that they
are such that the probability that the financial institution will lose customers because of its own credit
risk is very small. The analysis of why a financial institution benefits from risk management therefore
suggests that the focus of the risk management effort has to be on computing, monitoring, and
Chapter 4, page 6
managing a measure of risk that corresponds to the probability that the financial institution will lose
customers because of credit risk.
To figure out an appropriate measure of risk, let’s consider the type of event that could make
the financial institution risky for its customers. Let’s say that this financial institution is a bank. Most
of its assets are loans and traded securities; it has deposits and debt in its liabilities. An asset or
liability is marked-to-market when its value in the firm’s balance sheet corresponds to its market
value and when changes in its market value affect the firm’s earnings. Some assets and liabilities, the
traded securities, are marked-to-market; other assets and liabilities are kept at book value. For assets
and liabilities marked-to-market, their change in value can be observed directly and corresponds to
a gain or loss for the institution that affects its earnings directly. If the net value of the traded
securities falls sharply, the institution is likely to have difficulties in meeting its obligations. Financial
institutions also typically have derivative positions that at a point in time may have no value, yet could
require them to make large payments in the future. We can think of all the positions in traded
securities and derivatives as the bank’s portfolio of traded financial instruments.
The probability of a value loss that creates serious problems for an institution cannot be
reduced to zero. As long as the institution has a portfolio of risky traded financial instruments and
is levered, it cannot be completely safe. A key goal of risk management then is to keep the probability
that the firm will face an exodus of its customers low enough that its customers are not concerned.
This means that the probability of losses above a critical size from traded financial instruments has
to be kept low. To insure that this is the case, the firm can specify this critical loss size and the
probability that it will be exceeded that it can tolerate. For instance, the firm could decide that the
probability of having a loss exceeding 10% of the value of its traded financial instruments over a day
Chapter 4, page 7
must be 0.05. What this means is that 95% of the time, the return of the portfolio of traded financial
instruments must be higher than -10%. Given the firm’s existing portfolio of traded financial
instruments, it can find out the probability of a loss of 10% or more over one day. To compute this
probability, it has to know the distribution of the gains and losses of the portfolio. Let’s assume that
the portfolio’s return has a normal distribution. As a portfolio includes derivatives that have option
components, the distribution of its return is generally not normal. Consequently, evaluating the risk
of portfolios whose returns are not normal is an important issue in risk management. We will learn
how to do this in detail in later chapters. If returns are not normally distributed, however, the
computations in this chapter relying on the normal distribution would be changed but the general
principles we focus on here would not.
Let’s see how we can use the analysis of chapter 2 to find the probability that the portfolio
will make a loss of at least 10% over one day. The probability that the return will be lower than some
number x over a day, Prob[Return < x], is given by the cumulative normal distribution function
evaluated at x. To use the cumulative normal distribution, we need to know the expected return and
its volatility. Let’s assume that the expected return is 10% and the volatility is 20%. Figure 4.1. shows
how we can use the cumulative normal distribution to find the probability that the portfolio will make
a loss of at least 10% over one day. We pick the return of -10% and read the probability on the
vertical axis corresponding to the level of the cumulative normal probability distribution. We find that
the probability is 0.16. Alternatively, by setting the probability equal to 0.05, we can get on the
horizontal axis the number x such that the return will be less than x with probability of 0.05. For the
probability 0.05, x is equal to -33%, so that the firm has a 0.05 probability of losing at least 33%. If
the return of the portfolio of traded financial instruments is distributed normally, then knowing the
Chapter 4, page 8
volatility of the return and the expected return provides us with all the information we need to
compute any statistic for the distribution of gains and losses.
Put Figure 4.1. here
Suppose that the financial institution follows the approach just described. It has a portfolio
of traded financial instruments worth $2B, so that a 10% loss is equal to $200M. The institution finds
that it has a probability of 0.05 of making a loss of at least 23%, or $460M. If the bank had decided
that it could only afford to have a loss of $200M or more with a probability of 0.05, the computation
of the risk of the portfolio of traded assets tells us that the financial institution has too much risk.
Consequently, the financial institution has to decide what to do to reduce its risk. An obvious solution
is to reduce the size of the portfolio of traded financial instruments. Alternatively, the financial
institution can hedge more if appropriate hedging instruments are available. Hence, the risk measure
allows the financial institution to decide how to manage the risk its portfolio of traded financial
instruments so that shareholder wealth is maximized.
The risk measure defined as the dollar loss x that will be exceeded with a given probability
over some measurement period is called value at risk or VaR. With this definition, the VaR can be
computed for a firm, a portfolio, or a trading position. The VaR at the probability level of z% is the
loss corresponding to the z-th quantile of the cumulative probability distribution of the value change
at the end of the measurement period.1 Consequently, we define:
Value at Risk
Value at risk or VaR at z% is the dollar loss that has a probability z% of being exceeded over the
1
The z-th quantile of a random variable x is the number q that satisfies Prob(x < q) = z.
Chapter 4, page 9
measurement period. Formally, VaR is the number such that Prob[Loss > VaR] = z%.
Note that this definition makes no assumption about the distribution function of the loss. In the
financial institution example, the measurement period was one day, so that VaR was $460M at the
probability level of 5% using the normal distribution. Throughout the book, we will use z% as 5%
unless we specify otherwise. If VaR is a loss that is exceeded with probability of z%, there is a (1-z)%
probability that the loss will not be exceeded. We can therefore consider an interval from -VaR to
plus infinity such that the probability of the firm’s gain belonging to that interval is (1-z)%. In
statistics, an interval constructed this way is called a one-sided confidence interval. As a result, one
can also think of VaR as the maximum loss in the (1-z)% confidence interval, or in short, the
maximum loss at the (1-z)% confidence level. If we compute the VaR at the 5% probability level, it
is therefore also the maximum loss at the 95% confidence level. We are 95% sure that the firm’s gain
will be in the interval from -VaR to plus infinity.
The VaR risk measure came about because the CEO of JP Morgan, Dennis Weatherstone at
the time, wanted to know the bank’s risk at the end of the trading day. He therefore asked his staff
to devise a risk measure that would yield one number that could be communicated to him at 4:15 pm
each trading day and would give him an accurate view of the risk of the bank. His perspective on risk
was similar to the one we developed in this chapter. He wanted to have a sense of the risk of bad
outcomes that would create problems for the bank. In chapter 2, we considered three risk measures.
These measures were volatility, systematic risk, and unsystematic risk. None of these three measures
provides a direct answer to the question that Dennis Weatherstone wanted answered, but VaR does.
In general, there is no direct relation between VaR and these three risk measures. (We will see below
Chapter 4, page 10
that the normal distribution is an exception to this statement.) In particular, with complicated
securities, it is perfectly possible for volatility to increase and VaR to fall at the same time.
The use of volatility as a risk measure leads to a paradox that demonstrates neatly why it is
inappropriate.2 Suppose that a corporation has the opportunity to receive a lottery ticket for free that
pays off in one year. This ticket has a small chance of an extremely large payoff. Otherwise, it pays
nothing. If the firm accepts the free lottery ticket, its one-year volatility will be higher because the
value of the firm now has a positive probability of an extremely large payoff that it did not have
before. A firm that focuses on volatility as its risk measure would therefore conclude that taking the
lottery ticket makes it worse off if the volatility increase is high enough. Yet, there is no sense in
which receiving something that has value for free can make a firm worse off. The firm’s VaR would
not be increased if the firm accepts the free lottery ticket.
Figure 4.2. provides an illustration of how return volatility can fail to convey the information
that Dennis Weatherstone wanted. We show the return frequency distribution for two different
portfolios. These two portfolios are constructed to have the same return volatility of 30%. One
portfolio holds common stock and has normally distributed returns. The other portfolio holds the riskfree asset and options and does not have normally distributed returns. Even though these two
portfolios have the same return volatility, the portfolio with options has a very different VaR from
the portfolio without options. (To learn how to compute the VaR of a portfolio with options, we will
have to wait until chapter 14.) Let’s assume we invest $100M in each portfolio and hold this position
for one year. The stock portfolio has a VaR of $41.93M, while the portfolio holding the risk-free
asset and options has a VaR of $15M.
2
This example is due to Delbaen, Eber, and Heath (1997).
Chapter 4, page 11
In our example, volatility is not useful to evaluate the risk of bad outcomes. This is because
the portfolios have very different lower-tail returns since the worst returns of the portfolio with
options are much less negative than the worst returns of the stock portfolio. Consequently, bad
returns for the stock portfolio might bankrupt the bank but bad returns for the portfolio with options
might not. Systematic or unsystematic risk capture the downside risk that matters for the bank even
less than volatility. To construct Figure 4.2., we only need the volatility and the expected return of
the stock. Knowledge of the betas of the portfolios is not required. Hence, these betas could be
anything. This means that beta cannot help us understand the distribution of bad outcomes. With
different betas, these portfolios would have very different unsystematic risk, but we could change
betas so that Figure 4.2. would remain the same. Hence, had JP Morgan’s staff provided Dennis
Weatherstone with the systematic risk of the bank or its unsystematic risk, the information would not
have answered the question he asked and would not have been useful to him.
The plots in Figure 4.2. show that it is crucially important to understand the distribution of
the returns of the positions of the bank to compute its VaR. To answer Dennis Weatherstone’s
question, therefore, his staff had to know the value of all the marked-to-market positions of the bank
at the end of the trading day and had to have a forecast of the joint distribution of the returns of the
various securities held by the bank. As we will see in later chapters, forecasting the distribution of
returns and computing VaR can be a challenging task when the securities are complicated.
Computing the VaR is straightforward when the returns are normally distributed. We know
from chapter 2 that the fifth quantile of a normally distributed random variable is equal to the
expected value of that variable minus 1.65 times the volatility of that random variable. Hence, if the
return of a portfolio is distributed normally, the fifth quantile of the return distribution is the expected
Chapter 4, page 12
return minus 1.65 times the return volatility. For changes computed over one day, this number is
generally negative. The return loss corresponding to the fifth quantile is therefore the absolute value
of this negative change. Consider a bank whose portfolio of traded assets has an expected return of
0.1% and a volatility of 5%. The fifth quantile of the return distribution is 0.1% - 1.65*5%, or 8.15%. Hence, if the bank’s value is $100M, the VaR is 8.15% of $100m or $8.15M.
In general, the expected return over one day is small compared to the volatility. This means that
ignoring the expected return has a trivial impact on an estimate of the VaR for one day. In practice,
therefore, the expected return is ignored. With normally distributed returns and zero expected change,
the formula for VaR is:
Formula for VaR when returns are distributed normally
If the portfolio return is normally distributed, has zero mean, and has decimal volatility F over the
measurement period, the 5% VaR of the portfolio is:
VaR = 1.65* F *Portfolio value
The VaR in the example would then be 1.65*0.05*$100M, or $8.25M. Note that this formula shows
that, for the normal distribution, there is a direct relation between volatility and VaR since VaR
increases directly with volatility. In general, however, as shown in Figure 4.2., portfolios with same
return volatilities can have different VaRs and portfolios with the same VaRs can have different
volatilities.
An important issue is the time period over which the VaR should be computed. Remember
Chapter 4, page 13
that the reason for computing the VaR is that the financial institution wants to monitor and manage
the size of potential losses so that the probability that it will face financial distress is low. The
economically relevant distribution of losses is the one that corresponds to losses that the financial
institution can do nothing about after having committed to a portfolio of financial instruments. If a
financial institution can measure its risk and change it once a day, the relevant measure is the one-day
VaR. At the end of a day, it decides whether the VaR for the next day is acceptable. If it is not, it
takes actions to change its risk. If it is acceptable, there is nothing it can do about the VaR until the
next day. At the end of the next day, it goes through the process again. For such an institution,
a VaR cannot be computed over a shorter period of time because of lack of data or because there is
nothing the institution can do with the estimate if it has it. It is not clear how one would compute a
one-year VaR and what it would mean for such an institution. The bank could compute a one-year
VaR assuming that the VaR over the next day will be maintained for a year. However, this number
would be meaningless because the financial institution’s risk changes on a daily basis and because,
if the financial institution incurs large losses or if its risk increases too much, it will immediately take
steps to reduce its risk. If a financial institution cannot do anything to change the distribution of the
value changes of its portfolio for several days because it deals in illiquid markets, the risk measure
has to be computed over the time horizon over which it has no control over the distribution of losses.
Hence, the period over which the VaR is computed has to reflect the markets in which the firm deals.
Generally, the VaR computed by financial institutions takes into account only the derivative
price changes due to market changes, ignoring counterparty risks. For instance, if a bank has a
forward currency contract on its books, it will use the change in the value of the forward contract for
Chapter 4, page 14
the VaR computation using a forward currency contract valuation model that ignores the
creditworthiness of the counterparty to the forward contract and computes the forward exchange rate
without paying attention to the counterparty. Consequently, VaR generally ignores credit risks. This
means that VaR does not measure all risks the bank faces. Credit risks are generally a substantial
source of risk for banks because of their loan portfolio. However, banks do not have the information
to compute changes in the riskiness of their loan portfolio on a daily basis. Loans are not marked to
market on a daily basis and firms release accounting data that can be used to evaluate the credit
worthiness of loans generally on a quarterly basis. This means that a bank typically estimates credit
risks over a different period of time than it measures market risks. We will discuss in detail how credit
risks are measured in chapter 18. The VaR measure plays an important role in financial institutions
not only as a risk management tool, but also as a regulatory device. Box 4.1. VaR and regulators
shows that large banks have to use VaR.
Put Box 4.1. here
Financial firms have other risks besides market and credit risks. Lots of things can go wrong
in the implementation of firm policies. An order might be given to an individual to make a specific
transaction, perhaps to hedge to reduce the VaR, and the individual can make a mistake. A
standardized contract used by a financial institution might have a clause that turns out to be invalid
and this discovery creates large losses for the financial institution. An individual might find a problem
in the accounting software of the bank which allows her to hide losses in a trading position. These
types of problems are often included under the designation of operational risks. These risk are harder,
if not impossible, to quantify. Yet, they exist. Generally, these risks cannot be reduced through
financial transactions. The best way to reduce such risks is to devise procedures that prevent mistakes
Chapter 4, page 15
and make it easier to discover mistakes. Though the design of such procedures is very important, we
have little to say about this topic in this book. The one dimension of operational risk we focus on has
to do with risk measurement.
Section 4.1.2. The choice of a risk measure for a nonfinancial institution.
Let’s now consider a manufacturing firm that exports some of its production. The firm has
foreign currency receivables. Furthermore, an appreciation of the domestic currency reduces its
international competitiveness. Consequently, the firm has exposure to foreign exchange rate risks. For
simplicity, it does not currently hedge, does not have derivatives, and does not have a portfolio of
financial assets. To start the discussion, we assume that it cannot raise outside funds. This firm will
view its main risk as the risk that it will have a cash flow shortfall relative to expected cash flow that
is large enough to endanger the firm’s ability to remain in business and finance the investments it
wants to undertake. Generally, a bad cash flow for a week or a month is not going to be a problem
for such a firm. Cash flows are random and seasonal. Some weeks or some months will have low cash
flows. A firm has a problem if bad cash flows cumulate. This means that a firm is concerned about
cash flows over a longer period of time, a year or more. A poor cash flow over a long period of time
will be a serious problem. Consequently, in measuring risk, the firm wants to know how likely it is
that it will have a cash flow shortfall that creates problems over a relevant period of time.
Let’s say that a firm is concerned about the risk of its cash flow over the coming fiscal year.
It decides to focus on cash flow over the coming fiscal year because a low cash flow by the end of
that year will force it to change its plans and will therefore be costly. It has the funds available to
carry on with its investment plans for this year and it has enough reserves that it can ride out the year
Chapter 4, page 16
if cash flow is poor. To evaluate the risk that the firm’s cash flow will be low enough to create
problems, the firm has to forecast the distribution of cash flow.
If a specific cash flow level is the lowest cash flow the firm can have without incurring costs
of financial distress, the firm can use the cumulative distribution of cash flow to find out the
probability of a cash flow lower than this threshold. Alternatively, the firm can decide that it will not
allow the probability of serious problems to exceed some level. In this case, it evaluates the cash flow
shortfall corresponding to that probability level. If the cash flow shortfall at that probability level is
too high, the firm has to take actions to reduce the risk of its cash flow. This approach is equivalent
to the VaR approach discussed in the first part of this section, except that it is applied to cash flow.
By analogy, the cash flow shortfall corresponding to the probability level chosen by the firm is called
cash flow at risk, or CaR, at that probability level. A CaR of $100m at the 5% level means that there
is a probability of 5% that the firm’s cash flow will be lower than its expected value by at least
$100m. We can therefore define cash flow at risk as follows:
Cash flow at risk
Cash flow at risk (CaR) at z% or CaR is a positive number such that the probability that cash flow
is below its expected value by at least that number is z%. Formally, if cash flow is C and expected
cash flow is E(C), we have:
Prob[E(C) - C > CaR] = z%
Throughout the book, we use z% as 5% unless we specify otherwise. Let’s look at an
Chapter 4, page 17
example of these computations. Consider a firm that forecasts its cash flow for the coming year to
be $80M. The forecasted volatility is $50M. The firm believes that the normal distribution is a good
approximation of the true distribution of cash flow. It wants to make sure that the probability of
having to cut investment and/or face financial distress is less than 0.05. It knows that it will be in this
unfortunate situation if its cash flow is below $20M. Hence, the firm wants the probability that its
cash flow shortfall exceeds $60M (expected cash flow of $80M minus cash flow of $20M) to be at
most 0.05. Using our knowledge of the normal distribution, we know that the CaR is equal to 1.65
times volatility, or 1.65*$50M, which corresponds to a cash flow shortfall of $82.5M. This means
that there is a probability 0.05 that the cash flow shortfall will be at least $82.5M or, alternatively,
that cash flow will be lower than -$2.5M (this is $80M minus $82.5M). Since the CaR exceeds the
firm’s target, the cash flow is too risky for the firm to achieve its goal. It must therefore take actions
that reduce the risk of cash flow and ensure that it will earn at least $20M 95% of the time.
CaR is a risk measure for nonfinancial firms that is very much in the spirit of VaR. Whereas
the use of VaR is well-established and standardized, the same is not the case for CaR. The reason for
this is that there is much more diversity among nonfinancial firms than there is among financial firms.
With financial firms, it is generally clear that one has to focus on their value and the risk that their
value will fall because of adverse changes in the trading positions they have. The CaR measure
addresses the risk of cash flow and is generally computed for one year. This raises the question of
whether it makes sense for a firm to compute the risk of cash flow rather than the risk of firm value.
A corporation that hedges its whole value effectively hedges the present value of cash flows.
Consequently, it hedges not only the cash flow of this year but also the cash flow of future years.
There is much debate about whether nonfinancial corporations should hedge their value or their cash
Chapter 4, page 18
flows over the near future. This amounts to a debate on whether nonfinancial corporations should
compute CaR or a measure like VaR for the whole firm based on the risk of the present value of
future cash flows.
Section 4.1.3. CaR or VaR for nonfinancial firms?
To analyze whether a corporation should hedge cash flow or value, first consider a firm for
which focusing on the cash flow of the coming year is the solution that maximizes firm value. This
will be the case for a firm whose ability to take advantage of its growth opportunities depends solely
on its cash flow because it cannot access external capital markets. In this simple situation, the firm
has to manage the risk of its cash flow for the coming year to ensure that it can invest optimally. One
would expect such a firm to be more concerned about unexpectedly low cash flow than unexpectedly
high cash flow. With an unexpectedly low cash flow, the firm can incur costs of financial distress and
may have to cut investment. CaR provides a measure of the risk of having a cash flow shortfall that
exceeds some critical value. It is therefore an appropriate risk measure for such a firm.
Let’s consider how the reasoning is changed if the firm has other resources to finance
investment. This will be the case if (a) the firm has assets (including financial assets) that it can sell
to finance investment and/or (b) it has access to capital markets. It is straightforward that if the firm
has assets it can liquidate, especially financial assets, then it may choose to do so if its cash flow is
low. Consequently, when it computes risk, it has to worry about the risk of the resources it can use
to invest at the end of the year. The firm will have to reduce investment if it simultaneously has a low
cash flow from operations and the value of the assets that can be liquidated to finance capital
expenditures is low. The CaR can be extended in a straightforward way in this case. Instead of
Chapter 4, page 19
focusing on cash flow from operations, the firm computes cash flow as the sum of cash flow from
operations plus the change in value of the assets that can be used to finance investment. It then
computes the CaR on this measure of cash flow. Note, however, that the firm’s earnings statement
cannot be used to compute this cash flow measure if financial assets are not marked to market.
If a firm has access to capital markets, then it has resources to finance next year’s investment
in addition to this year’s cash flow. Such a firm has to worry about its ability to raise funds in public
markets. If its credit is good enough, it will be able to raise funds at low cost. If its credit deteriorates,
the firm may find it too expensive to access capital markets. To the extent that the firm’s credit does
not depend only on the coming year’s cash flow, it is not enough for the firm to measure the risk of
this year’s cash flow. The firm also has to focus on the risk of the assets that it can use as collateral
to raise funds. Generally, firm value will be an important determinant of a firm’s ability to raise funds.
The firm’s risk measure has to take into account the change in its ability to raise funds. If the firm
can freely use the capital markets to make up for cash flow shortfalls as long as its value is sufficiently
high, then the firm’s relevant risk measure is firm value risk. A firm’s value is the present value of its
cash flows. This means that firm value risk depends on the risk of all future cash flows. The
appropriate measure of firm value risk is firm VaR, namely the loss of firm value that is exceeded with
probability 0.05. Chapter 8 provides techniques to compute risk measures for nonfinancial firms when
the normal distribution cannot be used.
We considered two extreme situations. One situation is where the firm has free access to
capital markets, so that it computes a VaR measure. The other situation is where the firm’s ability to
carry on its investment program next year depends only on the cash flow of this year. In this case, the
firm computes CaR. Most firms are between these extreme situations, so that a case can be made to
Chapter 4, page 20
use either one of the risk measures depending on the firm’s situation.
Section 4.2. CaR, VaR, and firm value.
Let’s now look at a firm that has figured out how to measure its risk. All actions this firm
takes have the potential to affect its risk. This means that the firm has to evaluate its actions in light
of their impact on its risk measure. If the firm cares about risk measured by CaR or VaR, a project
that a firm indifferent to such risk might take may not be acceptable. Also, the firm might choose to
take some projects because they reduce its risk measure. This means that computing the NPV of a
project using the CAPM and taking all positive NPV projects is not the right solution for a firm that
is concerned about CaR or VaR. In this section, we consider how firms choose projects when CaR
or VaR is costly. We then show how firms should evaluate the profitability of their activities when
Car or Var is costly.
Section 4.2.1. The impact of projects on VaR.
Consider the case where a firm has a portfolio with securities that have normally distributed
returns. In this case, the VaR is equal to the dollar loss corresponding to the return equal to the
expected return of the portfolio minus 1.65 times the volatility of the portfolio. In this context, one
can think of a project as a trade which involves investing in a new security and financing the
investment with the sale of holdings of a security in the portfolio. This trade affects both the expected
return of the portfolio and its volatility.
In chapter 2, we showed how to compute the expected return and the volatility of a portfolio.
We saw that the change in the expected return of the portfolio resulting from buying security i and
Chapter 4, page 21
selling security j in same amounts is equal to the expected return of security i minus the expected
return of security j times the size of the trade expressed as a portfolio share )w, (E(Ri) - E(Rj))*)w.
We discussed in chapter 2 also how a small change in the portfolio share of a security affects the
volatility of the portfolio. We saw that the impact of the small change on the volatility depends on
the covariance of the return of the security with the return of the portfolio. We demonstrated there
that a trade that increases the portfolio share of a security that has a positive return covariance with
the portfolio and decreases the portfolio share of a security that has a negative return covariance with
the portfolio increases the volatility of the portfolio. Hence, such a trade has a positive volatility
impact. Denote the portfolio we are considering by the subscript p, remembering that this is an
arbitrary portfolio. The volatility impact of a trade that increases the portfolio share of security i in
the portfolio by )w and decreases the portfolio share of security j by the same amount has the
following impact on the volatility of the portfolio:
( β − β ) * ∆w * Vol(R ) = Volatility impact of trade
ip
jp
p
(4.1.)
(Technical Box 4.1. gives the exact derivation of the formula if needed.) $ip is the ratio of the
covariance of the return of security i with the return of portfolio p and of the variance of the return
of portfolio p. If portfolio p is the market portfolio, $ip is the CAPM beta, but otherwise $ip differs
from the CAPM beta. Let’s assume that the return of the portfolio is normally distributed. If the
expected return of the portfolio is assumed to be zero, the VaR impact of the trade is then 1.65 times
the volatility impact of the trade times the value of the portfolio since VaR is 1.65 times volatility
times the value of the portfolio. If the trade increases the expected return of the portfolio, this
Chapter 4, page 22
increase decreases the VaR because all possible portfolio returns are increased by the increase in
expected return. Let W be the initial value of the portfolio. The VaR impact of the trade is therefore:
VaR impact of trade
= − (E(R i ) − E(R j )) * ∆w * W + ( βip − β jp ) *1.65* Vol(R p ) * ∆w * W
(4.2.)
In other words, a trade increases the VaR of a portfolio if the asset bought has a greater beta
coefficient with respect to the portfolio than the asset sold and if it has a lower expected return. The
role of beta is not surprising since we know that beta captures the contribution of an asset to the risk
of a portfolio.
When deciding whether to make the trade, the firm has to decide whether the expected return
impact of the trade is high enough to justify the VaR impact of the trade. To make this decision, the
firm has to know the cost it attaches to an increase in VaR. We assume that the increase in the total
cost of VaR can be approximated for small changes by a constant incremental cost of VaR per unit
of VaR, which we call the marginal cost of VaR per unit. A firm that knows how much it costs to
increase VaR can then make the decision based on the expected gain of the trade net of the increase
in the total cost of VaR resulting from the trade:
Expected gain of trade net of increase in total cost of VaR =
Expected return impact of trade*Portfolio value
- Marginal cost of VaR per unit*VaR impact of trade
(4.3.)
From the discussion in the first part of this section, an increase in VaR makes it more likely that the
Chapter 4, page 23
firm will face financial distress costs if it does not take any actions. The marginal cost of VaR per unit
captures all the costs the firm incurs by increasing VaR by a dollar. These costs might be the greater
costs of financial distress or might be the costs of the actions the firm takes to avoid having an
increase in the probability of financial distress. We consider these actions in the next section.
Let’s consider an example. A firm has a portfolio of $100M consisting of equal investments
in three securities. Security 1 has an expected return of 10% and a volatility of 10%. Security 2 has
an expected return of 20% and a volatility of 40%. Finally, security 3 has an expected return of 25%
and a volatility of 60%. Security 1 is uncorrelated with the other securities. The correlation coefficient
between securities 2 and 3 is -0.4. Using the formula for the expected return of a portfolio, we have:
(1/3)*0.1 + (1/3)*0.2 + (1/3)*0.25 = 0.1833
The formula for the volatility of a portfolio gives us:
[(1 / 3) * 0.1 + (1 / 3) * 0.4 + (1 / 3) * 0.6 − 2 * (1 / 3) * (1 / 3) * 0.4 * 0.4 * 0.6]
2
2
2
2
2
2
0.5
= 0.1938
The VaR of the firm is $13.647M. It is 13.647% of the value of the portfolio because the return
corresponding to the fifth quantile of the distribution of returns is 0.1833 - 1.65*0.1938 or -13.647%.
Consider now the impact of a trade where the firm sells security 3 and buys security 1 for an amount
corresponding to 1% of the portfolio value, or $1m. The expected return impact of the trade is:
(0.1 - 0.25)*0.01 = -0.0015
Chapter 4, page 24
This means that the trade decreases the expected return of the portfolio by 0.15%. To compute the
VaR impact, we have to compute the beta of each security with respect to the portfolio. This requires
us to know the covariance of each security with respect to the portfolio. The covariances of the
securities with the portfolio are Cov(R1,Rp) = 0.0033 and Cov(R3,Rp) = 0.088.3 To compute the VaR
impact, we can obtain the beta of assets 1 and 3. The beta of asset 1 is Cov(R1,Rp)/Var(Rp), or
0.0033/0.19382, which is 0.088. The beta of asset 3 is 0.088/0.19382, or 2.348. The VaR impact is
therefore:
[-(0.10 - 0.25)*0.01 + (0.088 - 2.348)*1.65*0.1936*0.01]*$100M = -$571,934
The trade reduces the firm’s VaR by $571,934. It also makes an expected loss of $150,000. Suppose
that the firm believes that the trade would allow it to reduce its capital by the VaR impact of the
trade. To evaluate whether reducing the VaR is worth it, we have to evaluate the savings made as a
result of the reduction in the capital required by the firm resulting from the trade. To do that, we need
to know the total cost to the firm of having an extra dollar of capital for the period corresponding to
the VaR measurement period. This cost includes not only the cost charged by the investors discussed
in chapter 2 but also the firm’s transaction costs and deadweight costs (some of these costs were
discussed in chapter 3 and further analysis of these costs is given in section 4.3.1.) associated with
3
The covariance is computed as follows:
Cov(R 2 , R m ) = Cov(R 2 ,(1/ 3)* R1 + (1 / 3)* R 2 + (1 / 3)* R 3 )
= (1/ 3)* (Var(R1 ) + Cov(R1, R 2 ) + Cov(R1 , R 3 ))
= (1/ 3)*(0.01+ 0 + 0)
= 0.0033
Chapter 4, page 25
the use of external capital markets. Say that the total cost of capital is 14% for the firm and is
unaffected by the transaction. In this case, the net impact of the trade on the firm’s profits would be
-$150,000 + 0.14*$571,081 = -$70,049. Based on this calculation, the firm would reject the trade.
The approach developed in this section to evaluate the impact of a trade or a project on VaR
works for marginal changes. This is because the formula for the volatility impact of the trade uses the
covariances of security returns with the portfolio return. A large trade changes these covariances,
which makes the use of the formula inappropriate. This is a general difficulty with volatility and VaR
measures. These measures are not linear in the portfolio weights. In the case of VaR, this means that
the VaR of a portfolio is not the sum of the VaRs of the investments of that portfolio in individual
securities. This nonlinearity of VaR also implies that the VaR of a firm is not the sum of the VaRs of
its divisions. Therefore, one cannot simply add and subtract the VARs of the positions bought and
sold. Instead, one has to compare the VaR with the trade and the VaR without the trade to evaluate
the impact of large trades on VaR. Figure 4.3. plots the firm’s VaR as a function of the trade size.
This plot shows that if the firm wants to minimize its VaR, a trade much larger than the one
contemplated would be required.
Put Figure 4.3. here.
Section 4.2.2. CaR and project evaluation.
Let’s now see how a firm that finds CaR to be costly should decide whether to adopt or reject
a project. To do that, we have first to understand how the project affects the firm’s CaR. New
projects undertaken by a firm are generally too big relative to a firm’s portfolio of existing projects
to be treated like small trades for a bank. Consequently, the impact of a new project on CaR cannot
Chapter 4, page 26
be evaluated using a measure of marginal impact like the measure of marginal impact on VaR
presented in the previous section. Instead, the impact of a new project on CaR has to be evaluated
by comparing the CaR with the project to the CaR without the project. Let CE be the cash flow from
the existing projects and CN be the cash flow from a new project being considered. Assuming that
cash flow is normally distributed, the 5% CaR without the project is given by:
1.65*Vol(CE ))
(4.4.)
The CaR after the project is taken becomes:
1.65* Vol(C E + C N )
= 165
. [ Var(C E ) + Var(C N ) + 2Cov(C E , C N )]
0.5
(4.5.)
The impact of taking the project on CaR depends on its variance and on its covariance with existing
projects. A project with a higher variance of cash flow increases CaR more because such a project
is more likely to have large gains and large losses. Finally, a new project can have a diversification
benefit, but this benefit decreases as the covariance of the project’s cash flow with the cash flow of
the other projects increases.
Let’s consider now how the firm can evaluate a project when CaR has a cost. In the absence
of the cost of CaR, we already know from Chapter 2 how to evaluate a project using the CAPM. We
compute the expected cash flow from the project and discount it at the appropriate discount rate. If
CaR has a cost, we can treat the cost of CaR as another cost of the project. Hence, the NPV of the
project is decreased by the impact of the project on the cost of CaR. The cost of CaR is the cost as
of the beginning of the year resulting from the probability of having a low cash flow that prevents the
firm from taking advantage of valuable investment opportunities. This cost is therefore a present value
Chapter 4, page 27
and should be deducted from the NPV of the project computed using the CAPM. Consequently, the
firm takes all projects whose NPV using the CAPM exceeds their impact on the cost of CaR.
Let’s consider an example. We assume that cash flow is normally distributed. Suppose a firm
has expected cash flow of $80m with volatility of $50m. It now considers a project that requires an
investment of $50m with volatility of $50m. The project has a correlation coefficient of 0.5 with
existing projects. The project has a beta computed with respect to the market portfolio of 0.25. The
project has only one payoff in its lifetime and that payoff occurs at the end of the year. The expected
payoff of the project before taking into account the CAPM cost of capital for the initial investment
is $58m. We assume a risk-free rate of 4.5% and a market risk premium of 6%. Consequently, the
cost of capital of the project using the CAPM is 4.5% plus 0.25*6%. With this, the NPV of the
project using the CAPM is $58M/1.06 - $50M, or $4.72M. A firm that does not care about total risk
takes this project. However, the volatility of the firm’s cash flow in million dollars with the project
is:
(50 + 50 + 2 * 0.5 * 50 * 50) = 86.6025
2
2
0.5
Hence, taking the project increases the volatility of the firm’s cash flow by $36.6025M. The CaR
before the project is 1.65*50M, or $82.5M. The CaR after the project becomes 1.65*86.6025M, or
$142.894M. A firm that ignores the cost of CaR and uses the traditional NPV analysis would take
this project. A firm for which the cost of CaR is $0.10 per dollar of CaR would reject the project
because the NPV of the project adjusted for the cost of CaR would be $4.72M minus 0.10*60.394M,
or -$1.32M. More generally, a firm that assigns a cost of 8.28 cents per unit of CaR or higher would
Chapter 4, page 28
not take the project since in this case 8.28 cents times the change in CaR exceeds $5M.
If the project has cash flows over many years, it is not enough for the firm to subtract the cost
of CaR associated with the first year of the project. However, in that case, it may be difficult to
estimate the contribution of the project to the CaR of the firm in future years. To the extent that the
cost of CaR impact of the project in future years can be assessed, this impact has to be taken into
account when computing the value of the project. Each year’s cash flow of the project must therefore
be decreased by the impact of the project that year on the firm’s cost of CaR. If the impact of the
project on the cost of CaR is evaluated at the beginning of each year and takes place with certainty,
the project’s contribution to the firm’s cost of CaR is discounted at the risk-free rate to today.
Section 4.2.3. Allocating the cost of CaR or VaR to existing activities.
A firm has to evaluate the profitability of its current activities. This way, it can decide how
to reward its managers and whether it should eliminate or expand some activities. A popular
approach to evaluating profitability is EVA™, or economic value added. The idea of this approach
is straightforward. Over a period of time, an activity makes an economic profit. Computing this
economic profit can be complicated because economic profit is neither accounting profit nor cash
flow. Economic profit is the contribution of the activity to the value of the firm. Hence, it depends
on market value rather than accounting numbers. For instance, to compute economic profit we have
to take into account economic depreciation, which is the loss in value of the assets used by the
activity. In many cases, however, operating earnings of an activity are the best one can do to evaluate
its economic profits except for one crucial adjustment. When an activity’s accounting profits are
computed, no adjustment is made for the opportunity cost of the capital that finances the activity.
Chapter 4, page 29
When economic profits are computed, however, we must take this opportunity cost of the capital that
finances an activity into account. To see this, suppose that a firm has two divisions, division Light and
division Heavy. Both divisions have accounting profits of $20m. Based on accounting profits, the
CEO should be pleased. Yet, the key insight of EVA™ is that a firm might make accounting profits
but decrease shareholder wealth. This is because the accounting profits do not take into account the
opportunity cost of the capital used by the two divisions. Suppose that division Light uses $100M
of firm capital. For instance, it could have plants and working capital that are worth $100M. Suppose
that we assess a capital charge equal to the cost of capital of the firm of 10%. This means that the
firm could expect to earn 10% on the capital used by the division Light if it could invest it at the
firm’s cost of capital. The opportunity cost of the capital used by the division Light is therefore
$10M. Economic profits of division Light are $20M minus $10M, or $10M. Let’s now look at
division Heavy. This division has $1B of capital. Hence, the opportunity cost of this capital using the
firm’s cost of capital is $100M. Consequently, this division makes an economic loss of $80M, or
$20M minus $100M. A division that makes an economic profit before taking into account the
opportunity cost of capital can therefore make a loss after taking this cost into account.
The computation of economic profit that assigns a capital charge based on the assets of a
division is based on the capital budgeting tools derived in chapter 2. It ignores the fact that any
activity of the firm contributes to the risk of the firm. If the firm finds CaR to be costly, the
contribution of an activity to the cost of CaR is a cost of that activity like any other cost. An activity
might use no capital in the conventional sense, yet it might increase the firm’s CaR or VaR. If the firm
can access capital markets, one can think of that cost as the cost of the capital required to offset the
impact of the activity on the firm’s CaR or VaR. This capital is often called the risk capital. When one
Chapter 4, page 30
evaluates the profitability of an activity, one has to take into account not only the capital used to run
the activity but also the capital required to support the risk of the activity.
Consider a firm where CaR has a cost that the firm can estimate for the firm as a whole. This
means that the activities of the firm together have this cost of CaR. When evaluating a specific
activity, the firm has to decide how much of the cost of CaR it should assign to this activity. For the
firm to do this, it has to allocate a fraction of CaR to each activity. Let’s assume that the firm has N
activities. These activities could be divisions. Allocating 1/N of CaR to each activity would make no
sense because some activities may be riskless and others may have considerable risk. The firm
therefore has to find a way to allocate CaR to each activity that takes into account the risk of the
activity and how it contributes to firm risk. We have seen repeatedly that the contribution of a
security to the risk of a portfolio is measured by the covariance of its return with the return of the
portfolio or by the beta of the return of that security with the return of the portfolio. Using this insight
of modern portfolio theory, we can measure the contribution of the risk of an activity to the firm’s
CaR by the cash flow beta of the activity. An activity’s cash flow beta is the covariance of the cash
flow of the activity with the cash flow of the firm divided by the variance of the cash flow of the firm.
Let’s see how using cash flow betas allows the firm to allocate CaR to the various activities.
The cash flow of the i-th activity is Ci. The firm’s total cash flow is the sum of the cash flows of the
activities, which we write C. We assume that the cash flows are normally distributed. Consequently,
the CaR of this firm is 1.65Vol(C). Remember that the variance of a random variable is the covariance
of the random variable with itself, so that Var(Ci) is equal to Cov(Ci,Ci). Further, covariances of cash
flows of the various activities with the cash flow of activity i can be added up so that the sum of the
Chapter 4, page 31
covariances is equal to the covariance of the cash flow of activity i with the firm’s cash flow, Cov(Ci,
C). The variance of the firm cash flow is therefore the sum of the covariances of the cash flow of the
activities with the cash flow of the firm:
Firm cash flow variance = Var(C)
N
N
= ∑ ∑ Cov(C i , C j )
i=1 j=1
N
N
i=1
j=1
= ∑ Cov(C i , ∑ C j )
N
= ∑ Cov(C i , Firm cash flow C)
(4.6.)
i=1
Equation (4.6.) shows that the variance of firm cash flow can be split into the covariance of the cash
flow of each activity with the firm’s cash flow. Increasing the covariance of the cash flow of an
activity with the firm’s cash flow increases the variance of the firm’s cash flow. If we divide
Cov(Ci,C) by Var(C), we have the cash flow beta of activity i with respect to firm cash flow, $i. The
cash flow betas of the firm’s activities sum to one since adding the numerator of the cash flow betas
across activities gives us the variance of cash flow which is the denominator of each cash flow beta:
N
N
i=1
i=1
∑ βi = ∑ Cov(C i , C) / Var(C)
(4.7.)
= Var(C) / Var(C)
=1
Because the cash flow betas sum to one, they allow us to decompose CaR into N components, one
for each activity. The firm’s CaR is then equal to 1.65 times the sum of the cash flow betas times the
volatility of cash flow:
Chapter 4, page 32
Firm CaR = 1.65Vol(C)
= 1.65Var(C) / Vol(C)
N
= 1.65∑ Cov(C i , C ) / Vol(C)
i=1


 ∑ Cov(C i , C) 
 1.65Var(C) / Vol(C)
=  i=1


Var(C)




N
(4.8.)
N
= ∑ βi CaR
i=1
This decomposition attributes a greater fraction of CaR to those activities that contribute more to the
volatility or the variance of firm cash flow. Consequently, the cash flow beta of the i-th activity is the
fraction of the firm’s CaR that we can attribute to project i:
Contribution of activity to CaR
The fraction of CaR that is due to activity i is equal to the beta of activity i with respect to firm cash
flow, $i=Cov(Ci,C)/Var(C).
Consequently, the contribution of activity i to the cost of CaR is equal to the cash flow beta of the
activity times the firm’s cost of CaR.
Let’s consider an example. A firm has a CaR of $100M. An activity has an expected cash flow
of $12M. The beta of the cash flow is 0.5. The activity uses $40M of capital before taking into
account the CaR. The cost of capital is 10% p.a. In this case, the expected economic profit of the
Chapter 4, page 33
activity before taking into account CaR is $12M minus 10% of $40M, which amounts to $8M. The
project therefore has a positive expected economic profit before taking into account CaR. The
contribution of the activity to the risk of the firm is 0.5*$100M, or $50M. If the cost of CaR is $0.20
per dollar, the activity contributes $10M to the firm’s cost of CaR. As a result, the project is not
profitable after taking into account the cost of CaR.
Consider next a firm that uses VaR. Such a firm will want to evaluate the performance of
positions or activities. It must therefore be able to allocate VaR across these positions or activities.
Remember that with VaR, we use the distribution of the change in the value of the portfolio or firm
instead of the distribution of cash flow. However, if changes in value are normally distributed, the
analysis we performed for CaR applies to VaR. The change in value of the portfolio is the sum of the
changes in value of the positions in the portfolio. The variance of the change in the value of the
portfolio is equal to the sum of the covariances of the changes in the value of the positions with the
change in the value of the portfolio. Consequently, we can compute the beta of a position with
respect to the portfolio, namely the covariance of the change in value of the position with the change
in value of the portfolio divided by the variance of the change in value of the portfolio. This position
beta measures the contribution of the position to the VaR.
When considering VaR examples, we assumed that returns instead of dollar changes were
normally distributed. Let’s see how we can decompose VaR in this case among N positions. The
investment in position i is the portfolio share of the position times the value of the portfolio, wiW. The
return of the position is Ri. Consequently, the change in value due to position i is wiRiW. Let Rm be
the return of the portfolio. In this case, the variance of the dollar change in the portfolio can be
written as:
Chapter 4, page 34
Variance of change in portfolio value
N
= ∑ Cov(w i R i W, R m W)
i=1
N
= ∑ w i Cov(R i , R m )W 2
i=1
N
= ∑ w i (Cov(R i , R m ) / Var(R m ))Var(R m )W 2
i=1
N
= ∑ wi β im Var(Change in portfolio value)
(4.9.)
i=1
Consequently, the contribution of a position to the variance of the change in portfolio value is the
portfolio share of the position times the beta of the return of the position with respect to the return
of the portfolio. Using the same reasoning as with CaR, it follows that:
Contribution of position to VaR
The fraction of VaR that is due to position i is the portfolio share of that position times the beta of
its return with respect to the return of the portfolio.
Let’s look at an example. Suppose that a firm has 50% of its portfolio invested in a stock
whose return has a beta of 3 with respect to the portfolio. In this case, the portfolio share is 0.5 and
the fraction of VaR contributed by the position is 3*0.5, or 1.5. This means that 150% of the cost of
VaR should be charged to the position. If a position contributes more than 100% of VaR, it means
that the other positions reduce risk. Hence, contributions to VaR can exceed 100% or can be
negative. Consider now a firm where VaR has a cost of $10M per year. For the i-th position to be
worthwhile, it must be that its expected gain exceeds $15M, which is its contribution to the cost of
Chapter 4, page 35
VaR.
Rather than thinking of a position in a portfolio, however, we could think of an investment
bank where a trading post contributes to the VaR of the bank. Suppose that at the end of the year the
capital used by the trader has increased. The trader had $50M at the start of the year and has $75M
at the end of the year, so that he earned $25M. To find out whether the trading was profitable, we
have to take into account the opportunity cost of $50M. Say that the firm’s cost of capital is 15% and
that this is a reasonable measure of the opportunity cost of $50M. After this capital allocation, the
trader has a profit of only $17.5M, or $25M minus 15% of $50M. This capital allocation does not
take into account the impact of the trader’s activity on the VaR of the firm and its cost. Suppose that
the VaR of the firm is $2B and that the beta of the return of the trader with the return of the firm is
1. The firm’s value at the beginning of the year is $1B. The portfolio share of the trader is therefore
$50M/$1B, or 0.05. The trader also contributes 5% of VaR. Let’s now assume that the cost of VaR
is $100m. In this case, the trader contributes $5M of the cost of VaR. His contribution to the
economic profits of the firm is therefore $17.5M minus $5M, or $12.5M.
Firms that evaluate economic profits taking into account the contribution of activities to the
risk of the firm do so in a number of different ways and give different names to the procedure. One
approach that is often used, however, is called RAROC among financial institutions. It stands for riskadjusted return on capital. Box 4.2. RAROC at Bank of America discusses the application of that
approach at one financial institution.
Put Box 4.2. here
Section 4.3. Managing firm risk measured by VaR or CaR.
Firm risk as measured by VaR or CaR is costly. Consequently, the firm can increase its value
Chapter 4, page 36
if it succeeds in reducing its cost of risk. A firm can do so in two ways. First, it can reduce risk. This
amounts to reducing VaR or CaR. Second, it can reduce the cost of risk for a given level of CaR or
VaR. The firm has numerous tools available to change the cost of risk. The simplest way to reduce
risk is to sell the firm’s projects and put the proceeds in T-bills. To the extent that the projects are
more valuable within the firm than outside the firm, this approach to risk reduction is extremely
costly. Hence, the firm has to find ways to manage the cost of risk so that firm value is maximized.
Three approaches represent the firm’s main options to manage the cost of risk: increase capital, select
projects according to their impact on firm-wide risk, use derivatives and other financial instruments
to hedge. We discuss these approaches in turn.
Section 4.3.1. Increase capital.
Consider first a financial institution. This institution is concerned that a fall in value will
increase its credit risk and lead to an exodus of its customers. The institution’s credit risk is inversely
related to its equity capital since equity capital provides funds that can be used to compensate
customers in the event that the firm’s traded assets lose value. The same reasoning applies for a
nonfinancial corporation that focuses on CaR. The corporation is concerned about its CaR because
a bad cash flow outcome means that the firm cannot invest as much as would be profitable. If the firm
raises more capital, it can use it to create a cash reserve or to decrease debt. In either case, the firm’s
ability to pursue its investment plan increases with its equity capital.
If capital has no opportunity cost, there is no reason for a firm to be concerned about its CaR
or VaR risk. In this case, the firm can simply increase its capital up to the point where a bad outcome
in cash flow or in the value of its securities has no impact on credit risk. Let’s make sure that we
Chapter 4, page 37
understand how increasing the firm’s equity impacts CaR and its cost. Remember that CaR measures
a dollar shortfall. If the firm raises equity and uses it to just expand the scale of its activities, CaR
increases because the cash flow of the firm after the equity issue is just a multiple of the firm’s cash
flow before the equity issue. To issue equity in a way that reduces the firm’s risk, the proceeds of
the equity issue have to be invested in projects that have negative cash flow betas. Investing the
proceeds of the equity issue in risk-free investments or using them to pay back debt with fixed debt
service does not change the firm’s CaR. Remember that a project’s contribution to CaR is its cash
flow beta. If debt payments and interest earned on the risk-free asset are non-stochastic, they have
a cash flow beta of zero and therefore do not increase or decrease CaR. However, investing the
proceeds in the risk-free asset or using them to reduce the firm’s debt reduces the cost of CaR. As
the firm builds up slack, a bad cash flow outcome has less of an impact on investment because the
firm can use its slack to offset the cash flow shortfall. Hence, an equity issue reduces the cost of CaR
by enabling the firm to cope more easily with a cash flow shortfall. One might argue that the firm does
not have to raise equity now to cope with a cash flow shortfall that might occur in the future. This
would be correct if the firm faces the same cost of raising equity after the cash flow shortfall has
occurred than before. In general, this will not be the case. After a cash flow shortfall has occurred,
outsiders have trouble figuring out whether the cash flow shortfall occurred because of bad luck,
because of managerial incompetence, or because of management pursuing objectives other than the
maximization of shareholder wealth. As a result, equity capital will tend to be most expensive or even
impossible to obtain precisely when obtaining it has most value for the firm.
As a firm raises equity to reduce its cost of CaR, it must keep its operations unchanged and
use the proceeds to pay back debt, which reduces debt service, or to invest in risk-free assets.
Chapter 4, page 38
Consider now the firm that issues equity and buys back debt. The equity holders expect a return on
their equity commensurate with the risk of equity. By issuing equity to buy back debt, the firm makes
the remaining debt less risky and generally gives a windfall to existing debtholders whose debt is
bought back by making it unexpectedly risk-free. Hence, shareholders do not capture the whole
benefit of increasing the firm’s equity capital. Some of that benefit is captured by the debtholders.
Further, shareholders lose the tax shield on the debt bought back and replaced by equity. As a result,
issuing equity to reduce the cost of CaR has costs for the shareholders. The costs of equity are
increased further by the fact that management can do anything with this money that it wants unless
the firm becomes bankrupt or is taken over. Hence, management must convince equity holders that
it will invest the new money profitably. Doing so is difficult for management because of information
asymmetries and because of agency costs.
Management generally benefits from having more resources under its control. With more
assets, the firm is less likely to become bankrupt and management has more perks and is paid more.
Hence, if management could raise equity easily, it would keep doing so. The problem is that since
management benefits from having more resources under its control, it wants to raise equity even when
it does not have good uses for the new funds. The possibility that the projects the firm has might not
be as good as management claims reduces the proceeds from new equity issues. By lowering the price
they are willing to pay for shares, investors make it more likely that they will receive a fair return.
This makes new equity expensive. In fact, new equity can be so expensive that a firm with valuable
new projects to finance may decide not to issue equity and give up its investment opportunities.
Having enough equity to make risk management irrelevant is generally not an option.
Financial economists have provided measures of the cost of raising equity capital. First, there
Chapter 4, page 39
is a transaction cost. Investment bankers have to be paid. This cost can be of the order of 5% of the
value of the new equity. Second, there is a stock-price impact. When a firm announces an equity
issue, the value of its existing equity falls by about 2.5%. There is some debate as to whether some
or all of this fall in the equity price represents a cost of issuing equity or is just the result that the
market learns about the true value of the firm when it issues equity, information that the market
would have learned anyway. This fall, however, means that if a firm with one billion dollars worth of
equity sells $100m of new equity, its net resources increase by about $75m. In other words, the
increase in net resources associated with an equity issue is only a fraction of the equity raised. This
evidence is only for firms that issue equity. It does not include firms that give up profitable projects
instead of issuing equity. That firms give up valuable projects instead of issuing equity is documented
by a literature that shows the impact of firm liquidity on investment. This literature shows that firms
with fewer liquid assets typically invest less controlling for the value of the investment opportunities.
In other words, if two firms have equally valuable investment opportunities, the one with fewer liquid
assets invests less.
Section 4.3.2. Project choice.
When firm risk is costly, the firm can choose projects so that its risk is low. This means that
the firm gives up projects with high profits if these projects increase firm risk substantially and that
it chooses poor projects because they might reduce firm risk. This approach of dealing with risk is
quite expensive. It means that the firm avoids investing in projects that might have high payoffs simply
because they contribute to firm risk. It also means that it tailors its strategy so that it invests in
activities not because they are the most profitable but because they have a favorable impact on firm
Chapter 4, page 40
risk.
Traditionally, firms have viewed diversification across activities as a way to manage their risk.
By investing in a project whose return is imperfectly correlated with existing projects instead of
investing in a project whose return is perfectly correlated with them, the firm increases risk less. It
could be, however, that the project with greater diversification benefits has lower expected profits.
In fact, a firm might prefer a project that has a negative NPV using the CAPM to a project with a
positive NPV if the project with the negative NPV decreases firm risk. The problem with managing
the firm’s risk this way is that it is also very expensive. The reason for this is that managing projects
that are not closely related is costly. A firm that has firm-specific capital that gives it a comparative
advantage to develop some types of projects does not have that comparative advantage for
diversifying projects. Diversifying projects require another layer of management.
Empirical evidence demonstrates that typically the costs of diversification within firms are
high. The simplest way to evaluate these costs is to compare a diversified firm to a portfolio of
specialized firms whose assets match the assets of the diversified firm. The empirical evidence
indicates that the portfolio is on average worth about 14% more than the diversified firm. This
empirical evidence therefore documents a “diversification discount,” in that a diversified firm is worth
less than a matching portfolio of specialized firms.
Section 4.3.1.C. Derivatives and financing transactions.
We saw that firms could reduce their cost of risk by increasing their equity or by choosing
projects that reduce risk. Both of these solutions to managing firm risk involve substantial costs. In
general, one would expect that firms could reduce their risk through the use of financial instruments
Chapter 4, page 41
at less cost. The reason is that transaction costs associated with trades in financial instruments are
generally small. They are measured in basis points or pennies per dollar. In the most liquid markets,
such as the spot foreign exchange market, they are even smaller than that. By using financial
instruments to manage risk, the firm therefore achieves two outcomes. First, it reduces its risk, which
means that it increases its value. Second, when the firm contemplates a new project, it can structure
the new project so that its hedged risk is minimized. This means that the relevant measure of the
project’s contribution to firm-wide risk is its contribution net of hedging activities. It is perfectly
possible for an unhedged project to contribute too much to the risk of the firm to be worthwhile, but
once the project is hedged, it becomes a worthwhile project that the firm can take profitably.
Using financial instruments to reduce risk can therefore allow a firm to grow more and to be
worth more. The firm can use existing financial instruments or new ones. As the firm uses financial
instruments, however, the risk of these instruments has to be taken into account when measuring the
firm’s risk. This is feasible when the firm uses VaR since then the financial instruments that the firm
uses become part of its portfolio.
Taking into account the risk of financial instruments is more difficult if the firm uses CaR as
a risk measure. The reason for this is that the firm should monitor the value of its position in financial
instruments frequently and the risk of financial instruments is best measured with a VaR measure.
Hence, a firm could compute a VaR for the financial instruments it has taken positions in. Existing
SEC rules on the reporting of derivatives in fact encourage firms to report such a VaR measure. The
problem with such a measure, though, is that it is easy to misinterpret. Consider a firm that has
important risks that are hedged with financial instruments. The portfolio of financial instruments is
risky as a stand-alone portfolio. However, since the portfolio is a hedge, firm risk is less with the
Chapter 4, page 42
portfolio than without. The stand-alone risk of the portfolio of financial instruments is therefore not
instructive about the contribution of these financial instruments to the risk of the firm.
Section 4.3.3. The limits of risk management.
There are several reasons why firms cannot eliminate all risks and may not find it profitable
to eliminate some risks they could eliminate. It only makes sense to get rid of risks if doing so
increases firm value. This means that risks that have little impact on firm value but are expensive to
eliminate are kept. There are three key limits to risk management:
1. It is not rocket science. Financial engineering, derivatives and parts of risk management
are often viewed as the domain of rocket scientists. It is perfectly true that there are many
opportunities for scientists from the hard sciences to make major contributions in financial
engineering and risk management. At the same time, however, finance is part of the social sciences.
This means that there are no immutable laws such that every action always leads to the same
outcome. Markets are made of people. The way people react can change rapidly. Hence, models used
for risk management are always imperfect, between the time they were devised and the time they are
used, things have changed. A keen awareness of this intrinsic limitation of risk management is crucial.
This means that all the time we must seek to understand where the models we use have flaws. We
can never stop from asking “What if...” The risk measures we have developed are tremendously
useful, but no risk manager can ever be sure that he is taking into account all factors that affect the
risk of the firm and that he is doing so correctly. Even if he did everything perfectly when he
estimated the firm’s risk, the world may have changed since he did it.
2. Credit risks. For the firm to be able to receive payments in bad times, it must either make
Chapter 4, page 43
payments now or be in a position to make offsetting payments in good times. If there are no
information asymmetries about the financial position of the firm, the firm will be able to enter
contracts that achieve its risk management objectives. However, counterparties may be uncertain
about the firm’s ability to pay off in good times or even about what good times are for the firm. In
this case, counterparties will require collateral or will only agree to prices that are unfavorable to the
firm. The firm may find it too expensive to trade at the prices it can trade at.
Let’s look at an example. Suppose that the firm is an exporter to Germany. For constant sales,
the exporter receives more dollars as the dollar price of the Euro increases. If the exporter has
constant Euro sales, selling Euros forward eliminates all income risk. Hence, if this is the only risk
the firm faces, it should be easy for it to eliminate risk. However, the firm may have other risks. For
instance, it could be that sales in Euros are uncertain, so that in some situations the company might
not have enough Euros to deliver to settle the forward contract and has to default. If outsiders are
uncertain about the risks the firm has, they may only be willing to transact at a forward price that is
highly unfavorable to the firm given management’s information about the firm’s risks. In this case,
the firm may choose not to hedge unless management can convince outsiders about the nature of the
risks the firm faces. Alternatively, the firm may choose not to export to Germany if the unhedged
risks are too high.
3. Moral hazard. Let’s go back to the above example. Suppose now that the sales in
Germany depend on the efforts of management. Management sells a given quantity of Euros forward,
but to obtain that quantity of Euros, management will have to work hard. Whether management
works hard or not cannot be observed by outsiders, however. If management sold Euros forward and
the Euro appreciates, management may choose to work less hard because it does not get as much
Chapter 4, page 44
benefit from working hard as it would have if it had not sold the Euros forward. This is because the
Euros it gets are delivered to settle the forward contract at a lower price than the spot price. Hence,
management might be tempted not to work when the Euro appreciates and to work hard when it
depreciates since then it gets to sell the Euros at a higher price than the spot price. The fact that
management can take actions that affect the value of contracts adversely for the counterparty creates
what is called moral hazard. Moral hazard is the risk resulting from the ability of a party to a
contract to take unobserved actions that adversely affect the value of the contract for the other party.
The existence of moral hazard may make it impossible for the firm to take the forward position it
wants to. This is because the counterparty expects the firm to default if the Euro appreciates because
management does not work hard. Even if management wanted to work hard in that case, it may not
be able to convince the counterparty that it would do so because it would always benefit from
convincing the counterparty that it will work hard and then not doing so.
There is one aspect of the moral hazard problem that we want to stress here. Remember from
the discussion in the previous chapter that the firm can sell debt at a higher price if the buyers believe
that the firm will reduce its risk. Hence, it is tempting for management to promise to reduce risk and
then forget about its promise. However, if the firm’s creditors expect management to behave this
way, they buy debt at a low price. It is therefore important for the firm to be able to commit credibly
to a policy of risk management. For instance, it may want to write in the bond indenture provisions
about the risks it will eliminate and how, so that if it does not follow through with its risk
management promises, it will be in default. Interestingly, debt in leveraged buyouts, where firms
typically have extremely high leverage often, has covenants specifying that the firm has to manage
some risks, for instance interest rate risks.
Chapter 4, page 45
As we learn more about risk management and financial engineering, we will also learn more
about how to cope and extend the limits of risk management.
Section 4.4. An integrated approach to risk management.
In chapter 3, we showed how firms can benefit from risk management. In this chapter, we
showed how the analysis of chapter 3 implies that firms will want to manage specific risk measures.
We therefore introduced VaR and CaR. Estimating a firm’s risk is the starting point of using risk
management to increase firm value. Once a firm’s risk is estimated, it has to figure out the cost of
bearing this risk and assess how best to bear the optimal amount of risk. We argued that when
evaluating new projects, firms have to evaluate their impact on the cost of bearing risk. We then
showed that a firm has a number of tools available to manage firm risk. Firms can increase their equity
capital, take projects that are less risky, take projects that decrease firm-wide risk, or use financial
instruments to hedge risks. Derivatives often provide the most efficient tools to manage firm risk. The
reason for this is that this tool is often cheap and flexible. Derivatives can be designed to create a
wide range of cash flows and their transaction costs are generally extremely low compared to the
costs of using other risk management tools.
Chapter 4, page 46
Literature note
For an analysis of risk management in financial institutions and the role of risk capital, see
Merton (1993) and Merton and Perold (1993). VaR is presented in the Riskmetrics technical manual
and in Jorion (). EVA™ is presented extensively in Stewart (). For a discussion of RAROC, see
Zaik, Walter, Kelling, and James (1996). Litterman () discusses the marginal VaR and plots VaR as
a function of trade size. Matten (1996) provides a book-length treatment of the issues related to
capital allocation. Saita (1999) discussed a number of organizational issues related to capital
allocation. Froot and Stein (1998) provide a theoretical model where they derive optimal hedges and
capital budgeting rules. Stoughton and Zechner (1998) extend such an approach to take into account
information asymmetries. The internal models approach for regulatory capital is presented in
Hendricks and Hirtle (1997). Santomero (1995) provides an analysis of the risk management process
in commercial banks.
Smith (1986) reviews the literature on the stock-price impact of selling securities. Myers and
Majluf (1984) discuss the case where a firm does not issue equity because doing so is too expensive.
Stulz (1990) examines the implications of the agency costs of managerial discretion on the firm’s cost
of external finance. Fazzari, Hubbard, and Petersen (1988) provide a seminal analysis of how
investment is related to liquidity. Opler, Pinkowitz, Stulz and Williamson (1999) show how
investment is related to slack. Lang and Stulz (1995) and Berger and Ofek (1996) discuss the costs
of diversification.
Chapter 4, page 47
Key concepts
VaR, CaR, VaR, VaR impact of trade, CaR impact of project, cash flow beta, contribution of activity
to CaR, contribution of position to VaR.
Chapter 4, page 48
Review questions
1. Why do firms require a risk measure?
2. What is VaR?
3. How does VaR differ from variance?
4. How is the VaR of a portfolio computed if returns on assets are normally distributed?
5. Why is VaR an appropriate risk measure for financial institutions?
6. What is CaR?
7. When would a firm use CaR?
8. How do you choose a project if VaR is costly?
9. How do you choose a project if CaR is costly?
10. How do you measure a division’s contribution to the firm’s CaR?
11. Should the contribution of a division to the firm’s CaR affect the firm’s estimate of its
profitability?
12. Why and how should you use VaR to evaluate the profitability of a trader?
13. How can the firm reduce CaR or VaR?
14. How can the firm reduce its cost of CaR or VaR for given levels of CaR or VaR?
15. How does moral hazard limit the ability of a firm to manage its risk?
Chapter 4, page 49
Problems
1. Consider a firm with a trading book valued at $100M. The return of these assets is distributed
normally with a yearly standard deviation of 25%. The firm can liquidate all of the assets within an
hour in liquid markets. How much capital should the firm have so that 99 days out of 100, the firm’s
return on assets is high enough that it does not exhaust its capital?
2. Consider the firm of question 1. Now the firm is in a situation where it cannot liquidate its portfolio
for five days. How much capital does it need to have so that 95 five-day periods out of 100, its capital
supports its trading activities ignoring the expected return of the firm’s trading book?
3. How does your answer to question 2 change if the firm’s trading book has an annual expected
return of 10%?
4. A firm has a trading book composed of two assets with normally distributed returns. The first
asset has an annual expected return of 10% and an annual volatility of 25%. The firm has a position
of $100M in that asset. The second asset has an annual expected return of 20% and an annual
volatility 20% as well. The firm has a position of $50m in that asset. The correlation coefficient
between the return of these two assets is 0.2. Compute the 5% annual VaR for that firm’s trading
book.
5. Consider a firm with a portfolio of traded assets worth $100M with a VaR of $20M. This firm
considers selling a position worth $1m to purchase a position with same value in a different asset. The
Chapter 4, page 50
covariance of the return of the position to be sold with the return of the portfolio is 0.05. The asset
it acquires is uncorrelated with the portfolio. By how much does the VaR change with this trade?
Hint: Remember how to go from return VaR to dollar VaR.
6. Going back to question 4, consider a trade for this firm where it sells $10m of the first asset and
buys $10m of the second asset. By how much does the 5% VaR change?
7. A firm has a yearly cash flow at risk of $200M. Its cost of capital is 12%. The firm can expand the
scale of its activities by 10%. It wants to increase its capital to support the new activities. How much
capital does it have to raise? How much must the project earn before taking into account the capital
required to protect it against losses to be profitable?
8. Consider the choice of two mutually exclusive projects by a firm. The first project is a scale
expanding project. By investing $100M the firm expects to earn $20M a year before any capital
charges. The project is infinitely lived, so that there is no depreciation. This project also increases
cash flow at risk by $50M. The second project requires no initial investment and is expected to earn
$25M. This project increases the cash flow at risk by $200M. Which project has the higher expected
economic profit in dollars?
9. The treasurer of a firm tells you that he just computed a one-year VaR and a one-year CaR for his
firm. He is puzzled because both numbers are the same. How could that be the case?
Chapter 4, page 51
10. You sit on the board of corporation and you believe that management hedges too much. You
want to make sure that management hedges so as to maximize the value of the firm only. You do not
know which hedges will achieve that outcome. What are the options available to you to make the
desired outcome more likely given the empirical evidence?
Chapter 4, page 52
Figure 4.1. Using the cumulative distribution function.
The figure graphs the cumulative distribution function of a normally distributed return with expected
value of 10% and volatility of 20%. From this graph, the probability of a loss of 33% or greater is
0.05.
Probability
0.16
0.05
-33%
-10%
Chapter 4, page 53
Return in
percent
Figure 4.2. Frequency distribution of two portfolios over one year horizon.
The stock portfolio is invested in a stock that has an expected return of 15% and a volatility of return
of 20% over the measurement period. The other portfolio has 1.57 calls on the stock with exercise
price of $100M that cost $23.11M and the rest invested in the risk-free asset earning 6%.
P o r t f o lio v a lu e s
Chapter 4, page 54
403
367
332
296
260
224
189
153
117
81.5
46.6
Probability
D i s t r i b u t i o n fo r p o r t f o l i o v a l u e s
Figure 4.3. VaR as a function of trade size.
This figure graphs the VaR as a function of trade size for the example used in the text.
Chapter 4, page 55
Box 4.1. VaR, banks, and regulators
The U.S. regulatory agencies adopted the market risk amendment to the 1988 Basle Capital Accord
(which regulates capital requirements for banks to cover credit risk) in August 1996. This amendment
became effect in January 1998. It requires banks with significant trading activities to set aside capital
to cover market risk exposure in their trading accounts.
The central component of the regulation is a VaR calculation.The VaR is computed at the 1% level
for a 10-day (two weeks) holding period using the bank’s own model. The capital the firm must set
aside depends on this VaR in the following way. Let VaRt(1%,10) be the VaR of the trading accounts
computed at the 1% level for a ten day trading period at date t. The amount of capital the bank has
to hold for the market risk of its trading accounts is given by:


1 59
Required capital for day t + 1 = Max VaR t (1%,10 ); S t * ∑ VaR t-i (1%,10) + SR t
60


i= 1
where St is a multiplier and SRt is an additional charge for idiosyncratic risk. The terms in square
brackets are the current VaR estimate and an average of the VaR estimate over the last 60 days. The
multiplier St depends on the accuracy of the bank’s VaR model. The multiplier uses the banks VaR
for a one-day period at the 1% level over the last 250 days. If the bank exceeds its daily VaR 4 times
or less, it is in the green zone and the multiplier is set at 3. If the bank exceeds its daily VaR 5 to 9
times, it is in the yellow zone and the multiplier increases with the number of cases where it exceeds
the VaR. If the bank exceeds its daily VaR 10 times or not, its VaR model is deemed inaccurate and
the multiplier takes a value of 4. Hence, by having a better VaR model, the bank saves on regulatory
capital.
Banks routinely provide information on their VaR for trading accounts in their annual reports. They
did so even before the market risk amendment. In 1996, JP Morgan reported an average one-day VaR
at the 5% level of $21M. In contrast, Bankers Trust reported an average one-day VaR of $39M at
the 1% level.
Chapter 4, page 56
Box 4.2. RAROC at Bank of America
In November 1993, a Risk and Capital Analysis Department was formed at Bank of America and
charged with developing a framework for risk-adjusted profitability measurement. The requirement
was that the system would be operational within four months. The bank decided to measure risk over
a one-year horizon. Four risks were identified, namely credit risk, country risk, market risk, and
business risk. Credit risk is the risk of borrower default. Country risk is the risk of loss in foreign
exposures arising from government actions. Market risk is the risk associated with changes in market
prices of traded assets. Finally, business risk corresponds to operational risk associated with business
units as ongoing concerns after excluding the other three risks. For each of these risks, the bank is
concerned about unexpected risk and how it will affect its own credit rating. For instance, it regularly
makes provisions for credit losses. Consequently, normal credit losses do not affect earnings but
unexpected credit losses do.
Bank of America decided to have an amount of capital such that the risk of default is 0.03% per year,
which guarantees an AA rating. It concluded that to ensure a probability of default no greater than
0.03% across its various businesses, it had to allocate capital of 3.4 standard deviations to market
risks and 6 standard deviations to credit risks. The reason it allocates different amount of capital to
different risks is that it views market risks as being normally distributed whereas credit risks are not.
All capital allocated is charged the same hurdle rate, which is the corporate-wide cost of equity
capital. A project is evaluated based on its economic profit, calculated as earnings net of taxes,
interest payments, and expected credit losses. The capital required for the project is then determined
based on the credit risk, market risk, country risk, and business risk of the project. The risk-adjusted
expected economic profit is then the expected economic profit minus the hurdle rate times the
allocated capital.
Bank of America applies the above approach to evaluate its business units. For each business unit,
it computes its RAROC as follows:
RAROC of business unit '
Economic profit of unit & Capital allocated(Hurdle rate
Capital allocated
Source: Christoper James, RAROC based capital budgeting and performance evaluation: A case study
of bank capital allocation, working paper 96-40, Wharton Financial Institutions Center, The Wharton
School, Philadelphia, PA.
Chapter 4, page 57
Technical Box 4.1. Impact of trade on volatility.
Remember the definition of the variance of the portfolio return given in equation (2.3.):
N
N
N
i= 1
i= 1
j≠ i
∑ w 2i Var(R i ) + ∑ ∑ w i w jCov(R i , R j ) = Var(R p )
Taking the derivative of the formula for the variance with respect to wi, the impact of an increase in
the portfolio share of security i of )w on the variance is:
N
2w i Var(R i )∆w + 2∑ w jCov(R i , R j )∆w = Impact of trade ∆w on Var(R p )
j≠ i
The volatility is the square root of the variance. Consequently, taking the derivative of volatility with
respect to the variance, a change in the variance of )Var(Rp) changes the volatility by
0.5*)Var(Rp)*[Var(Rp)]-0.5 = 0.5*)Var(Rp)/Vol(Rp). We substitute in this expression the change in
the variance brought about by the increase in the holding of security i to get the impact on portfolio
return volatility:
N
0.5[2w i Var(R i )∆w + 2∑ w jCov(R i , R j )∆w] / Vol(R p )
j≠ i
N
= [w i Var(R i )∆w + ∑ w jCov(R i , R j )∆w] / Vol(R p )
j≠ i
= Cov(R i , R p )∆w / Vol(R p )
= βip Vol(R p )∆w
Equation (4.1.) follows from computing the impact of increasing the portfolio share of security i by
Chapter 4, page 58
)w and decreasing the portfolio share of security j by the same amount.
Chapter 4, page 59
Chapter 5: Forward and futures contracts
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section 5.1. Pricing forward contracts on T-bills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Section 5.1.A.Valuing a forward position at inception using the method of pricing
by arbitrage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Section 5.1.B. A general pricing formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Section 5.1.C. Pricing the contract after inception . . . . . . . . . . . . . . . . . . . . . . . 11
Section 5.2. Generalizing our results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Section 5.2.A. Foreign currency forward contracts. . . . . . . . . . . . . . . . . . . . . . 17
Section 5.2.B. Commodity forward contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Section 5.3. Futures contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Section 5.3.A. How futures contracts differ from forward contracts . . . . . . . . . 25
Section 5.3.B. A brief history of financial futures . . . . . . . . . . . . . . . . . . . . . . . 29
Section 5.3.C. Cash versus physical delivery. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Section 5.3.D. Futures positions in practice: The case of the SFR futures contract
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Section 5.4. How to lose money with futures and forward contracts without meaning to.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Section 5.4.A. Pricing futures contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Section 5.4.B. The relation between the expected future spot price and the price
for future delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Section 5.4.C. What if the forward price does not equal the theoretical forward
price? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Section 5.5. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Literature Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Table 5.1. Arbitrage trade when F = $0.99 per dollar of face value . . . . . . . . . . . . . . . . 51
Table 5.2. Long SFR futures position during March 1999 in the June 1999 contract . . . 52
Exhibit 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 5.1. Time-line for long forward contract position . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 5.2. Difference between the forward price and the spot price of the underlying
good . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 5.3. Margin account balance and margin deposits . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 5.4. Daily gains and losses from a long futures position in the SFR in March 1999
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 5.5. Marking to market and hedging.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Box 5.1. Repos and borrowing using Treasury securities as collateral . . . . . . . . . . . . . . 59
Box 5.2. Interest rate parity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Box 5.2. Figure. Term structures in the home and foreign country used to obtain the term
structure of forward premia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Box 5.3. The cheapest to deliver bond and the case of Fenwick Capital Management . . 63
Chapter 5: Forward and futures contracts
December 3, 1999
© René M. Stulz 1998, 1999
Ch. 1 Pg. 1
Chapter objectives
1. Explain how forward contracts are priced.
2. Describe how futures contracts differ from forward contracts.
3. Analyze the determinants of the price of futures contracts.
Ch. 1 Pg. 1
We now understand that risk management can be used to increase firm value and the welfare
of investors. In the remainder of this book, we show how to do it. Much of our focus is on how to
use existing derivatives and create new ones to increase firm value through risk management.
However, we also discuss how to use derivatives in portfolio strategies. There are too many different
types of derivatives for us to provide a complete catalogue of their properties and uses. Instead, we
will learn how to understand and use derivatives in general.
We start in this chapter with the simplest kinds of derivatives, forward and futures contracts.
In chapter 3, we saw examples where a firm could increase its value by changing its risk. To use
derivatives to increase shareholder wealth, one has to know how they are priced as well as how they
can be used to transform a firm’s risk. A firm that takes a position in a mispriced derivative can lose
the value gain it expects from risk management even though it changes its risk as theory recommends.
Therefore, in this chapter we show how to price forward and futures contracts. In the four chapters
that follow we examine the uses of forward and futures contracts in risk management.
To price forward and futures contracts, we apply the fundamental method used to price
derivatives in general. This method is called pricing by arbitrage. Let’s see how this method works
in principle for a derivative that is purchased now, has no cash flows until maturity, and makes a
payoff at maturity. For this derivative, the method of pricing by arbitrage requires us to find a
portfolio strategy that does not use the derivative, that has no cash flows until maturity, and has the
same payoff as the derivative. Since the portfolio strategy has exactly the same payoff as the
derivative, the current value of the derivative must be the same as the current value of the portfolio.
Otherwise, the portfolio strategy and the derivative yield the same payoffs for different prices. If the
same payoffs can be obtained for different prices, there is an arbitrage opportunity: one can
Chapter 5, page 1
simultaneously buy the payoff at the low price and sell it at the high price, making money for sure an unlikely and ephemeral event if traders actively seek out such opportunities and make them
disappear. Consequently, the value of the portfolio is the price of the derivative in the absence of
arbitrage opportunities.
In forward and futures markets, transaction costs are generally of little importance. We
therefore use the convenient simplifying assumption of perfect financial markets except at the end of
this chapter. Importantly for our analysis, this means that there are no transactions costs, no taxes,
no difference between borrowing and lending rates, and no restrictions on short sales. We start with
the simplest case: forward contracts on T-bills. In Section 5.2 we extend the analysis to forward
contracts on commodities, stocks, and currencies. With Section 5.3 we introduce futures contracts
and explain how and why they differ from forward contracts. We give a detailed example of how a
futures position evolves through time. In Section 5.4 we discuss how forward and futures prices
relate to each other and to future spot prices. Section 5.5 concludes.
Section 5.1. Pricing forward contracts on T-bills.
We introduced forward contracts in chapter 1 and have been using them ever since.
Remember that a forward contract is a purchase contract with delivery and payment taking place at
maturity of the contract, on terms agreed upon when the parties enter into the contract. We now want
to understand how the forward price is determined at the origination of a forward contract. For that
purpose, we study a forward contract on T-bills.
Let’s consider a forward contract where the buyer agrees on March 1 to buy on June 1 T-bills
maturing on August 30 at the price of 97 cents per dollar of face value for a total face value of $1M.
Chapter 5, page 2
Money changes hands only at maturity of the contract. $0.97 is the forward price per dollar of face
value, i.e., the price agreed upon on March 1 for delivery on June 1 of bills that mature on August
30. In this case, the forward contract specifies:
1. The goods to be exchanged for cash: $1M of face value of T-bills maturing on August
30.
2. The date when the exchange takes place: June 1.
3. The price to be paid: $0.97 per dollar of face value.
Figure 5.1. shows a time-line of the various actions taken for a long position in the forward
contract, which is a position where you promise to buy the bills.
Section 5.1.A.Valuing a forward position at inception using the method of pricing by arbitrage.
Suppose we consider taking a long position in this contract in perfect capital markets. This
means that we promise to buy the bills at the forward price. It is common to call the investor with a
long position the long and the one with a short position the short. We would like to know whether
$0.97 is an advantageous forward price for us. The spot or cash market is the market where
transactions take place for delivery for the nearest possible settlement date. T-bills trades are settled
the next day, so that the spot market for T-bills involves next day delivery. Denote the spot price (also
called the cash market price) on June 1 of the T-bills delivered to the buyer on that day by P(June 1,
August 30) per dollar of face value. The payoff of the long position on June 1 will be (P(June 1,
August 30) - $0.97)*$1M. This is the payoff of receiving bills for $1M of face value, worth P(June
1, August 30)*1M, and paying a fixed amount for them, $0.97M. To know whether the forward price
is advantageous, we need to know the net present value of taking a long forward position. Since no
Chapter 5, page 3
money changes hands before maturity of the forward contract, the net present value of the position
is the present value of the payoff at maturity.
The problem with computing the present value of the payoff is that we do not know today
what the spot price of the delivered bills will be on June 1, since between now and then interest rates
will almost certainly change. One way that we can compute the present value of the payoff is to use
the capital asset pricing model. This approach requires us to compute the expected payoff, and then
discount it at the required discount rate implied by the beta of the payoff as discussed in chapter 2.
With this approach, different investors reach different conclusions if they have different expectations
about the payoff.
There is, however, a simpler way to compute the value of the forward position that does not
require us to know the statistical distribution of the payoff and that provides a unique price on which
all investors agree. This is because we can create a portfolio today without using the forward contract
that has the same payoff as the forward position. Because we can create such a portfolio, we can use
the method of pricing by arbitrage. A portfolio that has the same payoff as a financial instrument but
does not contain that financial instrument is called a replicating portfolio.
We now construct a replicating portfolio for the forward contract. The forward contract pays
(P(June 1, August 30) - $0.97)*$1M on June 1. We therefore have to construct a portfolio that pays
this amount at that date. We know from chapter 2 that the payoff of a portfolio is the sum of the
payoffs of the financial instruments included in the portfolio. Consequently, we can look at the
payoffs P(June 1, August 30)*1M and - $0.97m separately. We start with creating a portfolio that
pays -$0.97M on June 1 since this is the easiest task and then create a portfolio that pays P(June 1,
August 30)*1m at the same date:
Chapter 5, page 4
1) Creating a portfolio that pays -$0.97M on June 1. A negative payoff, like -$0.97M,
means a cash outflow. Here, we have a cash outflow of a fixed amount on June 1. To create a
negative payoff at a future date, one borrows today and repays the principal and interest at maturity.
The negative payoff is then the amount one has to repay. With our assumptions of perfect financial
markets, we can borrow at the risk-free rate. To have to repay $0.97M, we borrow today the present
value of $0.97M, which is P(June 1)*$0.97M, where P(June 1) is the price of a T-bill on March 1 that
matures on June 1. A simple way to borrow is to sell T-bills short. Selling short a T-bill means
borrowing the T-bill and selling it. When the short sale is reversed, we buy the T-bill and deliver it
to the investor who lent us the T-bill initially. The gain from the short sale is the negative of the
change in the price of the T-bill. If the T-bill increases in value, we lose money with the short sale,
since we have to buy the T-bill back at a higher price than we sold it. Box 5.1. Repos and borrowing
using Treasury securities as collateral discusses the details of how short sales of Treasury securities
are usually implemented. Remember that we assume we can borrow at the risk-free rate because
financial markets are perfect.
Let P(June 1) be $0.969388. We therefore receive today
0.969388*0.97*$1M, which is $0.940306M.
2) Creating a portfolio that pays P(June 1, August 30)*$1M on June 1. T-bills are
discount bonds, so that they have no cash flows until maturity. This means that a bill that matures on
August 30, worth P(August 30) today, is worth P(June 1, August 30) on June 1, and can be delivered
on the forward contract at that date. Let’s assume that P(August 30) is equal to $0.95, so that we pay
$0.95*1M, or $0.95M to create the portfolio.
We have now completed the construction of our replicating portfolio. It has a long position
in T-bills that mature on August 30 which costs $0.95M and a debt with current value of
Chapter 5, page 5
$0.940306M, for a total value of $0.95M - $0.940306M or $0.009694M. Since T-bills that mature
on August 30 can be delivered on June 1 to fulfill the obligations of the forward contract, the
replicating portfolio is long the deliverable asset. Holding the portfolio until June 1, we then have
P(June 1, August 30)*1M - $0.97M, which is exactly the payoff of the long forward position.
We find that the replicating portfolio is worth $0.009694M. This means that a long forward
position has the same value. Yet, if we enter the forward contract, we pay nothing today. This creates
an arbitrage opportunity. Since we have two strategies that yield the same payoff but have different
costs, we would like to buy the payoff at a low price and sell it at a high price to make riskless profits.
Let’s figure out how we can do this. In our example, the forward contract has value. We therefore
go long the contract. However, doing so involves taking a risk. We want to find a hedge that allows
us to eliminate this risk. Since the replicating portfolio has the same payoff as the forward contract,
it is a perfect hedging instrument. We want to sell that portfolio short, so that at maturity we owe the
payoff that we receive from the forward contract. To sell short a portfolio, we sell short the securities
that are long in the portfolio and buy the securities that are short in the portfolio. By taking a short
position in the replicating portfolio and a long position in the forward contract, we end up with
positions that cancel out at maturity so that we owe nothing then because going short the replicating
portfolio has the same payoff as going short the forward contract. Yet, today, we receive money for
going short the replicating portfolio and pay no money for going long the forward contract. We
therefore make money today without owing money in the future.
Let’s make sure we understand exactly how the numbers work out if we go short the
replicating portfolio and long the forward contract to obtain a risk-free profit. Going short the
replicating portfolio involves a short sale of T-bills that mature on August 30 for $1M of face value
Chapter 5, page 6
and a risk-free investment that pays $0.97M on June 1. A short sale of T-bills maturing on August
30 for $1M face value means that we borrow $1M face value of these bills and sell them for $0.95M
today. When we reverse the short sale on June 1, we have to buy $1M face value of the T-bills for
P(June 1, August 30)*$1M and deliver the bills to our counterparty in the short sale. Therefore, the
short position in the T-bills requires us to pay P(June 1, August 30)*$1M on June 1. The investment
in the risk-free asset is worth $0.97M then by construction. Hence, this portfolio pays $0.97M P(June 1, August 30)*$1M on June 1, or using F as the forward price, F*$1M - P(June 1, August
30)*$1M, which is the payoff of a short forward position since with such a position we get F and
deliver the bills. The cash flow received on March 1 from establishing a short position in the
replicating portfolio is our risk-free profit:
Cash flow on March 1 from establishing a short position in the replicating portfolio
= [P(August 30) - P(June 1)F]*1M
= $0.95M - $0.969388*0.97*1M
= $0.95M - $0.940306M
= $0.009694M
Going short the replicating portfolio therefore replicates the payoff of a short forward position and
generates a positive cash flow on March 1. However, at maturity, the payoffs from the short position
in the replicating portfolio and the long position in the forward contract cancel out:
Payoff of short position in the replicating portfolio + Payoff of long forward position
Chapter 5, page 7
= [F - P(June 1, August 30)]*$1M + [P(June 1, August 30) - F]*$1M
=0
This confirms that there is an arbitrage opportunity at the forward price of $0.97 since we make
money today and have to pay nothing at maturity. As long as the replicating portfolio has a value
different from zero, money can be made without risk. Infinite profits are for the taking if the forward
price and the T-bill prices do not change! If the forward contract is priced so that the replicating
portfolio has negative value, then we make money by buying that portfolio and taking a short forward
position! Table 5.1. shows that if the forward price is $0.99, we receive a positive cash flow today
of $9,694.12 without owing money later. The only case where we make no riskless profits is when
the replicating portfolio has zero value.
It follows from our analysis that the forward contract must have zero value at inception. If
a forward contract does not have zero value at inception, it means that one party to the contract
makes a gift to the other. For instance, if the contract has positive value for the long, it must have
negative value for the short, because the payoff of the short is the negative of the payoff of the long.
In real world markets, gifts get taken by traders who live for such opportunities and hence disappear
quickly. Consequently, the forward price must satisfy:
The replicating portfolio has zero value at inception of the contract
[P(August 30) - P(June 1)F]*$1M = 0
Solving this equation for F gives the following result:
Chapter 5, page 8
F=
P(August 30)
P(June 1)
With the numbers of our example, this means that:
[$0.95 - $0.969388*F]*$1M = 0
Solving this equation for the unknown F, we get F = $0.98. With this forward price, the replicating
portfolio for the long position has no value and there is no arbitrage opportunity.
The condition that the forward price must be such that the replicating portfolio for the
forward contract has zero value has an important economic interpretation. With the forward contract,
we pay a price known today, F, for the bills at maturity of the contract. Suppose that the forward
contract does not exist and we want to pay a known price on June 1 for $1M face value of bills
maturing on August 30. The way to do that is to buy the bills maturing on August 30 today and
finance the purchase. We pay P(August 30)*$1M for the bills. If we finance them, at maturity we
have to repay Exp[r*0.25]*P(August 30)*$1M, where r is the continuously compounded annual rate
of interest on a bill that matures on June 1. By the definition of T-bills, the price of a T-bill that
matures on June 1 is P(June 1) = Exp[-r*0.25]. Consequently, the cost to us of buying the bills
maturing on August 30 on credit is that we have to pay P(August 30)*$1M/P(June 1) on June 1 when
we repay the loan, which is equal to the price we would pay using the forward contract with the price
obtained with our formula. This is not surprising since the two ways of buying the bills and paying
for them on June 1 should have the same price. Viewed this way, the replicating portfolio can be
Chapter 5, page 9
thought of as a portfolio where we buy the bills and finance them with a loan. On June 1, the cost of
owning the bills with this strategy is that we have to pay an amount equal to the initial cost of the bills
plus the financing costs. This amount has to be the forward price to avoid arbitrage opportunities.
Section 5.1.B. A general pricing formula.
Using our notation, we have shown that the forward price and the T-bill prices must be such
that P(August 30)*1M - P(June 1)F*1M = 0, which implies that F = P(August 30)/P(June 1). Our
analysis holds equally well if the forward contract has a different maturity or requires delivery of a
bill with a different maturity. Consequently, we have a general formula for the pricing of forward
contracts on T-bills:
Formula for the pricing of forward contracts on T-bills
The forward price per dollar of face value, F, of a contract entered into at t for delivery of bills at t+i
that mature at t+j must be such that the replicating portfolio for a forward position has no value at
inception:
Price at time t of deliverable T-bill with maturity at time t+j - Present value of forward price to be
paid at time t+i = 0
P(t+j) - P(t+i)F = 0
(5.1.)
Dividing the formula by P(t+i) shows that:
F=
P(t + j)
P(t + i)
(5.2.)
In our above discussion, P(t+j) = P(August 30) = 0.95 and P(t+i) = P(June 1) = 0.969388, so that F
Chapter 5, page 10
= 0.98.
Section 5.1.C. Pricing the contract after inception
Consider now the following situation. Suppose we entered a short forward position in the
contract discussed in section 5.1.A. with a forward price satisfying equation (5.2.) of $0.98, so that
we committed to sell $1M face value of 90-day T-bills in 90 days for $0.98 per dollar of face value.
Time has now elapsed, so that we are on April 15. Let’s figure out the value of the position.
We showed that there is a replicating portfolio that has the same payoff as a short forward
position. This replicating portfolio has an investment in T-bills maturing on June 1 that pays $0.98M
then and a short position in T-bills that mature on August 30. This portfolio has zero value at
origination of the contract, but on April 15 its value may have increased or decreased. To price the
forward position, we therefore have to price the portfolio on April 15 to get the value of a portfolio
on that date that has the same payoff as the short forward position. Hence, the value of the short
forward position is given by:
Value of replicating portfolio for short forward position on April 15
= [P(April 15, June 1)F - P(April 15, August 30)]*$1M
= [P(April 15, June 1)*0.98 - P(April 15, August 30)]*$1M
This portfolio replicates the payoff of the short forward position since on June 1, it is worth $0.98M P(June 1, August 30)*$1M. Let’s check how this works with an example. Suppose that P(April 15,
June 1) = $0.965 and P(April 15, August 30) = $0.93605. In this case, we have that:
Chapter 5, page 11
Value of replicating portfolio for short forward position on June 15
= [P(June 15, June 1)F - P(June 15, August 30)]*1M
= 0.965*$0.98M - $0.93605M
= $0.00965M
It follows from this that the short forward position has gained in value from March 1 to June 15.
We can think of valuing the short forward position on April 15 in a different way. Suppose
that we want to find out how much we would receive or pay to get out of the short forward position.
The way to get out of a forward position is to enter a new forward position of opposite sign: the
short enters a long forward position for the same date and with the same bills to be delivered. A
contract originated on April 15 for delivery on June 1 of T-bills maturing on August 30 would have
to be priced using our formula. Our formula yields a price of $0.97. This price would have created
arbitrage opportunities on March 1, but since interest rates have changed, it is a price that creates no
arbitrage opportunities on April 15. We could therefore cancel our position on April 15 by taking a
long forward position to buy on June 1 T-bills maturing on August 30 for $0.97 per dollar of face
value. On June 1, we would receive $0.98M and pay $0.97M, using the bills received for delivery on
the short contract. Consequently, we would make a net gain of $0.01M irrespective of interest rates
on that date. Since we would receive $0.01M on June 1 if we cancel our position by entering an
offsetting contract, it must be that the value of the position on April 15 is the present value of
$0.01M. This means that the value of the position per dollar of face value on April 15 is the present
value of the difference between the price at inception (the amount you receive by delivering on your
long position entered on March 1) and the new forward price (the amount you pay by delivering on
Chapter 5, page 12
your short position entered on April 15), where the discounting takes place from the maturity date
of the contract to April 15. This value is $0.00965. Not surprisingly, this is exactly the value of the
replicating portfolio per dollar of face value.
Our analysis therefore shows that:
Formula for the value of a forward position
The value at date t+, of a forward position per dollar of face value maturing at date t+i (t+i > t+,)
requiring delivery of a bill maturing at date t+j is the value of the replicating portfolio at t+,:
P(t+,,t+i)F - P(t+,,t+j)
where j > i
(5.3.)
The value of a short position per dollar of face value is minus one times the value of the replicating
portfolio at that date:
P(t+,,t+j) - P(t+,,t+i)F
(5.4.)
Section 5.2. Generalizing our results.
In the previous section, we learned how to price a forward contract on a T-bill. The key to
our approach was our ability to construct a replicating portfolio. We were then able to use the
method of pricing by arbitrage which states that the value of the forward position must be equal to
the value of the replicating portfolio. The replicating portfolio consists of a long position in the
deliverable asset financed by borrowing.
Consider now a forward contract that matures June 1 on a stock that pays no dividends. Let
S(June 1) be price of the deliverable asset on the cash market on June 1, the stock price at maturity
of the contract, and F be the forward price per share. The payoff of a long forward position is S(June
1) - F. We already know that we replicate -F by borrowing the present value of F so that we have to
Chapter 5, page 13
repay F at date June 1. We have to figure out how to create a portfolio today that pays S(June 1) at
maturity. Because the stock pays no dividends, the current value of the stock to be delivered at
maturity of the forward contract is the stock price today. By buying the stock on March 1 and holding
it until June 1, we own the deliverable asset on June 1. Consequently, our replicating portfolio works
as follows: buy one unit of the stock today and sell short T-bills maturing on June 1 for proceeds
equal to the present value of the forward price. With this portfolio, we spend S(March 1) to purchase
the stock and sell short T-bills for face value equal to F. On June 1, we have to pay F to settle the
short-sale. After doing so, we own the stock. If we had used a forward contract instead, we would
have paid F on June 1 as well. The forward price must be such that the replicating portfolio has zero
value, therefore S(March 1) - P(June 1)F must be equal to zero. Solving for the forward price, we
have that F is equal to A(March 1)/P(June 1). If the forward price differs from the value given by our
formula, there exists an arbitrage opportunity.
We can now price a forward contract on a stock. However, we can do much more. In all cases
considered, we had a situation where the current value of the asset delivered at maturity of the
contract is the current spot price of the asset. As long as the deliverable asset’s current value is its
spot price, we can find the forward price by constructing a replicating portfolio long the deliverable
asset today financed by borrowing the present value of the forward price. We can therefore get a
general formula for the forward price F for a contract where S(t) is the current price of the asset
delivered at maturity of the contract at t+i:
General formula for the forward price F on an asset with current price S(t) and no payouts
before maturity of the contract at t+i
The replicating portfolio for the contract must have zero value, which implies that:
Chapter 5, page 14
S(t) - P(t+i)F = 0
(5.5.)
S(t)
P(t + i)
(5.6.)
Solving this expression for F yields:
F=
The assumption of no cash flows for the deliverable asset before maturity of the contract is extremely
important here. Let’s find out what happens when this assumption does not hold. Suppose we
construct a replicating portfolio for a forward contract on a stock that matures on June 1 and the
stock pays a dividend D on April 15. For simplicity we assume that the dividend is already known.
In this case, the forward contract requires delivery of the stock ex-dividend, but the current price of
the stock includes the right to the dividend payment. The stock today is not the deliverable asset. The
deliverable asset is the stock without the dividend to be paid before maturity of the contract.
Consequently, the cost today of buying the replicating portfolio that pays S(June 1) on June 1 is no
longer S(March 1), but instead is S(March 1) minus the present value of the dividend, S(March 1) P(April 15)D. The portfolio to replicate a payoff of -F on June 1 is the same as before. We can
generalize this to the case of multiple payouts and arbitrary dates:
Forward price for a contract on an asset with multiple payouts before maturity
The forward price F at t for delivery at date t+i of an asset with price S(t) at time t that has N
intermediate payouts of D(t+)h), h = 1,...,N, is given by:
S(t) - ∑ h= 1 P(t + ∆ h ) D(t + ∆ h )
h=N
F=
P(t + i)
Chapter 5, page 15
(5.7.)
Let’s look at an example. Consider a forward contract entered into at date t with delivery in
one year of a share of common stock that pays dividends quarterly. The current price of the stock is
$100. The dividends are each $2 and paid at the end of each quarterly period starting now. The
discount bond prices are $0.98 for the zero-coupon bond maturing in three months, $0.96 for the
bond maturing in six months, $0.93 for the bond maturing in nine months, and $0.90 for the bond
maturing in one year. Consequently, we have S(t) = 100, P(t+i) = P(t+1) = 0.90, P(t+0.75) = 0.93,
P(t+0.5) = 0.96, P(t+0.25) = 0.98, and D(t+1) = D(t+0.75) = D(t+0.5) = D(t+0.25) = 2. Using the
formula, we have:
F=
100 - 0.98 * 2 - 0.96 * 2 - 0.93* 2 - 0.90 * 2
= 102.73
0.90
If we had ignored the dividends, we would have obtained a forward price of $111.11. The forward
price is much higher when dividends are ignored because they reduce the cost of the replicating
portfolio - they are like a rebate on buying the replicating portfolio.
The analysis in this section extends naturally to a forward contract on a portfolio. The
deliverable asset could be a portfolio containing investments in different stocks in specified quantities.
For instance, these quantities could be those that correspond to a stock index like the S&P 500. In
this case, we would have a forward contract on the S&P 500. The S&P 500 index is equivalent to
the value of a portfolio invested in 500 stocks where the investment weight of each stock is its market
value divided by the market value of all the stocks in the index. One can therefore construct a
portfolio of stocks whose return exactly tracks the S&P 500.
Chapter 5, page 16
Section 5.2.A. Foreign currency forward contracts.
Consider a forward contract where you promise to purchase SFR100,000 on June 1 at price
F per unit of foreign currency agreed upon on March 1. The price of the deliverable for spot delivery,
the Swiss Franc, is S(June 1) at maturity of the contract. The payoff of the contract at maturity is
100,000S(June 1) - F. We already know that we can replicate F by borrowing the present value of
F. Let’s now replicate 100,000S(June 1). You can purchase today an amount of SFRs such that on
June 1 you have SFR100,000 by buying SFR T-bills for a face value of SFR100,000 maturing on June
1.1 The cost of buying the SFR T-bills in dollars is the cost of SFR T-bills maturing on June 1 per
SFR face value, PSFR(June 1), multiplied by the exchange rate today, S(March 1), times 100,000,
which is 100,000S(March 1)PSFR(June 1). The present value of the forward price times 100,000,
100,000P(June 1)F, must be equal to 100,000PSFR(June 1)S(March 1) to insure that the replicating
portfolio has no value. Consequently, the forward price F is S(March 1)PSFR(June 1)/P(June 1).
Generalizing the formula for the forward price to a contract entered into a t and maturing at
t+i, we obtain the following result:
Pricing formula for a foreign exchange forward contract
Let S(t) be the current spot price of the foreign currency and PFX(t+i) the price of a foreign currency
T-bill with one unit of foreign currency face value maturing at date t+i. A forward contract on the
foreign currency maturing at date t+i originated at date t must have a forward price F per unit of
foreign currency such that the replicating portfolio has zero value at origination:
1
A technical issue should be mentioned here. Not all countries have T-bills, but in many
countries there is the equivalent of T-bills, namely a default free asset that pays one unit of local
currency at maturity. Whenever we talk about a T-bill for a foreign country, we therefore mean an
instrument that is equivalent to a U.S. T-bill.
Chapter 5, page 17
S(t)PFX(t+i) - P(t+i)F = 0
(5.8.)
This formula implies that the forward price must be:
S(t)P FX (t + i)
F=
P(t + i)
(5.9.)
Let’s consider an example of a contract entered into at t that matures one year later. Suppose the SFR
is worth 70 cents, PFX(t+1) = SFR0.99005, P(t+1) = $0.951229. Using our formula, we have that F
= $0.7*0.99005/0.951229 = $0.72857. In this case, the forward price of the SFR is more expensive
than its current spot price. For foreign currencies, the current spot price is the spot exchange rate
while the forward price is called the forward exchange rate. If the forward exchange rate is above the
spot exchange rate, the currency has a forward premium. The forward exchange rate pricing formula
is often expressed using interest rates. Box 5.2. Interest rate parity shows that the interest rate
formulation can be used profitably to understand how interest rates across countries are related.
Section 5.2.B. Commodity forward contracts.
Let’s conclude this section by considering a forward contract on a commodity, say gold.
Suppose that on March 1 you know that you will want 1,000 ounces of gold on June 1. You can buy
the gold forward or you can buy it today and finance the purchase until June 1. There is a difference
between holding the gold now versus holding the forward contract: holding gold has a convenience
yield. The convenience yield is the benefit one derives from holding the commodity physically. In the
case of gold, the benefit from having the commodity is that one could melt it to create a gold chain
that one can wear. Hence, if the cash buyer has no use for gold, he could lend it and whoever borrows
Chapter 5, page 18
it would pay the convenience yield to the lender. The rate at which the payment is determined is called
the gold lease rate. This means that the cost of financing the gold position is not the T-bill yield until
maturity of the forward contract, but the T-bill yield minus the convenience yield. As before, let’s
denote the cash market or spot price of the deliverable on March 1 by S(March 1). In this case,
S(March 1) is the gold price on March 1. Further, let’s assume that the convenience yield accrues
continuously at a rate of c% p.a. To create a replicating portfolio for a long position in the forward
contract, one has to hold exp(-0.25c) units of gold and sell short T-bills of face value F. To create
a replicating portfolio for a short position in the forward contract, one has to sell the commodity
short, which involves compensating the counterparty for the loss of the convenience yield. To sell
gold short, one therefore has to pay the lease rate. Though the lease rate for gold is relatively
constant, the convenience yield of other commodities can vary dramatically as holders may gain a
large benefit from holding the commodity - chainsaws are very valuable the day following a hurricane,
a benefit that is not shared by chainsaws bought forward with delivery a month after the hurricane.
This approach gives us a formula for the forward price for a commodity:
Forward price formula on a commodity with a convenience yield
The forward price at date t, F, for delivery of one unit of a commodity at date t+i that has a price
today of S(t) and has a convenience yield of c% per unit of time is priced so that the replicating
portfolio has zero value:
exp(-c*i)S(t) - P(t+i)F = 0
(5.10.)
Consequently, the forward price must satisfy:
F=
exp(-c * i)S(t)
P(t + i)
Chapter 5, page 19
(5.11.)
Let’s look at an example of a contract entered into at t that matures 90 days later. Suppose that the
price of gold today is $40 an ounce, the convenience yield is 2% p.a. continuously compounded, and
the price for a 90-day discount bond is $0.98 per dollar of face value. We have S(t) = $40, c = 0.02,
i = 0.25, and P(t+i) = $0.98. Consequently:
F = S(t)exp(-c*i)/P(t+i) = $40exp(-0.02*0.25)/0.98 = $40.6128
Without the convenience yield, the replicating portfolio would have been more expensive and as a
result the forward price would have been higher. The forward price without the convenience yield
is $40.8163.
Often, it is costly to store the deliverable commodity. For instance, replicating a forward
contract on heating oil requires storing the oil. Storage costs are like a negative convenience yield:
they make the forward contract more advantageous because buying forward saves storage costs.
Whereas the convenience yield lowers the forward price relative to the spot price, the storage costs
increase the forward price relative to the spot price. To see this, let’s look at the case where the
storage costs occur at a continuous rate of v% per year and the contract matures in 90 days. We can
think of storage costs as a fraction of the holdings of oil that we have to sell to pay for storage or we
can think of oil evaporating because it is stored. To create a replicating portfolio for the long position
in heating oil, we have to buy more oil than we need at maturity for delivery because we will lose
some in the form of storage costs. Hence, to have one unit of oil at maturity, we need to buy
exp[0.25v] units of oil now. If oil has both storage costs and a convenience yield, we require
exp[0.25(v-c)] units of oil now. To get a general formula for the pricing of forward contracts, we also
Chapter 5, page 20
want to allow for cash payouts at a rate d. In this case, we need exp[0.25(v-c-d)] units of the
underlying in the replicating portfolio. For a contract entered into at t and maturing at t+i, this
reasoning leads to the following formula for the forward price:
General formula for the forward price
A forward contract for delivery of a good at date t+i with price S(t+i) at that date available today for
price S(t), with continuously computed payout rate of d% per year, convenience yield of c%, and
storage costs of v% must have a forward price F such that the replicating portfolio has zero value2:
exp[(v-c-d)i]S(t) - P(t+i)F = 0
(5.12.)
If the continuously-compounded yield on the T-bill maturing at date t+i is r% p.a., the forward price
then satisfies:
exp[(r+v-c-d)*i]*S(t) = F
(5.13.)
Let’s look at an example. Consider a commodity that costs $50. Suppose that the interest rate is 10%,
storage costs take place at the rate of 6%, and there is a convenience yield of 4%. The forward
contract is for 90 days. Using our formula, we have that exp[(0.10 + 0.06 - 0.04)*90/360]*$50 =
$51.5227. This means that the forward price exceeds the spot price of the commodity. Keynes, in his
work on forward pricing, defined the price for future delivery to be in contango when it exceeds the
spot price and in backwardation otherwise.3
2
Note that this general formula can be extended to allow time variation in r, v, c, and d.
Further, discrete payouts or storage costs can be accommodated in equation (5.13.) by
discounting these costs on the left-hand side using the appropriate discount rate. Viewed this way,
the current value of the good delivered at date t+i is its current price plus the present value of
storage costs minus the present value of holding the good (payouts and convenience yield).
3
The current usage among economists is to use backwardation to designate the excess of
the expected spot price over the forward. While everyone can agree whether or not a forward
price exhibits backwardation in the sense of Keynes, people can disagree as to whether a forward
Chapter 5, page 21
Assuming that r, v, c, and d do not depend on S(t), equation (5.13.) shows that the forward
price has the following properties :
(1) The forward price increases with the current price of the good to be delivered at maturity
of the contract. This results from the fact that the replicating portfolio of the long forward position
is long in that good.
(2) The forward price increases with the interest rate. In the replicating portfolio, one sells
T-bills short for a face amount F. As the interest rate increases, the short-sale proceeds fall which
increases the value of the portfolio. Hence, to keep the value of the portfolio at zero, one has to
increase short sales.
(3) An increase in storage costs increases the forward price. This is because one has to buy
more of the good to have one unit at maturity. As storage costs increase, it becomes more
advantageous to buy forward for a given forward price, and the forward price has to increase to
compensate for this.
(4) An increase in the convenience yield and the payout rate decrease the forward price. The
convenience yield and the payout rate provide an added benefit to holding the good as opposed to
buying it forward, thereby making the forward less attractive and requiring a fall in the forward price.
Figure 5.2. shows the difference between the forward price and the spot price as maturity
changes and as (v-c-d) changes. In this figure, the forward price can be greater or smaller than the
spot price. If (r + v - c - d) is positive, the forward price exceeds the spot price. The sum (r + v - c d) is generally called the cost of carry, in that it is the cost (in this case expressed as a continuouslycompounded rate) of financing a position in the underlying good until maturity of the contract. For
price exceeds the expected spot price.
Chapter 5, page 22
gold, c is small (lease rates are typically of the order of 2%) and v is trivial. For oil, c can be
extremely large relatively to v. Hence, typically the present value of the forward price of gold exceeds
the spot price, while the present value of the forward price for oil is lower than the spot price.
It is important to note that the arbitrage approach cannot always be used to price forward
contracts. In particular, it may not be possible to establish a short position. If selling short the
commodity is not possible, then the forward price can be lower than the formula given above. This
is because the only way to take advantage of a forward price that is too low relative to the one
predicted by the formula without taking risk is to buy forward and sell the commodity short to hedge
the forward position. However, if it is difficult to sell short, this means that one has to make an
extremely large payment to the counterparty in the short sale. Consequently, use of the commodity
is extremely valuable and the convenience yield is very high. This means that there is a convenience
yield for which the formula holds. A bigger difficulty is that for perishable commodities the replicating
portfolio approach becomes meaningless because holding the portfolio becomes more expensive than
the spot price of the commodity. Think of replicating a long position in a one-year forward contract
on roses! In this case, the forward contract has to be priced using a model that specifies the risk
premium that investors require to take a long position in the forward contract (e.g., the CAPM) rather
than the arbitrage approach.
Section 5.3. Futures contracts.
Let’s consider more carefully the situation of a firm that has a long forward position to buy
Euros. Suppose that the contract matures in one year. The forward position makes money if the Euro
at the end of the year exceeds the forward price, provided that the forward seller honors the terms
Chapter 5, page 23
of the contract at maturity. If the forward seller cannot deliver at maturity because of a lack of
financial resources or other reasons, the forward contract is useless and the gain the firm expected
to make does not materialize. The risk that the counterparty in a derivatives contract fails to honor
the contract completely is called counterparty risk. It means that there is a risk of default and, when
the firm enters the contract, it has to take that risk into account. If the risk of default is too large, the
forward contract is useless or worse. Note that if the firm makes losses on the contract, the seller
makes a profit by delivering the Euros and therefore will do so. Hence, for the buyer, default risk
means that the distribution of the gain from the forward contract changes so that losses become more
likely. To adjust for the impact of default risk, the buyer wants a lower forward price.
The problem with default risk for the buyer in our example is that it requires careful
examination of the forward seller’s business. This increases the costs of forward contracting. One
way to eliminate this difficulty is for the forward buyer to enter a contract only if the counterparty
has a high debt rating indicating a low probability of default. Another way to avoid this problem is
to require the seller to post a bond that guarantees performance on the contract. This can work as
follows. When entering the contract, the seller sets aside an amount of money with a third party,
possibly in the form of securities, that he forfeits if he does not deliver on the forward contract. If the
value of the assets deposited as collateral is large enough, default becomes unlikely. The difficulty is
that such an arrangement has to be negotiated and the party who holds the collateral has to be
designated and compensated. Further, if the losses that can occur are large over the life of the
contract, then the collateral will have to be quite large as well. We also saw in chapter 3 that firms
often hedge to avoid situations where they might lack funds to invest and face difficulties in raising
funds in the capital markets. Forcing a firm to post a large amount of collateral is therefore likely to
Chapter 5, page 24
prevent it from hedging since it would force the firm to set aside either funds that have a high
opportunity cost or funds that it does not have. Requiring the posting of a large amount of collateral
makes it less likely that firms and individuals will want to enter the contract.
Rather than posting a collateral that covers possible losses for the whole life of the contract,
it seems more efficient to require a smaller collateral that covers potential losses over a small period
of time, but then to transfer the gains and losses as they accrue. In this case, whenever the collateral
becomes insufficient to cover potential losses over a small period of time, it is replenished and if the
counterparty fails to do so, the contract is closed. Since gains and losses are transferred as they
accrue, the contract can be closed without creating a loss for one of the parties to the contract. With
such a way of dealing with default risk, the seller receives his gains as they accrue and if he starts to
make losses, he pays them as they accrue. This arrangement can be renewed every period, so that
over every period the forward seller has no incentive to default because his posted collateral is
sufficient to cover the potential losses during the period. If the forward seller is concerned about
default of the forward buyer, the same arrangement can be made by the buyer.
Section 5.3.A. How futures contracts differ from forward contracts.
The best way to view futures contracts is to think of them as forward contracts with added
features designed to make counterparty risk economically trivial. Futures contracts are contracts for
deferred delivery like forward contracts, but they have four important features that forward contracts
do not have. First and most importantly, gains and losses are paid by the parties every day as they
accrue. This procedure, called marking the contract to market, is equivalent to closing the contract
at the end of each day, settling gains and losses, and opening a new contract at a price such that the
Chapter 5, page 25
new contract has no value. Second, collateral is posted to ensure performance on the contract. Third,
futures contracts are standardized contracts that trade on organized exchanges. Fourth, the
counterparty in a long (short) futures contract position is not the short (long) but rather an institution
set up by the exchange called the clearinghouse that has enough capital to make default extremely
unlikely. We discuss the significance of these features of futures contract in the remainder of this
section.
To open a futures contract in the U.S., we have to open a commodity trading account with
a futures commission merchant regulated by the Commodity Futures Trading Commission and deposit
enough money in it to cover the required initial collateral, called the contract’s initial margin. The
initial margin is set by the broker, but has to satisfy an exchange minimum. Like forward contracts,
futures contracts have no value when entered into. Since the long does not make a payment to the
short when he opens a futures contract, the futures contract would have negative value for the short
at origination if it had positive value for the long. In this case, therefore, the short would not enter
the contract. The same reasoning explains why the contract cannot have positive value for the short
at origination. The long makes a gain if the price increases after opening the contract and the short
loses. If we make a loss on a day, our account is debited by the loss. The amount that our account
changes in value on a given day is called that day’s settlement variation. After having paid or
received the day’s settlement variation, our futures position has no value since gains and losses are
paid up. After opening the futures position, the balance in our account has to exceed the
maintenance margin before the start of trading each day. The maintenance margin is lower than the
initial margin. Following losses, we receive a margin call if the account balance falls below the
maintenance margin. In this case, we have to replenish the account to bring its balance to the initial
Chapter 5, page 26
margin. If we make gains, we can withdraw the amount by which the margin account exceeds the
initial margin.
Forward contracts are generally traded over the counter. Therefore, any contract that two
traders can agree on can be made. No exchange rules have to be satisfied. For instance, foreign
currency forward contracts are traded on a worldwide market of traders linked to each other through
phones and screens. The traders handle default risk by having position limits for the institutions they
deal with. They do not trade at all with institutions that have poor credit ratings. If they are willing
to trade, they are willing to trade contracts of any maturity if the price is right. In contrast, futures
contracts are traded on futures markets that have well-defined rules. The trading takes place in a
designated location and available contracts have standardized maturities and sizes. Because the
contracts are traded daily, a futures position can always be closed immediately. To do that, all we
have to do is enter the opposite futures position to cancel out the initial position. If we close the
contract this way in the middle of the day, we are still responsible for the change in the value of the
contract from the beginning of the trading day to the middle of the trading day.
With forward contracts the counterparty is always another institution or individual.
Consequently, when A and B enter in a forward contract where A promises to buy SFRs from B, A
is the counterparty to B. In contrast, if A enters a long futures position in SFR futures contract and
B takes the offsetting short position, A is not the counterparty to B. This is because, immediately
after A and B have agreed to the futures trade, the clearing house steps in. The clearinghouse will
take a short position with A so that it will be responsible for selling the SFR to A and it will take a
long position with B so that it will be responsible for buying the SFR from B. The clearinghouse
therefore has no net futures position - it has offsetting long and short positions. However, the long
Chapter 5, page 27
and the short in a futures contract pay or receive payments from the exchange’s clearinghouse. This
means that if one takes a position in a futures contract, the only default risk one has to worry about
is the default risk of the clearinghouse. Margins corresponding to a broker’s net positions (the
position left after the short positions are offset against the long positions to the extent possible) are
deposited with the clearinghouse. Further, clearinghouses are well-capitalized by their members and
their resources are well-known. Consequently, default risk in futures contracts is almost nil, at least
for well-known exchanges. Clearinghouses in the U.S. have been able to withstand successfully
dramatic shocks to the financial system. For instance, more than two billion dollars changed hands
in the clearinghouse of the Chicago Mercantile Exchange during the night following the crash of
October 19, 1987. Yet, the futures markets were able to open the next morning because all payments
to the clearinghouse had been made. The fact that the final payments were made on Tuesday morning
with a few minutes to spare and that the CEO of a major bank had to be woken up in the middle of
the night to get a transfer done shows that the markets came close to not opening on that day,
however.
To understand the importance of the clearinghouse, let’s look at an example. Nick Leeson of
Barings Securities was an apparently very successful futures and options trader for his firm since he
arrived in Singapore in the spring of 1992. However, by the beginning of 1995, he had accumulated
extremely large positions that were designed to make a profit as long as the Nikkei index stayed
relatively stable in the 19,000-21,000 range. Things started poorly for Leeson from the start of 1995,
but on January 17 they got much worse as an earthquake devastated the industrial heartland of Japan
around Kobe. On the day of the quake, the Nikkei was at 19,350. Two weeks later it was at 17,785.
Leeson’s positions had large losses. To recover, he pursued a strategy of taking a large long position
Chapter 5, page 28
in the Nikkei futures contract traded in Singapore. It is not clear why he did so. He may have believed
that the Nikkei had fallen too much so that it would rebound, or worse, that he himself could push
the Nikkei up through his purchases. In the span of four weeks, his position had reached 55,399
contracts. As these contracts made losses, margin calls were made and Barings had to come up with
additional money. By Friday, February 25, Barings had losses of 384 million pounds on these
contracts. These losses exceeded the capital of the Barings Group, the parent company. Barings was
effectively bankrupt and unable to meet margin calls. For each long position of Leeson, there was an
offsetting short position held by somebody else. Had Leeson’s contracts been forward contracts
without collateral, the forward sellers would have lost the gains they expected to make from the fall
in the Nikkei index since Barings, being bankrupt, could not honor the contracts. However, the
contracts were futures contracts, so that between the short and long positions, there was the
clearinghouse of the Singapore futures exchange. No short lost money. The clearinghouse made
losses instead.
Section 5.3.B. A brief history of financial futures.
Futures markets exist in many different countries and have quite a long history. For a long
time, these markets traded mostly futures contracts on commodities. In the early 1970s, exchange
rates became more volatile and many investors wanted to take positions to exploit this volatility. The
interbank market did not welcome individuals who wanted to speculate on currencies. With the
support of Milton Friedman, an economist at the University of Chicago, the Chicago Mercantile
Exchange started trading foreign currency futures contracts. These contracts were followed by
contracts on T-bills, T-bonds, and T-notes. Whereas traditional futures contracts required physical
Chapter 5, page 29
delivery if held to maturity, the exchanges innovated by having contracts with cash delivery. In the
beginning of the 1980s the Kansas City Board of Trade started the trading of a futures contract on
a stock index. Since indices are copyrighted, trading a futures contract on an index requires approval
of the holder of the copyright. The Kansas City Board of Trade was able to enter an agreement with
Value Line to use their index. The Value Line index is a geometric average of stock prices, so that
it does not correspond directly to a basket of stocks.4 There is no way to deliver the Value Line
Index. Consequently, a futures contract on the Value Line Index must be settled in cash.
Subsequently, the Chicago Mercantile Exchange started trading a contract on the S&P 500 index. As
explained earlier, the performance of the S&P 500 can be exactly replicated by holding a portfolio
that has the same composition as the index. Nevertheless, the exchange chose delivery in cash since
such a delivery simplifies the procedure considerably. Since then, many more financial futures
contracts have started trading all over the world.
The experience with stock index futures shows that our arbitrage approach is far from
academic. Many financial firms opened arbitrage departments whose role was to exploit discrepancies
between the S&P 500 futures price and the cash market price. These departments would take
advantage of automated trading mechanisms whereby they could sell or buy a large portfolio whose
return carefully tracked the return of the S&P 500 with one computer instruction. For a number of
years, these arbitrage transactions were profitable for the firms with the lowest transaction costs that
quickly managed to take advantage of discrepancies. In a paper evaluating the profitability of
arbitrage strategies during the 1980s on a futures contract on an index similar to the Dow Jones
index, Chung (1991) found that only firms that had low transaction costs and traded within a few
4
Let a, b, and c be three stock prices. A geometric average of these prices is (a*b*c)1/3.
Chapter 5, page 30
minutes from observing the discrepancy could make money. Whereas in the late 1980s index arbitrage
was often blamed for creating artificial volatility in stock prices, there is little discussion about stock
index arbitrage in the 1990s. Arbitrage opportunities seem much less frequent than they were in the
1980s.
Section 5.3.C. Cash versus physical delivery.
Typically, participants in futures markets do not take delivery. They generally close out their
position before maturity. The ability to close a position before maturity is especially valuable for the
contracts that do not have cash delivery - otherwise, the long in the pork bellies contract would have
to cope with a truck of pork bellies. Even though participants typically do not take delivery, contracts
where delivery of physicals takes place at maturity - i.e., pork bellies for the pork bellies contract or
T-bonds for the T-bond contract - have some additional risks compared to the contracts with cash
delivery. Consider a futures contract on a specific variety of grain. A speculator could be tempted,
if she had sufficient resources, to buy this variety of grain both on the futures market and on the cash
market. At maturity, she might then be in a position where the sellers on the futures market have to
buy grain from her to deliver on the futures market. This would be an enviable position to be in, since
she could ask a high price for the grain. Such a strategy is called a corner. If delivery were to take
place in cash for that variety of grain, the futures sellers would not need to have grain on hand at
maturity. They would just have to write a check for the change in the futures price over the last day
of the contract.
In our example, a corner is more likely to be successful if the variety of grain is defined so
narrowly that its supply is quite small. As the supply of the deliverable commodity increases, a corner
Chapter 5, page 31
requires too much capital to be implemented. If a variety of grain has a small supply, the solution is
to allow the short to deliver other varieties that are close substitutes, possibly with a price adjustment.
This way, the deliverable supply is extended and the risk of a corner becomes smaller.
Financial futures contracts that require physical delivery generally allow the short to choose
among various possible deliverable instruments with price adjustments. Obviously, there is no risk
that a long could effect a corner on the Euro. However, for T-bonds and T-notes, there would be
such a risk if only one issue was deliverable. Box 5.3. The cheapest to deliver bond and the case
of Fenwick Capital Management shows how this issue is handled for the T-bonds and T-notes
contracts and provides an example of a successful corner in the T-note futures contract.
Section 5.3.D. Futures positions in practice: The case of the SFR futures contract.
To understand better the workings of futures contracts, let’s look at a specific case where you
take a long position in a SFR futures contract on March 1, 1999, and hold this position during the
month of March. Exhibit 5.1. shows the futures prices for currencies on March 1, 1999, in the Wall
Street Journal. On that day, we must establish an account with a certified broker if we do not have
one already. When the position is opened, the broker determines the initial margin that we must
deposit and the maintenance margin. The initial and maintenance margins cannot be less than those
set by the exchange, but they can be more. Whenever we make a loss, it is withdrawn from our
account. The change in price that determines whether we made a loss or a gain over a particular day
is determined by the change in the settlement price. The settlement price is fixed by the exchange
based on prices at the end of the trading day. An increase in the settlement price means that we gain
with a long position. This gain is then put in the margin account. A decrease in the settlement price
Chapter 5, page 32
means that we committed to buy SFRs at a higher price than the one we could commit now, so we
made a loss.
On March 1, 1999, the settlement price on the June contract is $0.6908. The contract is for
SFR125,000. This means that the value of SFR125,000 using the futures price of the June contract
on March 1 is $86,350. Many futures price contracts have price limits. In the case of the SFR
contract, the limit is effectively a change of $0.04 from a reference price determined during the first
fifteen minutes of trading. If the limit is hit, trading stops for five minutes. After the five minutes, the
limit is expanded and trading resumes as long as the new limit is not hit immediately. There is some
debate as to the exact role of price limits. Some argue that they make it possible for providers of
liquidity to come to the markets and smooth price fluctuations; others argue that they are another tool
the exchange uses to limit default risk. Figure 5.3. shows how the June 1999 SFR futures contract
price evolves during the month of March 1999. The contract price at the end of the month is lower
than at the beginning. This means that having a long position in the contract during that month is not
profitable. The last price in March is $0.6792, so that the dollar value of the SFRs of the contract is
then $84,900. The net sum of the payments made by the long during the month of March is equal to
the difference between $86,350 and $84,900, namely $1,450.
Figure 5.4. provides the evolution of the margin account assuming that no withdrawals are
made and that the margin account earns no interest. In March 1999, the initial margin for the SFR
contract was $2,160 and the maintenance margin was $1,600. Three additional margin payments are
required when the account falls below the maintenance margin of $1,600. These margin payments are
of $827.5 on March 5, $712.5 on March 26, and finally $625 on March 30. The data used for the two
figures are reproduced in Table 5.2.
Chapter 5, page 33
Section 5.4. How to lose money with futures and forward contracts without meaning to.
In this section, we extend our analysis to consider several questions. First, we discuss the
pricing of futures and whether it differs from the pricing of forwards. Second, we evaluate whether
a difference between the futures price and the expected price of the deliverable asset or commodity
represents an arbitrage opportunity. Finally, we discuss the impact of financial market imperfections
on the pricing of forwards and futures.
Section 5.4.A. Pricing futures contracts.
If marking-to-market does not matter, there is no difference between a futures contract and
a forward contract in perfect markets as long as the margin can consist of marketable securities. This
is because, in this case, an investor can always deposit the margin with marketable securities so that
there is no opportunity cost of making the margin deposit. Traditionally, futures are treated like
forwards and the forward pricing results are used for futures. Treating futures like forwards for
pricing purposes provides a good first approximation, but one should be careful not to rely too much
on this approximation since it has some pitfalls.
If one treats futures like forwards, one is tempted to think that any difference between the
forward price and the futures price represents an arbitrage opportunity. Therefore, it is important to
examine this issue carefully. Suppose that a trader learns that the futures price for a contract maturing
in 90 days is higher than the forward price for a contract maturing on the same date with identical
delivery conditions. For instance, there are both forward contracts and futures contracts on the major
foreign currencies. At times, these contracts have identical maturities. With this example, one might
Chapter 5, page 34
be tempted to short the futures contract and take a long position in the forward contract. Surprisingly,
this strategy could lose money. Remember that there is a key difference between the forward contract
and the futures contract. With the forward contract, all of the gain or loss is paid at maturity. In
contrast, with the futures contract, the gains and losses are paid as they accrue.
We can always transform a futures contract into a contract with payoffs at maturity only
through reinvestment and borrowing. This transformation works as follows. As we make a gain, we
invest the gain in the risk-free asset until maturity. At maturity, we receive the gain plus an interest
payment. As we make a loss, we borrow to pay the loss and at maturity we have to pay for the loss
and for the cost of borrowing. The daily settlement feature of futures contracts affects the payoff at
maturity by magnifying both gains and losses through the interest payments.
In an efficient market, we do not know whether we will make gains or losses on the contract.
As a result, these possible gains and losses do not affect the futures price as long as interest rates are
constant and are the same for lending and borrowing. In this case, gains and losses are magnified in
the same way. Since we do not know whether we will gain or lose, the expected interest payments
are zero. Things get more complicated if interest rates change over time. Suppose that interest rates
are high when we make gains and low when we have to borrow because we make losses. In this case,
the reinvestment of gains makes us better off because the expected interest gain on investing profits
exceeds the expected interest loss in borrowing to cover losses on the futures position. Since, with
a forward contract, gains and losses are not paid when they accrue, we receive no interest payments
with the forward contract. If the reinvestment feature is advantageous to us, it must then be that the
futures price exceeds the forward price to offset the benefit of the reinvestment feature and make us
indifferent between a forward contract and a futures contract. The opposite occurs if the interest rate
Chapter 5, page 35
is higher when we have to borrow than when we get to lend. From our discussion, it is possible for
the futures price to exceed the forward price simply because interest rates are higher when the futures
buyer makes gains. If that is the case, taking a long position in the forward contract and a short
position in the futures contract does not make profits. It is a zero net present value transaction. In
fact, in this case, a long position in the forward contract and a short position in the futures contract
would make money if the forward price equals the futures price.
To make sure that we understand the impact of the daily settlement feature of futures
contracts on futures prices, let’s look at a simple example that is also shown on Figure 5.5. Suppose
that we have three dates, 1, 2 and 3. The futures price at date 1 is $2. At date 2, it can be $1 or $3
with equal probability. At date 3, the futures price for immediate delivery is equal to the spot price
S and delivery takes place. Suppose first that the interest rate is constant, so that a one-period
discount bond costs $0.909. With this simple example, reinvestment of the proceeds takes place only
once, at date 2. At date 2, if the price is $1, the futures buyer has to pay $1 and borrow that amount.
In this case, the futures buyer has to repay 1/0.909 = 1.1 at date 3. Alternatively, if the price is $3,
the futures buyer receives $1. Investing $1 for one period yields $1.1 at date 3. Since the futures
buyer has a probability of 0.5 of gaining $0.1 through reinvestment and of losing that amount, there
is no expected benefit from reinvestment. Suppose next that the price of a one-period bond at date
2 is $0.893 if the futures price is $3 and $0.926 if the price is $1. The futures buyer gains $0.12 from
investing the gain (1/0.893 -1) and loses $0.08 from paying interest on borrowing when the price is
$1. In this case, because the interest rate is higher when the futures buyer gains than when he loses,
there is an expected gain from reinvestment of $0.02 (0.5*0.12 - 0.5*0.08). The futures price has to
be higher than the forward price to insure that holding the futures contract does not have a positive
Chapter 5, page 36
net present value. Finally, if the price of the discount bond is $0.926 when the futures buyer makes
a profit and $0.893 when he makes a loss, the futures buyer loses from reinvestment since he receives
0.08 with probability 0.5 and has to pay 0.12 with the same probability. Therefore, the futures price
has to be lower than the forward price.
Keeping the futures price constant, the expected profits of the holder of a long futures
position increase when the correlation between interest rates and the futures price increases.
Consequently, for futures sellers to be willing to enter contracts where the futures price is positively
correlated with interest rates, the futures price of such contracts must be higher. Therefore, an
increase in the correlation between interest rates and futures prices results in an increase in the futures
price relative to the forward price. With this reasoning, one expects the futures price to exceed the
forward price when interest rates are positively correlated with the futures price. When interest rates
are negatively correlated with the futures price, the forward price should exceed the futures price.
A number of papers examine the relation between forward and futures prices. Some of these
papers focus on currencies and generally find only trivial differences between the two prices. In
contrast, studies that investigate the difference between forward and futures prices on fixed income
instruments generally find larger differences. This is not surprising in light of our theory since one
would expect the correlation between interest rates and the underlying instrument to be high in
absolute value for fixed income instruments. However, authors generally find that the theory
presented in this section cannot explain all of the difference, leaving a role for taxes and liquidity,
among other factors, in explaining it.
Section 5.4.B. The relation between the expected future spot price and the price for future
Chapter 5, page 37
delivery
Suppose that an investor expects the spot exchange rate for the SFR 90 days from now to be
at 74 cents and observes a forward contract price for delivery in 90 days of 70 cents. If the investor
is right, 90 days from now he can buy SFRs at 70 cents and he expects to resell them on the spot
market at 74 cents. Let’s consider how we can understand this expected profit and whether markets
could be efficient with such an expected profit.
Note first that it could simply be that the investor is misinformed. In other words, the market
could be right and he could be wrong. In this case, the strategy would not produce an expected gain
if the investor knew what the market knows. Second, it could be that the market expects the spot rate
to be 70 cents and the investor is right. In this case, in expected value, the strategy would make a
profit. Remember, however, that there is substantial volatility to exchange rates, so that even though
the true expected spot rate is 74 cents, 90 days from now the actual exchange rate could turn out to
be 60 cents and the strategy would make a loss.
We now consider whether it is possible that the investor is right in his expectation of a spot
exchange rate of 74 cents and that the forward contract is correctly priced at 70 cents per SFR.
Taking a long forward position because one believes that the forward price is lower than what the
spot price will be when the contract matures is not an arbitrage position. This position can make
losses. It can also make larger gains than expected. We saw in chapter 2 that investors are rewarded
by the market for taking some risks. In particular, if the capital asset pricing model holds, investors
are rewarded for taking systematic risks. Suppose that a foreign currency has a beta of one. In this
case, bearing currency risk is like bearing market risk. The investor expects to be rewarded for
bearing this type of risk. An investor who has a long forward position is rewarded for bearing risk
Chapter 5, page 38
by a lower forward price. This increases the expected profit given the investor’s expectation of the
future spot price. Hence, going back to our example, if the investor expects the spot price to be at
74 cents but the forward price is at 70 cents, he makes an expected profit of four cents per DM when
he enters a forward contract he expects to hold to maturity. This expected profit is compensation for
bearing systematic risk. If a long forward position has a negative beta, then the forward price exceeds
the expected spot price because the short position has a positive beta and expects to be compensated
for bearing systematic risk.
A lot of research has been conducted to study the relation between forward prices, futures
prices, and expected spot prices. Most of this research has focused on the foreign exchange market.
The bottom line of this research is that if we average contracts over very long periods of time, the
forward price of these contracts on average equals the spot price at maturity. Hence, on average, the
forward price is the expected spot price. At the same time, this literature shows that typically a
forward contract on a currency for which interest rates are high is on average profitable. The two
findings are not contradictory: countries will sometime have high interest rates and other times low
interest rates. This means that at any point in time the forward price is unlikely to be equal to the
expected spot price. There is some evidence that shows that the current spot price is a better forecast
of the future spot price for major currencies than the forward exchange change rate.
Section 5.4.C. What if the forward price does not equal the theoretical forward price?
Throughout this chapter, we have made the assumption that markets are perfect. Suppose
now that there are some transaction costs. In this case, the forward price may not be equal to the
theoretical forward price. The reason is straightforward. Remember that at the theoretical forward
Chapter 5, page 39
price, in the absence of transaction costs, the cost of the replicating portfolio of the buyer is equal to
the cost of the replicating portfolio of the seller. In this case, if the forward price differs from its
theoretical value, either the buyer or the seller can construct an arbitrage portfolio that takes
advantage of the discrepancy. For instance, if the forward price is higher than its theoretical value,
it pays to sell forward and hedge the transaction by buying the deliverable asset and financing the
purchase with a loan that matures with the contract.
Suppose now that there are some transaction costs. Say that cF is the proportional transaction
cost for the forward contract (for instance, a commission that has to be paid to the broker), cA the
proportional transaction cost on purchasing the deliverable asset, and cB the proportional transaction
cost on borrowing. In this case, taking advantage of a forward price that is too high requires paying
transaction costs of cFF on the forward position, cAS(t) on purchasing the deliverable asset, and of
cBS(t) on borrowing. The total transaction costs for the long to hedge his position are therefore equal
to cFF + cAS(t) + cBS(t). Hence, if the difference between the forward price and its theoretical value
is positive, there is an arbitrage opportunity only if this difference exceeds the transaction costs of
going long the contract and hedging the position. Similarly, if the difference between the forward
price and its theoretical value is negative, this difference has to exceed the transaction cost of shorting
the contract and hedging the short position for there to exist an arbitrage opportunity.
With our reasoning, we can construct an expanded formula for the forward price. Let SL be
the sum of the transaction costs that have to be paid in forming an arbitrage portfolio that is long the
forward contract. Remember that such a portfolio involves a long forward position and a short
position in the replicating portfolio. Let SS be the sum of the transaction costs that have to be paid
in forming an arbitrage portfolio that is short the forward contract. Remember that an arbitrage
Chapter 5, page 40
portfolio that is long the forward contract makes it possible to capture the value of the replicating
portfolio. To do that, one goes long the contract and short the replicating portfolio. One generates
a profit equal to the value of the replicating portfolio. Before transaction costs, the value of the
arbitrage portfolio must be less than the transaction costs required to create it. In this case, we must
have:
Forward price in the presence of transaction costs
S(t) - Ω L
S(t) + Ω S
≤ F ≤
P(t, t + i)
P(t, t + i)
(5.14.)
To check the formula, note it can be rewritten as:
-SS # S(t) - P(t,t+i)F # SL
If the forward price is lower than its theoretical value under perfect markets, then the term in the
middle is positive. Since transaction costs are positive, this means that the left-hand side inequality
is satisfied. If the right-hand side inequality is not satisfied, then the costs of forming an arbitrage
portfolio are less than the profit from forming the arbitrage portfolio before transaction costs. If the
forward price is too high, the middle term is negative and one would like to short the forward
contract, using a long position in the replicating portfolio to hedge. To avoid arbitrage opportunities,
the gain from doing this has to be less than the transaction costs.
It is perfectly possible in the presence of real world frictions for the forward price (and the
Chapter 5, page 41
futures price by extension) to be different from the theoretical price that must hold if markets are
perfect. In some cases, the costs of forming arbitrage portfolios are minuscule so that one would
expect the discrepancies to be extremely small. One such case would be where one tries to arbitrage
forward price discrepancies for short standard maturity contracts on foreign exchange for major
currencies. Short-term forward contracts for currencies have standard maturities of 30, 90, and 180
days. The transaction costs for arbitraging a ten-year foreign exchange forward contract would be
more significant. Whether the arbitrage costs are important or not depends on the liquidity of the
deliverable asset, on the trading expertise of the arbitrageur, and on the credit risk of the arbitrageur.
An AAA bank with an active trading room faces much lower transaction costs than a weakly
capitalized occasional participant in the foreign exchange market.
Section 5.5. Conclusion.
In this chapter, we learned how to price forward contracts. The key to pricing these contracts
is that they can be replicated by buying the underlying asset and financing the purchase until maturity
of the contract. Because of this, we can price the contracts by arbitrage. This means that we can price
a forward contract without having a clue as to the expected value of the underlying asset at maturity.
To price a forward contract on the SFR, we therefore do not need to know anything about exchange
rate dynamics. The forward price depends on the interest rate that has to be paid to finance the
purchase of the underlying asset, the cost to store it, and the benefit from holding it. As financing and
storage become more expensive, the forward price increases relative to the current price of the
underlying asset. As the benefit from holding the underlying asset increases, the forward price falls
relative to the price of the underlying asset. We then discussed how forward contracts can have
Chapter 5, page 42
substantial default risk and how futures contracts are similar to forward contracts with a built-in
mechanism to reduce and almost eliminate default risk. This mechanism is the daily settlement and
the posting of a margin. We then explained that the daily settlement allows participants in futures
markets to reinvest their gains. The futures price exceeds the forward price when this feature is
advantageous. We then discussed that forward and futures contracts may have systematic risk.
Systematic risk lowers the forward and futures prices to compensate the long for bearing the risk.
Finally, we showed how transaction costs can imply small deviations of forward prices from their
theoretical formula. The next step is to learn how to use forward and futures contracts for risk
management. We turn to this task in the next two chapters.
Chapter 5, page 43
Literature Note
The topics discussed in this chapter have generated an extremely large literature. Specialized
textbooks on futures, such as Duffie () or Siegel and Siegel () develop the material further and
provide references to the literature. Brennan (1986) and Kane (1980) have interesting theoretical
studies of the mechanisms used in futures markets to reduce counterparty risk. Melamed () provides
a good history of financial futures with a wealth of anecdotes. The comparison of futures and forward
is analyzed theoretically in Cox, Ingersoll, and Ross (1981) and Jarrow and Oldfield (1981). Richard
and Sundaresan (1981) provide a general equilibrium model of pricing of futures and forwards.
Routledge, Seppi, and Spatt (2000) derive forward prices when the constraint that inventories cannot
be negative is binding and show that this constraint plays an important role. When that constraint is
binding, spot has value that the forward does not have. Cornell and Reinganum (1981), French
(1983), Grinblatt and Jegadeesh (1996), and Meulbroek (1992) compare forwards and futures,
respectively, for currencies, copper, and Euro-deposits. Theoretical and empirical studies that
compare the future expected spot exchange rate and the forward exchange rate are numerous.
Hodrick reviews some of these studies for foreign exchange and Froot and Thaler (1990) provide an
analysis of this literature arguing that departures of the forward exchange rate from the future
expected spot exchange rate are too large to be consistent with efficient markets. For a recent study
of interest rate parity in the presence of transaction costs, see Rhee and Chang (1992). The Barings
debacle is chronicled in Rawnsley (1995).
Chapter 5, page 44
Key concepts
Forward contract, replicating portfolio, convenience yield, storage costs, futures contract, initial
margin, settlement variation, margin call, marking-to-market, cash delivery, physical delivery, price
limit, counterparty risk.
Chapter 5, page 45
Review questions
1. What is a forward contract?
2. What is the spot price?
3. What is a replicating portfolio for a forward contract?
4. What is required for the price of a forward contract to be such that there is an arbitrage
opportunity?
5. What is the impact of dividends on the forward price in a contract to buy a stock forward?
6. What is the convenience yield?
7. Why do storage costs matter in pricing forward contracts?
8. How do futures contracts differ from forward contracts?
9. What is the maintenance margin in a futures contract?
10. What is the difference between cash and physical delivery for a futures contract?
11. What happens if a price limit is hit?
12. Why would the futures price differ from the forward price when both contracts have the same
underlying and the same maturity?
13. Does the futures price equal the expected spot price at maturity of the futures contract?
14. Why would an arbitrage opportunity in perfect financial markets not be an arbitrage opportunity
in the presence of financial market imperfections?
Chapter 5, page 46
Problems
1. Suppose that on January 10, a one-year default-free discount bond is available for $0.918329 per
dollar of face value and a two-year default-free discount bond costs $0.836859 per dollar of face
value. Assuming that financial markets are perfect, suppose you want to own $1.5M face value of 1year discount bonds in one year and you want to pay for them in one year a price you know today.
Design a portfolio strategy that achieves your objective without using a forward contract.
2. Using the data of question 1, suppose that you learn that the forward price of 1-year discount
bonds to be delivered in one year is $0.912 per dollar of face value. What is the price today of a
portfolio that replicates a long position in the forward contract?
3. Using the replicating portfolio constructed in question 2, construct a strategy that creates arbitrage
profits given the forward price of $0.912. What should the forward price be so that you would not
be able to make arbitrage profits?
4. Suppose that you enter a long position in the forward contract at a price of $0.911285 per dollar
of face value for a total amount of face value of $1.5M. One month later, you find that the 11-month
discount bond sells for $0.917757 and that the 23-month discount bond sells for $0.829574. Compute
the value of your forward position at that point.
5. Given the prices of discount bonds given in question 4, how did the price of a one-year discount
bond for delivery January 10 the following year change over the last month? Did you make money
Chapter 5, page 47
having a long forward position?
6. Consider a Treasury bond that pays coupon every six months of $6.5 per $100 of face value. The
bond matures in 10 years. Its current price is $95 per $100 of face value. You want to enter a forward
contract to sell that bond in two years. The last coupon was paid three months ago. The discount
bond prices for the next two years are: P(t,t+0.25) = 0.977508, P(t,t+0.5) = 0.955048, P(t,t+0.75)
0.932647, P(t,t+1) = 0.910328, P(t,t+1.25) = 0.888117, P(t,t+1.5) = 0.866034, P(t,t+1.75) =
0.844102, P(t,t+2) = 0.822341. Using these discount bond prices, find the forward price per dollar
of face value for a contract for the purchase of $100M of face value of the ten-year bond such that
the value of the replicating portfolio is zero.
7. Using the discount bond prices of question 6, price a forward contract on oil. The current price
per barrel is $18. The contract has maturity in one year. The inventory cost is 3% at a continuously
compounded rate. The convenience yield associated with having oil in inventory is 5% per year.
Given this, what is the forward price per barrel?
8. Consider the contract priced in question 7. Suppose that immediately after having entered the
contract, the convenience yield falls to 2%. How is the forward price affected assuming that nothing
else changes?
9. You are short the SFr. futures contract for five days. Let’s assume that the initial margin is $2,000
and the maintenance margin is $1,500. The contract is for SFr. 100,000. The futures price at which
Chapter 5, page 48
you enter the contract at noon on Monday is $0.55. The Monday settlement price is $0.56. The
settlement prices for the next four days are $0.57, $0.60, $0.59, $0.63. You close your position on
Friday at the settlement price. Compute the settlement variation payment for each day of the week.
Show how your margin account evolves through the week. Will you have any margin calls? If yes,
for how much?
10. Suppose that on Monday at noon you see that the forward price for the SFr. for the same maturity
as the maturity of the futures contract is $0.57 and that you believe that the exchange rate is not
correlated with interest rate changes. Does this mean that there is an arbitrage opportunity? If not,
why not? Does it mean that there is a strategy that is expected to earn a risk-adjusted profit but has
risk?
Chapter 5, page 49
Exhibit 5.1. Currency futures prices for March 1, 1999, published in the Wall Street Journal on
March 2, 1999.
Chapter 5, page 50
Table 5.1. Arbitrage trade when F = $0.99 per dollar of face value.
In this case, the replicating portfolio for a long forward position is worth $0.95 - $0.969388*0.99
= -$0.00969412. To exploit this, we short the contract and go long the replicating portfolio for a
long forward position. The third column shows the cash flow on March 1 and the fourth column
shows the cash flow on June 1.
Positions on March 1
Cash flow on
March 1
Cash flow on June 1
Short forward position
0
$0.99 - P(June 1, August
30)
Long bill maturing
on August 30
-$0.95
P(June 1, August 30)
Borrow present
value of F
$0.969388*0.99
= $0.959694
-$0.99
$0.009694
0
Replicating portfolio for long position
Net cash flow
Chapter 5, page 51
Table 5.2. Long SFR futures position during March 1999 in the June 1999
contract
Date
3/1/99
Futures price F*contract size
(F)
(SFR125,000)
0.6908
Gain (Loss)
Margin account
(before add. deposit)
0
86,350.0
3/2/99
0.6932
3/3/99
0.6918
3/4/99
0.6877
3/5/99
0.6869
3/8/99
0.6904
3/9/99
0.6887
3/10/99
0.6919
3/11/99
0.6943
3/12/99
0.6883
3/15/99
0.6888
3/16/99
0.6944
3/17/99
0.6953
3/18/99
0.6918
3/19/99
0.6872
3/22/99
0.6897
3/23/99
0.6911
3/24/99
0.6891
3/25/99
0.6851
3/26/99
0.6812
3/29/99
0.6793
3/30/99
0.6762
3/31/99
0.6792
add. deposit
Margin account
(after add. deposit)
0
300.0
2,160.0
2,460.0
0
2,160.0
(175.0)
2,285.0
0
(512.5)
1,772.5
0
(100.0)
1,672.5
827.5
437.5
2,937.5
0
(212.5)
2,725.0
0
400.0
3,125.0
0
300.0
3,425.0
0
(750.0)
2,675.0
0
62.5
2,737.5
0
700.0
3,437.5
0
112.5
3,550.0
0
(437.5)
3,112.5
0
(575.0)
2,537.5
0
312.5
2,850.0
0
175.0
3,025.0
0
(250.0)
2,775.0
0
(500.0)
2,275.0
0
(487.5)
1,787.5
712.5
(237.5)
2,262.5
0
(387.5)
1,875.0
625
375.0
2,875.0
0
86,650.0
2,460.0
86,475.0
2,285.0
85,962.5
1,772.5
85,862.5
2,500.0
86,300.0
2,937.5
86,087.5
2,725.0
86,487.5
3,125.0
86,787.5
3,425.0
86,037.5
2,675.0
86,100.0
2,737.5
86,800.0
3,437.5
86,912.5
3,550.0
86,475.0
3,112.5
85,900.0
2,537.5
86,212.5
2,850.0
86,387.5
3,025.0
86,137.5
2,775.0
85,637.5
2,275.0
85,150.0
2,500.0
84,912.5
2,262.5
84,525.0
2,500.0
84,900.0
Source : The Wall Street Journal various issues (3/2/99 ~ 4/1/99)
Chapter 5, page 52
2,875.0
Exhibit 5.1
Wall Street Journal, March 1, 1999
Chapter 5, page 53
Figure 5.1. Time-line for long forward contract position
Chapter 5, page 54
Figure 5.2. Difference between the forward price and the spot price of the underlying good.
The term structure used for the example is the one used in Box 5.2. Figure for the home country.
Chapter 5, page 55
Figure 5.3. Margin account balance and margin deposits
4,000.0
1000
800
3,500.0
3,000.0
400
2,500.0
200
0
-200
2,000.0
1 2 3 4 5 8 9 10 11 12 15 16 17 18 19 22 23 24 25 26 29 30 31
1,500.0
-400
1,000.0
-600
500.0
-800
-1000
date
Chapter 5, page 56
margin account balance
gain(loss) and deposits
600
gain(loss)
deposits
balance
Figure 5.4. Daily gains and losses from a long futures position in the SFR in March 1999
800
0.7
600
0.695
400
200
0.685
0
0.68
-200
-400
0.675
-600
0.67
-800
0.665
-1000
1 2 3 4 5 8 9 10 11 12 15 16 17 18 19 22 23 24 25 26 29 30 31
date
Chapter 5, page 57
gain(loss)
futures price
0.69
futures price
gain(loss)
Figure 5.5. Marking to market and hedging.
To make a futures contract have payoffs only at maturity, one borrows marked-to-market losses and
invests marked-to-market gains. In this example, the futures price and the forward price are equal at
$2 at date 1. The futures price then falls to $1 at date 2 and stays there. Consequently, at date 3, the
spot price is $1. The payoff of the forward contract at date 3 is -$1 for a long position. The payoff
for the futures contract is -$1 at date 2. To make the futures contract have a payoff at date 3 only like
the forward contract, one can borrow $1 at date 2 which one repays at date 3. Hence, if the interest
rate is 10% for each period, one loses $1.1 with the futures contract at date 3 in this example.
Chapter 5, page 58
Box 5.1. Repos and borrowing using Treasury securities as collateral.
Typically, borrowing for those who hold inventories of Treasury securities is done in the repo market.
A repo is a transaction that involves a spot market sale of a security and the promise to repurchase
the security at a later day at a given price. One can therefore view a repo as a spot market sale of a
security with the simultaneous purchase of the security through a forward contract. A repo where the
repurchase takes place the next day is called an overnight repo. All repos with a maturity date in more
than one day are called term repos. A repo amounts to borrowing using the Treasury securities as
collateral. For this reason, rather than stating the price at which the underlying is bought back at a
later day, the contract states a rate that is applied to the spot price at origination to yield the
repurchase price. For instance, consider a situation where dealer Black has $100m worth of Treasury
securities that he has to finance overnight. He can do so using a repo as follows. He can turn to
another dealer, dealer White, who will quote a repo rate, say 5%, and a haircut. The haircut means
that, though White receives $100m of Treasury securities, he provides cash for only a fraction of
$100m. The haircut protects White against credit risk. The credit risk arises because the price of the
securities could fall. In this case, Black would have to pay more for the securities than they are worth.
If he could walk away from the deal, he would make a profit. However, since he has to pay for only
a fraction of the securities to get all of them back if there is a haircut, he will pay as long as the price
is below the promised payment by an amount smaller than the haircut. For instance, if the haircut is
1%, Black receives $100*(1/1.01) = $99.0099m in cash. The next day, he has to repay
$100*(1/1.01)*(1+0.05 /360 ) = $99.037m. Hence, if the securities had fallen in value to $99.05m,
Black would still buy them back. With this transaction, Black has funds for one day. Because this
transaction is a forward purchase, Black benefits if the price of the securities that are purchased at
maturity have an unexpectedly high value. With Treasury securities, therefore, the borrower benefits
if interest rates fall. Viewed from White’s perspective, the dealer who receives the securities, the
transaction is called a reverse repo. A reverse repo can be used to sell Treasury securities short in
the following way. After receiving the securities, the dealer can sell them. He then has to buy them
back to deliver them at maturity of the reverse repo. He therefore loses if the price of the securities
increases because he has to pay more for them than he receives when he delivers them at maturity.
Chapter 5, page 59
Box 5.2. Interest rate parity.
Suppose that an investor faces (continuously compounded) annual interest rates of 5% in the
U.S. and 1% in Switzerland for 90 days. He would like to know whether it is more advantageous to
invest in the U.S. or in Switzerland. The investment in Switzerland has exchange rate risk. One dollar
invested in SFRs might lose value because of a depreciation of the SFR. For instance, if the investor
invests $1m in Switzerland at the exchange rate of $0.7 per SFR, he might earn 1% p.a. on this
investment but exchange the SFRs into dollars in 90 days at an exchange rate of $0.8 per SFR. In this
case, he would gain 14.57% of his investment for an annualized rate of return of 54.41%!
This is because he gets 1/0.7 SFRs or SFR1.4286 initially that he invests at 1%, thereby having
SFR1.4321 at maturity (Exp[0.01*90/360]/0.7). He then converts his SFRs at $0.8, thereby obtaining
$1.1457 in 90 days. Hence, despite earning a lower interest rate in Switzerland than in the U.S., the
investor gains by investing in Switzerland. The exchange rate gain is an exchange rate return of 53.41% (Ln[0.8/0.7]*360/90) measured at an annual continuously compounded rate. The total return
is the sum of the exchange rate return plus the interest earned, or 53.41% + 1% = 54.41%.
We know, however, that the investor could sell the proceeds of his SFR investment forward
at an exchange rate known today. Doing this, the investor is completely hedged against exchange
rate risk. He gets F*Exp[0.01*90/360]/0.7 for sure in 90 days, where F is the forward exchange rate,
irrespective of the spot exchange rate at that date. Consequently, F has to be such that investing at
the risk-free rate in the U.S. would yield the same amount as investing in SFRs and selling the
proceeds forward to obtain dollars. Let’s define the annualized exchange rate return from buying
SFRs and selling them forward as the forward premium and use the notation f for this exchange rate
return. We use continuous compounding, so that for our example, the forward premium solves the
equation F = $0.7*Exp[f*90/360], so that we have f = Ln[F/0.7]*360/90. It must be the case, since
a hedged investment in SFRs is risk-free, that the return of this investment must equal 5% p.a. The
return of investing in SFRs is 1% + f. Hence, it must be that f = 4% computed annually. We can then
solve for the forward exchange rate such that 0.04 = Ln[F/0.7]*360/90. This gives us F = $0.7070.
The result that the foreign interest rate plus the forward premium must equal the domestic interest
rate to avoid arbitrage opportunities is called the interest rate parity theorem:
Interest rate parity theorem
When the domestic risk-free rate, the forward premium, and the foreign risk-free rate are for the same
maturity and continuous compounding is used, it must be true that:
Domestic risk-free rate = Forward premium + Foreign risk-free rate (IRPT)
The interest rate parity theorem states that, after hedging against foreign exchange risk, the risk-free
asset earns the same in each country. There is a considerable amount of empirical evidence on this
result. This evidence shows that most observed forward prices satisfy the interest rate parity theorem
if there are no foreign exchange controls and if transaction costs are taken into account. We discuss
the impact of transaction costs in the last section of this chapter.
The interest rate parity theorem as well as the result for the pricing of foreign exchange
forward contracts have a powerful implication. If one knows interest rates at home and in the foreign
country, we can compute the forward exchange rate. Alternatively, if one knows the forward
premium and the interest rates in one country, we can compute interest rates in the other country.
Chapter 5, page 60
Box 5.2. Figure shows graphically how one can do this. (Note that the result in equation (IRPT)
holds only approximately when one does not use continuously-compounded returns.)
Chapter 5, page 61
Box 5.2. Figure. Term structures in the home and foreign country used to obtain the term
structure of forward premia.
The interest rate parity result states that the continuously-compounded yield difference is equal to the
forward premium. Consequently, we can obtain the forward premium that holds in the absence of
arbitrage opportunities, whether a contract exists or not, if we know the yield difference.
Chapter 5, page 62
Box 5.3. The cheapest to deliver bond and the case of Fenwick Capital Management
The Chicago Board of Trade started trading the T-bond futures contract in August 1977. The
contract requires delivery of T-bonds with a face value at maturity of $100,000. To limit the
possibility of a successful corner, the contract defines the deliverable T-bonds broadly as long-term
U.S. Treasury bonds which, if callable, are not callable for at least 15 years or, if not callable, have
a maturity of at least 15 years. The problem with such a broad definition is that the short always
wants to deliver the bonds which are cheapest to deliver. This means that the short will find those
bonds with $100,000 face value that cost the least to acquire on the cash market. Without some price
adjustments to make the bonds more comparable, the bond with the longest maturity would generally
be the cheapest bond to deliver if interest rates have increased. This is because this bond will have a
low coupon for a long time to maturity so that it well sell at a low price compared to par. If the
supply of that bond is small, there could still be a successful short squeeze for that bond. Such a
squeeze might force the short to deliver other bonds than the one it would deliver without the
squeeze.
The exchange allows a great number of bonds to be deliverable, but makes these bonds
comparable through price adjustments. When a short delivers bonds, the amount of money the short
receives is called the invoice price. This invoice price is computed as follows. First, the settlement
futures price is determined. Second, this futures price is multiplied by a conversion factor. The
conversion factor is the price the delivered bond would have per dollar of face value if it were priced
to yield 8% compounded semiannually. Finally, the accrued interest to the delivery date is added to
this amount.
The use of the conversion factor is an attempt to make all deliverable bonds similar. This
works only imperfectly. The ideal method is one that would make the product of the futures price and
the conversion factor equal to the cash price of all deliverable bonds. This does not happen with the
method chosen by the exchange. At any point in time, there are bonds whose cash market price is
higher than their invoice price. Which bond is the cheapest to deliver depends on interest rates.
Typically, if interest rates are low, the cheapest to deliver bond will be a bond with a short maturity
or first call date because this bond will have the smallest premium over par. In contrast, if interest
rates are high, the cheapest bond to deliver will be a bond with a long maturity or first call date.
The futures price reflects the price of the cheapest bond to deliver. However, since there is
some uncertainty as to which bond will be the cheapest to deliver at maturity, the pricing issue is a
bit more complicated when interest rates are highly volatile. As a first step in pricing the T-bond
contract, however, one can use as the deliverable instrument the T-bond that is currently cheapest to
deliver. More precise pricing requires taking into account that the cheapest bond to deliver could
change over the life of the contract.
The T-note market works in the same way as the T-bond market and offers a recent example
showing that one should not completely dismiss the risk of manipulation in these markets. In July
1996, the Commodity Futures Trading Commission fined Fenwick Capital Management for having
cornered the market of the cheapest to deliver note on the June 1993 contract. The cheapest to
deliver note was the February 2000 note with a coupon of 8.5%. Although the supply of that note
was initially of $10.7 billion, there had been a lot of stripping. Fenwick bought $1.4 billion of that
issue and had a long position in the T-note contract of 12,700 contracts for $1.27 billion notional of
notes. In June 1993, it was very difficult to find the February 2000 note. Consequently, most shorts
had to deliver the next-cheapest note, an 8.875% May 2000 note. Delivering that note involved an
Chapter 5, page 63
extra cost of $156.25 a contract. Fenwick got that note delivered on 4,800 of its contracts. This
yielded a profit of $750,000.5
5
See “The S.E.C. says it happened: A corner in the Treasury market” by Floyd Norris,
The New York Times, July 11, 1996, page C6.
Chapter 5, page 64
Chapter 6: Risk measures and optimal hedges with forward and futures contracts
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Section 6.1. Measuring risk: volatility, CaR, and VaR . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Section 6.2. Hedging in the absence of basis risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1. The forward contract solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. The money market solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. The futures solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Section 6.3. Hedging when there is basis risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Section 6.3.1. The optimal hedge and regression analysis . . . . . . . . . . . . . . . . . 33
Section 6.3.2. The effectiveness of the hedge. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Section 6.4. Implementing the minimum-variance hedge. . . . . . . . . . . . . . . . . . . . . . . . 41
Section 6.4.1. The relevance of contract maturity . . . . . . . . . . . . . . . . . . . . . . . 42
Section 6.4.2. Hedging with futures when the maturity of the contract is shorter
than the maturity of the exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Section 6.4.3. Basis risk, the hedge ratio, and contract maturity. . . . . . . . . . . . . 48
Section 6.4.4. Cross-hedging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Section 6.4.5. Imperfect markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Section 6.4.6. Imperfect divisibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Section 6.4.7. The multivariate normal increments model: Cash versus futures
prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Section 6.5. Putting it all together in an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1. Forward market hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2. Money market hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3. Futures hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.6. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Questions and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 6.1. Relation between cash position and futures price when the futures price
changes are perfectly correlated with the cash position changes . . . . . . . . . . . . 67
Figure 6.2. Relation between cash position and futures price changes when the futures
price changes are imperfectly correlated with the cash position changes . . . . . . 68
Figure 6.3. Regression line obtained using data from Figure 6.2. . . . . . . . . . . . . . . . . . 69
Figure 6.4. This figure gives the SFR futures prices as of March 1, 1999 for three
maturities as well as the spot price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure 6.5. Variance of hedged payoff as a function of hedge ratio and variance of basis
risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Table 6.1. Market data for March 1, 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Technical Box 6.1. Estimating Mean and Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Technical Box 6.2. Proof by example that tailing works . . . . . . . . . . . . . . . . . . . . . . . . 74
A) Futures hedge without tailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B) Futures hedge with tailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Technical Box 6.3. Deriving the minimum-volatility hedge . . . . . . . . . . . . . . . . . . . . . . 76
Technical Box 6.4. The statistical foundations of the linear regression approach to
obtaining the minimum-variance hedge ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 6: Risk measures and optimal hedges with forward
and futures contracts
December 13, 1999
© René M. Stulz 1998, 1999
Chapter objectives
1. Measure risk of forward and futures positions.
2. Derive optimal hedges with forward and futures contracts when perfect hedge is possible.
3. Derive optimal hedges with futures contracts in the presence of basis risk.
4. Examine the implementation of optimal hedging strategies.
5. Review the practical obstacles to successful implementation of optimal hedging strategies.
Chapter 6, page 2
In this chapter, we investigate how to minimize risk using forward and futures contracts. A
simple example is as follows: suppose that a U.S. firm expects to receive 1 million Swiss Francs
(SFR1M) in three months. This cash flow is risky because the firm does not know what the exchange
rate of the dollar for the SFR will be in three months. Because of this receivable, firm cash flow in
three months is sensitive to changes in the dollar price of the SFR. The source of risk is the SFR
exchange rate. We call an identifiable source of risk a risk factor and a cash flow’s sensitivity to a
risk factor its exposure to that risk factor. Similarly, the sensitivity of the value of a position to a risk
factor is the exposure of that position to that risk factor. Here, an increase in the dollar price of the
SFR of x increases the firm’s cash flow in three months by 1M*x. Consequently, the firm’s exposure
to the dollar price of the SFR is 1M.
To choose its hedging policy with respect to its cash flow, the firm first has to measure the
cash flow’s risk and then figure out how much of that risk it wants to bear. In chapter 2, we saw that
individuals care about volatility risk. In chapter 4, we introduced value at risk (VaR) and cash flow
at risk (CaR) as risk measures for firms. In this chapter, we first show how to obtain volatility, CaR,
and VaR measures for a SFR exposure like the one described in the above paragraph. Since a long
forward position of SFR1M has the same exposure as this SFR cash flow, we also compute the risk
of such a position. In chapter 4, we discussed that a firm often finds increases in CaR or VaR to be
costly. We therefore consider optimal hedges of a foreign currency position when risk is measured
by volatility, CaR, and VaR. We first address the case where the firm can eliminate all risk resulting
from its SFR exposure. In this case, the optimal hedge eliminates all risk when such a hedge is feasible
and is costless. Hedging decisions are generally more complicated. This is especially the case when
no hedge eliminates all risk or when hedging costs are important. When hedging costs are
Chapter 6, page 3
unimportant but hedging cannot eliminate all risk, the firm has to find out which hedge minimizes risk.
We show how to find and implement this hedge. When hedging costs are important, the firm may
choose not to eliminate risk even though it can do so. In this case, the firm’s hedge depends on how
it trades off hedging costs and benefits. We address this problem in chapter 7.
Section 6.1. Measuring risk: volatility, CaR, and VaR.
Consider the simple situation where, on March 1, 1999, a firm expects to receive SFR1M in
three months. We assume for simplicity that there is no default risk on the part of the firm’s debtor,
so that there is no uncertainty as to whether the SFRs will be received. Further, interest rates are
assumed to be constant. The only uncertainty has to do with the dollar value of the SFRs when they
are received, which is our risk factor.
Let’s call the firm Export Inc. and assume that it exported goods to Switzerland for which
it will receive SFR1M in three months. The SFR1M is the only cash flow of Export Inc. and the firm
will dissolve after receiving the SFRs. Suppose that the firm incurs a deadweight cost if the cash flow
is low. This could be the result of debt that the firm has to repay in three months, so that there might
be a bankruptcy cost if the firm defaults. We saw in chapter 4 that an appropriate risk measure in this
case is CaR. The CaR at the x% level is the cash flow shortfall relative to expected cash flow such
that the probability that the cash flow shortfall exceeds CaR is x%. To compute the CaR for Export
Inc., we need the probability distribution of the cash flow in three months.
Let’s assume that the increments of the exchange rate are identically independently distributed
and that their distribution is normal. We use the abbreviation i.i.d to denote random variables that are
identically independently distributed. If increments are i.i.d, past increments provide no information
Chapter 6, page 4
about future increments. This means that a large increase in the exchange rate during one month
means nothing about the exchange rate change for the next month. With the assumption that
increments are i.i.d., we can estimate the mean and the variance of the increments. Technical Box
6.1. Estimating mean and volatility shows how to do this. Using monthly data for the dollar price
of the SFR, we obtain a mean monthly change of -$0.000115 and a variance of 0.000631.
With our assumptions, the variance for each period is the same and the increments are
uncorrelated. To see what this implies for the variance over multiple future periods, let’s look at the
variance of the exchange rate as of May 1 viewed from March 1. This is a two-month interval to
make things easier to follow. The exchange rate on May 1 is the current exchange rate. Remember
that we denote a spot price by S. The spot price of the SFR, its exchange rate, on May 1 is S(March
1), plus the monthly increments, the change in March, )S(March), and the change in April,
)S(April). The variance of S(May 1) is therefore obtained as follows:
Var(S(May 1)) = Var(S(March 1) + )S(March) + )S(April))
= Var()S(March)) + Var()S(April)) + 2Cov()S(March), )S(April))
= Var()S(March)) + Var()S(April))
= 2Var()S(March))
The Var(S(March 1)) is equal to zero since we know the exchange rate on that day. We argued in
chapter 2 that assuming serially uncorrelated increments for financial prices is a good first assumption.
We therefore make this assumption here, so that Cov()S(March), )S(April)) = 0. Finally, we
assume that the distribution of each increment is the same, so that Var()S(March)) = Var()S(April)).
Chapter 6, page 5
The variance of the change in the exchange rate over 2 periods is twice the variance of the change
over one period. The same reasoning leads to the result that the variance of the exchange rate in N
periods is N times the variance of a one-period increment to the exchange rate. Since the volatility
is the square root of the variance, it follows that the volatility of the exchange rate over N periods is
/N times the volatility per period. This rule is called the square root rule for volatility:
Square root rule for volatility
If a random variable is identically independently distributed with volatility per period F, the volatility
of that random variable over N periods is F*/N.
This rule does not work if increments are correlated because, in that case, the variance of the sum of
the increments depends on the covariance between the increments.
Let’s now look at an horizon of three months. The monthly volatility is the square root of
0.000631, which is 0.0251197. To get the three-month volatility, we multiply the one-month volatility
by the square root of three, 0.0251197*/3, which is 0.0435086. The volatility of the payoff to the
firm, the dollar payment in three months, is therefore 0.0435086*1M, or $43,508.6. The expected
change of the exchange rate over three months is three times the expected change over one month,
which is 3*(-.000115) = -0.000345. Since the monthly exchange rate increment is normally
distributed, the sum of three monthly increments is normally distributed also because the sum of
normally distributed random variables is normally distributed. We know that with the normal
distribution, there is a probability of 0.05 that a random variable distributed normally will have a value
lower than its mean by 1.65 times its standard deviation. Using this result, there is a 0.05 probability
Chapter 6, page 6
that the exchange rate will be lower than its expected value by at least 1.65*0.0435086 = 0.0717892
per SFR. In terms of the overall position of Export Inc., there is a 0.05 probability that it will receive
an amount that is at least 0.0717892*1,000,000 = $71,789.2 below the expected amount. Suppose
that the current exchange rate is $0.75 and we use the distribution of the exchange rate we have just
estimated. In this case, the expected exchange rate is $0.75 - $0.00325, or $0.7496550. The expected
cash flow is 1m*0.749655 = $749,655. As derived above, there is a 5% probability that the firm’s
cash flow shortfall relative to its expected value will exceed $71,789.2. Consequently, there is a 5%
chance that the cash flow will be lower than $749,655 - $71,789.2, or $677,865.8. The CaR is
therefore $71,789.2. Let’s suppose that the firm wants to have a 95% chance that its cash flow
exceeds $710,000. In this case, the firm has taken on too much risk because its target CaR is $39,655,
namely $749,655 - $710,000.
We now consider a firm, Trading Inc., that has the same exposure measured but different
concerns. For simplicity, let’s assume that interest rates are constant. This firm owns SFR T-bills with
face value of SFR1M that mature in three months so that, like Export Inc., it will receive SFR1M on
June 1. The exposure of Trading Inc. to the price of the SFR on June 1 is 1M. Because the firm has
limited capital, it has to finance c% of this position with a three-month loan, so that a drop in the
value of the position by 1 - c% before the firm can trade would make the firm insolvent. From the
previous chapter, we know that a position in SFR T-bills of SFR1M financed by selling short T-bills
in dollars maturing at the same date is the replicating portfolio for a long forward position to buy
SFR1M. Hence, the case where c% is equal to 100% corresponds to the case where Trading Inc. has
a long forward position. Such a firm cares about the value of its positions and the change in their
value over a short internal of time, say one day. If the position loses value over the next day, the firm
Chapter 6, page 7
can change the risks it takes. Suppose that the firm and its creditors are willing to tolerate at most
a 5% chance of insolvency. We saw in chapter 4 that such a firm would want to compute a VaR
measure.
To compute the VaR, one has to have the market value of the long SFR position. Because
the return of the US T-bill position is not random with constant interest rates, its value does not
matter for the VaR! The current exchange rate is $0.75 per SFR and the price of a 3-month discount
bond in SFR is SFR0.96. In this case, the current dollar value of the SFR payoff is 0.75*0.96x1M =
$720,000. Over the next day, the expected change in the exchange rate is trivial and can be ignored.
The exposure of Trading Inc. to the price of the SFR tomorrow is 960,000, or 0.96*1M since an
increase in the price of the SFR tomorrow by x increases the value of the position of Trading Inc. by
960,000*x. To get the volatility of the exchange rate change over one day, we use the square root
rule. Let’s use 21 trading days per month. This means that the volatility over one day is the square
root of 1/21 times the monthly volatility. We therefore have a daily exchange rate volatility given by
/(1/21)*0.0251107, which is 0.0054816. If it were to liquidate the position today, Trading Inc.
would get SFR960,000, which it then multiplies by the exchange rate to get the dollar value of the
position. The daily dollar volatility is therefore 960,000*0.0054816 = $5,262.3. Let’s assume that the
daily change in the dollar price of the SFR is normally distributed. There is therefore a 5% chance that
the position will lose at least 1.65*5,262.3 = $8,683.
Remember that Trading Inc. finances c% of its position at inception. Suppose that c% is equal
to 99.5%. In this case, Trading Inc. has equity on March 1 corresponding to 0.005*$720,000. This
amounts to $3,600. Consequently, if the VaR is $8,683, there is more than a 5% chance that the
equity of Trading Inc. could be wiped out in one day. This means that Trading Inc. has more risk than
Chapter 6, page 8
it wants. It can reduce that risk by scaling down its position, by hedging, or by increasing its equity
so that it requires less financing. In this chapter, we focus on hedging as a way to solve this problem.
Note again that, because the return of the short U.S. T-bill position is nonstochastic, it does not affect
the VaR. Consequently, the VaR of Trading Inc. is the same whether it finances all or nothing of its
position in SFR T-bills. Hence, even though a forward contract has no value at inception, it has a
positive VaR because there is a risk of loss. In this case, a forward contract to buy SFR1M on June
1 has a daily VaR of $8,683.
Let’s go back to our formula for the replicating portfolio of a forward contract to check that
we computed the VaR correctly. Remember that the value of the forward contract at inception is
equal to the present value of SFR T-bills with face value equal to the amount of SFR bought forward
minus the present value of US T-bills with same maturity for a present value equal to the present
value of the forward price per SFR times the amount of SFR bought forward:
Value of forward at inception = S(March 1)*PSFR(June 1)*1M - F*P(June 1)*1M
where P(June 1) (PSFR(June 1)) is the March 1 value of a discount bond that pays $1 (SFR1) on June
1. The only random variable in this expression is the spot exchange rate. Hence, the volatility of the
change in value is:
Volatility of one day change in value =
Vol[S(March 2)*PSFR(March 2, June 1)*1M - S(March 1)*PSFR(June 1)*1M] =
PSFR(June 1)*Vol[One day change in spot exchange rate]*1M
Chapter 6, page 9
where PSFR(March 2, June 1) is the March 2 price of a discount bond that pays SFR1 on June 1. We
neglect the change in value of the discount bond over one day in the last line since it is small. Since
the change in the exchange rate is normally distributed, the VaR of the forward contract is 1.65 times
the volatility of the change in value of the position.
Both Export Inc. and Trading Inc. have taken on too much risk. In the case of Export Inc.,
the CaR over three months is computed as:
Cash flow at risk over three months =
1.65*Cash flow volatility =
1.65*Exchange rate change volatility over three months*1M
= 1.65*0.00435086*1M
= $71,789.2
In the case of Trading Inc., the relevant measure of risk is VaR over one day and it is computed as:
Value at Risk over one day
= 1.65*Volatility of value of position over one day
= 1.65*Exchange rate volatility over one day*SFR value of position
= 1.65*0.0054816*960,000
= $8,683
In this case, both CaR and VaR depend on the exchange rate volatility linearly. If the exchange rate
Chapter 6, page 10
volatility doubles, VaR and CaR double. This is because the only source of risk for both firms is the
price of the SFR. This exposure is fixed in that the firms receive a fixed amount of SFRs at a future
date. We call the price of the SFR the risk factor for our analysis. Whenever the exposure is a
constant times the risk factor and the distribution of the risk factor is normal, volatility, CaR, and VaR
depend linearly on the volatility of the risk factor.1
Section 6.2. Hedging in the absence of basis risk.
Consider now the situation of Export Inc. and Trading Inc. These firms have to figure out
how to reduce their risk because they have taken on too much risk. In this section, we explore how
these firms can eliminate risk using the forward market, the money market, and the futures market:
1. The forward contract solution. Suppose that Export Inc. sells its exposure to the risk
factor, SFR1M, forward on March 1. Let’s simplify our notation and denote the forward price today
on a forward contract that matures in three months by F. The forward contract is a derivative that
pays F - S(June 1) per unit of SFR sold forward, so that Export Inc. will have on June 1:
Cash position + payoff of forward position =
S(June 1)*1M + [F*1M - S(June 1)*1M] = F*1M
Since the forward price is known today, there is no risk to the hedged position as long as there is no
1
The proof is straightforward. The payoff is Constant*Risk factor. The volatility of the
payoff is Constant*Volatility of risk factor. For CaR, the payoff is cash flow, so that the CaR is
1.65*Constant*Volatility of risk factor. For VaR, we have that the VaR is
1.65*Constant*Volatility of risk factor. In all three cases, the risk measure is a constant times the
volatility of the risk factor.
Chapter 6, page 11
counterparty risk with the forward contract. In the absence of counterparty risk, the dollar present
value of the hedged SFRs is P(June 1)*F*1M. We define a hedge to be a short position in a hedging
instrument put on to reduce the risk resulting from the exposure to a risk factor. Here, the hedge is
SFR1M since we go short SFR1M on the forward market. We can also compute the hedge for a one
unit exposure to a risk factor. Computed this way, the hedge is called the hedge ratio. In our
example, the hedge ratio is one because the firm goes short one SFR forward for each SFR of
exposure.
Consider now the impact of a forward hedge on our risk measures. We can compute the
volatility of the hedged payoff. When the firm goes short SFR1M forward for delivery on June 1 at
a price of F per unit, the volatility of the hedged payoff is equal to zero:
Volatility of hedged payoff = Vol($ value of SFR payment - Forward position payoff)
= Vol(S(June 1)*1M - (S(June 1) - F)*1M)
= Vol(F)
=0
Since the hedged payoff is the hedged cash flow, going short the exposure makes the CaR equal to
zero:
CaR of hedged payoff = 1.65*Vol(Hedged payoff)
=0
Chapter 6, page 12
Further, since the value of the position is the present value of the hedged payoff, this hedge makes
the VaR equal to zero:
VaR of hedged position = 1.65*Vol(PV(Hedged payoff))
= 1.65*Vol(P(June 1)*F*1M)
=0
In the context of this example, therefore, the same hedge makes all three risk measures equal to zero.
This is because all three risk measures depend linearly on the volatility of the same risk factor when
the risk factor has a normal distribution.
Though a three-month forward contract eliminates all risk for Trading Inc., Trading Inc. cares
only about the risk over the next day. With constant SFR interest rates, Trading Inc. has a
nonstochastic exposure of SFR960,000 tomorrow. Consequently, a one-day forward contract
eliminates all risk over that day also. This means that if Trading Inc. goes short SFR960,000 on the
forward market, its one-day VaR becomes zero. With perfect financial markets, it does not matter
which solution Trading Inc. chooses since both solutions achieve the same objective of eliminating
the one-day VaR risk and no transaction costs are involved. If SFR interest rates are random, the
exposure tomorrow is random because the present value of SFR1M to be paid on June 1 is random
while the exposure on June 1 is certain. In this case, the three-month forward contract eliminates all
risk for Trading Inc. while the one-day forward contract does not. We address the issue of hedging
a random exposure in chapter 8.
It is important to understand that using a succession of one-day forward contracts cannot
Chapter 6, page 13
make the CaR of Export Inc. equal to zero if interest rates change randomly over time. This is
because, in this case, the one-day forward exchange rate is not a deterministic fraction of the spot
exchange rate. For instance, if the SFR interest rate increases unexpectedly, the forward exchange
rate falls in relation to the spot exchange rate. Consequently, the amount of dollars Export Inc. would
receive on June 1 using one-day forward contracts would be random, since it would depend on the
evolution of interest rates between March 1 and June 1.
2. The money market solution. We know that there is a strategy that replicates the forward
contract with money market instruments. In three months, Export Inc. receives SFR1M. The firm
could borrow in SFR the present value of SFR1m to be paid in 90 days. In this case, Export Inc. gets
today PSFR(June 1)*1M in SFRs. These SFRs can be exchanged for dollars at today’s spot exchange
rate, S(March 1). This gives Export Inc. S(March 1)*PSFR(June 1)*1m dollars today. We know from
the forward pricing formula that S(March 1)*PSFR(June 1) is equal to P(June 1)*F if the forward
contract is priced so that there are no arbitrage opportunities.
3. The futures solution. Suppose that there is a futures contract that matures on June 1. In
this case, the futures contract’s maturity exactly matches the maturity of the SFR exposure of Export
Inc. In the absence of counterparty risk, going short the exposure eliminates all risk with the forward
contract. This is not the case with the futures contract. Remember that futures contracts have a daily
settlement feature, so that the firm receives gains and losses throughout the life of the contract. In
chapter 5, we saw that we can transform a futures contract into the equivalent of a forward contract
by reinvesting the gains at the risk-free rate whenever they occur and borrowing the losses at the riskfree rate as well. The appropriate risk-free rate is the rate paid on a discount bond from when the gain
or loss takes place to the maturity of the contract. Since a gain or loss made today is magnified by
Chapter 6, page 14
the interest earned, we need to adjust our position for this magnification effect. Let’s assume that
interest rates are constant for simplicity. A gain (loss) incurred over the first day of the futures
position can be invested (borrowed) on March 2 until June 1 if one wants to use on June 1 to offset
a loss (gain) on the cash position at that date. This means that over the next day the firm has to only
make the present value of the gain or loss that it requires on June 1 to be perfectly hedged. Therefore,
the futures position on March 1 has to be a short position of SFR 1M*P(June 1).
Remember that $1 invested in the discount bond on March 1 becomes 1/P(June 1) dollars on
June 1. Any dollar gained on March 1 can be invested until June 1 when the gain made the first day
plus the interest income on that gain will correspond to what the gain would have been on a futures
position of SFR1M:
Gain on June 1 from change in value of futures position on March 1
= Change in value of futures position on March 1/P(June 1)
= P(June)*1M*Change in futures price of 1 SFR on March 1/P(June 1)
= Change in futures price of 1 SFR on March 1*1M
This change in the hedge to account for the marking to market of futures contracts is called tailing
the hedge. Remember that the forward hedge is equal to the exposure to the risk factor. To obtain
the futures hedge, we have to take into account the marking-to-market of futures contracts by tailing
the hedge. To tail the hedge, we multiply the forward hedge on each hedging day by the present value
on the next day (when settlement takes place) of a dollar to be paid at maturity of the hedge. The
tailed hedge ratio for a futures contract is less than one when the hedge ratio for a forward contract
Chapter 6, page 15
is one. Remember from the chapter 5 that the March 1 price of a T-bill that matures on June 1 is
$0.969388. Therefore, the tailed hedge consists of selling short SFR969,388 for June 1 even though
the exposure to the SFR on June 1 is SFR1M. Note now that on the next day, March 2, the gain on
the futures position can only be invested from March 3 onwards. Hence, tailing the hedge on March
2 involves multiplying the exposure by the present value on March 3 of a dollar to be paid on June
1. This means that the tailed hedge changes over time. For constant interest rates, the tailed hedge
increases over time to reach the exposure at maturity of the hedge. With this procedure, the sum of
the gains and losses brought forward to the maturity of the hedge is equal to the change in the futures
price between the time one enters the hedge and the maturity of the hedge. Technical Box 6.2. Proof
by example that tailing works provides an example showing that this is indeed the case.
Through tailing, we succeed in using a futures contract to create a perfect hedge. Tailing is
often neglected in practice. Our analysis makes it straightforward to understand when ignoring tailing
can lead to a large hedging error. Consider a hedge with a short maturity. In this case, the present
value of a dollar to be paid at maturity of the hedge is close to $1. Hence, tailing does not affect the
hedge much and is not an important issue. Consider next a situation where a firm hedges an exposure
that has a long maturity. Say that the firm wants to hedge a cash flow that it receives in ten years. In
this case, the present value of a dollar to be paid at maturity of the hedge might be fifty cents on the
dollar, so that the optimal hedge would be half the hedge one would use without tailing. With a hedge
ratio of one, the firm would effectively be hedging more exposure than it has. This is called overhedging.
Let’s consider an example where over-hedging occurs. Suppose that Export Inc. receives the
SFR1M in ten years and does not tail the hedge. Let’s assume that the futures price is the same as the
Chapter 6, page 16
ten-year forward price on March 1, which we take to be $0.75 for the sake of illustration, and that
the present value of $1 to be received in ten years is $0.50. Consider the hypothetical scenario where
the SFR futures price increases tomorrow by 10 cents and stays there for the next ten years. Without
tailing the hedge, Export Inc. goes short SFR1M on the futures market. In ten years, Export Inc. will
receive $850,000 by selling SFR1M on the cash market. However, it will have incurred a loss of 10
cents per SFR on its short futures position of SFR1M after one day. If Export Inc. borrows to pay
for the loss, in ten years it has to pay 0.10*1M*(1/0.50), or $200,000. Hence, by selling short
SFR1M, Export Inc. will receive from its hedged position $850,000 minus $200,000, or $650,000,
in comparison to a forward hedge where it would receive $750,000 in ten years for sure. This is
because its futures hedge is too large. The tailed hedge of 0.5*SFR1M would have been just right.
With that hedge, Export Inc. would receive $750,000 in ten years corresponding to $850,000 on the
cash position minus a loss of $100,000 on the futures position (an immediate loss of $50,000 which
amounts to a loss of $100,000 in ten years). The greater the exchange rate at maturity, the greater
the loss of Export Inc. if it goes short and fails to tail its hedge. Hence, the position in futures that
exceeds the tailed hedge Export Inc. should have represents a speculative position. Over-hedging
when one is long in the cash market amounts to taking a short speculative position.
Note that in addition to affecting the size of the position, the marking-to-market feature of
the futures contract creates cash flow requirements during the life of the hedge. With a forward
hedge, no money changes hands until maturity of the hedge. Not so with the futures hedge. With the
futures hedge, money changes hands whenever the futures price changes. One must therefore be in
a position to make payments when required to do so. We will talk more about this issue in the next
chapter.
Chapter 6, page 17
The position in the futures contract maturing on June 1 chosen by Export Inc. would eliminate
the risk of Trading Inc. also. With constant interest rates, a position in a futures contract maturing
tomorrow would also eliminate the VaR risk of Trading Inc. for the same reason that a position in
a one-day forward contract would eliminate the VaR risk of Trading Inc. There is no difference
between a futures contract that has one day to maturity and a one-day forward contract.
Section 6.3. Hedging when there is basis risk.
We assumed so far that the SFR futures contract matures when Export Inc. receives the SFR
cash payment. Remember that if the contract matures when Export Inc. receives the SFRs, the futures
price equals the spot price on that date. This is no longer true if the futures contract does not mature
on June 1. Suppose that there is no contract maturing on June 1 available to Export Inc., so that it
uses a futures contract maturing in August instead. With this contract, Export Inc. closes the futures
position on June 1. Since Export Inc. receives the cash payment it hedges on June 1, the adjustment
factor for tailing the hedge is the price of a discount bond that matures on June 1, the maturity date
of the hedge, and not the price of a discount bond that matures when the futures contract matures.
In this case, assuming a tailed hedge, the payoff of the hedged position on June 1 is:
Payoff of cash position + Payoff of hedge
= 1M*S(June 1) - 1M*[G(June 1) - G(March 1)]
= 1M*G(March 1) + 1M*{e(June 1) - G(June 1)}
where G(March 1) is the futures price when the position is opened on March 1. The term in curly
Chapter 6, page 18
brackets is the basis on June 1. The basis is usually defined as the difference between the spot price
and the futures price on a given date.2 There is basis risk if the value of the cash position at maturity
of the hedge is not a deterministic function of the futures price at maturity. With our example, there
would be no basis risk if the futures contract were to mature when Export Inc. receives the SFR1M
since, in that case, the term in curly brackets is identically equal to zero and the value of the cash
position is the futures price times 1M. The lack of a contract with the right maturity is not the only
reason for the existence of basis risk, however. Suppose that, for the sake of illustration, there is no
SFR contract available and that Export Inc. has to use a Euro futures contract instead. The futures
strategy would have basis risk in this case even if the Euro contract matures when Export Inc.
receives the SFRs since on that date the difference between the price of the Euro and the price of the
SFR is random. Consequently, a hedge can have basis risk because the available contract has the
wrong maturity or the wrong deliverable good.
We will now show that when the spot price is a constant plus a fixed multiple of the futures
price, a perfect hedge can be created. In this case, the basis is known if the futures price is known.
The method used to find the hedge when the spot price is a linear function of the futures price will
be useful when there is basis risk also. Let’s assume that the futures price and the spot exchange rate
are tied together so that for any change in the futures price of x, the spot exchange rate changes
exactly by 0.9*x. This means that if we know the change in the futures price from March 1 to June
1, we also know the change in the spot exchange rate and, hence, we know the change in the basis
(it is -0.1x since the change in the basis is the change in the spot exchange rate minus the change in
2
This is called more precisely the spot-futures basis. Authors also sometimes call the basis
the difference between the futures price and the spot price, which is called the futures-spot basis.
Chapter 6, page 19
the futures price).
To eliminate the risk caused by its exposure to the SFR, Export Inc. wants a hedge so that
the firm is unaffected by surprises concerning the SFR exchange rate. This means that it wants the
unexpected change in the value of its futures position to offset exactly the unexpected change in the
value of its cash market position. The value of Export Inc.’s cash market position on June 1 is
SFR1M*S(June 1). With our assumption about the relation between the futures price and the spot
exchange rate, an unexpected change of x in the futures price is associated with a change in the value
of the cash position of SFR1M*0.9*x. The exposure of the cash position to the futures price is
therefore SFR futures 1M*0.9, or SFR futures 900,000, since a change in the futures price of x is
multiplied by 900,000 to yield the change in the cash position. Another way to put this is that the
change in value of the cash position is exactly equivalent to the change in value of a futures position
of SFR900,000. Suppose that Export Inc. goes short h SFRs on the futures market. In that case, the
impact of an unexpected change in the futures price of x and of the associated change in the spot
exchange rate of 0.9*x on the value of its hedged position is equal to the impact of the spot exchange
rate change on the cash position, 0.9*1M*x, minus the impact of the change in the futures price on
the value of the futures position, - h*x. For h to eliminate the risk associated with Export Inc.’s
exposure to the SFR, it has to be that 0.9*1M*x - h*x is equal to zero irrespective of the value of x.
Consequently, h has to be equal to a short futures position of SFR900,000. With h = 900,000, we
have:
Unexpected change in payoff of hedged position
= 0.9*1M*x - h*x = 0.9*1M*x - 900,000*x = 0
Chapter 6, page 20
With h = 900,000, the volatility of the payoff of the hedged position is zero, so that a short futures
position equal to the exposure of the cash position to the futures price is the volatility-minimizing
hedge. Since the exposure of Export Inc. to the futures price is 900,000, Export Inc. goes short
SFR900,000 futures. Had Export Inc. gone short its exposures to the risk factor instead of its
exposure to the futures price, a change in the futures price of x would change the cash market
position by 0.9*1M*x dollars and would change the futures position by -1M*x dollars, so that the
value of the hedged position would change by -0.1*1M*x dollars. Hence, if the firm went short its
exposure to the risk factor on the futures market, the value of its hedged position would depend on
the futures price and would be risky. In contrast, the hedged position has no risk if the firm goes short
its exposure to the futures price.
In the absence of basis risk, we can go short a futures contract whose payoff is perfectly
correlated with the risk factor, so that the optimal hedge is to go short the exposure to the risk factor
as we saw in the previous section. In the presence of basis risk, we cannot go short the risk factor
because we have no hedging instrument whose payoff is exactly equal to the payoff of the risk factor.
In our example, the risk factor is the SFR on June 1, but we have no way to sell short the SFR on
June 1. Consequently, we have to do the next best thing, which here works perfectly. Instead of going
short our exposure to the risk factor, we go short our exposure to the hedging instrument we can use.
Here, the hedging instrument is the futures contract. Our exposure to the futures price is SFR900,000
in contrast to our exposure to the risk factor which is SFR1M. Since we are using the futures contract
to hedge, the best hedge is the one that makes our hedged position have an exposure of zero to the
futures price. If the exposure of the hedged position to the futures price were different from zero, we
would have risk that we could hedge since our hedged position would depend on the futures price.
Chapter 6, page 21
In the above analysis, the optimal hedge is equal to the hedge ratio, which is the futures hedge
per unit of exposure to the risk factor, 0.9, times the exposure to the risk factor, 1M, or a short
futures position of SFR900,000. Suppose the exposure to the risk factor had been SFR100M instead.
In this case, the hedge would have been h = 0.9*100M, or a short position of SFR90M. Generally,
for a given exposure to a risk factor, the hedge is the hedge ratio times the exposure. The hedge ratio
would be the same for any firm hedging an SFR exposure in the same circumstances as Export Inc.
even though each firm would have a different hedge because the size of its exposure would be
different. In other words, if there was another firm with an exposure of 100M, that firm could use the
same hedge ratio as Export Inc., but its hedge would be a short SFR futures position of 0.9*100M.
The hedge can be computed as the product of the hedge ratio and the exposure to the risk factor
whenever the exposure to the risk factor is known with certainty at the start of the hedging period.
Here, the firm knows how many SFRs it will receive on June 1. In this chapter, we assume the
exposure to the risk factor is known with certainty. We can therefore compute the hedge ratio and
then multiply it by the exposure to get the actual hedge. To compute the hedge ratio, we have to
analyze the relation between the changes in the spot exchange rate and the futures price since the
hedge ratio is the exposure to the futures price of one unit of exposure to the risk factor, but we can
ignore the size of the firm’s exposure to the spot exchange rate.
Suppose that the relation in the past between changes in the spot exchange rate and changes
in the futures price over periods of same length as the hedging period is the same as what it is
expected to be during the hedging period. We can then plot past changes in the spot exchange rate
against past changes in the futures price. For simplicity, we assume that these changes have an
expected value of zero, so that they are also unexpected changes. If their expected value is not zero,
Chapter 6, page 22
these changes can be transformed into unexpected changes by subtracting their expected value. If
there is an exact linear relation between these changes, the plot would look like it does in Figure 6.1.
Figure 6.1. shows the case where the spot exchange rate changes by 0.9x for each change in the
futures price of x. (The plot uses simulated data which satisfy the conditions of our example.)
Because the change in the spot exchange rate is always 0.9 times the change in the futures price, all
the points representing past observations of these changes plot on a straight line.
With basis risk, things are more complicated since the change in the futures price is randomly
related to the change in the spot exchange rate. Let’s assume again that the distribution of changes
in the spot exchange rate and the futures price is the same in the past as it will be over the hedging
period. We assume that the futures contract is the only hedging instrument available. Suppose,
however, that the relation that holds exactly in Figure 6.1. no longer holds exactly. Instead of having
past observations plot along a straight line as in Figure 6.1., they now plot randomly around a
hypothetical straight line with a slope of 0.9 as shown in Figure 6.2. (using simulated data). It is
assumed that for each realization of the change in the futures price, the change in the spot exchange
rate is 0.9 times that realization plus a random error that is not correlated with the futures price.3 This
random error corresponds to the basis risk with our assumptions, since it is the only source of
uncertainty in the relation between the spot exchange rate and the futures price.
With our change in assumptions about the relation between the spot price and the futures
price, we can no longer know for sure what the spot price will be given the futures price as we could
3
We are therefore assuming a linear relation between changes in the spot exchange rate
and the futures price. The relation does not have to be linear. For instance, the basis could be
smaller on average when the futures price is high. We will see later how the analysis is affected
when the relation is not linear, but in this chapter we stick to a linear relation.
Chapter 6, page 23
with our example in Figure 1. The best we can do is to forecast the spot price assuming we know the
futures price. Since the spot price is 0.9 times the futures price plus noise that cannot be forecasted,
our best forecast of the exposure of Export Inc. to the futures price is 0.9*1M, or 0.9 per unit of
exposure to the risk factor. In this case, 0.9*1M is our forecasted exposure to the futures price.
Export Inc.’s volatility-minimizing hedge is to go short the forecasted exposure to the hedging
instrument because with this hedge the unexpected change in the cash position is expected to be equal
to the unexpected change in the futures position plus a random error that cannot be predicted and
hence cannot affect the hedge. If the futures price changes unexpectedly by x, the expected change
in the value of the cash position of Export Inc. per unit exposure to the risk factor is 0.9*x plus the
random error unrelated to the futures price. The expected value of the random error is zero, so that
the futures gain of 0.9*x is expected to offset the cash position loss per SFR but does so exactly only
when the random error is zero.
Let’s now examine why, in the presence of basis risk, we cannot find a hedge that decreases
the volatility of the hedged cash flow more than a hedge equal to the forecasted exposure of the cash
position to the futures price. We assume that Export Inc. knows the forecasted exposure of the spot
exchange rate to the futures price. Let )S be the change in the spot exchange rate and )G be the
change in the futures price over the hedging period. The hedged cash flow per SFR is therefore:
Hedged cash flow
= Cash market position on March 1 + Change in value of cash position + Hedge position payoff
= S(March 1) + [S(June 1) - S(March 1)] - h*[G(June 1) - G(March 1)]
= S(March 1) + )S - h*)G
Chapter 6, page 24
We know, however, that the change in the cash market position per SFR is forecasted to be 0.9 times
the change in the futures price. Using this relation, we get the following expression for the hedged
cash flow per unit of exposure to the SFR:
Hedged cash flow when forecasted exposure of SFR to futures price is 0.9
= 0.9*)G + Random Error - h*)G + S(March 1)
= (0.9 - h)*)G + Random Error + S(March 1)
Consequently, the hedged cash flow is risky for two reasons when the forecasted exposure of the cash
position to the futures price is known. If the hedge differs from the forecasted exposure, the hedged
cash flow is risky because the futures price is random. This is the first term of the last line of the
equation. Irrespective of the hedge, the hedged cash flow is risky because of the random error which
represents the basis risk. To get the volatility of a sum of random variables, we have to take the
square root of the variance of the sum of random variables. With this argument, therefore, the
volatility of the hedged cash flow per unit exposure to the SFR is:
Volatility of hedged cash flow per unit exposure to the SFR
= {(0.9 - h)2*Var()G) + Var(Random Error)}0.5
Remember that the volatility of k times a random variable is k times the volatility of the random
variable. Consequently, the volatility of Export Inc.’s hedged cash flow is simply 1M times the
volatility of the hedged cash flow per unit of SFR exposure. There is no covariance term in the
Chapter 6, page 25
expression for the volatility of the hedged cash flow because the random error is uncorrelated with
the change in the futures price, so that the two sources of risk are uncorrelated. Setting the hedge
equal to the expected exposure to the futures price of 0.9*1M, we eliminate the impact of the
volatility of the futures price on the volatility of the hedged cash flow. There is nothing we can do
about the random error with our assumptions, so that the volatility of hedged cash flow when the
hedge is 0.9*1M is minimized and is equal to the volatility of the random error. Hence, going short
the forecasted exposure to the hedging instrument is the volatility-minimizing hedge ratio.
So far, we assumed that Export Inc. knows the forecasted exposure of the risk factor to the
hedging instrument. Suppose instead and more realistically that it does not know this forecasted
exposure but has to figure it out. All Export Inc has available is the information contained in Figure
6.2. In other words, it knows that the distribution of past changes in the spot exchange rate and the
futures price is the same as the distribution of these changes over the hedging period and has a
database of past observations of these changes. Therefore, Export Inc. must use the data from Figure
6.2. to find the optimal hedge.
To minimize the volatility of its hedged cash flow, Export Inc. wants to find a position in
futures such that the unexpected change in value of that position matches the unexpected change in
value of its cash position as closely as possible. (Remember that expected changes, if they differ from
zero, do not contribute to risk, so that to minimize risk one has to offset the impact of unexpected
changes in the cash position.) This turns out to be a forecasting problem, but not the usual one. The
usual forecasting problem would be one where, on March 1, Export Inc. would try to forecast the
change in the futures price and the change in the value of the cash position from March 1 to June 1.
Here, instead, Export Inc. wants to forecast the change in the value of the cash position from March
Chapter 6, page 26
1 to June 1 given the change in the futures price from March 1 to June 1. Per unit of exposure to the
risk factor, this amounts to finding h so that h times the unexpected change in the futures price from
March 1 to June 1 is the best forecast of the unexpected change in the value of the spot exchange rate
from March 1 to June 1 in the sense that the forecasting error has the smallest volatility. The
volatility-minimizing hedge is then h per unit of exposure to the risk factor. If the change in the
futures price is useless to forecast the change in value of the risk factor, then h is equal to zero and
the futures contract cannot be used to hedge the risk factor. This is equivalent to saying that for the
futures contract to be useful to hedge the risk factor, changes in the futures price have to be
correlated with changes in the risk factor.
We already know how to obtain the best forecast of a random variable given the realization
of another random variable. We faced such a problem in chapter 2. There, we wanted to know how
the return of IBM is related to the return of the stock market. We used the S&P500 as a proxy for
the stock market. We found that the return of IBM can be expected to be IBM’s beta times the return
of the market plus a random error due to IBM’s idiosyncratic risk. If past returns are distributed like
future returns, IBM’s beta is the regression coefficient in a regression of the return of IBM on the
return of the market. To obtain beta using an ordinary least squares regression (OLS), we had to
assume that the return of IBM and the return of the S&P500 were i.i.d. Beta is the forecast of the
exposure of the return of IBM to the return of the market.
If the past changes in the spot exchange rate and the futures price have the same distribution
as the changes over the hedging period, the forecasted exposure and the volatility-minimizing hedge
ratio is the regression coefficient of the change in the spot price on a constant and the change in the
futures price over past periods. Export Inc. can therefore obtain the hedge ratio by estimating a
Chapter 6, page 27
regression. Figure 6.3. shows the regression line obtained using data from Figure 6.2. If we denote
the change in the spot exchange rate corresponding to the i-th observation by )S(i) and the change
in the futures price by )G(i), Export Inc. estimates the following regression:
)S(i) = Constant + h*)G(i) + ,(i)
(6.1.)
where ,(i) is the regression error or residual associated with the i-th observation. A negative value
of the residual means that the change in the exchange rate is lower than the predicted change
conditional on the change in the futures price. Consequently, if we are short the futures contract and
the futures price falls, the gain we make on the futures contract is insufficient to offset the loss we
make on our cash position. Figure 6.3. shows the regression line obtained by Export Inc. using the
100 observations of Figure 6.2. The slope of this regression line gives Export Inc.’s estimate of the
exposure to the futures price per SFR and hence its hedge ratio. If the expected changes are different
from zero, this does not matter for the regression coefficient. It would only affect the estimate of the
constant in the regression. The estimate of the constant will be different from zero if the expected
value of the dependent variable is not equal to the coefficient estimate times the expected value of
the independent variable. Consequently, our analysis holds equally well if expected changes are
different from zero, so that the regression would be mis-specified without a constant term. From now
on, therefore, we do not restrict expected changes to be zero.
The regression coefficient in a regression of changes in the spot exchange rate on a constant
and on changes in the futures price is our volatility-minimizing hedge for one unit of exposure. The
formula for the volatility-minimizing hedge for one unit of exposure is therefore:
Chapter 6, page 28
Volatility-minimizing hedge for one unit of exposure to the risk factor
h=
Cov( ∆S, ∆G)
Var( ∆G)
(6.2.)
If the exposure to the risk factor is 1M, we can obtain the optimal hedge by multiplying the optimal
hedge per unit of exposure to the risk factor by 1M. Alternatively, we can note that the covariance
in the formula would be multiplied by 1M if we regressed changes in the value of the firm’s cash
position on the futures price changes since the numerator of the hedge formula would be
Cov(1M*)S,)G) or 1M*Cov(1M*)S,)G), which would be the covariance of an exposure of 1M
to the risk factor with the futures price, and the denominator would be unaffected. Hence, in this case,
we would get a hedge equal to 1M times the hedge per unit exposure.
To obtain our formula for the volatility-minimizing hedge ratio, we assumed that the relation
between the change in the futures price and the change in the spot exchange rate is linear and that we
could use regression analysis to estimate that relation. To estimate a regression, we need a database
of past changes for the exchange rate and the futures price and we need the distribution of these past
changes to be the same as the distribution of the changes over the hedging period. We now show that
the formula for the volatility-minimizing hedge ratio holds even if we cannot estimate a regression
and there is no linear relation between the change in the futures price and the change in the spot
exchange rate. This is because the volatility-minimizing hedge has to be such that the hedged cash
flow has to be uncorrelated with the change in the futures price over the hedging period. To see this,
suppose that Export Inc. picks a hedge ratio of 0.8 instead of 0.9. In this case, the hedge position is
too small. If the SFR futures price falls by x, Export Inc. loses 0.9x on its cash position but gains only
Chapter 6, page 29
0.8x on its futures position per SFR of cash position. Hence, Export Inc. loses money when the
futures price falls and makes money when the futures price increases. To reduce the volatility of its
hedged cash flow, Export Inc. would therefore want to increase its hedge ratio. However, if it
increases its hedge ratio so that it exceeds 0.9, then the opposite happens. In this case, it expects to
lose money when the futures price increases and to make money when the futures price falls. The only
case where Export Inc. does not expect to make either a gain or a loss when the futures price
unexpectedly increases is when it chooses the hedge ratio given by h in equation (6.2.). All other
hedge ratios have the property that the hedged cash flow is more volatile and is correlated with the
futures price.
Let’s see what a choice of h that makes the hedged cash flow uncorrelated with the change
in the futures price implies. To do this, we compute the covariance of the hedged cash flow per unit
of cash position with the futures price and set it equal to zero:
Cov(Hedged cash flow per unit of cash position, )G)
= Cov()S - h*)G + e(March 1), )G)
= Cov()S,)G) - h*Var()G)
Setting the covariance of the hedged cash flow with the futures price equal to zero, we get the hedge
ratio of equation (6.2.). However, we did not use a regression to obtain the hedge ratio this way. This
volatility-minimizing hedge ratio is the ratio of the covariance of the change in the spot exchange rate
with the change in the futures price divided by the variance of the change in the futures price. As long
as we know the distribution of the change of the spot exchange rate and the change of the futures
Chapter 6, page 30
price over the hedging period, we can compute the optimal hedge ratio since we could compute the
covariance and the variance from the distribution of the changes.
Though we derived the hedge ratio for the case where the cash price is an exchange rate,
nothing in our derivation limits it to the case where the cash price is an exchange rate. To see this,
note that we can write the hedged cash flow as:
General formula for hedged cash flow
Hedged cash flow =
Cash position at maturity of hedge h*[Futures price at maturity of hedge - Futures price at origination of hedge]
(6.3.)
The covariance of the hedged cash flow with the futures price has to be zero. A non-zero covariance
creates risk since it implies that when the futures price changes unexpectedly, there is some
probability that the hedged cash flow does too. We can reduce this risk by changing our futures
position to eliminate the covariance. In computing this covariance, we can ignore the futures price
at origination of the hedge since it is a constant which does not affect the covariance. Consequently,
we require that:
Cov(Cash position at maturity of hedge - h*G(Maturity of hedge), G(Maturity of hedge))
= Cov(Cash position at maturity of hedge, G(Maturity of hedge)) - h*Var(G(Maturity of hedge))
=0
Chapter 6, page 31
where G(Maturity of hedge) denotes the futures price of the contract used for hedging at the maturity
of the hedge. Solving for the optimal volatility-minimizing hedge, we get:
General formula for volatility-minimizing hedge of arbitrary cash position
h=
Cov(Cash position at maturity of hedge, G(Maturity of hedge))
Var(G(Maturity of hedge))
(6.4.)
This formula is the classic formula for the volatility-minimizing hedge. To hedge an arbitrary cash
position, we go short h units of the hedging instrument. In Technical Box 6.3. Deriving the
volatility-minimizing hedge, we provide a mathematical derivation of the volatility-minimizing
hedge. The hedge obtained in equation (6.2.) is a special case of the hedge obtained in equation (6.4.).
If the cash flow in equation (6.4.) is one unit exposure to the SFR, we get the hedge in equation
(6.2.). Importantly, we made no assumption about exposure being constant to obtain (6.4.). As a
result, this formula holds generally for any random cash flow we wish to hedge. Further, we derived
equation (6.4.) without making the assumption that there is a linear relation between the change in
value of the cash position and the change in the futures price. This assumption is not needed to obtain
the formula for the hedge, but it is needed if one wants to obtain the hedge from a regression of
changes in the cash position on changes in the futures price. This means that the formula for the
hedge obtained in equation (6.4.) is extremely general. However, generality in a hedging formula is
not as important as the ability to implement the formula. In the rest of this chapter, we focus on the
implementation of the hedge ratio when it is obtained from a regression.
Chapter 6, page 32
Though our analysis has focused on Export Inc.’s problem, there is no difference between the
problem of Export Inc. and the problem of Trading Inc. This is because both firms are minimizing the
volatility of the hedged position. The only difference is that for Trading Inc., the hedging period is
one day while it is three months for Export Inc. Because the exposure of Trading Inc. is the value of
its SFR position the next day, it is 960,000 instead of 1M. However, for each SFR it is exposed to
tomorrow, it sells short 0.9 futures if the exposure of the SFR spot exchange rate to the futures price
is 0.9 for a period of one day also.
Section 6.3.1. The optimal hedge and regression analysis.
With our assumptions, the optimal hedge is the regression coefficient in a regression of the
cash position increment on the futures price increment when the joint distribution of the unexpected
increments in the cash position and the futures price over periods of same length as the hedging
period are the same in the past as over the hedging period. This raises the question of whether the
length of the period over which we hedge matters. Specifically, Export Inc. hedges over three months
and Trading Inc. hedges over one day. We would like to know when these two firms can estimate
their hedge ratios using the same regression and when they use the same hedge ratio before tailing.
This will be the case as long as an unexpected futures price increment of $x has the same impact on
our forecast of the cash position increment irrespective of the time interval over which the unexpected
futures price increment takes place and irrespective of the sign or magnitude of $x. For instance, this
will be the case if we expect the cash position increment per unit to be 0.9 times the futures price
increment for any time interval over which the futures price increment takes place - in other words,
whether this time interval is one minute, two days, five weeks, or two months - and whether the
Chapter 6, page 33
futures price increment is $1 or -$10. We call this assumption here the assumption of a constant
linear relation between futures price and cash position increments. To insure that we can apply the
statistical analysis of linear regression without additional qualifications, we assume that over any time
interval of identical length, the futures price increments are independently identically normally
distributed, that the cash position increments are independently identically normally distributed, and
that there is a constant linear relation between the futures price increments and the cash position
increments. Technical Box 5.4. The statistical foundations of the linear regression approach to
obtaining the minimum-variance hedge ratio provides a detailed discussion of conditions that lead
to this assumption and what this assumption implies. We call this assumption the multivariate
normal increments model.
Let’s now implement the regression analysis. We want to find the optimal hedge ratio for
hedging a SFR cash position with a futures SFR contract. Let’s assume that the constant linear
relation assumption holds. To estimate the relation between futures price changes and cash price
increments, we can therefore use weekly increments over a period of time, the sample period. Let’s
assume that we use the shortest maturity contract to hedge. Regressing the weekly increment of the
SFR futures contract of shortest maturity on the weekly increment in the spot exchange rate from
9/1/1997 to 2/22/1999, we obtain (t-statistics in parentheses):
)S(i) = Constant + h*)G(i) + ,(i)
0.00
0.94
(0.03)
(32.91)
Chapter 6, page 34
(6.5.)
In other words, if the SFR futures price unexpectedly increases by one cent, the SFR spot exchange
rate increases by 0.94 cents. The regression coefficient for the futures price increment is estimated
with great precision. The regression suggests that the futures price and the spot price move very
closely together. To hedge one unit of spot SFR, one should short 0.94 SFRs on the futures market.
In our example in this chapter, we have an exposure to the risk factor of SFR1M. We should
therefore short SFR940,000.4
The regression equation we estimate is one where the dependent and independent variables
are increments. One might be tempted to use the spot exchange rate and the futures price instead.
This could yield highly misleading results. The reason for this is that a random variable with i.i.d.
increments follows a random walk. This means that while the increments have a well-defined mean,
the random variable itself does not. The exchange rate level does not have a constant mean and
neither does the level of the futures price. The expected value of a random variable with i.i.d.
increments increases over time if the increment has a positive expected value. Hence, the average of
the random variable depends on the period that we look at. Since the random variable trends up, the
longer the period over which we compute the average, the higher the average. We could therefore
easily find that two random variables which follow random walks are positively correlated because
they happen to trend up over a period of time and hence move together even though their increments
are uncorrelated. As an example, one might find that the average height of U.S. residents and the U.S.
national income per capita are correlated over long periods of time because both have positive time
trends. However, the unexpected change in the national income per capita for one year is uncorrelated
4
If you remember from chapter 5 that the size of the SFR contract is SFR125,000, please
note that for the moment, we abstract from the details of the contract and assume that we can
short SFR940,000.
Chapter 6, page 35
with the unexpected change in height for that year.
In our analysis, we used weekly increments in the SFR spot exchange rate and the SFR futures
price. To the extent that the relation between unexpected increments in the SFR spot exchange rate
and unexpected increments in the SFR futures price is constant and increments are i.i.d. irrespective
of the measurement interval, we could use any measurement interval we want. Hence, since more
observations would allow us to estimate the relation between the two random variables more
precisely, we might want to use shorter measurement intervals - days or even minutes. The problem
that arises as we use shorter intervals is that we get better estimates of the relation as long as the price
increments are measured accurately. Price data is noisy for a number of reasons. An important source
of noise is the existence of market imperfections, such as transaction costs. As a result, if the change
in price is small compared to the noise in the data, we end up getting poor measures of the relation.
This means that using very short measurement intervals might give poor results. At the same time,
however, using a long measurement interval means that we have few data points and imprecise
estimates. In highly liquid markets with small bid-ask spreads, it is reasonable to use daily data. In less
liquid markets, weekly or monthly data are generally more appropriate. Though we used weekly data
for our example, we would have obtained almost the same estimate using daily data (0.91), but the
estimate would have been more precise - the t-statistic using daily data is 52.48.
Section 6.3.2. The effectiveness of the hedge.
Since the optimal hedge is the slope coefficient in a linear regression, we can use the output
of the regression program to evaluate the effectiveness of the hedge. The R2 of a regression tells us
Chapter 6, page 36
the fraction of the variance of the dependent variable that is explained by the independent variable
over the estimation period. Hence, in our IBM example of chapter 2, the R2 of the regression
measures how much of the return of IBM is explained by the return of the market over the estimation
period. If IBM’s return is perfectly correlated with the return of the market, then the R2 is one. In
contrast, if the return of IBM is uncorrelated with the return of the market, then the R2 is zero.
Let’s look at our optimal hedge per unit of exposure to the SFR. To obtain it, we regress the
increments of the spot exchange rate on the increments of the futures price:
)S(i) = a + h*)G(i) + ,(i)
Taking the variance of the exchange rate increments, we have:
Var()S(i)) = h2Var()G(i)) + Var(,(i))
This equation has a regression interpretation as well as a portfolio interpretation. The regression
interpretation is that the variance of the dependent variable is equal to the square of the regression
slope coefficient times the variance of the independent variable plus the variance of the regression
residual. The ratio h2Var()G(i))/Var()S(i)) equals R2. Since h is the hedge ratio, this equation also
states that the variance of the cash position is equal to the variance of the hedge instrument plus the
variance of the unhedgeable payoff of the cash position. Hence, rearranging the equation, the
variance of the hedged position over the estimation period when the optimal hedge ratio is used is
equal to the variance of the regression residual in the regression of the spot exchange rate on the
Chapter 6, page 37
futures price:
Var()S(i)) - h2Var()G(i)) = (1 - R2)Var()S(i)) = Var(,(i))
This equation shows again that minimizing the variance of the forecasting error of the spot exchange
rate conditional on the futures price minimizes the variance of the hedged cash flow. If the regression
has an R2 of one, the hedged position has no variance. In contrast, if the R2 is equal to zero, the
variance of the hedged position equals the variance of the unhedged position. In a regression with a
single independent variable, the R2 equals the square of the correlation coefficient between the
dependent variable and the independent variable. Consequently, the effectiveness of the hedge is
directly related to the coefficient of correlation between the cash position and the futures contract
used to hedge. A hedging instrument with increments uncorrelated with the increments in the value
of the cash position is useless. Since the volatility is the square root of the variance, the square root
of R2 tells us the fraction of the volatility explained by the hedging instrument.
The R2 measures the fraction of the variance of the cash position we could have eliminated
had we used the optimal hedge during the estimation period. In the regression for the SFR
reproduced above, the R2 is equal to 0.934. This means that using the optimal hedge eliminates 93.4%
of the variance of the cash position. Usefully, R2 measures the effectiveness of the hedge irrespective
of the size of the exposure to the risk factor, so that Export Inc. would have been able to eliminate
93.4% of the variance of its cash position had it used the optimal hedge during the estimation period.5
5
To check this, let Var()SFR) be the variance for one unit of cash position. The hedged
variance is (1 - R2)Var()SFR), so that the ratio of hedged variance to unhedged variance is 1 R2. Suppose now that the exposure is n units instead of one unit. This means that we have n
Chapter 6, page 38
Note that the ratio of the variance of the hedged position and the variance of the unhedged position
is equal to one minus R2 . Taking the square root of the ratio gives us the ratio of the volatility of the
hedged position and the volatility of the unhedged position. Consequently, the volatility of the hedged
position as a fraction of the volatility of the unhedged position is the square root of one minus R2 .
Since R2 is 0.934, the square root of one minus R2 is the square root of 1 - 0.934, or 0.26.
Consequently, the volatility of the hedged position is 26% of the volatility of the unhedged position.
Through hedging, we therefore eliminated 74% of the volatility of the unhedged position.
Remember that we have assumed that the joint distribution of the cash position increments
and the futures price increments does not change over time. Hence, with that assumption, the
performance of the hedge during the estimation period is a good predictor of the performance of the
hedge over the period when the hedge ratio is implemented. In the case of our SFR example, the
optimal hedge ratio turns out to be about the same for the hedging and the estimation periods. If we
estimate our regression over the hedging period using weekly increments, the regression coefficient
is 0.94 also and the R2 is 0.983. Using the hedge ratio of 0.94, we end up with a volatility of the
hedged payoff of 0.00289 compared to a volatility of the unhedged payoff of 0.02099. The hedge
therefore removes 87% of the volatility of the unhedged payoff or more than 98.3% of the variance.
Consequently, the hedge performs well during the hedging period, but not as well as it does during
the estimation period.
It is important to understand that to hedge in the presence of basis risk, we exploit a statistical
identical cash positions and n identical hedged positions. The variance of n times a random
variable is n2 times the variance of the random variable. Therefore, the variance of the cash
position is n2Var()SFR) and the hedged variance for n units hedged is n2(1 - R2)Var()SFR). The
ratio of the hedged variance to the unhedged variance is therefore 1 - R2 for an exposure to the
risk factor of n irrespective of the size of n.
Chapter 6, page 39
relation between futures price increments and cash position increments. Consequently, the
performance of the hedge depends on the random error associated with this statistical relation. When
R2 is high, the volatility of the random error is low, so that the cash position increment must typically
be close to its predicted value conditional on the futures price increment. However, as R2 falls, the
volatility of the random error increases and large differences between the cash position increment and
its predicted value given the futures price increment are not unlikely. As R2 falls, it even becomes
possible that hedging does more harm than good ex post. This is because the volatility of the random
error becomes larger, so that we might predict a positive relation between futures price increments
and spot exchange rate increments, but ex post that relation might turn out to be negative. In this
case, if the spot exchange rate falls, Export Inc. makes losses both on its cash position and its futures
position.
There are two different key sources of poor hedge performance. One source is that the futures
price explains little of the variation of the cash market price. This is the case of a low R2. We could
know the true exposure of the cash price to the futures price, yet have a low R2. The other source of
poor performance is that we estimate the hedge imprecisely, so that our hedge is the true exposure
plus an error. The precision of the estimate is given by the t-statistic and the standard error of the
estimate. In our example, the t-statistic is quite large and therefore the standard error is very small.
Since the t-statistic is the estimated coefficient divided by the standard error, the standard error is
0.94 divided by 32.91, or 0.03 for the regression with weekly data. There is 5% chance that the true
hedge ratio exceeds 0.97 by 0.014*1.65 or 0.022 and there is a 5% chance that it is below 0.97 by
more than 0.022. Alternatively, there is a 90% confidence interval for the true hedge ratio of (0.94 0.0495, 0.94 + 0.0495). When a hedge ratio is estimated this precisely, the probability of going short
Chapter 6, page 40
when we would be going long if we knew the true hedge ratio is less than one in a trillion. However,
this probability increases as the precision of the estimate falls. A conjunction of a low t-statistic (a rule
of thumb is less than 2) and a low R-square (a rule of thumb is less than 0.5 for this type of analysis)
means that hedging is unlikely to be productive. This is because the probability of having the wrong
hedge makes large losses more likely and hence increases CaR and VaR. To see this, note that if we
are not hedged, the CaR and VaR measures are the exposure times 1.65 the volatility of the changes
in the exchange rate. However, if we are hedged and there is some chance we have the wrong hedge
ratio, then one has to take into account the fact that with the wrong hedge ratio we make a bigger
loss if the exchange rate falls since we also lose on our hedge. This means that if the hedge ratio is
wrong, we have a bigger loss which translates into larger CaRs and VaRs when we take into account
the risk of having the wrong hedge.
Our hedge could fail for a more fundamental reason. We have assumed that the multivariate
normal increments model holds for the increments of the cash price and the futures price. If this is not
true, we could have precise estimates of the statistical model over a period of time by chance, yet this
would be the wrong statistical model. Hence, the relation between increments of the cash price and
of the futures price might be completely different during the hedging period than we expect. We will
explore different models for the joint distribution of the increments. One important check to perform,
however, is whether the relation we estimate holds throughout the estimation period. For instance,
it makes sense to divide the estimation period into two subperiods and estimate the relation for each
subperiod. A relation that holds for only part of the estimation period is suspicious.
Section 6.4. Implementing the minimum-variance hedge.
Chapter 6, page 41
In this section, we focus on the detailed implementation issues that arise when one wants to
hedge an exposure. The implementation details of hedging with futures are essential in actual futures
hedges but some readers will prefer reading about these details when they actually face problems in
implementing hedging strategies. These readers can jump to section 6.5. without worrying about
losses in continuity. All the issues discussed in this section arise because the basis is not zero for the
hedges we contemplate. The basis can be different from zero for a number of reasons. Not all of these
reasons involve basis risk as defined in the previous section. For instance, it could be that there is no
contract that matures on June 1 to hedge the exposure of Export Inc., but that nevertheless the
relation between the futures price on June 1 on a contract that matures later and the spot exchange
rate is fully predictable conditional on the futures price. In this case, the hedge has to take this
predictable relationship into account and we show how to do that.
Section 6.4.1. The relevance of contract maturity.
Remember that Export Inc. receives SFR1M on June 1. At any point in time, there are several
different SFR contracts traded at the Chicago Mercantile Exchange. These contracts are all for
SFR125,000, but they have different maturities. The available maturities are for the months of March,
April, September, and December. The maturing contract stops trading two business days before the
third Wednesday of the contract month. Hence, to figure out how to hedge Export Inc.’s exposure,
we have to decide which contracts we will use to hedge. One contract, the March contract, ends
before our exposure matures, and three contracts, the June, September, and December contracts, end
afterwards. We could proceed in two different ways. First, we could take a position in a contract that
expires after June 1 and close it on that date. Second, we could take a position in the March contract.
Chapter 6, page 42
Close to maturity of the March contract, we close our March position and open a June position.
Since we have contracts of different maturities, we can plot their prices in the same way that
we would plot the term structure of interest rates. Figure 6.4. plots the term structure of futures
prices on March 1, 1999. In the figure, the price on March 1, 1999 is the spot price. As can be seen
from the figure, the term structure is upward-sloping. We know that with forward currency contracts
an upward-sloping term structure can only occur if foreign interest rates are lower than domestic
interest rates.
Let’s assume constant interest rates. Our notation must now specify both the maturity of the
futures contract and the date when the price is observed. Consequently, we write G(t,T) as the price
at t for a futures contract that matures at T. If domestic and foreign interest rates are constant, we
know that futures and forward prices are the same. From chapter 5, we know that the futures price
on June 1 for a contract that matures at date T is:
G(June 1, T) = PSFR(June 1,T)*S(June 1)/P(June 1,T),
where PSFR(June 1,T) is the price of a SFR default-free discount bond on June 1 that matures at date
T. We can substitute this expression in the optimal hedge per SFR, the hedge ratio (remember that
our optimal hedges are computed before tailing):
Chapter 6, page 43
h'
'
Cov(S(June 1),G(June 1,T))
Var(G(June 1,T))
Cov(S(June 1),P SFR(June 1,T)(S(June 1)/P(June 1,T)))
Var(P SFR(June 1,T)(S(June 1)/P(June 1,T))
'
P(June 1,T)
P SFR(June 1,T)
If it is reasonable to assume that interest rates are constant and that the pricing formula holds exactly,
it does not matter which contract we use as long as we make the above adjustment to the hedge ratio.
All contract prices at maturity of the exposure are equal to a constant, the ratio of the foreign bond
price to the domestic bond price that matures at the same time as the futures contract, times the
exchange rate at that date. A change in the exchange rate over the hedging period equal to )S of the
exposure has therefore the following impact on the futures price:
)G = [PSFR(June 1,T)/P(June 1,T)]*)S
Suppose that for each SFR of cash position, we go short one futures SFR. In this case, if the spot
exchange rate of the SFR is $0.1 higher than expected, the futures price is [PSFR(June 1,T)/P(June
1,T)] higher than expected. If the futures price is at a premium relative to the spot price, this means
that the futures price increases more than the spot exchange rate because [PSFR(June 1,T)/P(June 1,
T)] exceeds one. The opposite holds if the futures price is at a discount relative to the spot price. A
hedge ratio of one (before tailing) is therefore not a good solution unless the futures contract matures
on the day that the exposure matures or interest rates are the same in both countries. If the futures
Chapter 6, page 44
contract is at a premium, a hedge ratio of one means that if the spot exchange rate increases, we lose
more on the short futures position than we gain on the long cash position. We should therefore
decrease the hedge ratio. If we use a hedge ratio of P(June 1,T)/PSFR(June 1,T) before tailing, we have
a perfect hedge which insures that our hedged cash position is worth [P(June 1,T)/PSFR(June
1,T)]*S(March 1). The payoff of the hedged position does not depend on the exchange rate at
maturity of the hedge and is therefore riskless! With constant interest rates, there is no basis risk for
foreign exchange futures contracts as long as markets are perfect. Hence, in the absence of basis risk,
we can eliminate all risk in a cash position with a position in any futures contract that matures later
than the exposure. The size of the position, i.e., the hedge ratio, depends on the maturity of the
contract as described above.
The previous analysis leads to a general formula for the hedge ratio in the absence of basis
risk. Suppose then that we are at date t and hedge a payoff occurring at date t+i with a futures
contract that matures at date T, where T > t+i. Using the yield formulation for the futures price with
a convenience yield developed in chapter 5, we have that the futures price at date t+i is:
G(t+i,T) = exp{[r(t+i,T) - c(t+i,T)](T-(t+i))}A(t+i)
where the futures contract is on an underlying asset with price A(t) at t and expires at T. r(t+i,T) is
the continuously compounded yield from t+i on a discount bond maturing at T and c(t,T) is the
continuously compounded convenience yield over the interval of time from t+i to T. With this
formula, a change of ) in the spot price increases the futures price by exp{[r(t+i,T) - c(t+i,T)](T(t+i))}). Hence, to balance the change of ) in the cash position, we need h = exp{-[r(t+i,T) Chapter 6, page 45
c(t+i,T)](T-(t+i))}. In this case, we have:
) - h*exp{[r(t+i,T) - c(t+i,T)](T-(t+i))}) = 0.
Solving for the hedge ratio, we have:
h = exp{-[r(t+i,T) - c(t+i,T)](T-(t+i))}
(6.6.)
In the case of a currency, the convenience yield of that currency is the rate of return on a bond
denominated in that currency. Hence, this formula is the yield equivalent formula to the formula for
the hedge for foreign currency futures using the discount bond prices. With this hedge ratio, there is
no risk to the hedged position as long as the interest rate and the convenience yield are constant.
Random changes in interest rates and convenience yields create basis risk .
Section 6.4.2. Hedging with futures when the maturity of the contract is shorter than the
maturity of the exposure.
In the absence of basis risk, we saw that the maturity of the futures contract does not matter
for the case where the maturity of the futures contract exceeds the maturity of the cash market
exposure. This result also holds if we roll contracts over that have a shorter maturity than the cash
market exposure. To see this, suppose that we use the March contract and immediately before
maturity of that contract we switch to the June contract. March therefore denotes that the maturity
date of the March contract and G(March) denotes the futures price for the contract maturing in
Chapter 6, page 46
March on March 1. We assume that interest rates are constant since otherwise the basis is random.
For each dollar of cash position, we go short a tailed hedge of [P(March)/PSFR(March)] March
contracts on March 1 at the price of G(March) = [PSFR(March)/P(March)]*S(March 1). Remember,
though, that we are hedging an exposure that matures on June 1. Hence, at that date we want to have
S(March 1) - S(June 1). This means that the tailing factor is a discount bond that matures on June 1.
Suppose that we get out of this position on March 12, which is shortly before maturity of the March
contract. At that date, we will have gained P(March 12,June 1)*[S(March 1) - S(March 12)], but
this gain becomes S(March 1) - S(March 12) on June 1. Let’s now go short a tailed hedge of
[P(March 12,June 1)/PSFR (April 15,June 1)] of the June contract on March 12. The tailing factor is
again a discount bond that matures on June 1. On June 1, we exit our futures position after having
gained S(March 12) - S(June 1). Adding up the gains from the futures position, we have S(March 1) S(June 1), which exactly offsets the losses from the cash position. Hence, in the absence of basis risk,
we can create a perfect hedge by rolling contracts over.
In the presence of basis risk, we have basis risk when we roll over in that we do not know the
relation between the G(March 12, June) and S(March 12) when we put on the hedge. Once we have
the position in the contract maturing in June, we have the basis risk discussed earlier. Consequently,
by rolling the contract over, we incur more basis risk than by using the contract maturing in June or
less depending on the properties of basis risk. The simplest case, though, is the one where, since we
hold the March contract almost to maturity, it has trivial basis risk. In this case, our basis risk is the
same as holding the June contract from March 1. This suggests that it makes no sense to first take
a position in the March contract. Such a conclusion is right in the setting discussed here but wrong
when one allows for market imperfections. We discuss how market imperfections affect the choice
Chapter 6, page 47
of contract used later in this section.
Section 6.4.3. Basis risk, the hedge ratio, and contract maturity.
In perfect markets, basis risk can occur because interest rates change randomly and because
the convenience yield is random. Market imperfections lead to situations where the pricing formula
does not hold exactly which adds additional basis risk. Let’s consider how basis risk affects the hedge
ratio and the volatility of the hedged position, so that we understand better which contract to use. Let
B(June 1,T) be the basis of the SFR futures contract that matures at T on June 1. Remember that we
define the basis as the difference between the cash market price and the futures price. Consequently,
the futures price on June 1, G(June 1,T), is equal to S(June 1) - B(June 1,T). The optimal hedge ratio
is still the one we obtained earlier, but now we replace the futures price by the spot price minus the
basis to see directly the impact of basis risk on the hedge ratio:
h'
Cov(S(June 1),G(June 1,T))
Var(G(June 1,T))
(6.7.)
Cov(S(June 1),S(June 1)&B(June 1,T))
'
Var(S(June 1)&B(June 1,T))
If the basis is not correlated with the spot exchange rate, the numerator is simply the variance of the
spot exchange rate. In this case, the optimal hedge ratio is the variance of the spot exchange rate
divided by the sum of the variance of the spot exchange rate and of the basis. Since variances are
always positive, the greater the variance of the basis the smaller the optimal hedge ratio when the
basis is uncorrelated with the spot exchange rate.
To understand why basis risk decreases the optimal hedge ratio when the basis is uncorrelated
Chapter 6, page 48
with the spot price, it is useful to consider the payoff of the hedged position per SFR:
S(June 1) - h*[G(June 1,T) - G(March 1, T)]
= S(June 1) - h[S(June 1) - B(June 1,T) - G(March 1, T)]
= (1 - h)*S(June 1) + h*B(June 1,T) + h*G(March 1, T)
Suppose that we choose h = 1. In this case, the payoff of the hedged position does not depend on the
exchange rate at maturity. Yet, we do not have a perfect hedge because the payoff of the hedged
position is then equal to the basis. Computing the variance of the hedged position, we have:
Variance of hedged position = (1-h)2[Variance of spot exchange rate] + h2[Variance of basis]
Figure 6.5. shows how the variance of the hedged position changes as h and the variance of basis risk
change. As the hedge ratio increases from zero, the contribution of the variance of the spot exchange
rate to the variance of the hedged position falls. However, at the same time, the contribution of the
variance of the basis to the variance of the hedged position increases. Close to h = 1, the contribution
of the variance of the cash position to the variance of the hedged position is quite small (because (1h)2 is small) in comparison to the contribution of the variance of the basis (because h2 is close to one).
Consequently, close to h = 1, one can decrease the variance of the hedged position by decreasing the
hedge ratio slightly.
If the basis covaries with the cash position, then a basis that covaries negatively with the cash
price increases h relative to the case where the basis is independent of the cash position. The opposite
Chapter 6, page 49
takes place when the basis covaries positively with the cash price. This is not surprising given the
results we have already discussed. A basis that is high when the spot price is high means that the
futures price has a smaller covariance with the spot price. A larger futures position is therefore
needed to achieve the same covariance of the futures position with the cash position as in the case
of no basis risk. The opposite takes place if the basis is low when the spot price is high.
With basis risk, the contract maturity we choose is no longer a matter of indifference.
Contracts of different maturities have different variances of basis risk at the date when our cash
exposure matures. We know that a contract that matures then has no basis risk. It makes sense to
think that a contract that matures close to that date has little basis risk. In contrast, the futures price
of a contract that matures in one year reflects the term structure of interest rates for the 270
remaining days in 90 days. Since this term structure is random, one would expect a contract that
matures in 360 days to have more basis risk than one that matures close to 90 days. With this
reasoning, the best contract is the one that expires the closest to the maturity of the cash position.
Section 6.4.4. Cross-hedging.
Most of our discussion has focused on the example of hedging a SFR exposure with a SFR
futures contract. Much of the hedging taking place is not as straightforward. There might be no
futures contract on the cash position we want to hedge. For instance, one might want to hedge a
portfolio of highly rated corporate bonds. There is no futures contract for which this portfolio could
be delivered. We might, however, conclude that the changes in the price of the T-bond futures
contract are correlated with the changes in the value of our portfolio and that therefore we can use
that contract to hedge. When one uses a futures contract to hedge a cash position which is not
deliverable with the futures contract, one uses a cross-hedge. Our approach to finding the optimal
Chapter 6, page 50
hedge ratio is not different in the case of a cross-hedge. We find that ratio by regressing the changes
in the cash position on the changes in the futures price of the contract we plan to use assuming that
the multivariate normal increments model holds. If we have a choice of contracts, we use the contract
that provides the most effective hedge using our measure of hedging effectiveness. Often, one might
choose to use several different futures contracts. Suppose that one would like to hedge a portfolio
of unrated corporate bonds. There is no futures contract on such corporate bonds, so one must use
a cross-hedge. For instance, one could use the T-bond futures contract and the S&P500 futures
contract together. The T-bond futures contract would help to hedge interest rate risk while the
S&P500 futures contract would help to hedge credit risk. We will see more about this in the next
chapter.
Section 6.4.5. Imperfect markets.
In perfect markets, we would always hold the contract that matures when our exposure
matures if such a contract is available. Otherwise, we would be taking on unnecessary basis risk.
However, in the presence of transaction costs, this need not be the case. When we open a futures
position, we have to pay a commission and deposit a margin. In addition, our transactions may affect
prices: we may have to move prices some so that somebody will be willing to take the opposite side
of our transaction. This is called the market impact of our trade.
The extent of the market impact depends on the liquidity of the market. In highly liquid
markets, most trades have no impact on prices. In futures markets, the shortest maturity contract has
the greatest liquidity. Contracts that have the longest maturity are often extremely illiquid. This lack
of liquidity means that when we try to trade, we end up getting very unfavorable prices because we
Chapter 6, page 51
have to move prices substantially to make the trade. A measure of the liquidity of a futures market
is the open interest. With a futures contract for a given maturity, we can add up all the long positions.
The sum of the long positions is the open interest of the contract. Alternatively, we could add up the
short positions and get the same number. Generally, the open interest falls as one moves from
contracts with short maturities to contracts with long maturities. Contracts with long maturities
sometimes do not trade at all during a day. For instance, on March 1, 1999, the open interest for the
March SFR contract was 56,411 contracts, but only 6,093 for the June contract and 322 for the
September contract (as reported by the Wall Street Journal on 3/3/99). Because of the low liquidity
of contracts with long maturities, it might be worth it to take a position in the shortest maturity
contract and then roll it over in the next contract as the shortest maturity contract matures. While it
is true that proceeding this way implies rollover risk, it may well be that the benefit of trading in more
liquid markets more than offsets the cost of the rollover risk.
Market imperfections matter for reasons other than for rollover risk, however. With market
imperfections, transactions costs for forward and futures hedges need not be the same, so that one
hedge may be cheaper than the other. Further, with market imperfections, there can be counterparty
risks. Futures contracts have typically trivial counterparty risk for the reasons discussed in chapter
5. This is not always the case for forward contracts. Finally, in imperfect markets it is easy to change
one’s futures position, but not necessarily easy to change one’s forward position. Changing a forward
position involves a negotiation with the counterparty of the contract or the opening of a new position
with a different counterparty. With all this, therefore, futures are flexible but imperfect hedges
because of basis risk. Forwards require bank relationships and are inflexible but generally offer perfect
hedges at least for fixed foreign exchange exposures in the absence of counterparty risks.
Chapter 6, page 52
Section 6.4.6. Imperfect divisibility.
Throughout our discussion, we have assumed that futures contracts are perfectly divisible.
They are not. For instance, for the SFR contract, we have to buy or sell SFR futures in units of
SFR125,000. The fact that we have to trade contracts of fixed size is a disadvantage of futures
relative to forward contracts. Let’s see how we handle this problem in figuring out the optimal hedge.
If the optimal hedge is not an integer, we have to round out the hedge. In some cases, however, this
rounding out creates a situation where we might be better off not hedging at all. Consider the case
where we have an exposure of SFR62,500 expiring in three months and there is a futures contract
expiring in three months. In this case, the volatility of the hedged position where we go short one
contract is the same as the volatility of the unhedged position:
Volatility of unhedged position = Vol(62,500S(t+0.25))
Volatility of hedged position = Vol(62,500S(t+0.25)-125,000{S(t+0.25) - G(t,S(t+0.25)})
=Vol(-62,500S(t+0.25)) = Vol(62,500S(t+0.25))
With this example, we are indifferent between hedging and not hedging if the exposure is SFR62,500.
Suppose that the exposure is SFR50,000. In this case, the volatility of the unhedged position is
Vol(50,000S(t+0.25)). In contrast, the volatility of the hedged position is Vol(75,000S(t+0.25))!
Therefore, one is better off not hedging at all.
If one wants to minimize the risk of the hedged position, the solution to the rounding out
problem is to compare the volatility of the hedged position for the integers adjacent to the optimal
hedge ratio. For instance, if the optimal hedge requires us to go short 5.55 contracts, then one
Chapter 6, page 53
compares the volatility of the hedged position for six contracts and the volatility of the hedged
position for five contracts. The optimal integer futures position is the one that yields the lowest
volatility of the hedged position.
Section 6.4.7. The multivariate normal increments model: Cash versus futures prices.
Models are approximations and the normal increment model is no different from any model.
As one looks more closely at the multivariate normal increments model, one finds many reasons why
it might not hold exactly. One reason has to do with how futures contracts are priced. Remember
from chapter 5 that if the interest rate and the convenience yield are constant, the futures contract is
priced like a forward contract. In this case, the futures price is equal to the price of the deliverable
good times a factor that depends on the interest rate and time to maturity. If the futures price depends
only on the price of the deliverable good, the interest rate and the convenience yield, it must satisfy
the following formula:
G(t+i,T) = exp{[r - c](T-(t+i))}S(t+i)
where G(t+i,T) is the futures price at t+i for a contract maturing at T, r is the annual continuously
compounded interest rate, c is the convenience yield, and S(t+i) is the spot price of the deliverable
good at t+i.6 The futures price increment from t+i to t+i+*, where * is a period of time such that
6
We ignore payouts and storage costs for simplicity. Taking payouts and storage costs
into account changes the formula as shown in chapter 5 but has no impact on the rest of the
analysis.
Chapter 6, page 54
t+i+* < T, is:
G(t+i+*,T) - G(t+i,T) = exp{[r - c](T-(t+i+*))}S(t+i+*) - exp{[r - c](T-(t+i))}S(t+i)
For futures price increments to be i.i.d., we would need the change in exp{[r - c](T-(t+i))} to be zero
if the increment of the price of the deliverable good is i.i.d. This happens only if r is equal to c so that
exp{[r - c](T-(t+i))} is always equal to one. Otherwise, if r is greater than c, exp{[r - c](T-(t+i))} is
greater than one and falls over time to become closer to one as maturity approaches. Consequently,
the futures price increment falls over time as a proportion of the increment of the price of the
deliverable good. The opposite occurs if r is smaller than c. This implies that the distribution of the
futures price increments changes over time if the distribution of the increments of the price of the
deliverable good are i.i.d. In our regression analysis, we effectively assume that basis risk due to the
fact that we do not hold the futures contract to maturity is a first-order phenomenon while the impact
of predictable changes in the distribution of the futures price is not. If we hold a contract to maturity
or if basis risk due to the fact that we do not hold it to maturity is unimportant, we may be better off
to use cash price increments of the deliverable good rather than futures price increments in our
regression because the multivariate normal increments model is more likely to hold exactly for
increments in the price of the deliverable good than increments in the futures price. Since the futures
price is not the cash price, we then need to use the futures price formula to adjust the hedge if we do
not hold the futures contract to maturity (similarly to the analysis of section 6.4.1.). Hence, if the
regression coefficient is $, we use as hedge ratio h the ratio $/exp{[r - c](T-(t*))}, where t* is the
maturity of the hedge. This is because an increment of ) in the cash price changes the futures price
Chapter 6, page 55
by )exp{[r - c](T-(t*))}, so that on average the futures price changes more than the cash price if r
is greater than c and we have to take that into account when constructing our hedge.
Let’s consider an example. Suppose that we face the following situation. We hold a contract
close enough to maturity that the contract’s basis risk with respect to the deliverable good is trivial.
However, the deliverable good is not the good we are trying to hedge. For instance, we are holding
a wheat contract to maturity but the wheat grade we hedge is different from the wheat grade that is
delivered on the contract. In this case, our hedge has basis risk in that the prices of the two grades
of wheat at maturity might be different from what we expect. In this case, what we care about at
maturity of the hedge is not how the futures price differs from the cash price, but how the cash price
we hedge differs from the cash price of the deliverable good. Hence, to estimate that relation, we are
better off to estimate a regression of changes in the cash price of our exposure on the cash price of
the deliverable good if we believe that cash prices follow the multivariate normal increments model.
Section 6.5. Putting it all together in an example.
Consider now the situation of the exporter on March 1, 1999. Table 6.1. shows market data
for that day for the three types of hedging instruments discussed in this chapter. It includes interest
rate data, currency market data, and futures data. Our exporter will receive the SFR1M on June 1.
Foreign exchange contracts have a two-day settlement period. This means that our exporter must
enter contracts that mature two days before June 1 so that he then delivers the SFR on that day for
these contracts. Let’s look at the various possible hedges:
1. Forward market hedge. The 90 day forward price is at $0.6902. Selling forward the SFRs
would therefore yield at maturity $690,200.
Chapter 6, page 56
2. Money market hedge. Using Eurocurrency rates, the exporter could borrow for three
months the present value of SFR1m at the annual rate of 1 3/8 and invest these proceeds in dollars
at the rate of 4 15/16. Euro-rates are add-on rates, so the amount borrowed is SFR1m minus the
interest to be paid, which is 1 3/8* 1/4*Amount borrowed. So, we have:
Amount borrowed = SFR1M - [(1 3/8* 1/4)/100] * Amount borrowed
Solving for the amount borrowed, we get SFR996574.3. This amount in dollars is 0.6839*996,574.3
= $681,557.2. Investing this at the rate of 4 15/16% yields interest of $8,413. Consequently, we end
up with $681,557.2 + $8,413 = $689,970.2. This is slightly worse than what we would get using the
forward market. This is unlikely to be an arbitrage opportunity or even a profit opportunity, however.
Notice that the euro-rates are London rates whereas the foreign currency data are U.S. afternoon
data. This mismatch can be sufficient to create pseudo-arbitrage opportunities. Another issue we do
not take into account is that only one foreign currency price is quoted for each maturity. This means
that we ignore the bid-ask spread on foreign exchange.
3. Futures hedge. Suppose we could enter futures positions at the daily settlement price. In
this case, we could have taken a long position in the March contract or a long position in the
September contract. If we had taken the position in the March contract, we would have needed to
roll the position over in March. Irrespective of what we do, we have to remember that the units of
the contract are SFR125,000. The optimal hedge is to go short 0.94 contracts before tailing. This
means going short SFR940,000. This corresponds to 7.52 contracts. The June T-bill sells for
$98.8194 , so that the tailing factor is slightly more than 0.9881. The optimal tailed hedge involves
Chapter 6, page 57
7.44 contracts. We have to decide whether to go short 8 or 7 contracts. To do that, we estimate the
variance of the hedged position with 7 contracts and compare it to the variance of the hedged position
with 8 contracts. Over our estimation period, the weekly standard deviation of the SFR increment is
0.01077, the weekly standard deviation of the continuous series of the futures price increment is
0.0107, and the weekly covariance between the two increments is 0.000115. Suppose we evaluate
the hedging position weekly. Remember that marking to market increases the payoffs of a futures
contract by the inverse of the tailing factor. We can take into account the marking-to-market by
dividing the futures volatility by 0.9881 for the first week. We can compute the variance of the
hedged position using 8 contracts and the variance of the hedged position using 7 contracts, using
the formula that the variance of the hedged position is the variance of the cash position increment
minus the increment in the hedge position. For 7 contracts, the variance of the hedged position is:
Variance of hedged position with 7 contracts for first week = (1,000,000*0.01077)2 +
(7*125,000*0.01107/0.98881)2 - 2*1,000,000*7*125,000*0.000115/0.9881 = 8.41612*107
We also find that the variance of the hedged position using 8 contracts is 8.73731*107. Based on this,
we choose to go short 7 contracts initially. Going short 7 contracts turns out to be the optimal
position except for the last week of the hedging strategy, when going short 8 contracts has a lower
variance. Note that the tailing factor falls over time, so that it is not surprising that the number of
contracts would increase as one gets close to the maturity of the hedging strategy. In our example,
tailing turns out to matter because before tailing we are close to the point where we are indifferent
between going short 7 contracts or going short 8 contracts.
Chapter 6, page 58
Consider the case where the firm’s cash accumulates in a bank account that earns no interest.
In this case, tailing is irrelevant and we go short 8 contracts. The June futures price is 0.6908 on
March 1, 1999. On May 27, 1999, the June contract settled at 0.6572. This means that we gain
$33,600 on our futures position (-(0.6572-0.6908)*8*125,000). On Thursday May 27, 1999, the spot
rate for the SFR is 0.6559.7 Hence, our cash position is equal to $655,900. The value of our hedged
position is $655,900 - $33,600 = $689,500. Note that the spot exchange rate falls during the period
over which we hedge. This means that the value of our cash position falls. Consequently, if we try
to eliminate the volatility of the hedged position, it must be that the loss in the cash position is offset
by a gain in the futures position. The opposite would have happened had the cash position gained
money.
When the firm takes the futures position, it has to make margin deposits. In March 1999, the
initial margin was $2,160 for each contract (note, however, that the firm could have had a lower
margin because of its classification as a hedger or a higher margin because of concerns about credit
risk on the part of its broker). Consequently, the firm would have needed to make a deposit of
somewhat less than $24,000 to open the futures position. Since the futures price decreases over time,
the firm would have benefitted from daily marking to market.
How would the exporter decide which instrument to use? Three key considerations will matter
the most in the decision:
(A) Credit risk. Suppose that the exporter has significant credit risk. This will most likely
make the forward market hedge and the money market hedge impractical, since counterparties will
7
We have to use Thursday instead of Friday because the following Monday is a holiday in
the U.S.
Chapter 6, page 59
be concerned about Export Inc.’s credit risk.
(B) Demand for flexibility. If the exporter foresees situations where it might want to change
the hedge, this will make the forward market hedge and the money market hedge impractical as well.
(C) Cost of risk. Futures hedges have basis risk, so that a situation where risk is very
expensive makes perfect hedges obtainable with forward contracts more advantageous than futures
hedges.
Our discussion of this example has focused on the situation of Export Inc. As we saw in
Section 6.2., Trading Inc. could hedge its position by eliminating the risk of its position as of June
1. The problem with this approach, though, is that Trading Inc. might have a completely different
position in two days. In perfect markets, this would not matter, but in real world markets it does since
there are transactions costs. Since Trading Inc. values flexibility, a forward contract that matures on
June 1 is unattractive. A futures strategy that minimizes risk as of June 1 might not be the best
strategy to minimize risk as of March 2 because of basis risk. For instance, if Trading Inc. uses a
futures contract that matures in June, there might be considerable randomness in the relation between
this future price on March 2 and the spot exchange rate on that date. This is due to the fact that the
June contract is less liquid as well as to the fact that its pricing depends on discount bonds that mature
in more than 3 months. Consequently, the optimal hedge for Trading Inc. is likely to involve the
futures contract closest to maturity. Other possible strategies involving the short-sale of SFR discount
bonds of very short maturity using the repo market or one-day forward contracts have the
disadvantage that they require large foreign exchange transactions, so that typically they would be
more expensive than the futures strategy.
Chapter 6, page 60
6.6. Summary.
In this section, we learned how to hedge with forward and futures contracts. We started with
a computation of the risk of an exposure to a risk factor using our three risk measures, volatility, CaR
and VaR. We showed that going short the exposure to a risk factor sets the volatility, the CaR, and
the VaR associated with an exposure equal to zero when a perfect hedge is feasible. We then moved
on to situations where a perfect hedge is not feasible because of basis risk. In that case, the optimal
hedge is to go short the exposure of the cash position to the futures price if transaction costs are not
significant. This optimal hedge is the minimum volatility hedge. The exposure generally has to be
estimated. When the assumptions for linear regression analysis are met, the estimate of the exposure
of the cash position to the futures price is the slope of a regression of the increments of the cash price
on the increments of the futures price. We examined the issues that arise in implementing the
minimum volatility hedge in the context of an example involving hedging a SFR cash position.
Whereas hedging with futures contracts generally involves basis risk, futures hedging has the
advantage of flexibility and of minimal counterparty risk compared to forward hedging. As we will
see later, the flexibility advantage of futures hedging becomes more important when the optimal
hedge changes over time.
Chapter 6, page 61
Literature note
The method of using regression ratios to hedge was first proposed by John (1960) and Stein (1961).
Much recent work has addressed the issue of how to estimate the minimum-volatility hedge when the
i.i.d. assumption made in this chapter does not hold. See Cechetti, Cumby, and Figlewski (1988) and
Baillie and Myers (1991) for approaches that use econometric techniques that take into account
changes in the joint distribution of the cash market price and the hedging instrument. See Kawaller
(1997) for a discussion of tailing.
Chapter 6, page 62
Key concepts
Risk factor, exposure to a risk factor, exposure to a futures price, i.i.d., square root rule, minimum
volatility hedge, basis, basis risk, tailing.
Chapter 6, page 63
Review questions
1. How do you compute the volatility of the exchange rate change over three months when you have
the volatility of the change over one month and you know that the exchange rate increments are i.i.d.?
2. How do you compute the volatility of a foreign exchange exposure?
3. How do you compute the VaR of a foreign exchange exposure?
4. How do you compute the CaR of a foreign exchange exposure?
5. What is the impact of a perfect forward hedge on the CaR of a foreign exchange exposure?
6. When does tailing improve a futures hedge significantly?
7. What is the optimal hedge in the absence of basis risk?
8. What is the optimal hedge in the presence of basis risk?
9. How do you forecast exposure in the presence of basis risk?
10. What are the two key reasons why a hedge based on forecasted exposure might perform poorly?
11. How do you estimate the effectiveness of a hedging strategy with past data?
12. How do you decide the number of contracts in your hedge when divisibility is an issue?
13. What are the advantages of a futures hedge over a forward hedge?
14. When is a forward hedge strategy preferable?
Chapter 6, page 64
Questions and exercises
1. Consider a firm that expects to sell 10 million barrels of oil at the cash market price in one year.
It wants its CaR to be at most $5M. The price of a barrel is currently $15 with a one-year standard
deviation of $2. What is the CaR of the firm in the absence of hedging assuming that the price change
is distributed normally?
2. Consider now the situation where there is a forward contract that the firm can use to sell oil
forward. The forward price is $16 a barrel. How many barrels must the firm sell forward to reduce
its CaR to $5M?
3. Suppose now that there is a futures contract that matures exactly in one year that requires delivery
of the type of oil that the firm produces. The contract is assumed to be perfectly divisible for
simplicity. The interest rate for a one-year discount bond is 10%. Interest rates are assumed to be
constant. How many barrels must the firm sell today on the futures market to minimize its CaR?
4. With the assumptions of question 3, assume now that the firm sells 10 million barrels on the futures
market with delivery in one year and does not change its position over the year. How does the interest
rate affect the firm’s CaR?
5. Suppose now that the futures contract requires delivery of a different grade of oil than the one the
firm produces, but maturity of the contract is in one year. What information would you need to
compute the volatility-minimizing hedge for the firm?
Chapter 6, page 65
6. You are told by a consultant that the slope of a regression of the change in the price of the oil the
firm produced on the change in the price of the oil that is delivered with the futures contract is 0.9
with a t-statistic in excess of 10. What would be your minimum-volatility hedge before tailing? How
would tailing affect that hedge?
7. The R-square in the regression discussed in question 6 is 0.85. What can you learn from this
number?
8. Suppose now that there is no futures contract that matures in exactly one year. There is, however,
a futures contract that matures in 18 months. A regression of changes in the cash market price of the
oil the firm produces on the changes in the futures price of the contract that matures in 18 months
using past history yields a slope of 0.8 with an R-square of 0.6. How many barrels should the firm
sell on the futures market if it wants to minimize its CaR?
9. Does your answer to the previous question change if you are told that the deliverable grade of oil
for the contract that matures in 18 months is not the grade of oil that the firm produces?
10. How would the optimal hedge derived in question 8 change over time?
11. What could go wrong with the optimal hedge derived in question 8?
Chapter 6, page 66
Figure 6.1. Relation between cash position and futures price when the futures price changes
are perfectly correlated with the cash position changes.
In this example, the change in the cash price, here the spot exchange rate, is 0.9 times the change in
the futures price. We have 100 observations. These observations are generated assuming that the
Unexpected change in
spot exchange rate
change in the futures price has an expected value of zero and a standard deviation of one.
-4
-2
3
2
1
0
-1 0
-2
-3
2
Unexpected change in futures price
Chapter 6, page 67
4
Figure 6.2. Relation between cash position and futures price changes when the futures price
changes are imperfectly correlated with the cash position changes.
In this example, the change in the cash price, here the spot exchange rate, is 0.9 times the change in
the futures price plus a normally distributed random error with expected value of zero and standard
error of 0.5. We have 100 observations. These observations are generated assuming that the change
Unexpected change in
spot exchange rate
in the futures price has an expected value of zero and a standard deviation of one.
-4
-2
3
2
1
0
-1 0
-2
-3
2
Unexpected change in futures price
Chapter 6, page 68
4
Figure 6.3. Regression line obtained using data from Figure 6.2.
The regression in this figure is estimated using the data from Figure 6.2. The changes in the spot
Spot exchange rate
change
exchange rate are regressed on a constant and the change of the futures price.
3
2
1
0
-4
-2
-1 0
-2
-3
Futures price change
Chapter 6, page 69
2
4
Figure 6.4. This figure gives the SFR futures prices as of March 1, 1999 for three maturities
as well as the spot price.
0.7
0.698
0.696
0.694
Price
0.692
0.69
0.688
0.686
0.684
0.682
0.68
01-Mar-99
Mar-99
Jun-99
Maturity
Chapter 6, page 70
Sep-99
Figure 6.5. Variance of hedged payoff as a function of hedge ratio and variance of basis risk.
In this figure, basis risk is assumed to be uncorrelated with the futures price.
Chapter 6, page 71
Table 6.1. Market data for March 1, 1999.
This is a summary of the market data that the expoter discussed in this
chapter would have considered to decide how to hedge. This data is
obtained from the Wall Street Journal and the Financial Times of March
2, 1999.
Panel A. Treasury bill data
Maturity
Days to
Maturity
4-Mar-99
2
11-Mar-99
9
1-Apr-99
30
8-Apr-99
37
3-Jun-99
93
10-Jun-99
100
Maturity
Overnight
7 days
1 month
3 months
US$ Bid
4 15/16
4 27/32
4 27/32
4 15/16
Bid
Ask
Ask Yield
4.50
4.52
4.51
4.49
4.59
4.56
4.42
4.44
4.47
4.45
4.57
4.54
4.48
4.51
4.55
4.53
4.69
4.66
Panel B. Euro-rates
US$ Ask
SFR Bid
5 1/32
1 1/8
4 31/32
1 3/16
4 31/32
1 5/32
5 1/16
1 7/32
SFR Ask
1 5/8
1 9/32
1 5/16
1 3/8
Panel C. Spot and forward exchange rates for SFR
Maturity
price
spot
0.6839
1-m forward
0.6862
3-m forward
0.6902
Panel D. Futures contracts
Maturity
Open
March
0.6907
June
0.6973
September
0.6995
High
0.6923
0.6974
0.6995
Low
0.6840
0.6902
0.6980
Chapter 6, page 72
Settlement
0.6845
0.6908
0.6974
Technical Box 6.1. Estimating Mean and Volatility
Suppose that we have T different observations of past increments and are forecasting the increment
and its volatility for T+1. Let’s use the notation )S(i) = S(i+1) - S(i). With this notation, the
expected increment is simply the average of the increments over the estimation period:
i'T
E[)S(T%1)]')S'(1/T)'i'1 )S(i)
(B6.1.1.)
Suppose we use as our estimation monthly quotes of the dollar price of the SFR from March 1988
to February 1999. The mean of the monthly change in the exchange rate over our estimation period
is -0.000115. This reflects the fact that, over our estimation period, the SFR depreciates against the
dollar. Having obtained the expected increment, the expected variance of the increment T+1 is:
Var()S(T%1))'
1 i'T
'i'1 [)S(i)&)S]2
T&1
(B6.1.2.)
Note that the estimate of the variance differs from the average of the squared deviations of the
increments from their mean. Since we have T squared deviations, the average would divide the sum
of the squared deviations by T. Instead, here we divide the sum by T-1 to avoid a bias in the estimate
of the variance. As T gets large, it makes little difference whether we divide by T or T-1. In the case
of the SFR, we have 131 observations, so that the adjustment is 131/(131-1). The historical variance
is 0.000626 whereas the variance after the adjustment is 0.000631.
Chapter 6, page 73
Technical Box 6.2. Proof by example that tailing works.
Let’s make sure that tailing works to create a perfect hedge when the futures contract matures
at the same time as the exposure by using a three-period example similar to the one used in section
5.4.A. We have three dates: 1, 2, and 3. The futures contract matures at date 3. At that date the
futures price equals the spot market price since it is the price determined at date 3 to buy the foreign
currency at that date. We use the numbers of section 5.4.A. The futures price is $2 at date 1. It goes
to either $3 or $1 at date 2 and stays there at date 3. For simplicity, the spot price and the futures
price are assumed to be the same. A discount bond that pays $1 in one period costs $0.909 at date
1 and at date 2. We are trying to hedge an exposure of 100 units.
A) Futures hedge without tailing. We go short 100 units at date 1 on the futures market.
There are two possible outcomes with equal probability: (1) the futures price falls to $1 at date 2, and
(2) the futures price increases to $3 at date 3. Suppose first the futures price falls to $1 at date 2. We
make a profit of $100 on the futures market that we get to invest at the risk-free rate. We earn $10
of interest, so that at date 3 the futures profit is $110. At date 3, the spot price is $1. We lost $100
on the cash market since our position was worth $200 at date 1 but is worth only $100 at date 3. Our
hedged position at date 3 is worth $100 plus $110, or $210. Suppose that instead the futures price
goes to $3 at date 2. We lose $100 on the futures contract. To bring all the cash flows to date 3 to
compute the value of the hedged position at that date, we have to borrow $100 at date 2. At date 3,
we have to repay $110, but we made a profit of $100 on the cash market. As a result, our hedged
position at date 3 is worth $300 - $110, or $290. The variance of our hedged position is 0.5*(290 300)2 + 0.5*(310 - 300)2, or 100.
Chapter 6, page 74
B) Futures hedge with tailing. The tailing factor is 0.909. We therefore sell short 90.9 units
at date 1. At date 2, if the futures price is $1, we gain $90.9 which we invest for one period to get
$100 at date 3. At date 3, the value of our hedged position is the sum of our $100 cash market
position and our $100 futures gain, for a total of $200. At date 2, the tailing factor is 1 since we will
not be able to invest profits we make on our futures position at date 3 because it is the end of our
hedging period. Therefore, at date 2, we increase our short position to 100 contracts. If the futures
price is $2 at date 2, we lose $90.9. We borrow that amount at date 2 and have to repay $100 at date
3. Therefore, at date 3, our hedged position is worth a $300 cash market position and a $100 futures
loss, for a total of $200. Since our hedged position always has the same value at date 3, it has no
variance.
In this example, the hedged position with a tailed hedge has no risk while the hedged position with
a hedge that is not tailed is risky. The risk of the position that is not tailed comes from the fact that
at date 1 we do not know whether we will make a gain at date 2 and hence have interest income at
date 3 or make a loss at date 2 and hence have to pay interest at date 3.
Chapter 6, page 75
Technical Box 6.3. Deriving the minimum-volatility hedge.
The minimum-volatility futures hedge given by equation (6.4.) can be obtained directly by minimizing
the volatility of the hedged payoff at maturity. Consider hedging an exposure with a futures contract.
At maturity of the exposure, Cash position denotes the value of the exposure and G denotes the
futures price of the contract used for hedging. The variance of the hedged cash position at maturity
of the hedge using a hedge h is:
Variance of payoff of hedged position = Var(Cash position) - hG)
= Var(Cash position) + h2Var(G) - 2hCov(Cash position,G)
We can compute how a change in h affects the variance of the hedged position by taking the
derivative of the function with respect to h:
d[Variance of hedged position]
'2hVar(G)&2Cov(Cash position,G)
dh
By setting this derivative equal to zero, we obtain the hedge ratio that minimizes the variance of the
hedged position:
h'
Cov(Cash position,G)
Var(G)
Remember that h is the optimal hedge ratio before tailing, so that the number of units we need to
short on the futures market is h times the current value of a discount bond that pays $1 at maturity
of the hedge.
Chapter 6, page 76
Technical Box 6.4. The statistical foundations of the linear regression approach to obtaining
the minimum-variance hedge ratio.
In this box, we examine the conditions that must be met to use linear regression analysis to estimate
the optimal hedge ratio. To do that, we need to find a period in the past where the joint distribution
of the increments in the cash price and in the futures price is the same as what we expect that joint
distribution to be over the hedging period. Let’s call this period the hedge estimation period. It is
important to note that this period does not have to be the immediate past. It might be that the
immediate past is unusual. For instance, the immediate past might be a period of unusual turbulence
and one does not expect such turbulence in the future. In this case, one would not use data from the
immediate past in the estimation period. We then require that the joint distribution of the increments
in the cash price and in the futures price is such that we can estimate their relation using linear
regression. It is a property of jointly normally distributed variables that their increments are linearly
related (6.1.).8 A linear relation between increments holds under more general conditions.
In the previous section, we estimated our hedge using ordinary least squares, or OLS. To be
able to use OLS, the distribution of the increments over time must satisfy some important conditions
that we now address. For simplicity, let’s have the cash price be the spot exchange rate, remembering
that our results are not limited to this case. )S(i) is the cash price increment over the i-th period and
)G(i) is the futures price increment over the i-th period. We assume that )S(i) and )G(i) are jointly
normal for any i. To use ordinary least-squares, we require that:
8
Any statistics textbook that discusses the multivariate normal distribution generally has
this result. See, for instance, Mood, Graybill, and Boes ().
Chapter 6, page 77
A1) The distribution of the cash price increment is i.i.d.
A2) The distribution of the futures price increment is i.i.d.
A3) Cov()S(i), )G(i)) = Constant, for all i’s.
A4) Cov()S(i), )G(j)) = 0, for all i … j.
These assumptions require that the cash price increment for a given period is uncorrelated with the
cash price increment and the futures price increment for any other period. When two or more random
variables each are normal i.i.d., and in addition satisfy assumptions (A3) and (A4), we say that they
follow the multivariate normal increments model.
Given that the increments of the cash price and the futures price follow the multivariate
normal increments model, we want now to find out how to compute the volatility-minimizing hedge
ratio. We know that our assumptions allow us to use regression analysis to find the relation between
the cash price increments and the futures price increments. However, we would like to know how the
hedge ratio depends on the length of the period of time over which the hedge is maintained and on
the interval over which increments are measured.
Let’s assume that the multivariate normal increment model holds from the perspective of
Trading Inc., so that it holds using daily changes. We now investigate whether it also holds from the
perspective of Export Inc. which hedges over three months. The change in the cash price over any
period longer than one day is just the sum of the daily increments and the same is true for the change
in the futures price. We know that the sum of normally distributed random variables follows a normal
distribution. Consequently, if the cash price increments follow a normal distribution over days, the
change in the cash price over any period of time is the sum of these increments and hence is normally
Chapter 6, page 78
distributed. The same applies to the futures price increments. Let’s look more closely at the
implications of our assumptions for the distribution of the increments measured over a period of time.
With our assumptions, the expected cash price increment is the same over any day. Consequently, the
expected cash price increment over N days is simply N times the daily expected cash price increment.
Let Var(S(one day)) and Var(G(one day)) be the variance of the one-day increment of the exchange
rate and of the futures price, respectively. The increment of the spot exchange rate over N trading
days is then distributed normally with a variance equal to N times the variance of the one-day spot
exchange rate increment, or N*Var(S(one day)). The same applies for the variance of the change in
the futures price. The square root rule can be used to get the volatilities over N days since the
increments are independent.
Let’s now look at the covariance between the increments of the spot exchange rate and the
futures price over N days. Define )S(i) to be the increment over day i and )S(N days) the increment
over N days. The covariance over the N trading days is:
Cov( ∆ S(N days), ∆ G(N days))
N
N
i= 1
i= 1
= Cov( ∑ ∆ S(i), ∑ ∆ G(i))
Note now that Cov()S(i), )G(j)) is equal to zero because increments are independent over nonoverlapping periods. Further, since the increments are i.i.d., their distribution is the same for each day.
Consequently:
Chapter 6, page 79
N
N
i= 1
i= 1
Cov( ∑ ∆ S(i), ∑ ∆G(i))
N
= ∑ Cov( ∆S(i), ∆G(i))
i= 1
= N * Cov( ∆S(i), ∆ G(i))
= N * Cov[ ∆S(j), ∆G(j)]
We already know that the variance computed over N daily increments is N times the variance over
one daily increment. Consequently, the hedge ratio over N days is:
N * Cov( ∆ S(i), ∆ G(i))
N * Var( ∆ G(i))
Cov( ∆ S(i), ∆ G(i))
=
Var( ∆ G(i))
Hedge ratio =
In that equation, day i could be any day since the covariance and the variance are the same for any
day. It follows from this that the hedge ratio is the same over any hedging period as long as our
assumptions are satisfied. This is a far-reaching result, because it tells us that the estimate of the
hedge ratio is not affected by the measurement interval or by the length of the hedging period when
the multivariate normal increment model applies. If our assumptions hold, therefore, how we estimate
the hedge ratio is not dictated by the length of the hedging period. We should not expect to find a
different estimate for the hedge ratio if we use daily or weekly data. Hence, Export Inc. and Trading
Inc. can estimate the hedge ratio using the same regression and use the same hedge ratio despite
hedging over different periods. The precision with which we estimate the hedge ratio increases as we
have more observations, but the regression approach yields an unbiased estimate of the hedge ratio
irrespective of the measurement interval.
Chapter 6, page 80
Chapter 7: Hedging costs and the portfolio approach to hedging
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Section 7.1. The costs of hedging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Section 7.2. Multiple cash flows occurring at a point in time. . . . . . . . . . . . . . . . . . . . . 16
1. The SFR futures contract helps hedge the yen exposure . . . . . . . . . . . . . . . . 20
2. The yen futures contract helps hedge the SFR and yen exposures . . . . . . . . . 21
Section 7.3. Cash flows accruing at different dates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Section 7.4. The i.i.d. returns model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Section 7.5. The log increment model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Section 7.5.1. Evaluating risk when the log increments of the cash position are
normally distributed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Section 7.5.2. Var and Riskmetrics™ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Section 7.5.3. Hedging with i.i.d. log increments for the cash position and the
futures price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Section 7.6. Metallgesellschaft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Section 7.7. Conclusion and summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Questions and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 7.1. Expected cash flow net of CaR cost as a function of hedge. . . . . . . . . . . . . 53
Figure 7.2. Marginal cost and marginal gain from hedging. . . . . . . . . . . . . . . . . . . . . . . 54
Figure 7.3. Optimal hedge ratio as a function of the spot exchange rate and " when the
cost of CaR is "*CaR2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 7.4. Short position to hedge Export Inc.’s exposure. . . . . . . . . . . . . . . . . . . . . . 56
Figure 7.5. Normal and lognormal density functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Box 7.1. The costs of hedging and Daimler’s FX losses. . . . . . . . . . . . . . . . . . . . . . . . . 58
Technical Box 7.2. Derivation of the optimal hedge when CaR is costly . . . . . . . . . . . . 60
Chapter 7: Hedging costs and the portfolio approach to
hedging
December 14, 1999
© René M. Stulz 1996, 1999
Chapter objectives
1. Understand how hedging costs affect hedging policies.
2. Find out how one can reduce hedging costs by taking advantage of diversification.
3. Extend the analysis to the i.i.d. returns model used by Riskmetrics™.
4. Examine the Metallgesellschaft case to understand why a widely publicized hedging strategy failed.
Chapter 7, page 2
In the previous chapter, we learned how to hedge a single exposure using forwards and
futures when hedging is costless. In this chapter, we extend the analysis to consider the costs of
hedging and how they affect the hedging decision. When hedging is costly, the firm faces a tradeoff
between the benefit from reducing risk and the cost from doing so. We define the costs of hedging
and explore this tradeoff. Firms almost always have exposures to multiple risk factors. Firm with
exposures to multiple risk factors can reduce hedging costs by treating their exposures as a portfolio.
Typically, some diversification takes place within this portfolio, so that a firm with multiple exposures
puts on a smaller hedge than if it hedges each exposure separately.
To consider portfolios of exposures, we first investigate the case where all exposures have
the same maturity. We show that the approach of the previous chapter can be adapted to construct
a minimum-volatility hedge for the portfolio of exposures. We then turn to the case where a firm has
both multiple cash flows accruing at a point in time and cash flows accruing at different points in time.
In this case, diversification can take place both across exposures at a point in time and across
exposures with different maturities. We show how the firm should measure the risk and hedge
portfolios of exposures with different maturities. One important case where exposures have the same
maturity is a firm with a portfolio of financial securities that manages risk measured by a one-day
VaR. In this case, the firm hedges the change in value over one day, so that exposures effectively
have a one-day maturity. For that case, it generally turns out that it is often more appropriate to
assume that log price increments are identically independently normally distributed in contrast to the
analysis of the previous chapter where price increments were assumed to be identically independently
normally distributed. We therefore consider how our analysis of chapter 6 is affected by making the
assumption of normally distributed log price increments. Finally, we conclude with a detailed study
Chapter 7, page 3
of the Metallgesellschaft case that enables us to understand better the practical difficulties of hedging.
Section 7.1. The costs of hedging.
In chapter 6, we considered two firms, Export Inc. and Trading Inc. Each firm will receive
SFR1M in three months, on June 1. Export Inc. measures risk using CaR and Trading Inc. using VaR.
Each firm views risk to be costly. Consequently, in the absence of costs of hedging, each firm
chooses to minimize risk. In the absence of basis risk, we saw that each firm can eliminate risk
completely. Throughout our analysis, we completely ignored the impact of hedging on the expected
cash flow for Export Inc. or the expected value of the position for Trading Inc. This is correct if
hedging is costless and risk is costly. In this case, the firm just eliminates risk. Though it is possible
for hedging to be costless, there are often costs involved in hedging. Sometimes, these costs are so
small compared to the benefits from hedging that they can be ignored. In other cases, the costs cannot
be ignored. In these cases, the firm trades off the costs and benefits of hedging. We focus only on the
marginal costs associated with a hedge in our analysis. For a firm, there are also costs involved in
having a risk management capability. These costs include salaries, databases, computer systems, and
so on. We take these costs as sunk costs which do not affect the hedging decision.
For the two firms, exposure is known exactly. In a case where exposure is measured exactly,
hedging can alter the expected payoff for two reasons. First, ignoring transaction costs, the hedge can
affect the expected payoff because the price for future delivery is different from the expected spot
price for the date of future delivery. Second, putting on the hedge involves transaction costs. We
discuss these two effects of hedging on the expected payoff in turn. We go through the analysis from
the perspective of Export Inc., but the analysis is the same from the perspective of Trading Inc.
Chapter 7, page 4
From the earlier chapters, we already know that if the exposure has systematic risk, the price
for future delivery differs from the expected cash market price because of the existence of a risk
premium. Suppose that the CAPM accurately captures differences in priced risk across securities. In
this case, if the beta of the SFR is positive, investors who take a long SFR position expect to be
rewarded for the systematic risk they bear. This means that the forward price of the SFR is below the
expected spot price for the maturity date of the forward contract. Since Export Inc. takes on a short
SFR position, it follows that it expects to receive fewer dollars if it hedges than if it does not. Though
hedging decreases the expected payoff in this case, there is no sense in which this reduction in the
expected payoff is a cost of hedging. If the firm does not hedge, it pays the risk premium in the form
of a lower value on its shares. If the firm hedges, it pays the risk premium in the form of a lower
forward price. Either way it pays the risk premium. To avoid treating compensation for risk as a cost,
it is generally best to focus on expected cash flows net of the premium for systematic risk as discussed
in chapter 4.
From the analysis of chapter 2, if one knows the beta of the SFR, one can compute the
market’s expected spot exchange rate for the maturity of the forward contract. For instance, if the
beta of the SFR is zero, then the market’s expected spot exchange rate is equal to the forward
exchange rate. Let’s suppose that this is the case. If Export Inc. has the same expectation for the spot
exchange rate as the market, then there is no cost of hedging except for transaction costs. Suppose,
however, that Export Inc. believes that the expected price for the SFR at maturity of the exposure
is higher than the forward exchange rate. In this case, Export Inc. believes that the forward exchange
rate is too low. With our assumptions, this is not because of a risk premium but because the market
is wrong from Export Inc.’s perspective. By hedging, the firm reduces its expected dollar payoff by
Chapter 7, page 5
the exposure times E (S(June 1)) - F, where E(S(June 1)) is Export Inc.’s expectation of the price of
the SFR on June 1, S(June 1), as of the time that it considers hedging, which is March 1. This
reduction in expected payoff is a cost of hedging. Hence, when the firm decides to eliminate all risk,
it has to think about whether it is worth it to do so given this cost. Box 7.1. The cost of hedging and
Daimler-Benz’s FX losses shows how one company thought through this issue at a point in time.
Besides having different expectations than the market for the cash price, the firm faces costs
of hedging in the form of transaction costs. In the case of Trading Inc., it would have to consider the
fact that the bid-ask spread is smaller on the spot market than on the forward market. This means that
if there is no systematic risk, the expected spot exchange rate for a sale is higher than the forward
exchange rate for a short position. The difference between these two values represents a transaction
cost due to hedging. We can think of the transaction cost of hedging as captured by a higher forward
price for a firm that wants to buy the foreign currency forward and as a lower forward price for a firm
that wants to sell the foreign currency forward. There may also exist commissions and transaction
taxes in addition to this spread.
When the manager of Export Inc. responsible for the hedging decision evaluates the costs and
benefits of hedging, she has to compare the benefit from hedging, which is the reduction in the cost
of CaR, with the cost of hedging. Suppose the manager assumes that one unit of CaR always costs
$0.5 and that hedging is costly because the forward exchange rate is too low by two cents. The
manager could conclude that the forward exchange rate is lower than the expected spot exchange rate
by 2 cents either because of transaction costs or because the market has it wrong. Remember that
from Table 6.1. the forward exchange rate on March 1 for delivery three months later is $0.6862.
Hence, if the forward exchange rate is too low by 2 cents, the firm expects the spot exchange rate to
Chapter 7, page 6
be $0.7062 on June 1. In this case, going short SFR1M on the forward market saves the costs of
CaR. In chapter 6, CaR for Export Inc. was estimated to be $71,789.20. Consequently, hedging
increases firm value by 0.5*$71,789.20 = $35,894.60. At the same time, however, the manager of
Export Inc. expects to get $20,000 more by not hedging than by using the forward hedge. In this
case, the cost of CaR is so high that Export Inc. hedges even though the forward contract is
mispriced. If the forward contract is mispriced by four cents instead, the firm loses $40,000 in
expected payoff by hedging and gains only $35,894.60 through the elimination of the cost of CaR.
It obviously does not pay for the firm to hedge in this case even though risk reduces firm value.
One can argue whether the firm might have valuable information that leads it to conclude that
the market’s expectation is wrong. If markets are efficient, such information is hard to get but this
does not mean that it does not exist or that a firm cannot get it. The problem is that it is too easy to
believe that one has such information. Everybody has incentives to correct mistakes that the market
might have made because this activity makes money. Some individuals and institutions have
tremendous resources at their disposal and spend all their time searching for such mistakes. This
makes it hard to believe that individuals and institutions for whom watching the markets is at best a
part-time activity are likely to have information that is valuable. Having information publicly available,
such as newspaper articles, analyst reports or consensus economic forecasts, can’t be worth much.
Since everybody can trade on that information, it gets incorporated in prices quickly. A test of
whether information is valuable at least from the perspective of the manager is whether she is willing
to trade on that information on her own account. If she is not, why should the shareholders take the
risk? Remember that here, the risk involves a cost, namely the impact of the risk on CaR. The bet the
manager wants to take has to be good enough to justify this cost.
Chapter 7, page 7
Let’s go back to the case where the forward exchange rate is mispriced by $0.04 when the
cost of CaR is $0.5 per unit of CaR. In this case, it does not pay for Export Inc. to hedge. However,
suppose that Export Inc. not only does not hedge but in addition takes on more foreign exchange risk
by going long SFRs on the forward market to take advantage of the mispricing on the forward
market. Each SFR it purchases on the forward market has an expected profit of $0.04 and increases
the CaR by $0.0717892. Each dollar of CaR costs the firm $0.5. So, the net expected gain from
purchasing a SFR forward is $0.04 - 0.5*$0.0717892, or $0.0041054. Hence, with this scenario, the
firm can keep increasing its risk by buying SFRs forward with no limits. In this example, therefore,
the firm is almost risk-neutral. When a risk has enough of a reward, there is no limit to how much the
firm takes of this risk. One should therefore be cautious about assuming that the cost of CaR is fixed
per unit of CaR. Eventually, the cost of CaR has to increase in such a way that the firm does not take
more risks.
In general, one would therefore expect the cost of CaR per unit to increase as CaR increases.
In this case, there is always an optimal hedge position. The optimal hedge is the one that maximizes
the firm’s expected cash flow taking into account the cost of CaR. Let Export Inc. go short h SFR
units on the forward market. Suppose further that the cost associated with CaR is " times the square
of CaR, "(CaR)2, so that the cost per unit of CaR increases with CaR. In this case, the expected
payoff net of the cost of CaR for Export Inc. is:
Expected cash flow net of the cost of CaR
= Expected cash flow - Cost of CaR per unit of CaR*CaR
= Expected cash flow - "(CaR)2
Chapter 7, page 8
= (1M - h)E(S(June 1)) + h*F - " [(1M - h)1.65Vol(S(June 1))]2
Figure 7.1. shows the expected cash flow net of the cost of CaR as a function of h with our
assumptions and shows that it is a concave function. Since the expected cash flow net of the cost of
CaR first increases and then falls as h increases, there is a hedge that maximizes this expected cash
flow. We can obtain this hedge by plotting the expected cash flow net of the cost of CaR and
choosing the hedge that maximizes this expected cash flow. Assuming an expected spot exchange rate
of $0.7102, a forward exchange rate of $0.6902, and a volatility of the spot exchange rate of
0.0435086, we find that the optimal hedge is to go short SFR980,596 if the cost of CaR is $0.0001
per unit, and to go short SFR805,964 if the cost of CaR is $0.00001 per unit. Since hedging has a
positive cost because the forward exchange rate is too low, the firm hedges less as the cost of CaR
falls.
An alternative approach to finding the optimal hedge is to solve for it directly. At the margin,
the cost of hedging has to be equal to the benefit of hedging. Consequently, we can find it by setting
the cost of hedging slightly more (the marginal cost of hedging in the following) equal to the benefit
from doing so (the marginal benefit).
By hedging, one gives up the expected spot exchange rate to receive the forward exchange
rate for each SFR one hedges. Consequently, the expected cost of hedging ) more unit of the cash
position, where ) represents a very slight increase in the size of the hedge, is:
Marginal cost of hedging = )(E(S(June 1)) - F)
Chapter 7, page 9
For given ), the marginal cost of hedging does not depend on h. Further, the marginal cost of
hedging depends only on the difference between the spot exchange rate the firm expects and the
forward exchange rate as long as neither transaction costs nor the forward exchange rate depend on
the size of the hedge. Note that in our example, we assume that the spot exchange rate has no
systematic risk. If the spot exchange rate has a positive beta, then we know that the forward exchange
rate is lower than the expected spot exchange rate before transaction costs. In that case, we would
want to consider the marginal cost of hedging after taking into account the risk premium, so that we
would subtract the risk premium from the cash position to compute a risk-adjusted cash flow. Once
cash flow is adjusted for risk, hedging has no cost in the absence of transaction costs when the firm
has the same expectation as the market. When the firm’s expectation of the spot exchange rate differs
from the market’s expectation, hedging can have a negative cost if the firm believes that the forward
exchange rate is too high after taking into account transaction costs.
The marginal benefit of hedging is the decrease in the cost of CaR resulting from hedging )
more units of the cash position when ) represents a very slight increase in the size of the hedge:1
Marginal benefit of hedging for firm with hedge h
= cost of CaR for hedge h - cost of CaR for hedge (h + ∆ )
= α ((1M - h)1.65Vol(S(June1))2 - α ((1M - h) - ∆ ) *1.65* Vol(S(June1))2
= 2α (1M - h)∆ (1.65Vol(S(June 1))2 - α∆21.65Vol(S(June 1))2
= 2α (1M - h)∆ (CaR per unit of exposure)2
To go to the last line, we use the fact that as long as ) is small enough, the square of ) is so small
1
This marginal benefit is obtained by taking the derivative of the cost of CAR with respect
to the size of the hedge.
Chapter 7, page 10
that the second term in the third line can be ignored. Because the cost of risk for the firm is " times
CaR squared, the marginal benefit of hedging turns out to have a simple form. The marginal benefit
of hedging depends on ", the unhedged exposure of SFR1M - h, and the CaR per unit of exposure
squared. The CaR per unit of exposure is fixed. Consequently, the marginal benefit of hedging falls
as the unhedged exposure falls. In other words, the more hedging takes place, the less valuable the
next unit of hedging. Since the firm’s CaR falls as the firm hedges more, it follows that the lower the
CaR the lower the marginal benefit of hedging. Figure 7.2. shows the marginal cost and the marginal
benefit curves of hedging. The intersection of the two curves gives us the optimal hedge. Increasing
the marginal cost of hedging moves the marginal cost of hedging curve up and therefore reduces the
extent to which the firm hedges. Increasing the cost of CaR for a given hedge h moves the marginal
benefit curve of hedging up and leads to more hedging. We can solve for the optimal hedge by
equating the marginal cost and the marginal benefit of the hedge. When we do so, ) drops out
because it is on both sides of the equation. This gives us:
h = 1M -
E(S(June 1)) - F
2α (CaR per unit of exposure) 2
Suppose that the forward exchange rate is $0.6902, the expected spot exchange rate is 2 cents higher,
" is equal to 0.0001, and the CaR per SFR is $0.0717892. In this case, the optimal hedge applying
our formula is to sell forward SFR998,596. Not surprisingly, the optimal hedge is the same as when
we looked at Figure 7.1. Figure 7.3. shows the optimal hedge for different values of the expected spot
exchange rate at maturity of the exposure and different values of ". In this case, the hedge ratio
becomes one as " increases and the optimal hedge ratio is below one if the forward exchange rate is
Chapter 7, page 11
below the expected spot exchange rate and above one otherwise. If the forward exchange rate is
equal to the expected spot exchange rate, the hedge is to go short SFR1M. As the expected spot
exchange rate increases, the hedge falls and can become negative. The hedge falls less if the cost of
CaR is higher because in that case risk is more expensive. As the forward exchange rate increases,
the optimal hedge increases because it is profitable to sell the SFR short when the forward exchange
rate is high relative to the expected spot exchange rate.
The approach we just presented to hedge when CaR is costly gives a very general formula for
the optimal hedge:
Optimal hedge when CaR is costly
The optimal hedge (the size of a short position to hedge a long exposure) when the cost of CaR is
"*CaR2 and when the cost of hedging does not depend on the size of the hedge is given by the
following expression:
 Expected cash price per unit − Price for future delivery per unit 
h = Exposure − 
 (7.1.)
α *2 *(CaR per unit of exposure)2


To understand the expression for the optimal hedge, suppose first that the cost of CaR is very large.
In this case, " is very large so that the second term in the expression is trivial. This means that h is
about equal to the total exposure and hence close to the hedge that we would take in the absence of
hedging costs. (Remember from the previous chapter that in the absence of basis risk, the minimumvolatility hedge is to go short the exposure, so that h is equal to the exposure.) As " falls, so that CaR
becomes less expensive, it becomes possible for the firm to trade off the cost and benefit of hedging.
Chapter 7, page 12
The numerator of the second term is the marginal cost of hedging. As this marginal cost increases,
the firm hedges less. The extent to which the firm hedges less depends on the benefit of hedging. As
the benefit from hedging increases, the expression in parentheses becomes closer to zero in absolute
value, so that the firm departs less from the minimum-volatility hedge.
Let’s consider how the analysis would be different if we looked at Trading Inc. instead. This
firm uses VaR which is measured over a one-day period. This means that it focuses on the value of
the position tomorrow. So far, when we computed the one-day VaR, we assumed that there was no
expected gain in the position over one day. In this case, it is optimal to reduce VaR to zero if VaR
is costly because there is no offsetting gain from taking on risk. If there is an offsetting gain, then
Trading Inc. trades off the impact of hedging on the expected gain from holding the position
unhedged with the gain from reducing the VaR through hedging. The resulting optimal hedge is one
where the firm equates the cost of hedging one more unit on the expected gain of its position with
the reduction in the cost of VaR from hedging one more unit. Since the difference between the VaR
of Trading Inc. and the CaR of Export Inc. is that the VaR depends on the value tomorrow of
receiving spot exchange rate on June 1 rather than on the spot exchange rate on June 1, the formula
for the optimal hedge when CaR is costly extends to the case when VaR is costly as follows:
n = Exposure −
E[PV(Spot)] − E[PV(F)]
2 * α * (VaR per unit of unhedged spot) 2
where E[PV(Spot)] denotes the expected present value of the SFR unhedged position at the end of
the VaR measurement period (the next day if VaR is computed over the next day) and E[PV(F)]
Chapter 7, page 13
denotes the expected present value of the position at the end of the VaR measurement period if it is
hedged. Remember that the SFR position pays SFR1M in three months, so that E[PV(Spot)] is the
expected present value tomorrow of receiving SFR1M in three months. A forward contract that
matures in three months is a perfect hedge for the position, so that PV(F) denotes the present value
tomorrow of the forward exchange rate to be received in three months.
Let’s apply our formula to the case of Trading Inc. Suppose that Trading Inc. computes VaR
over one day. From our earlier analysis, we know that each SFR of exposure has a present value of
0.96SFR. Say that we expect the spot SFR to be at $0.76 and the forward SFR to be at $0.75
tomorrow. The PV(Spot) is then 0.96*$0.76 = $0.7296 and PV(Forward) is $0.969388*$0.75 =
$0.727041. The exposure is SFR1M. Vol(PV(Spot)) is the volatility of 0.96 units of spot over one
day, or 0.96*0.0054816 = 0.005262336. We choose " = 0.001. Putting all this in the formula, we
obtain:
PV(Spot) − PV(F)
2 *1.65* α * (VaR per unit of spot) 2
0.7296 − 0.727041
= 1M −
2 * 0.001* (1.65 * 0.005262336) 2
= 983,029
n = Exposure −
Hence, in this case, the firms goes short just about all its exposure because VaR is sufficiently costly
that eliminating the foreign exchange rate risk is the right thing to do. This is not surprising because
with our assumption of " = 0.001, the cost of VaR if the firm does not hedge is large. With our
formula, the cost is $75,392, or 0.001*(1M*1.65*0.00526233)2. Suppose that we make the cost of
Chapter 7, page 14
VaR smaller, so that " = 0.0001. The cost of VaR if the firm does not hedge becomes $7,539.2. In
this case, the hedge becomes one where we sell short SFR830,287. Hence, as the cost of VaR falls,
the firm hedges less when hedging has a cost (the cost here is that the firm sells something it thinks
is worth $0.7269 per SFR and gets for it something worth $0.727041 per unit of SFR).
As with the case of Export Inc., we can extend our formula for Trading Inc. to a general
formula for the optimal hedge when the cost of VaR is "(VaR)2:
Optimal hedge when VaR is costly
The optimal hedge when the cost of VaR is "VaR2 is given by the following expression:
n = Exposure
 Expected cash value per unit − Expected price for future delivery per unit  (7.2.)
−

2α ( VaR per unit of exposure)2


This expression has the same interpretation as the expression for the optimal hedge for CaR.
However, now the expected cash value is the one at the end of the VaR horizon and so is the price
for future delivery.
Both formulas for the optimal hedge, equations (7.1.) and (7.2.), have the same important
feature: the optimal hedge is to go short the exposure minus an adjustment factor reflecting the cost
of the hedge. The adjustment factor has the striking property that it does not depend on the exposure
directly. This is because the decision of how much to hedge depends on the marginal benefit and
marginal cost of CaR rather than on the level of the firm’s CaR. If the adjustment factor is positive,
so that it is costly to hedge, one has to treat the formula with care if the adjustment factor leads to
Chapter 7, page 15
an optimal hedge so extreme that it has the opposite sign from the exposure. For instance, this could
be a case where the firm is long in SFRs and the optimal hedge given by the formula involves buying
SFRs for future delivery. Such a situation could arise for two different reasons. First, suppose that
there are no transaction costs of hedging, but the firm believes that the expected spot price is high
compared to the forward price after adjusting for systematic risk. In this case, going long on the
forward market represents a profit opportunity for the firm if it does not have an exposure and it
should take advantage of it. The exposure means that the firm does not go long as much as it would
without it. Second, suppose that the spot exchange rate expected by the firm is equal to the forward
price before transaction costs, but that the forward price net of transaction costs is low for the short
position because of these costs. In this case, going long forward does not present a profit opportunity
for the firm because the forward price for going long would be different from the forward price for
going short and would be higher than the expected spot price. Hence, if the formula leads to a
negative h with a positive exposure because of transaction costs, this means that the firm should take
no position because going long would create an economic loss. When the expected spot price differs
from the forward price because of transaction costs, therefore, only nonnegative (negative) values
of h should be considered when the exposure is a long (short) position.
Section 7.2. Multiple cash flows occurring at a point in time.
Let’s extend the example of the previous chapter to consider the impact on optimal hedges
of multiple exposures. We still consider Export Inc. Now, however, the firm has two exposures that
mature on June 1. One cash flow is the one the firm already has, the cash flow of SFR1M. The other
exposure is a cash flow of ¥1M. Let’s write Syen(June 1) for the spot exchange rate of the yen on
Chapter 7, page 16
June 1 and SSFR(June 1) for the spot exchange rate of the SFR at the same time. The variance of the
unhedged cash flow is now:
Variance of unhedged cash flow = Var[1M*Syen(June 1) + 1M*SSFR(June 1)] =
Var[1M*Syen(June 1)] + 2Cov[1M*SSFR(June 1),1M*Syen(June 1)] + Var[1M*SSFR(June 1)]
This formula is similar to the formula we had in chapter 2 for the variance of the return of a portfolio
with two stocks. Not surprisingly, the variance of the total cash flow depends on the covariance
between the cash flows. Consequently, the general formula for the variance of unhedged cash flow
when one has m cash flows accruing at date T is:
Variance of unhedged cash flow accruing at time T '
ji'1 jj'1 Cov[Ci(T),Cj(T)]
i'm
j'm
(7.3.)
where Ci(T) is the i-th cash flow accruing at time T. (Remember that the covariance of a random
variable with itself is its variance.) When a firm has several distinct cash flows accruing at the same
point in time, there is a diversification effect like in portfolios. As in our analysis of portfolio
diversification, the diversification effect is due to the fact that in general the covariance of two
random variables is less than the product of the standard deviations of two random variables because
the coefficient of correlation is less than one. The variance of total unhedged cash flow falls as the
correlation coefficient between the two cash flows falls. If the correlation coefficient between the two
Chapter 7, page 17
cash flows is negative, the firm may have very little aggregate risk. This diversification effect can be
used by the firm to reduce the extent to which it has to hedge through financial instruments to reach
a target CaR.
Let’s look at an extreme example. Suppose that the yen and the SFR cash flows have the same
variance but are perfectly negatively correlated. In this case, the aggregate unhedged cash flow of the
firm has no risk, so that no hedging is required. This is because in our formula for the unhedged cash
flow, the two variance terms are equal and the covariance term is equal to the variance term times
minus one. Nevertheless, if the firm looked at the cash flows one at a time, it would conclude that it
has two cash flows that need to be hedged. It would hedge each cash flow separately. If it used
forward contracts for each, it could eliminate all risk. Yet, this is risk that it does not need to
eliminate because in the aggregate it cancels out.
When hedging is costly, the diversification effect can reduce hedging costs by reducing the
number and size of the hedges the firm puts on. To take this diversification effect into account, the
firm should compute optimal hedges from the aggregate cash flow rather than individual cash flows.
Using our example, the aggregate cash flow is:
CUnhedged(June 1) = 1M*Syen(June 1) + 1M*SSFR(June 1)
Suppose that we can use both a SFR futures contract and a yen futures contract that mature on June
1 or later. In this case, the hedged cash flow is:
Chedged = CUnhedged(June 1) - hSFR[GSFR(June 1) -GSFR(March 1)] - hyen[Gyen(June 1) - Gyen(March 1)]
Chapter 7, page 18
where hSFR is the number of units of the SFR futures and hyen is the number of units of the yen futures
we short before tailing. The SFR futures contract has price GSFR(June 1) on June 1 and the yen futures
contract has price Gyen(June 1). In the presence of tailing, GSFR(June 1) - GSFR(March 1) is the gain
from a long position in the SFR futures contracts from March 1 to June 1 equal every day to the
present value of a discount bond the next day that pays SFR1 on June 1. The problem of minimizing
the volatility of hedged cash flow is similar to the one we faced in chapter 6. Since we now have
exposures in two currencies, we want to find a portfolio of positions in futures contracts that evolves
as closely as possible with the value of the cash position. This amounts to finding the portfolio of
futures positions such that changes in the value of these positions best predict changes in the value
of the cash position. We can use a regression to solve our problem.
Let’s assume that the exchange rate and futures price increments have the same joint
distribution over time and that they satisfy the multivariate normal increments model presented in
chapter 6. This means that we can use ordinary least squares regression to obtain the volatilityminimizing hedges. We know Export Inc.’s exposures, so that we can compute what the value of the
cash position would have been for past values of the exchange rates. Hence, we can regress changes
in the value of the cash position using historical exchange rates on a constant and changes in the
futures prices. The regression coefficients provide us with optimal volatility-minimizing hedges since,
with our assumptions, the distribution of past changes is the same as the distribution of future
changes. The coefficients of the independent variables are the optimal hedges for the dependent
variable. Here, the dependent variable is the change in the value of the cash position. Consequently,
the coefficient for the change in the SFR futures price is the number of SFR units for future delivery
one goes short and the coefficient for the change in the yen futures price is the number of units of yen
Chapter 7, page 19
for future delivery one goes short. The regression we estimate is therefore:
)SSFR(t) *1M + )Syen(t)*1M = constant + hSFR *)GSFR(t) + hyen*)Gyen(t) + ,(t)
where ) denotes a change over a period, so that )SSFR(t) is the change in the SFR spot exchange rate
over a period starting at t. The period is the measurement interval for the regression, so that it is a
day if we use daily observations. ,(t) is the random error of our regression. To hedge the cash flow,
Export Inc. should go short hSFR (hyen) SFR (yen) for future delivery.
Let’s try to understand how the number of SFRs for future delivery Export Inc. should go
short differs from the hedge when Export Inc. has only a SFR exposure. The hedge differs because
the SFR futures contract helps hedge the yen exposure and because the yen futures contract helps
hedge the SFR exposure. We discuss these two reasons for why the SFR futures position differs in
this chapter from what it was in the previous chapter in turn:
1. The SFR futures contract helps hedge the yen exposure. To discuss this, let’s consider
the situation where Export Inc. hedges using only the SFR futures contract. In this case, treating each
exposure separately, we could estimate a regression of changes in the SFR cash position on changes
in the SFR futures price to get the hedge for the SFR exposure and we could estimate a regression
of changes in the yen cash position on changes in the SFR futures price to obtain the hedge for the
yen exposure. Instead of treating each exposure separately, we can consider them together. In this
case, we estimate a regression where we regress the change in the total cash position on the change
in the SFR futures price. This regression is equivalent to taking the regressions for the SFR and yen
exposures and adding them up. The coefficient of the SFR futures price in this regression is the sum
Chapter 7, page 20
of the coefficients in the regressions for the SFR and yen exposures. The hedge is therefore the sum
of the hedges for the individual exposures. This means that the hedge for the aggregate cash flow
differs from the hedge for the SFR exposure if the regression of the yen on the SFR futures price has
a slope different from zero. This is the case if the SFR futures contract can be used to reduce the risk
of a yen exposure.
The regression coefficient in the regression of the yen position on the SFR futures price has
the same sign as the covariance between changes in the yen spot exchange rate and changes in the
SFR futures price. If the covariance between changes in the yen spot exchange rate and changes in
the SFR futures price is positive, going short the SFR futures contract helps hedge the yen exposure
so that Export Inc. goes short the SFR futures contract to a greater extent than if it had only the SFR
exposure. If the covariance is negative, it has to buy SFRs for future delivery to hedge the yen
exposure. Since it is optimal to sell SFRs for future delivery to hedge the SFR exposure, this means
that Export Inc. sells fewer SFRs for future delivery than it would if it had only a SFR exposure and
hence takes a smaller futures position. In the extreme case where the two exchange rate increments
are perfectly negatively correlated and have the same volatility, the firm takes no futures positions
because the yen exposure already hedges the SFR exposure.
2. The yen futures contract helps hedge the SFR and yen exposures. The SFR futures
contract is likely to be an imperfect hedge for the yen exposure. Suppose, for instance, that the yen
futures contract is a perfect hedge for the yen exposure and that the SFR futures price is positively
correlated with the yen spot exchange rate. In this case, our analysis of the SFR futures hedge of
Export Inc. if it does not use the yen futures contract would lead to the result that it goes short more
than SFR1M for future delivery because the SFR futures contract helps hedge the yen exposure.
Chapter 7, page 21
However, if Export Inc. uses the yen contract also, it will be able to hedge the yen exposure perfectly
with that contract. This means that it will take a smaller short position in the SFR futures contract
than it would if it used only the SFR futures contract. It might even turn out that the yen futures
contract is useful to hedge the SFR exposure in conjunction with the SFR futures contract, in which
case the SFR futures short position would drop even more. The yen futures contract will be helpful
to hedge the SFR exposure if it can be used to reduce the volatility of the hedged SFR exposure when
only the SFR futures contract is used.
From this discussion, it follows that the hedge for the SFR exposure will be the same when
Export Inc. has multiple exposures as when it has only the SFR exposure if (a) the SFR futures
contract does not help to hedge the yen exposure and (b) the volatility of the hedged SFR exposure
when only the SFR futures contract is used cannot be reduced by also using the yen futures contract.
This is the case if the SFR spot exchange rate is uncorrelated with the yen futures price and if the yen
spot exchange rate is uncorrelated with the SFR futures price.
The optimal hedges can be obtained using a similar regression irrespective of the number of
different futures contracts we use to hedge an aggregate cash flow. Hence, if Export Inc. had a cash
flow at date T corresponding to fixed payments in x currencies and we had m futures contracts to
hedge the total cash flow, we could obtain positions in these m futures contracts by regressing the
increment of the total cash flow on the increments of the m futures contracts. Since the total cash
flow is the sum of known payments in foreign currencies, the hedges can be obtained from a
regression using historical data on exchange rates and futures prices assuming that the joint
distribution of futures prices and exchange rates is constant over time. We provide an application of
the use of multivariate regression to compute optimal hedge ratios in section 7.4.
Chapter 7, page 22
Focusing on exposures of aggregate cash flows makes it possible to take advantage of the
diversification effect across cash flows in a way that is not apparent in our discussion so far but is
very important in practice. Consider a firm that has lots of different exposures. It could be that it finds
it difficult to construct good hedges for each of these exposures because the right futures contracts
are not available. In other words, possible hedges would have too much basis risk. It could be,
however, that when the firm considers the aggregate cash flow, there are some good hedges because
the basis risks get diversified in the portfolio. A good analogy is to consider hedging a stock against
market risk. Stock betas are generally estimated imprecisely for individual stocks. Often, they are not
significantly different from zero and the R-squares of the regressions are low. However, once the
stocks are put in a portfolio, the unsystematic risk gets diversified away, beta is estimated much more
precisely, and the R-square is larger. Hence, risk management may be feasible at the level of
aggregate cash flow when it does not seem promising at the level of individual exposures.
Section 7.3. Cash flows accruing at different dates.
The discussion in section 7.2. showed the benefit of using a total cash flow approach to
obtaining the optimal hedge ratios. Applying this approach, it can turn out that a firm needs very little
hedging even though it has large cash flows in a number of different currencies. However, in general,
payments accrue at different dates, thereby creating exposures that have different maturities. Let’s
now investigate the case where payments occur at different dates.
If all payments are made at the same date, there is only one cash flow volatility to minimize.
When payments occur at different dates, it is not clear which cash flow volatility we should minimize.
We could try to minimize the cash flow volatility at each date. Doing so ignores the diversification
Chapter 7, page 23
that takes place among payments at different dates. For instance, suppose that a firm must make a
payment of SFR0.9 in 90 days and receives a payment of SFR1M in 100 days. Hedging each cash
flow separately ignores the fact that the payment the firm must make is already to a large extent
hedged by the payment it will receive.
The only way we can take diversification across time into account is if we bring all future
payments to a common date to make them comparable. One solution is to use borrowing and lending
to bring the cash flows to the end of the accounting period and consider the resulting cash flow at that
time. Another solution is to bring all future payments back to the present date. This second solution
amounts to taking present values of future payments we will receive and will have to make. In this
case, we become concerned about the risk of the increments of the present value of the firm over
some period of time. We saw that one measure of such risk is VaR. If all cash flows to the firm are
taken into account in this case and the multivariate normal increments model holds for the cash flows,
VaR is directly proportional to the volatility of firm value increments since firm value is the present
value of future cash flows and since volatility and VaR are proportional to each other with normally
distributed increments. Instead of minimizing the volatility of cash flow at a point in time, volatility
minimization now requires that we minimize the volatility of the increments of the present value of
the firm for the next period of time. With this approach, the optimal hedges change over time because
payments are made, so that the present value of the payments exposed to exchange rate fluctuations
changes over time.
To understand this, let’s look at Export Inc. assuming now that it receives a payment of
SFR1M on June 1 and must make a payment of SFR1M on date T. Suppose that Export Inc. wants
to hedge with a futures contract maturing later than date T. The present value of the payments is:
Chapter 7, page 24
Present value of the payments = (PSFR(June 1) - PSFR(T))*S(March 1)*1M
where PSFR(T)) is the price on March 1 of a discount bond paying SFR1 at T. With this equation, the
firm has an exposure of SFR1M*(PSFR(June 1) - PSFR(T)). Viewed from March 1, the firm has little
exposure if T is close to June 1 since the two discount bonds have similar values. This is because
until June 1 the positive gain from an appreciation of the SFR on the dollar value of the payment to
be received offsets the loss from an appreciation of the SFR on the dollar value of the payment to be
made. One way to understand this limited exposure is to compute the VaR over one day. Suppose
that T is September 1, so that it is six months from March 1, and PSFR(March 1, September 1)) = 0.92.
Using the data for the SFR exchange rate, we have:
One-day VaR = 1.65Vol()S(March 1))[1M*(PSFR(March 1, June 1) - PSFR(March 1, September 1))]
= 1.65*0.0054816*1M**0.96 - 0.92*
= $361.8
where *a* denotes the absolute value of a and Vol()S(March 1)) denotes the volatility of the one-day
increment of the SFR spot exchange rate. The one-day VaR is dramatically smaller than if Export Inc.
received only the payment of SFR1M on June 1, which is $8,683, or 1.65*0.0054816*1M*0.96.
Note, however, that if Export Inc. makes the payment a long time in the future, its VaR becomes
closer to what it would be absent that payment. For instance, if the payment were to be made in ten
years and the price of a discount bond maturing in ten years is 0.2, the one-day VaR is $6,874.
To find the hedge that minimizes the volatility of hedged firm value, we need to find a
minimum-volatility hedge ratio for a one-day SFR exposure and then multiply this hedge ratio by our
Chapter 7, page 25
exposure. We already know that the volatility-minimizing hedge ratio over one day can be obtained
by regressing the daily change in the SFR on the daily change in the futures price. Letting )S(t) and
)G(t) be respectively the change in the spot exchange rate and the change in the futures price from
date t to the next day, the regression coefficient is Cov()S(t),)G(t))/Var()G(t)). This gives us the
hedge ratio per unit of exposure. The exposure is SFR1M*(PSFR(June 1) - PSFR(T)). Consequently,
multiplying the exposure with the hedge ratio, we have the optimal hedge h:
h = 1M * (P SFR (May 2, August 1) − PSFR (May 2,September 1)) *
Cov( ∆S(t), ∆G(t))
Var( ∆G(t))
Using a regression coefficient of 0.94, a maturity for the payment to be made of September 1, and
PSFR(March 1, September 1) = 0.92, the optimal hedge consists in going short SFR37,600 as opposed
to going short SFR902,400 if there was no payment to be made. Over the next day, the two SFR
exposures almost offset each other so that the VaR is small and only a small hedge is required.
Exposures change over time. This was not an important issue when we analyzed Export Inc.’s
SFR exposure in the previous chapter because at some point the exposure matured and disappeared.
Here, however, things are dramatically different. As of June 1, the offsetting effect of the two
exposures disappears. Consequently, on June 2, the exposure of Export Inc. is the exposure resulting
from having to make a SFR1M payment at date T. Hence, the exposure increases sharply on June 1,
which makes the hedge change dramatically then also. Before the payment is received on June 1,
Export Inc. goes short SFRs for future delivery by a small amount. After the payment is received, the
only SFR exposure of Export Inc. is the payment it has to make later. As of that time, an unhedged
Export Inc. is therefore short SFRs without any offset from receiving a SFR payment. This means
Chapter 7, page 26
that after June 1, Export Inc. must switch its hedge from being short SFRs for future delivery to being
long SFRs for future delivery. Figure 7.4. shows how the hedge changes over time. The VaR also
evolves over time, in that it is very small until June 1, when it increases dramatically. Since, after the
payment is received on June 1, Export Inc.’s exposure is a payment to be made in three months of
SFR1M, the VaR on June 2 is essentially the same as the VaR on March 1 in the previous chapter.
This is because the VaR of a short SFR1M exposure is the same as the VaR of a long SFR1M
exposure.2
It is important to note that the reasons that a firm finds it advantageous to manage risk are
sometimes such that they preclude it from taking advantage of diversification in cash flows across
time. This is because a firm might be concerned about the cash flow in a given period of time, for
instance a specific year. If that cash flow is low, the firm might not be able to invest as much as it
wants. The fact that there is some chance that a cash flow might later be unexpectedly high is not
relevant in this case. The firm will have to hedge cash flows that accrue before the investment has to
be financed separately to prevent a liquidity crunch.
Section 7.4. The i.i.d. returns model.
So far, we have assumed that the increments of the cash position and of the futures price have
the same joint distribution over time and are serially uncorrelated. In the case of increments to
exchange rates, this made sense. There is little reason, for instance, to believe that the volatility of the
2
This statement is correct only if the VaR is computed ignoring the expected return,
which is standard practice for computing the one-day VaR.
Chapter 7, page 27
exchange rate increments changes systematically with the level of the exchange rate. In the last
section, however, we focused on the increments of present values. In this case, it becomes harder to
assume that these increments have a constant distribution irrespective of the level of the present
values. A simple way to check whether the i.i.d. increments model is appropriate is to perform the
following experiment. Since, with the i.i.d. increments model, the distribution of increments is always
the same, we can put increments in two bins: those increments that take place when the price or
present value is above the sample median and the other increments. The mean and volatility of
increments in the two bins should be the same if the i.i.d. increments model holds.
To understand the issue, think about a common stock, say IBM. The statistical model we have
used so far would state that the distribution of the dollar increment of a share of IBM is the same
irrespective of the price of a share. This implies that if the price of an IBM share doubles, the
expected return of the share falls in half. Why so? Suppose that we start with a price of $50 and say
an expected increment of $1 over a month. This corresponds to a monthly expected return of 2%
(1/50 times 100). Suppose now that the price doubles to become 100. In this case, the assumption
of i.i.d. increments implies that the expected increment is also $1. Consequently, the expected return
on the share becomes 1% (1/100 times 100). With i.i.d. increments, therefore, the expected return
on a share falls as its price increases! We know that this cannot be the case if the CAPM holds since
with the CAPM the expected return depends on beta rather than on the stock price. Another
implication of i.i.d. dollar increments is that the standard deviation of the increments becomes a
smaller fraction of the price as the price increases. One way to put that is that increments of prices
that have increased a lot experience a fall in volatility.
With stock prices, a more sensible approach is to assume that the returns are i.i.d. To see the
Chapter 7, page 28
implications of this assumption, suppose that you want to hedge IBM with a position in the market
portfolio. The beta of IBM is the regression coefficient of a regression of the return of IBM on the
return of the market portfolio. Using the logic of our hedging analysis, this regression coefficient
gives us the best forecast of the return of IBM for a given return on the market portfolio. A return
is a dollar increment on one dollar. Hence, if the beta of IBM is 1.1, for each $1 gain on the market
portfolio, a dollar invested in IBM earns $1.1. This means that to hedge IBM we go short $1.1. on
the market. Suppose that we have $1M invested in IBM. Then, we go short $1.1M in the market. If
the market falls by 10%, we expect our IBM investment to fall by 11%. Consequently, we expect to
lose $110,000 on our IBM investment and gain $0.10*1.1M, or $110,000 on our short position in
the market.
After the market fell by 10%, our investment in IBM is worth $890,000 if IBM fell by the
amount predicted by our regression, namely a return equal to IBM’s beta times the market return.
Let’s assume that this is the case. To hedge this, we need to go short 1.1*$890,000 in the market.
If the joint distribution of returns is i.i.d., the joint distribution of dollar increments is not. The hedge
therefore has to change as the value of the position changes. To see how important it is to change the
hedge, suppose we forget to change it. We therefore have a short position in the market of $1.1M.
This position is too large compared to the short position we should have of 1.1*$890,000, or
$979,000. Suppose now the market increases by 10% and IBM increases by 11% as predicted by the
regression equation. In this case, we gain 0.11*$890,000 on IBM, or $97,900, and we lose
0.10*$1.1M, or $110,000, on our short position on the market. On net, therefore, if we forget to
change the hedge, we lose $12,100 as the market increases by 10%. If we had adjusted the hedge,
we would have lost 0.10*$979,000 on our short position on the market and this loss would have
Chapter 7, page 29
offset the gain on IBM.
When the distribution of the returns is i.i.d. instead of the distribution of the dollar increments,
the regression of the dollar increments of the cash position on the dollar increments of the hedging
instrument is misspecified. Remember that the regression coefficient in such a regression is the
covariance of the dollar increments of the cash position with the dollar increments of the hedging
instrument divided by the variance of the dollar increments of the hedging instrument. If returns are
i.i.d., the covariance term in the regression coefficient depends on the value of the cash position and
on the price of the hedging instrument. Therefore, it not constant. The regression of the returns of
the cash position on the returns of the hedging instrument is well-specified because, if returns are
i.i.d., the covariance between the returns of the cash position and the returns of the hedging
instrument is constant. The regression of the returns of the cash position on the returns of the hedging
instrument gives us the optimal hedge ratio for a dollar of exposure. In the case of IBM, the
regression coefficient is 1.1. which means that we need to be short the market for $1.1 for each $1
we are long in IBM. Multiplying the regression coefficient by the dollar exposure gives us the dollar
short position we need. If the hedging instrument is a futures contract, we have no investment and
hence there is no return on the futures contract. However, we can still apply our reasoning in that we
can use as the return in our regression analysis the change in the futures price divided by the futures
price.
Our discussion so far in this section gives us a general result for the hedge ratio when returns
are jointly i.i.d.:
Volatility-minimizing hedge when returns are i.i.d.
Chapter 7, page 30
Consider a cash position worth C dollars. The return on the cash position is r(cash). The return on
the hedging instrument is r(hedge). With this notation, the volatility-minimizing hedge is given by:
Volatility - minimizing hedge of cash position =
Cov[r(cash), r(hedge)]
* Cash position (7.4.)
Var[r(hedge)]
Equation (7.4.) makes clear that the optimal hedge in this case depends on the size of the cash
position. As the cash position increases, the optimal hedge involves a larger dollar amount short in
the hedge instrument.
With our IBM example, the regression coefficient of IBM’s return on the market’s return is
1.1. With a cash position of $1M, using equation (7.4.) tells us that we have to go short the market
$1M times 1.1., or $1.1M.
Section 7.5. The log increment model.
An issue that we have not addressed so far is the measurement interval for returns. One choice
that has substantial advantages over others is to use continuously compounded returns. Consider a
stock price P(t) at t. The price becomes P(t+)t) after an interval of time )t has elapsed. The
continuously compounded return over )t is the log price increment: log(P(t+)t)) - log(P(t)). We
have used the assumption that the dollar increment is i.i.d. and normally distributed. The dollar
increment is P(t+)t) - P(t). If we assume that the log price increment is i.i.d. and normally distributed
instead, the sum of log price increments is normally distributed. Since the log price increment is the
continuously compounded return over the period over which the increment is computed, this means
that continuously compounded returns are i.i.d. and normally distributed. We assume that the log
Chapter 7, page 31
price increments are i.i.d. and normally distributed as the period over which the increment is
computed becomes infinitesimally small. As a result, log price increments are normally distributed
over any measurement period.
A random variable is said to have a lognormal distribution if its logarithm has a normal
distribution: X follows a lognormal distribution when X = Exp(y) and y follows a normal distribution
since log(X) = y, so that the log of the random variable X is normally distributed. If y = log (P(t+)t))
- log(P(t)), we have that Exp(log(P(t+)t))-log(P(t))) =P(t+)t)/P(t). Consequently, P(t +)t)/P(t) is
lognormally distributed. Since Exp(minus infinity) is the lowest value a lognormally distributed
random variable can take, the price cannot have a negative value. This makes the lognormal
distribution attractive for stock prices since it precludes negative stock prices. Figure 7.5. shows the
density function of a normally distributed random variable and of a lognormally distributed random
variable. If we assume that returns are normally distributed when measured over a different time
period, it is possible for the stock price to be negative because with the normal distribution, returns
can take very large negative values.
If log increments are normally distributed over some period of time, say )t, they are normally
distributed over any measurement interval. Consider the log increment over a period of k*)t. This
is the sum of log increments computed over k periods of length )t. However, a sum of normally
distributed random variables is itself normally distributed. Consequently, if the log increment is i.i.d.
and normally distributed over a period of length )t, we can obtain its distribution over intervals of
any other length from its distribution over the interval of length )t.
If the simple return over an interval of )t is i.i.d. and normally distributed, it is not the case
that the simple return over an interval of k*)t is i.i.d. and normally distributed. The reason for this
Chapter 7, page 32
is that compounding is involved in getting the simple return over the longer interval. If r is the simple
return over one interval, the simple return over k intervals is (1+r)k. This compounding implies that
the simple return over multiple intervals involves products of the simple returns. If a simple return
is normally distributed, the simple return of a longer interval consists of products of simple returns
that are normally distributed. Products of normally distributed random variables are not normally
distributed. This means that if weekly simple returns are normally distributed, monthly simple returns
cannot be. In contrast, if continuously compounded returns measured over a week are normally
distributed, then continuously compounded returns measured over a month are normally distributed
also.
Section 7.5.1. Evaluating risk when the log increments of the cash position are normally
distributed.
Let’s now look at the case where the continuously compounded return of a position is
normally distributed. Since we know the distribution of the continuously compounded return, the
VaR over one day ignoring the expected return is simply 1.65 times the volatility of the continuously
compounded return over one day times the value of the position. The formula for the VaR of an asset
with value S(t) and normally distributed continuously-compounded returns is therefore:
Formula for VaR when continuously compounded returns are normally distributed
The formula for the VaR for a position with value at t of S(t) is:
VaR = S(t)*1.65*Vol[Continuously compounded return over one day]
Chapter 7, page 33
Suppose that you have an asset worth $50M and that the continuously compounded return has a oneday volatility of 1%. In this case, the VaR is $50M*1.65*0.01, or $825,000.
Consider the following example. You have a portfolio of $100 million invested in the world
market portfolio of the Datastream Global Index on September 1, 1999. You would like to know the
VaR of your portfolio at a one-month horizon assuming the continuously compounded returns are
normally distributed. The volatility of the monthly log return of the world market portfolio from
January 1, 1991 to September 1, 1999 is 0.0364 in decimal form. Multiplying the volatility by 1.65,
we have 0.0606. The expected log increment is 0.0093. Subtracting 0.0606 from 0.0093, we obtain
-0.05076. The VaR of our portfolio is therefore $5.076M. This means that one month out of twenty
we can expect to lose at least $5.076M.
It is straightforward to see why assuming that the increments to the world market portfolio
are normally distributed would be nonsensical. The world market portfolio was worth $6,818B on
January 1, 1991. By September 1, 1999, its value had almost quadrupled to $26,631B. If dollar
increments are normally distributed, this means that the expected dollar increment is the same in
January 1991 and in September 1999. This would mean that in September 1999, the expected rate
return of the world market portfolio and the standard deviation of return are about 1/4th of what they
were in 1991!
Section 7.5.2. Var and Riskmetrics™.
The procedure we just discussed to compute the risk measure is widely used. It is the
foundation of the approach to computing VaR proposed by Riskmetrics™. With that approach, daily
Chapter 7, page 34
log increments are assumed to be normally distributed and the mean of daily log increments is
assumed to be zero. However, distributions of increments change over time and these changes have
to be accommodated. For instance, in 1987, after the October Crash, the volatility of the stock
market increased dramatically for a period of time. If the distribution changes over time, at date t, we
are estimating the distribution of the next log increment based on the information available to us up
to that point. This distribution is the conditional distribution. For instance, we would choose a
different distribution of increments for tomorrow if we knew that volatility has been high over the last
week than if we knew that it has been low. Consequently, Riskmetrics™ does not assume that the
distribution of continuously compounded returns is i.i.d. It only assumes that the distribution of
increments looking forward from a point in time, the conditional distribution, is the normal
distribution.
The procedure used by Riskmetrics™ to obtain estimates of variances and correlations puts
more weight on recent observations. Consequently, this procedure effectively presumes that variances
and correlations are not constant over time. It uses no other data than the time-series of observations,
however. One could think of models to estimate variances and correlations that use other information.
For instance, one might forecast the market’s volatility using a model where the market’s volatility
depends on the level of interest rates or on the volume of trades. This is a topic of active research
among financial economists. A simple way to cope with the fact that variances and correlations might
change over time is to use recent data only to estimate variances and correlations. It is also a good
idea to examine the data for changes in distributions. If in a time-series plot of the data, one sees a
period that is quite different from the current period, it may make sense to start the sample after that
period if one believes that the period is somehow unique.
Chapter 7, page 35
Riskmetrics™ makes its estimates of the volatilities and correlations freely available over the
internet. It computes these volatilities and correlations for a large and growing number of assets - in
excess of 400. A large institution will have thousands of different assets. Consequently, to simplify
the problem of computing its VaR, it will choose to assign assets into bins. Assets that are similar will
be put into these bins. One way of creating the bins is to use the Riskmetrics™ assets. We will
consider this problem for fixed income portfolios in chapter 9.
Section 7.5.3. Hedging with i.i.d. log increments for the cash position and the futures price.
We already saw that when returns are i.i.d., the regression approach gives us an optimal hedge
for a cash position of one dollar. Nothing is changed in that analysis if continuously compounded
returns are i.i.d. Using log increments in the regression analysis gives us the optimal hedge per dollar
of cash position. The one question that arises is whether using log increments instead of simple
returns makes an important difference in the computation of the optimal hedge. The answer is that
generally it does not.
Consider the following problem. As in the VaR example, we have a position of $100M dollars
in the world market portfolio on September 1, 1999. We believe that there is a higher probability that
the world market portfolio will have a negative return during the next month than the market believes.
We therefore want to hedge the position for the coming month using futures. After the end of the
month, we will reconsider whether we want to hedge. The reason we hedge rather than sell is that
it is much cheaper to hedge with futures than to incur the transaction costs of selling the shares. As
we saw, it makes sense to assume that the log increments of the world market portfolio are i.i.d. By
the same token, we can assume that the log increments of index futures are also i.i.d. Let’s consider
Chapter 7, page 36
the case where we use two futures contracts to hedge: the S&P 500 contract and the Nikkei contract.
The Nikkei contract traded on the IMM gives payoffs in dollars.
Regressing the monthly log increments of the world market portfolio on a constant and the
log increments of the futures contracts from January 1, 1991 to September 1, 1999 gives us:
)Log(World index) = c + $)Log(S&P 500) + ()Log(Nikkei) + ,
0.0048
0.6905
0.2161
(0.26)
(12.91)
(7.72)
We provide the t-statistics below the coefficients. These coefficients show that the log increments of
the world index are positively related to the log increments of the S&P500 and of the Nikkei. The
exposure of the world market to the S&P 500 is greater than its exposure to the Nikkei. Both
exposure coefficients are estimated precisely. The standard error of the estimate of the S&P 500
futures coefficient is 0.054 and the standard error of the estimate of the ( coefficient is 0.028. The
regression explains a substantial fraction of the variation of the log increments of the world market
portfolio since the R2 is 0.7845. This R2 means that we can expect to hedge 78.45% of the variance
of the world market portfolio with the two futures contracts.
Consider now the implementation of this hedge where the hedge is adjusted every month.
Effectively, we hedge over a month, so that tailing is trivial and can be ignored. The Nikkei contract
is for $5 times the Nikkei. Consequently, the contract is for a position in the Nikkei worth $89,012,4
on September 1, 1995, since on that day the Nikkei stands at 17,802.48. The S&P contract is for 500
times the S&P 500. The S&P 500 is at 1,331.06 on September 1, 1999, so that the contract
Chapter 7, page 37
corresponds to an investment of 1,331.06*500 in the S&P 500, namely $665,531.5. The regression
coefficient for the S&P 500 is 0.69048. This means that one would like a position in the S&P 500
futures of $69,048,000 (i.e., 0.69048*$100M). A one percent increase in that position is $690,480.
The expected increase in the value of the portfolio if the S&P500 increases by one percent is
0.69048*$100m = $690,480 as well. To obtain the number of S&P 500 futures contracts, we divide
69,048,000 by 665,531.5. The result is 103.7, which we round out to 104 contracts. To get the
number of Nikkei contracts, we would like a Nikkei position worth $21,609,000. The best we can
do is 243 contracts since 21,609,000/89,012.4 = 242.8. On October 1, the hedged portfolio is worth:
Initial cash portfolio
$100,000,000
Gain on cash portfolio
-$1,132,026
($100M*(1059.4 - 1071.53)/1071.53)
Gain on S&P500 futures
$2,509,000
-104*500*(1281.81- 1331.06)
Gain on Nikkei position
$109,253
-243*5*(17,712,56 - 17,802.48)
New portfolio value
$101.486,227
Note that, in this case, in the absence of hedging we would have lost more than $1M. Because of the
hedge, we end up earning more than $1M. One might argue that this shows how valuable hedging
is since we can earn more than $1M instead of losing more than $1M. However, while hedging helps
us avoid a loss, the fact that we earn that much on the hedged portfolio is actually evidence of an
Chapter 7, page 38
imperfect hedge. Remember that if we hedge a portfolio of stocks completely, the hedged portfolio
is a risk-free asset that earns the risk-free rate. On September 1, 1999, the risk-free rate would have
been less than 0.5% a month, so that the risk-free return on our hedged portfolio would have been
less than $500,000. We therefore earned on the hedged portfolio close to a million dollars that we
would not have earned had the hedged portfolio been risk-free. This is because the world market
portfolio includes markets that have little correlation with the U.S. and Japanese markets. We could
therefore try to use additional futures contracts to hedge the world market portfolio. For instance,
there is a futures contract traded on the U.K. market which could help us hedge against European
market risks that are not correlated with the U.S. and Japanese market risks. Since this contract is
traded in pounds, the hedging portfolio would become more complicated.
Consider now the futures positions on October 1, 1999. If we change them monthly, then we
have to reexamine these positions at that time. Our portfolio now has $101,486,227 invested in the
world market portfolio assuming that we reinvest all the futures gains in the world market portfolio.
To hedge, assuming that the regression coefficients are still valid, we need a position in the S&P 500
of 0.69048*$101,486,227 = $70,074,210. At that date, the S&P 500 is at 1,282.81, so that the
number of contracts we would like is 70,074,210/(1,282.81*500) = 109. Hence, our S&P 500
position increases by 5 contracts. What about our Nikkei position? We need to compute
0.21609*101,486,227/(5*17,712.56). This gives us 248 contracts. We therefore want to go short 248
contracts, which is five more contracts than we had initially.
In this example, we readjust the hedge every month. If the cash position and the futures prices
are highly volatile, one will generally be better off readjusting the hedge more often. In this case,
however, not much happens during the hedging period. The best recommendation is therefore to
Chapter 7, page 39
adjust the hedge when significant changes occur.
Section 7.6. Metallgesellschaft.
A useful way to put all the knowledge we have acquired about hedging together is to consider
the experience of a firm with futures hedging and evaluate that experience. Metallgesellschaft AG is
a German firm that hedged with futures and, despite its hedges, faced important difficulties. It offers
a perfect case study to evaluate the problems that can arise with futures hedges and the mistakes that
can be made in hedging programs that use futures. In December 1993, Metallgesellschaft AG
headquartered in Frankfurt, Germany, faced a crisis. It had made losses in excess of one billion dollars
on futures contracts. The chief executive of Metallgesellschaft AG was fired, a package of loans and
debt restructuring was put together, and eventually large amounts of assets were sold. The losses
brought about a drastic reorganization of the firm. The losses of Metallgesellschaft are among the
highest reported by firms in conjunction with the use of derivatives. Are derivatives to blame? The
board of directors of the firm thought so and changed the firm’s strategy after becoming aware of the
losses. Others, especially Merton Miller of the University of Chicago, have accused the board of
directors of panicking when it had no reason to do so and of giving up a sound business strategy.
Even though the case of Metallgesellshaft has generated strong controversies, there is general
agreement about the facts of the case. Metallgesellschaft is a large German company with interests
in metal, mining, and engineering businesses. In 1993, it had sales in excess of $16B dollars and assets
of about $10B dollars. It had 15 major subsidiaries. MG Corp was the U.S. subsidiary of
Metallgesellschaft. Its equity capital was $50M dollars. It was mostly a trading operation, but it had
an oil business organized in a subsidiary called MG Refining and Marketing (MGRM). MGRM had
Chapter 7, page 40
a 49% stake in a refiner, Castle Energy. It had contracted to buy from this company its output of
refined products, amounting to approximately 46M barrels per year at guaranteed margins for up to
ten years. MGRM then turned around and offered long-term contracts in refined products at fixed
prices. With this business strategy, MGRM provided insurance to retailers by selling them oil at fixed
prices. It thought that its expertise would enable it to make this strategy profitable.
The fixed-price contracts offered by MGRM consisted of three types. The first type of
contract required the buyer to take delivery of a fixed amount of product per month. This type of
contract, most of them with a maturity of ten years, accounted for 102M barrels of future deliveries.
The second type of contract had fixed prices but gave considerable latitude to buyers concerning
when they would take delivery. This type of contract, again mostly running ten years, accounted for
52M barrels of future deliveries. Finally, there was a third type of contract, which had flexible prices.
Essentially, at the end of 1993, MGRM had therefore guaranteed delivery at fixed prices of 154M
barrels of oil.
To alleviate credit risk, MGRM allowed buyers to take fixed price contracts for a fraction of
their purchases of oil not exceeding 20%. It further included cash-out options. With the first type of
contracts, buyers could cash out and receive half the difference between the near-term futures price
and the fixed price times the remaining deliveries. With the second type of contracts, they would
receive the full difference between the second nearest futures contract price and the fixed price times
the amount not yet delivered.
With these contracts, MGRM took on a large exposure to fluctuations in oil prices. Since it
would pay spot prices to Castle and would receive fixed prices from retail customers, increases in oil
prices could potentially bankrupt the company. MGRM therefore decided to hedge. However, it did
Chapter 7, page 41
not do so in a straightforward way. Remember that the fixed-price contracts have a maturity of 10
years, so that MGRM would keep paying spot prices to Castle over the next ten years. To analyze
the issues involved in MGRM’s hedging, let’s assume for simplicity that the fixed price contracts have
no credit risk, no flexible delivery dates, and no cash-out options. In this case, the simplest way for
MGRM to eliminate all risks would be to buy a strip of forward contracts where each forward
contract has the maturity of a delivery date and is for a quantity of oil corresponding to the quantity
that has to be delivered on that date. If MGRM had done this, it would have been perfectly hedged.
It would no longer have had an exposure to oil prices.
MGRM chose not to buy a strip of forward contracts. It did so for two reasons. First, buying
a strip of forward contracts would have been expensive because it would have required using overthe-counter markets of limited liquidity. This issue made a strategy using short-term futures contracts
more attractive. Mello and Parsons (1995) argue that the minimum-volatility hedge would have
involved buying 86M barrels in the short maturity futures contract. Pirrong (1997) estimates the
minimum-volatility hedge differently and under most of his assumptions the minimum-volatility hedge
is lower than the one obtained by Mello and Parson. One can argue about some of the assumptions
of these studies, but it is certainly the case the minimum volatility hedge involved a long position in
the short maturity futures contract of much less than 154M barrels. This is the case for two important
reasons. In chapter 6, we talked about the necessity to tail the hedge and pointed out that tailing can
have a dramatic impact in a situation where we are hedging exposures that mature a number of years
in the future. Here, Metallgesellschaft is hedging some exposures that mature in 10 years. When
hedging these exposures, the tailing factor should therefore be the present value of a discount bond
that matures in ten years. This reduces sharply the size of the hedge. In addition, remember from our
Chapter 7, page 42
analysis that basis risk decreases the size of the hedge relative to the cash position. In this case, rolling
over short-term contracts exposes MGRM to substantial rollover risk that should decrease its
position. Despite these considerations, MGRM bought 154M barrels in short maturity contracts.
Some of these contracts were futures contracts, but others were traded over the counter. Hence, in
no sense did MGRM use a minimum-volatility hedge.
This brings us to the second reason why they did not buy a strip of forward contracts. For any
maturity, a forward price is the spot price plus a forward premium. The minimum volatility hedge of
86M barrels is computed ignoring expected returns altogether. In contrast, MGRM thought that,
whereas it did not have good information about future spot prices, it had good information on basis.
Its view was that the typical relation was one of backwardation, meaning that prices for future
delivery are lower than spot prices, and that such a relation implies that a long futures position would
have a positive expected return as futures prices converge to spot prices at maturity. With such a
view, even if MGRM was not exposed to oil prices, it would want to take a long position. The
magnitude of this speculative position in futures would depend on the willingness to bear risk and on
the magnitude of the expected profit. Since MGRM had an exposure to the price of oil, the positive
expected return on futures increased its long position relative to the optimal hedge. MGRM ended
up having a long position of 154M barrels of oil. The argument that makes sense of such a position
is that with this position MGRM was exposed to the basis but not to oil prices - eventually, it would
take delivery on every one of the contracts as it rolled them over.
From the perspective of the optimal hedging theory developed so far, the long position of
154M barrels can be decomposed in a pure hedge position of 86 million barrels of oil and a
speculative position of 68M barrels of oil. MGRM was net long in oil prices. As prices fell
Chapter 7, page 43
dramatically in 1993, it lost on its futures positions. Each dollar fall in the price of oil required margin
payments of $154M. Whereas the losses on the minimum-volatility position were exactly offset by
gains on the fixed-price contracts, the losses on the speculative positions had no offset and were
losses in the market value of Metallgesellschaft. In other words, assuming that the computation of
the minimum-volatility hedge is correct, the margin payment of $154M for a dollar fall in the price
of oil was offset by an increase in the value of the long-term contracts of $86M. The net loss for
Metallgesellschaft of a one dollar fall in the price of oil was $68M.
MGRM’s speculative position was based on gains from backwardation. Adding to the losses
of MGRM was the fact that in 1993 the markets were in contango. This means that futures prices
were above spot prices. Hence, the source of gains on which MGRM had counted on turned out to
be illusory that year. Instead of making rollover gains from closing contracts at a higher price than
the contracts being opened (i.e., with backwardation, the maturing contract is close to the spot price
which is above the futures price of the newly opened contract with a longer maturity),
Metallgesellschaft made rollover losses. These rollover losses were of the order of $50M every
month.
The massive losses on futures contracts grew to more than one billion dollars. Some of these
losses were offset by gains on the fixed price contracts, but the net loss to the corporation was large
because of the speculative position. The losses on futures contracts were cash drains on the
corporation since the gains on the fixed price contracts would only be recovered later. A hedging
program that generates a cash drain at its peak of about one billion dollars on a corporation with
assets of ten billion dollars can only create serious problems for that corporation. At the same time,
however, Metallgesellschaft could have raised funds to offset this liquidity loss. If Metallgesellschaft
Chapter 7, page 44
thought that the speculative position was sound at the inception of the program, it should have still
thought so after the losses and hence should have kept taking that position since doing so would have
maximized shareholder wealth. However, losses lead people to look at their positions more critically.
Hence, it might not have been unreasonable to expect a change in policy. In particular, it was harder
to believe that backwardation was a free lunch.
Was the MGRM debacle a hedge that failed or was Metallgesellschaft a gutless speculator?
After all, if it really believed that backwardation made money, the losses of 1993 were irrelevant. The
strategy had to be profitable in the long run, not necessarily on a month by month or year by year
basis. Losses could lead Metallgesellschaft to give up on the strategy only if it changed its position
as to the expected profits of the speculation or could not afford the liquidity costs of the speculation.
Since Metallgesellschaft had lines of credit that it did not use, the most likely explanation is that the
company gave up on the strategy. As it changed direction, the board wanted the new management
to start on a clean slate and so decided to take all its losses quickly. This practice of taking a bath
around management changes is not unusual, as discussed in Weisbach (1995).
What do we learn from the Metallgesellschaft experience? Mostly that large losing futures
positions create liquidity drains due to the requirement to maintain margins and that it pays to
compute correct minimum-volatility hedges. When implementing a hedging program with futures
contracts, it is therefore crucial to plan ahead for meeting the liquidity needs resulting from losing
futures positions. The VaR of the futures contract provides an effective way to understand the
distribution of the liquidity needs. With VaR, we know that over the period for which the VaR is
computed, we have a 5% chance of losing that amount. A hedge that develops losing futures
positions is not a bad hedge. On the contrary, if the hedge is properly designed, losing futures
Chapter 7, page 45
positions accompany winning cash positions, so that on net the company receives or loses nothing
when one uses present values. However, the gains and losses of the company’s positions might have
different implications for the company’s liquidity. With Metallgesellschaft, the gains from a fall in oil
prices would have to be paid over time by the holders of long-term contracts whereas the losses from
futures prices had to be paid immediately.
Even though a hedged position may have no economic risk, it can be risky from an accounting
perspective. This issue played a role in the case of Metallgesellschaft. In principle, a hedge can be
treated from an accounting perspective in two conceptually different ways. First, the cash position
and the hedge can be treated as one item. In this case, the losses on the hedge are offset by gains in
the cash position, so that losses on a hedge position that is a perfect hedge have no implications for
the earnings of the firm. Alternatively, the hedge position and the cash position can be treated
separately. If accounting rules prevent the gains in the cash position that offset the losses in the hedge
position from being recognized simultaneously, a firm can make large accounting earnings losses but
no economic losses on a hedged position.
An intriguing question with the Metallgesellschaft case is whether the firm would have done
better had it not hedged than with the hedge it used. From a risk management, the only way this
question makes sense is to ask whether Metallgesellschaft had more risk with the wrong hedge than
with no hedge at all. The answer to that question depends on the assumptions one makes. A least one
study, the one by Pirrong (1997) reaches the conclusion that an unhedged Metallgesellschaft would
have been less risky than the Metallgesellschaft with the hedge it used. Irrespective of how this
question is decided, however, the risk of Metallgesellschaft would have been substantially smaller had
it used the appropriate volatility-minimizing hedge. However, if one believes that backwardation is
Chapter 7, page 46
a source of profits, Metallgesellschaft with the volatility-minimizing hedge would have been less
profitable ex ante.
Section 7.7. Conclusion and summary.
Hedging can be expensive, but it need not be so. When one hedges with derivatives traded
in highly liquid markets, hedging generally has a low cost. Nevertheless, it is important to be able to
deal with situations where the costs of hedging are high. To do so, one has to have a clear
understanding of the costs of the risks that one is trying to hedge. We saw how this understanding
can be put in quantitative terms to derive an optimal hedge. We showed that firms with lower costs
of hedging or greater costs of risks hedge more. We then showed that treating exposures as portfolios
of exposures reduces the costs of hedging because doing so takes into account the diversification
across risks. In general, when a firm has exposures maturing at different dates, one has to focus on
their present value or their value at a terminal date. When one considers the present value of
exposures, one effectively uses VaR. We saw that with VaR it often makes more sense to think that
the return of a position is i.i.d. rather than the dollar increment. We therefore developed the hedging
model for positions that have i.i.d. returns. We concluded the chapter with a detailed discussion of
the difficulties faced by Metallgesellschaft in its hedging program.
Chapter 7, page 47
Literature note
Portfolio approaches to hedging take into account the expected gain associated with futures
contracts as well as the risk reduction resulting from hedging. A number of papers discuss such
approaches, including Rutledge (1972), Peck (1975), Anderson and Danthine (1981, 1983), Makin
(1978) and Benninga, Eldo, and Zilcha (1984). Stulz (1983) develops optimal hedges in the presence
of transaction costs. The analysis of this chapter which uses CaR and Var is new, but given our
assumptions it is directly related to the analysis of Stulz (1983). Witt, Schroeder, and Hayenga (1987)
compare different regression approaches to obtaining hedge ratios. Bailey, Ng, and Stulz (1992) study
how hedges depend on the estimation period when hedging an investment in the Nikkei 225 against
exchange rate risk. The Riskmetrics™ technical manual provides the details for their approach and
is freely available on the internet. Extending optimal hedging models to take into account the liquidity
of the hedging firms turns out to be difficult as demonstrated by Mello and Parsons (1997). However,
their work shows that firms with limited liquidity may not be able to hedge as much and that careful
attention has to be paid to liquidity.
Culp and Miller (1994) present the view that the hedging strategy of Metallgesellschaft was
sensible. Edwards and Canter (1995), Mello and Parsons (1995) and Pirrong (1997) provide estimates
of the minimum-volatility hedge for Metallgesellschaft.
Chapter 7, page 48
Key concepts
Hedging costs, cost of CaR, lognormal distribution, i.i.d. returns model, log increment model.
Chapter 7, page 49
Review questions
1. Why could hedging be costly?
2. Why hedge at all if hedging is costly?
3. What is the marginal cost of CaR?
4. What is the optimal hedge if hedging is costly?
5. Why does diversification across exposures reduce hedging costs?
6. How does diversification across exposure maturities reduce hedging costs?
7. When is the i.i.d. increments model inappropriate for financial assets?
8. What is a better model than the i.i.d. increments model for financial assets?
9. How can we find the optimal hedge if returns are i.i.d.?
10. How does the optimal hedge change through time if returns are i.i.d.?
10. What model for returns does Riskmetrics™ use?
11. What were the weaknesses of the hedge put on by Metallgesellschaft?
Chapter 7, page 50
Questions and exercises
1. Consider a firm that has a long exposure in the SFR for which risk, measured by CaR, is costly.
Management believes that the forward price of the SFR is low relative to the expected spot exchange
rate at maturity of the forward contract. Suppose that management is right. Does that necessarily
mean that the firm’s value will be higher if management hedges less than it would if it believed that
the forward price of the SFR is exactly equal to the expected spot exchange rate?
2. How would you decide whether the i.i.d. return model or the i.i.d. increment model is more
appropriate for the hedging problem you face?
3. Suppose that Metallgesellshaft had ten-year exposures and that the minimum-volatility hedge ratio
before tailing is 0.7. Assume that the 10-year interest rate is 10%. What is the effective long position
in oil of Metallgesellschaft per barrel if it uses a hedge ratio of 1 per barrel? Does Metallgesellshaft
have a greater or lower exposure to oil prices in absolute value if it uses a hedge ratio of 1 or a hedge
ratio of 0?
4. A firm will receive a payment in one year of 10,000 shares of company XYZ. A share of company
XYZ costs $100. The beta of that company is 1.5 and the yearly volatility of the return of a share is
30%. The one-year interest rate is 10%. The firm has no other exposure. What is the one-year CaR
of this company?
Chapter 7, page 51
5. Using the data of question 4 and a market risk premium of 6%, what is the expected cash flow of
the company in one year without adjusting for risk? What is the expected risk-adjusted cash flow?
6. Suppose the volatility of the market return is 15% and XYZ hedges out the market risk. What is
the CaR of XYZ after it hedges the market risk?
7. Suppose that the marginal cost of CaR is 0.001*CaR for XYZ and it costs 0.2 cents for each dollar
of the market portfolio XYZ sells short. Assuming that XYZ hedges up to the point where the market
cost of CaR equals the marginal cost of hedging, how much does XYZ sell short?
8. Suppose that in our example of hedging the world market portfolio, we had found that the tstatistic of the Nikkei is 1.1 instead. What would be your concerns in using the Nikkei contract in
your hedging strategy? Would you still use the contract to hedge the world market portfolio?
9. Suppose that you use the i.i.d. increment model when you should be using the return i.i.d. model.
Does your mistake imply that the VaR you estimate is always too low? If not, why not?
10. You hire a consultant who tells you that if your hedge ratio is wrong, your are better off if it is
too low rather than too high. Is this correct?
Chapter 7, page 52
Figure 7.1. Expected cash flow net of CaR cost as a function of hedge.
We use an expected spot exchange rate of $0.7102, a forward exchange rate of $0.6902, and a cost
of CaR of 0.0001 and 0.00001. With this example, hedging reduces the expected cash flow because
the spot exchange rate exceeds the forward exchange rate. Consequently, if " = 0, no hedging would
take place and the firm would go long on the forward market. The fact that risk is costly makes the
expected cash flow net of CaR a concave function of the hedge, so that there is an optimal hedge. As
" falls, the optimal hedge falls also.
Chapter 7, page 53
Figure 7.2. Marginal cost and marginal gain from hedging.
We use an expected spot exchange rate of $0.7102, a forward exchange rate of $0.6902, and a cost
of CaR of 0.0001 and 0.00001. With this example, hedging reduces the expected cash flow because
the spot exchange rate exceeds the forward exchange rate. Consequently, if " = 0, no hedging would
take place and the firm would go long on the forward market. The fact that risk is costly makes the
expected cash flow net of CaR a concave function of the hedge, so that there is an optimal hedge. As
" falls, the optimal hedge falls also.
Chapter 7, page 54
Figure 7.3. Optimal hedge ratio as a function of the spot exchange rate and " when the cost
of CaR is "*CaR2.
We use a forward exchange rate of $0.6902.
Chapter 7, page 55
Figure 7.4. Short position to hedge Export Inc.’s exposure.
The hedging period starts on March 1. Export Inc. receives a payment of SFR1m on June 1 and must
make a payment of SFR1m on November 1. The price on March 1 of a discount bond maturing on
June 1 is $0.96 and the price on June 1 of a discount bond maturing on September 1 is $0.92. The
optimal hedge ratio for an exposure of SFR1 is 0.94. March 1 is time 0 and September 1 is time 0.5.
Chapter 7, page 56
Figure 7.5. Normal and lognormal density functions.
The normal distribution has mean 0.1 and volatility 0.5. The logarithm of the lognormally distributed
random variable has mean 0.1 and volatility 0.5.
Chapter 7, page 57
Box 7.1. The costs of hedging and Daimler’s FX losses.
In the first half of 1995, Daimler-Benz had net losses of DM1.56 billion. These losses were
the largest in the group’s 109-year history and were due to the fall for the dollar relative to the DM.
Daimler-Benz Aerospace (DASA) has an order book of SFR20 billion. However, 80% of that book
is denominated in dollar. Consequently, a fall in the dollar reduces the DM value of the order book.
The company uses both options and forwards to hedge. On December 31, 1994, DaimlerBenz group had DM23 billion in outstanding currency instruments on its books and DM15.7 billion
of interest rate instruments. Despite these large outstanding positions, it had not hedged large
portions of its order book. German accounting rules require the company to book all expected losses
on existing orders, so that Daimler-Benz had to book losses due to the fall of the dollar. Had DaimlerBenz been hedged, it would have had no losses.
The explanation the company gave for not being hedged highlights the importance of costs
of hedging as perceived by corporations. Some analysts explained the lack of hedging by their
understanding that Daimler-Benz had a view that the dollar would not fall below DM1.55. This view
turned out to be wrong since the dollar fell to SFR1.38. However, the company blamed its losses on
its bankers. According to Risk Magazine, Daimler-Benz did so “claiming their forecasts for the
dollar/Deutschmark rate for 1995 were so diverse that it held off hedging large portions of its foreign
exchange exposure. It claims 16 banks gave exchange rate forecasts ranging from DM1.20 to
DM1.70 per dollar.”
If a company simply minimizes the volatility of its hedged cash flow, it is completely
indifferent to forecasts of the exchange rate. However, to the extent that it has a forecast of the
exchange rate that differs from the forward exchange rate, it bears a cost for hedging if it sells the
Chapter 7, page 58
foreign currency forward at a price below what it expects the spot exchange rate to be. Nevertheless,
one would think that if forecasts differ too much across forecasters, this means that there is a great
deal of uncertainty about the future spot exchange rate and that, therefore, the benefits of hedging
are greater. This was obviously not the reasoning of Daimler-Benz. It is unclear whether the company
thought that it could hedge a forecasted depreciation or whether it was concerned about its inability
to pin down the costs of hedging.
Source: Andrew Priest, “Daimler blames banks’ forex forecasts for losses,” Risk Magazine 8, No.
10 (October 1995), p. 11.
Chapter 7, page 59
Technical Box 7.2. Derivation of the optimal hedge when CaR is costly.
To obtain equation (7.1.), assume that the firm’s cash flow consists of receiving nS units of spot at
some future date and selling h units of spot at the forward price of F per unit for delivery at the same
date. (The analysis is the same for a futures price G for a tailed hedge assuming fixed interest rates.)
With this notation, the expected cash flow net of the cost of CaR for an exposure of n and a hedge
of h is:
Expected cash flow net of cost of CaR = Expected income minus cost of CaR
= (n - h)E(S) + hF - "CaR2
(Box 7.2.A)
This equation follows because we sell n - h units on the spot market and h units on the forward
market. Remember now that the CaR with a hedge h assuming that increments are normally
distributed is 1.65Vol((n - h)S) where S denotes the spot price at the delivery date. Substituting the
definition of CaR in the above equation, we have:
Expected cash flow net of cost of CaR = (n - h)E(S) + hF - "(1.65Vol((n - h)S))2 (Box 7.2.B)
To maximize the expected cash flow net of the cost of CaR, we take the derivative with respect to
h and set it equal to zero (remembering that Vol((n - h)S) is equal to (n - h)Vol(S)):
-E(S) + F + 2"1.65(n - h)Vol(S)*1.65Vol(S) = 0
Chapter 7, page 60
(Box 7.2.C)
Solving for h and rearranging, we get the formula:
h = n -
F - E(S)
2α1.652 Var(S)
(Box 7.2.D)
Remember now that 1.65Vol(S) is the CaR of one unit of unhedged exposure. Using the definition
of CaR therefore yields the formula in the text.
Chapter 7, page 61
Chapter 8: Identifying and managing cash flow exposures
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Section 8.1. Quantity and price risks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Section 8.1.1. Risky foreign currency cash flows: An introduction. . . . . . . . . . . 5
a) The quantity risk is uncorrelated with the futures price . . . . . . . . . . . . 8
b) The quantity risk is positively correlated with the futures price . . . . . . 8
c) The quantity risk is negatively correlated with the price risk . . . . . . . . 8
Section 8.1.2. An example of an optimal hedge with quantity risk. . . . . . . . . . . . 9
Section 8.2. The exposures of Motor Inc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Section 8.2.1. Time horizon and cash flow exposure. . . . . . . . . . . . . . . . . . . . . 15
Section 8.2.2. Exchange rate dynamics and competitive exposures. . . . . . . . . . . 17
Section 8.2.3. Competitive exposures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Section 8.3. Using the pro forma statement to evaluate exposures. . . . . . . . . . . . . . . . . 26
Section 8.4. Using past history to measure exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Section 8.5. Simulation approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Section 8.6. Hedging competitive exposures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Section 8.7. Summary and conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Questions and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Figure 8.1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Figure 8.1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Figure 8.2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
Figure 8.2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..56
Figure 8.3. Cash flow and exchange rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 8.4. Distribution of Motor Inc.’s cash flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Table 8.1.Cash flow statement of Motor Inc. for 2000 (in million) . . . . . . . . . . . . . . . . 59
Box 8.1.: Monte Carlo Analysis Example: Motor Inc. B. . . . . . . . . . . . . . . . . . . . . . . 60
Figure Box 8.1. Motor Inc. B Corporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 8: Identifying and managing cash flow exposures
December 14, 1999
© René M. Stulz 1996,1999
Chapter objectives
1. Understand how to hedge when exposures are not constant.
2. Show how to design hedges using pro forma statements.
3. Find out the determinants of exposures for importers and exporters.
4. Explain how to use economic models in the construction of hedging strategies.
5. Show how Monte Carlo analysis and regression analysis can be used to obtain estimates of
exposures.
Chapter 8, page 2
In the previous two chapters, we learned how to evaluate and hedge risks when exposures
are fixed. Our key example was Export Inc. The exposure of the firm was fixed at SFR1M on June
1. This type of situation arises, for instance, when a firm has receivables and payables in foreign
currencies. A firm’s transaction exposure to a currency is the exposure that results from having
receivables and payables in that currency. Transaction exposure results from past business deals, but
whether future business deals take place in a foreign country will depend on the exchange rate. In
addition to their transaction exposures, firms therefore often have foreign currency exposures that
depend on the exchange rate as well as on other variables. An example would be the case where
Export Inc.’s SFR exposure depends on how competitive the firm is in Switzerland and how its
competitiveness is affected by exchange rate changes. This chapter focuses on how to estimate and
hedge exposures that depend on risk factors such as the foreign exchange rate.
To understand why a firm’s total exposures might depend on risk factors, let’s stick to the
case of foreign exchange risks. If Export Inc. were a car manufacturer, the exposure to the SFR we
discussed in the previous two chapters might correspond to cars already sold but not yet paid for. The
company could also have contracts to sell a fixed number of cars in the future at a fixed SFR price.
There is no transaction at this point: the cars may not have been produced and certainly have not been
shipped. Nevertheless, the present value of these contracts depends on the SFR exchange rate. A
firm’s contractual exposure to a currency is its exposure to that currency resulting from contractual
commitments, both booked and unbooked. In general, however, the amount of business a firm does
in a foreign country is affected by foreign exchange rate changes. An appreciation of the SFR could
enable the company to decrease the SFR price of a car without reducing its dollar income from selling
the car. By reducing the SFR price of the car, the firm would become more competitive in
Chapter 8, page 3
Switzerland and could sell more cars. Hence, the firm’s exposure to the SFR will depend on the SFR
exchange rate because it affects the number of cars and the price of cars it sells in Switzerland. As
a result, rather than having a fixed exposure to the SFR, the firm has an exposure that depends on the
exchange rate. A firm’s competitive exposure to a risk factor is the impact on the firm’s cash flow
of a change in the risk factor resulting from changes in the firm’s competitive position. In this chapter,
we want to understand how to identify and quantify competitive exposures so that we can hedge
them.
In the hedging literature, a situation where the exposure is not fixed is often described as one
where the firm faces both price risk and quantity risk. The quintessential example is the case of a
farmer wanting to hedge a crop. The farmer’s problem is that crop size is uncertain. In the first
section, we discuss the problem of hedging with both quantity and price risks. We then turn to
identifying and managing exposures that depend on risk factors and are therefore random. We focus
our discussion on foreign exchange exposures, but it will be clear that the techniques we develop can
be used whenever exposures are random. In the next chapter, we will discuss the issues that arise with
interest rate exposures separately. We first show in this chapter how pro forma analysis can be used
to identify the determinants of a firm’s exposures. It is often the case that one has to carefully model
the firm’s cash flow to identify its exposures. Monte Carlo simulations are an important tool for such
analyses. We therefore discuss how to use Monte Carlo simulations to evaluate exposures and
construct hedges. We then go back to the regression approach developed in chapters 6 and 7 to see
how this approach can be used to evaluate total exposures. In general, a firm’s exchange rate
exposure is itself a function of the exchange rate. As a result, firm value becomes a nonlinear function
of the exchange rate. We develop a technique to hedge nonlinear exposures with futures and
Chapter 8, page 4
forwards. In chapter 10 we will see that options provide powerful tools to hedge nonlinear exposures.
Section 8.1. Quantity and price risks.
We first consider the optimal hedge when both quantity and price are random. Let’s go back
to the case of Export Inc. Remember that in chapter 6, Export Inc. knew on March 1 that it would
receive SFR1M on June 1. We now consider how our analysis changes when the cash flow of SFR1m
is risky in SFRs.
Section 8.1.1. Risky foreign currency cash flows: An introduction.
The cash flow could be risky in SFRs for many reasons. Suppose that Export Inc. plans to sell
widgets in Switzerland on June 1 and that it does not know how many widgets it will sell but the SFR
price of the widgets is fixed over time. The cash flow in SFRs on June 1 is simply the number of
widgets Export Inc. sells times the price of the widgets. In this case, the SFR cash flow is random,
so that the exposure to the SFR is itself random. Another example is one where Export Inc. has
already sold the widgets, but there is default risk. If the purchaser defaults, Export Inc. earns less than
the promised payment. Again, the exposure in SFRs is random because it depends on how much the
creditor pays.
Random exposures can make a firm that hedges worse off than a firm that does not hedge.
Let’s look at an example. Suppose that Export Inc. expects to receive SFR1M from having sold
widgets to Risque Cie and sells these SFRs forward. If it actually receives the SFR1M, it delivers
them on the forward contract and receives a fixed amount of dollars. However, if Risque Cie defaults
and pays nothing, Export Inc. not only does not receive the promised SFRs, but in addition it incurs
Chapter 8, page 5
a loss on the forward contract if the price of the SFR on June 1 is higher than the forward price. In
this case, not only Export Inc. makes a loss because it is not paid, but it could also make an additional
loss because it does not have the SFRs it was planning to deliver on the forward contract and hence
has to buy them on the spot market on June 1.
We now consider how Export Inc. can construct a futures hedge when its cash flow is random
in foreign currency. For this purpose, let c be the random SFR cash flow and S be the exchange rate
on June 1. In this case, Export Inc.’s dollar cash flow on June 1 is cS. The expected cash flow is
therefore E(cS) and the variance of the dollar cash flow is Var(cS). Whereas we can decompose the
variance of a sum of random variables into a sum of variances and covariances of these random
variables, there exists no such convenient decomposition for products of random variables.
Consequently, the only way to compute Var(cS) is to treat cS as a single random variable and
compute the variance as we would if we had to compute the variance of a single random variable.
This single random variable, which we write C, is the dollar cash flow, so that C = cS.
We assume that Export Inc. wants to take a futures position on March 1 that minimizes the
volatility of the hedged cash flow on June 1. At this point, we only consider hedges that involve a
position in a single SFR futures contract. Let G(June 1) be the price on June 1 of the SFR futures
contract Export Inc. chooses to use. To avoid the issue of basis risk, we assume that the contract
matures on June 1. (Basis risk does not introduce additional considerations here relative to those
discussed in chapter 6.) We want to find h, which is the number of units of the futures contract we
go short to hedge the dollar cash flow before tailing. If we go short h units of the futures contract,
we make a profit on the futures position of - h(G(June 1) - G(March 1)). Consequently, the hedged
cash flow is the cash flow on June 1 plus the futures profit:
Chapter 8, page 6
Hedged cash flow = C - h(G(June 1) - G(March 1))
We are back to the problem we solved in chapter 6. We have a random payoff, here C, that we are
trying to hedge. Before, the random payoff was the random price, the exchange rate, S, times a fixed
quantity - SFR1M in our example. Whether we call the random payoff S*1M or C, the optimal hedge
obeys the same formula. With the problem of chapter 6, we wanted to find a position in futures whose
unexpected gain in value over the life of the hedge would best match an unexpected cash flow
shortfall. We saw that the solution to the problem was to set h equal to the best predictor of the
relation between cash flow and the change in the futures price. If the multivariate normal increments
model holds for C and the futures price, the formula for this best predictor is given by the formula
for the regression coefficient in a regression of realizations of unexpected cash flow on realizations
of changes in the futures price. Hence, the optimal hedge of a random cash flow is:
Formula for the optimal hedge of a random cash flow
h'
Cov(C,G)
Var(G)
(8.1.)
where G is the futures price at maturity of the hedge. In chapter 6, C was equal to the dollar value
of SFR1M. Here, it is equal to cS. In both cases, the formula has in its numerator the covariance of
a random variable with the futures price. This formula holds whenever we want to use a specific
futures contract to hedge a random cash flow. The formula does not care about what C is. If we
wanted to hedge rainfall with the SFR futures contract, we would still use this formula and we could
Chapter 8, page 7
still find the optimal hedge ratio using regression analysis if the proper conditions are met.
Despite the fact that the same formula holds to find the optimal hedge when there is quantity
risk and when there is none, quantity risk can dramatically alter the volatility-minimizing hedge as well
as the effectiveness of the volatility-minimizing hedge. To understand how the optimal hedge is
affected when cash flow in foreign currency is random, one has to find out how the quantity risk is
related to the price risk. In our example, the quantity risk is the SFR cash flow and the price is the
foreign exchange rate. We can look at three different cases:
a) The quantity risk is uncorrelated with the futures price. In this case, the volatilityminimizing hedge is the same as if the quantity was fixed and was equal to the mean of the random
quantity. This is because the quantity risk is like uncorrelated noise added to the price risk. It does
not affect the covariance of the cash position with the futures price and the futures contract is useless
to hedge the quantity risk.
b) The quantity risk is positively correlated with the futures price. Since quantity is high
when the futures price is high, a short position in the futures contract provides a hedge for the
quantity risk. Hence, because of the quantity risk, one takes a larger short position in the futures
contract than without quantity risk.
c) The quantity risk is negatively correlated with the price risk. Because of this negative
correlation, the quantity risk hedges some of the price risk. This means that we do not have to hedge
as much. In the extreme case where the quantity risk is perfectly negatively correlated with the price
risk, one could have a situation where there is price risk and there is quantity risk, but there is no cash
flow risk.
Quantity risk can decrease the effectiveness of the hedge. To see this, consider the case in our
Chapter 8, page 8
example where quantity risk is uncorrelated with price risk and there is a perfect hedge for the price
risk. Remember that if one is concerned about the volatility of the hedged position, the effectiveness
of a hedge is measured by the decrease in volatility of the cash position achieved with the hedge. In
the absence of quantity risk, the hedge is perfect since the hedged cash flow has no volatility. With
uncorrelated quantity risk in our example, it will necessarily be that hedged cash flow is risky because
of the quantity risk that we cannot hedge. Consequently, in this case quantity risk decreases hedging
effectiveness. Quantity risk does not necessarily decrease the effectiveness of the hedge, however.
If the hedge for price risk is imperfect, it could be that quantity risk is highly negatively correlated
with that part of the price risk that cannot be hedged, so that with quantity risk the hedge could
become more effective.
In our discussion, we assumed that Export Inc. uses only one futures contract. We saw in
chapter 6 that this would be reasonable in the absence of quantity risk. However, because now there
is quantity risk, it could be that other futures contracts are helpful to hedge quantity risk even though
they are not helpful to hedge price risk. Suppose, for instance, that there is quantity risk because
Export Inc. does not know how many widgets it will sell and that it sells more widgets when interest
rates in Germany are low. Export Inc. might benefit from hedging against interest rate increases in
addition to hedging against an unexpected fall in the value of the SFR. Hence, hedging instruments
that are useless to hedge price risk can become useful hedges when price risk is combined with
quantity risk.
Section 8.1.2. An example of an optimal hedge with quantity risk.
Let’s look at a simple quantitative example where quantity risk affects the exposure of Export
Chapter 8, page 9
Inc. Suppose the SFR exchange rate in 90 days is either $0.5 or $1.5 with equal probability. The
futures contract matures in 90 days, so that the futures price in 90 days is the spot exchange rate, and
the futures price today is $1. The firm’s cash flow in foreign currency in 90 days is either SFR0.5M
or SFR1.5M with equal probability, so that its expected value is SFR1M. To find out how to hedge
the firm’s income, we need to know how cash flow covaries with the foreign exchange rate. Suppose
first that the cash flow is perfectly positively correlated with the foreign exchange rate. In other
words, the cash flow is SFR1.5M when the foreign exchange rate is $1.5 and it is SFR0.5M when
the exchange rate is $0.5. The dollar cash flow is therefore either $2.25M or $0.25M with equal
probability, so that the expected cash flow is $1.25M.
It follows from the formula for the optimal hedge that to compute the volatility-minimizing
hedge, we need to compute the covariance of the cash flow and the futures price and the variance of
the futures price. The covariance between cash flow and the futures price and the variance of the
futures price are respectively:
0.5*(2.25M - 1.25M)*(1.5 - 1) + 0.5*(0.25M - 1.25M)*(0.5 - 1) = 0.5M (Covariance)
0.5*(1.5 - 1)2 + 0.5*(0.5 - 1)2 = 0.25
(Variance)
We know from equation (8.1.) that the optimal hedge is given by the ratio of the covariance and the
variance. Consequently, the optimal hedge is a short position equal to 0.5M/0.25 = SFR2M. In this
example, the dollar cash flow depends only on the exchange rate because quantity and price risks are
perfectly correlated. As mentioned before, the formula does not care about what the random payoff
in the covariance formula represents. As long as that random payoff is perfectly correlated with a
Chapter 8, page 10
futures price, the futures contract can be used to eliminate all risk. As a result, we can eliminate all
risk because there is effectively only one source of risk, the exchange rate, and we can hedge that
source of risk.
Let’s now look at the case where quantity risk is uncorrelated with price risk. We have four
states of the world:
a) Exchange rate is $1.5 and cash flow is SFR1.5M.
b) Exchange rate is $1.5 and cash flow is SFR0.5M.
c) Exchange rate is $0.5 and cash flow is SFR1.5M.
d) Exchange rate is $0.5 and cash flow is SFR0.5M.
There are four distinct states of the world. Suppose that the probability of the exchange rate taking
value 1.5 is 0.5. In this case, each state of the world is equally likely and has probability 0.25. We
cannot construct a perfect hedge using only the currency futures contract. This is because the futures
contract has the same payoff in distinct states of the world. For instance, it pays the same when the
exchange rate is $1.5 irrespective of whether the SFR cash flow is SFR1.5M or SFR0.5M.
Consequently, dollar cash flow is not perfectly correlated with the exchange rate. One solution would
be to find additional hedging instruments whose payoff is correlated with the SFR cash flow. In the
absence of such instruments, the best we can do is to find the futures position that minimizes the
volatility of the hedged position, knowing that we cannot make this volatility zero. This position is
again given by the formula (8.1.). The expected unhedged cash flow is $1M. Computing the
covariance between the cash flow and the futures price in million, we have:
0.25*(2.25M - 1M)*(1.5 - 1) + 0.25*(0.75M - 1M)*(1.5 - 1) + 0.25*(0.75M - 1M)*(0.5 - 1) +
Chapter 8, page 11
0.25*(0.25M - 1M)*(0.5 - 1) = 0.25M
Solving for the optimal hedge, we divide the covariance of 0.25M by the variance of the futures price,
which is 0.25 as before. We therefore find that the optimal hedge is SFR1M. This means that the
futures position consists of a short position of SFR1M, so that in this case the firm goes short its
expected exposure.
Lastly, we consider the case where the futures price is perfectly negatively correlated with the
SFR cash flow. In this case, the exchange rate is $0.5 when the cash flow is SFR1.5M and is $1.5
when the cash flow is SFR0.5. The firm receives $0.75M when the exchange rate is low and the same
amount when the exchange rate is high. No hedge is required to minimize the volatility of the dollar
cash flow because the dollar cash flow in this case has no volatility.
The expected SFR exposure is the same for each one of our examples, namely SFR1M.
Despite this, however, the optimal hedge is to go short SFR2M when quantity risk and exchange rate
risk are perfectly positively correlated, to go short SFR1M when quantity risk and exchange rate risk
are uncorrelated, and finally to take no futures position when quantity risk and exchange rate risk are
perfectly negatively correlated. In general, the exact effect of quantity risk on the optimal hedge can
be quite complicated. This is particularly the case when the quantity is itself a function of the price
as we will see later in this chapter.
Section 8.2. The exposures of Motor Inc.
In this section, we consider the various risks that can affect a firm and how to measure their
impact on firm value. The best way to do this is to look at an example. Let’s suppose that Motor Inc.
Chapter 8, page 12
is a car producer in the U.K. that exports parts of its production to the US. Table 8.1. shows an
extremely simplified cash flow to equity statement for Motor Inc. for 2000. To simplify the numbers,
we assume that a pound is worth two dollars and that the tax rate is 25%. In 2000, the car producer
exports half of its production to the US and sells the remainder in the U.K.
We would like to understand the distribution of future cash flows. This is a complicated
undertaking because the cash flow for each year can be affected by a large number of variables. Some
of these variables are directly under the control of management of Motor Inc. Its management has
plans for the firm. For instance, it might have decided to expand production by building a new factory
that comes on stream in three years. However, other variables are not under the control of
management. Some of these variables are specific to the operations of Motor Inc. For instance, a
factory could be destroyed by fire or a strike could take place that would immobilize a factory for a
long period of time. Such risks are unique to Motor Inc. They can be quantified but there are no
financial instruments that Motor Inc. could use to hedge these risks. Motor Inc. can reduce these risks
in a variety of ways. For instance, it can install sprinklers and various fire alarms to minimize the
danger of losing a plant to a fire. Once it has taken such protective measures, the only way that Motor
Inc. can protect itself further against such risks is through insurance contracts that are written
specifically against contingencies faced by Motor Inc. alone.
Another possible risk that could affect Motor Inc. alone is if a creditor defaults. In this case,
Motor Inc.’s cash flow is lower than expected. If Motor Inc. has many different creditors and none
is too important to its cash flow, it is most likely that credit risks are diversifiable for Motor Inc. In
other words, whereas there is some risk that a creditor defaults, considering all creditors together the
number and size of defaults can be predicted fairly well. In this case, it makes no sense for Motor Inc.
Chapter 8, page 13
to try to protect itself against defaults of specific creditors because these defaults are not a source of
risk when one looks at Motor Inc.’s overall cash flow. If a firm has some large creditors, credit risks
are unlikely to be diversifiable. These risks are not market risks, however, in the sense that there is
no broad-based market where they are traded. The risk of default from an individual creditor depends
on circumstances unique to that creditor. Hence, to protect itself completely against the default of
a specific creditor, the firm would have to enter a contract that is akin to an insurance contract. We
discuss the use of credit derivatives to hedge against such a default later.
In addition to risks that affect Motor Inc. only, it faces risks that affect other firms as well.
For instance, a depreciation of the dollar means that, for a constant dollar price of its cars in the US,
Motor Inc. receives fewer pounds. An increase in the price of crude oil means that the demand for
cars falls and Motor Inc.’s sales drop. Broad-based risks corresponding to random changes in
macroeconomic variables, such as exchange rate changes, are called market risks. They are not under
the control of individual firms and correspond to market forces that affect the economy as a whole.
These risks can be managed with financial instruments that can be used for hedging purposes by a
large number of firms. An example of such a financial instrument would be a futures contract on the
dollar. The payoff of such a financial instrument does not depend at all on the actions of Motor Inc.
However, Motor Inc. can sell dollars using this futures contract to reduce the dependence of its
pound income on the dollar exchange rate. To do that, Motor Inc. has to figure out the size of the
futures position that hedges its exchange rate risks. This position depends on how exchange rate
changes affect Motor Inc.’s future cash flows.
Consider Motor Inc.’s cash flow for 2000. Let’s consider a base case of constant exchange
rates and no inflation. Let’s assume that Motor Inc. will export 22,000 cars to the US in 2002 and
Chapter 8, page 14
sell each car for $20,000. It will also sell 22,000 cars in the U.K. Suppose that there is no other
source of uncertainty in Motor Inc.’s cash flow. In this case, Motor Inc. will receive $440m in 2002.
The pound value of this amount will be $440m times the pound price of the dollar. If the only effect
of a change in the exchange rate for Motor Inc. is to change the pound value of the dollar sales, then
$440m is the dollar exposure of the cash flow. Multiplying the exchange rate change by the exposure
yields the cash flow impact of the change in the exchange rate.
Exposure of cash flow to a specific market risk measures the dollar change in the value of a
cash flow (ªcash flow) or a financial asset for a unit change in that market variable (ªM):
Definition of exposure of cash flow to risk factor M
Exposure of cash flow to M = ÎCash flow/ÎM
(8.2.)
With our example, exposure to the exchange rate is equal to $440m*ÎM/ÎM = $440m, where ÎM
is the change in the exchange rate. Suppose that the dollar appreciates 10%, so that its price in
pounds goes from $0.5 to $0.55. In this case, the cash flow impact of the appreciation is an increase
in cash flow equal to $440m*0.05, which is £22m. This number is obtained by multiplying the dollar
exposure by the pound change in the price of the dollar.
Section 8.2.1. Time horizon and cash flow exposure.
Since the value of a firm is the present value of its cash flow, we can understand the exposure
of firm value to the exchange rate by computing the exposure of discounted future cash flows to the
exchange rate. The nature of a firm’s exposure to the exchange rate changes as the cash flow
Chapter 8, page 15
measurement period is farther in the future. To see this, suppose first that we compute Motor Inc.’s
exposure to the dollar exchange rate for its next quarterly cash flow. Most of that exposure is fixed.
It depends on transactions that have already been undertaken. For instance, the cars that will be sold
in the US during this quarter are already there, the prices have already been agreed upon with the
dealers, and so on. In the short run, most cash flow exposure arises from existing transactions or from
financial positions that are on the books. The exposures arising from financial positions are generally
handled using the tools required to compute VaR if the financial positions are marked to market. We
are therefore going to ignore such exposures and focus on the firm’s other exposures. Transaction
exposures are measured quite precisely by the accounting system.
Next, suppose we consider the exposure of Motor Inc. for the quarterly cash flow four
quarters from now. The firm’s exposure will consist of two components. One component will be the
transaction exposure resulting from booked transactions. For instance, Motor Inc. might receive or
make payments in dollars that are already contracted for and booked today. It might have dollar debt
requiring coupon payments at that time. The other component will be the contractual exposure not
associated with booked transactions. Motor Inc. will have contractual agreements, implicit or
explicit, that affect its exposure at that horizon. For instance, it might have made commitments to
dealers about prices. Even though these commitments do not result in booked transactions, they
cannot be changed easily, if at all. Such commitments create contractual exposures which do not
show up on the firm’s balance sheet, yet any change in the exchange rate affects the value of the firm
through its effect on the domestic currency value of these commitments.
With contractual exposures, the foreign currency amount of the exposure is often fixed and
not affected by exchange rate changes. For instance, if Motor Inc. has promised to deliver a fixed
Chapter 8, page 16
number of cars to dealers in the US at a fixed dollar price, this creates a contractual exposure. The
pound value of the payments from the dealers is simply the dollar value of these payments translated
at the exchange rate that prevails when the payments are made. For such an exposure, the relevant
exchange rate is the nominal exchange rate at the time of these payments.
Let’s now look at the exposure of cash flow three years from now. Very little of that exposure
is contractual exposure. In fact, as the exchange rate changes, Motor Inc. can change its production,
marketing, and sourcing strategies. If the exchange rate changes so that keeping sales in the US
constant is not profitable, then Motor Inc. can change its sales in the US. It can develop new markets
or new products. The exposures at long horizons are competitive exposures. Changes in market
prices such as exchange rates affect the firm’s competitive position. The nature of the firm’s
competitive exposures therefore depends on the markets in which the firm does business. Others have
called competitive exposures either operating exposures or economic exposures. Contractual losses
due to exchange rate changes are economic losses, which makes the concept of economic exposures
not adequate to describe the exposures we have in mind. Operating exposure ignores the strategic
implications of exchange rate changes - exchange rate changes do not affect only the profitability of
current operations but what the firm does and where it does it. Although in our discussion of
competitive exposures we focus mostly on exchange rate competitive exposures, the firm’s
competitive position is generally affected by changes in other market prices.
Section 8.2.2. Exchange rate dynamics and competitive exposures.
The impact of an exchange rate change on a competitive exposure depends crucially on the
nature of the exchange rate change. Suppose that the dollar price of the pound falls. The simplest
Chapter 8, page 17
scenario is that the pound depreciates because the pound’s purchasing power in the UK falls because
of inflation, so that a buyer of pounds would pay less for them. In this case, if the exchange rate
stayed constant, one could make a profit by arbitraging between the two countries. Suppose that we
can have perfect markets for goods, in the sense that there are no frictions of any kind to arbitrage
goods across countries. In particular, there are no transportation costs, transportation is
instantaneous, there are no customs and no taxes. One could then buy goods in the US at a low price
and resell them immediately in the UK for a profit assuming that there are no arbitrage opportunities
before the change in prices. This cannot be an equilibrium. It violates the law of one price which
states that in perfect markets two identical goods or assets must have the same price to avoid
arbitrage opportunities. With perfect goods markets, the same bar of chocolate in the UK and the US
must sell for the same dollar price in each country. Consequently, with our example, the price of the
pound has to fall so that the same amounts of dollars buys the same amount of goods in the UK
before and after the increase in prices in the UK. This means that in this case the pound depreciates
to compensate for the impact of inflation in the UK. Since a pound buys fewer goods in the UK, the
pound is worth less. To avoid arbitrage opportunities, the offset has to be exact. The prediction that
the exchange rate changes to offset changes in the purchasing power of a currency is called
purchasing power parity:
The law of one price and purchasing power parity
If AUS(t) is the price of a basket of goods in the US at time t, AF(t) is the price of the same basket
of goods in the foreign country at time t, and e(t) is the dollar price of the foreign currency at time
t, with perfect goods markets the law of one price must hold:
AUS(t) = S(t)*AF(t)
Chapter 8, page 18
(8.3.)
This law of one price implies that the exchange rate is determined so that purchasing power parity
holds:
S(t) = AF(t)/AUS(t)
(8.4.)
Define (US to be the continuously-compounded inflation rate in the US, (F to be the continuouslycompounded inflation rate in a foreign country, and x to be the continuously-compounded rate of
increase of the price of the foreign currency in dollars. Purchasing power parity therefore implies
that:
x = (US - (F
(8.5.)
so that the rate of appreciation of the foreign currency is equal to the difference between the inflation
in the US and inflation in the foreign country. As inflation increases in the US (foreign countries),
foreign currencies appreciate (depreciate) relative to the dollar.
Let’s look at an example. Suppose that a bar of chocolate costs $2.50 in the US and that the
price of the bar of chocolate in pounds is £1b in the UK. If the pound costs $2, then the chocolate
bar purchased in the US costs $2.50/$2 in pounds, or £1.25. Since equation (8.4.) does not hold,
some entrepreneurs would import chocolate bars from the US to sell in the UK for a profit of about
£1b - £1.25, or £0.42 per chocolate bar. The only case where there is no arbitrage opportunity is if
the dollar price of the pound is $1.5. This is the purchasing power parity value of the pound given
by equation (8.3.): 2.5/1b = 1.5. In this case, the dollar price of the chocolate bar purchased in the
UK is 1.5*1b or $2.50. Suppose now that, at date t+1, the price of a chocolate bar in the UK
increases to £2. In this case, if the price in the US is still $2.50, the exchange rate must be 2.5/2 or
$1.25. Note now that the price in the UK increased by 18.23% (Ln(2/1b)). The rate of appreciation
of the exchange rate is also -18.23% (Ln(1.25/1.5)). This is exactly what we would have expected
with equation (8.5.), since we have 0 = x + 18.23%. To see that the purchasing power parity result
is true generally, we can divide AUS(t+1) by AUS(t) and take logs:
(US = Ln(AUS(t+1)/AUS(t))
Chapter 8, page 19
= Ln(S(t+1)*AF(t+1)/S(t)*AF(t))
= Ln(S(t+1)/S(t)) + Ln(AF(t+1)/AF(t))
= x + (F
Rearranging gives us equation (8.5.).
Purchasing power parity is not a result that holds on average or in an expected sense. Anytime
it does not hold there are arbitrage opportunities, so it should always hold in terms of realized
inflation rates. Ex ante, though, we can use it to forecast exchange rate changes. If we know
expected inflation in the UK and in the US, then we know what the exchange rate change
corresponding to the expected inflation differential must be. In other words, if inflation over a period
is 10% in the US and 5% in the UK, it must be that over that period the pound appreciates by 5%
relative to the dollar. Further, if we expect inflation to be 10% in the US and 5% in the UK in the
future, then we must expect the pound to appreciate by 5% relative to the dollar.
Equation (8.5.) does not work with simple arithmetic percentages. To see this, note that the
price increases by 0.5/1.5, or 20%, while the exchange rate falls by 0.25/1.5, or by 16.67%. Hence,
the percentage appreciation is not the negative of the percentage price increase. To see the reason
for the problem, let aUS be the US arithmetic inflation rate, aF be the foreign arithmetic inflation rate,
and a be the arithmetic appreciation rate of the foreign currency. Then, we have:
AUS(t+1)/AUS(t) = 1 + aUS
=(S(t+1)/S(t))*(AF(t+1)/*AF(t))
= (1 + a)*(1 + aF)
Working this out in terms of arithmetic percentages, we have:
aUS = a + aF + a*aF
Chapter 8, page 20
The cross-product term is not there when we take logs because when we take the log of a product
we end up with a sum of logs. Hence, using arithmetic percentages, the rate of appreciation of the
exchange rate is not exactly equal to the domestic rate of inflation minus the foreign rate of inflation.
How well does purchasing power parity hold? The answer is that it works poorly among
countries that have little inflation (e.g., less than 20%). The reason for that is that the assumption of
perfect goods markets is a poor description of goods markets. Some goods cannot be arbitraged
across countries at all. The Niagara Falls and the Matterhorn cannot be moved. Other goods can be
arbitraged, but it takes a long time. For instance, Italian haircuts are cheaper in Rome than in New
York. To equalize prices requires Italian barbers to move to New York. However, the cost of a good
espresso in New York is also much higher than in Rome. So, Italian barbers consume goods that are
cheap in Rome but expensive in New York, which means that they require a higher salary measured
in Lire to be in New York than in Rome. Markets for goods are often not perfectly competitive. The
same camera might sell at different dollar prices in different countries because of the market strategy
of the producer. Finally, exchange rates move every minute but goods prices do not. If the dollar
loses ten percent of its value overnight, one will not notice the change in stores tomorrow. When
goods trade in highly frictionless markets, purchasing power parity becomes a good approximation,
but most goods do not trade on such markets. As a result, purchasing power parity seems to be more
of a long-run result when inflation is low. When inflation is high, goods prices adjust to inflation
quickly and inflation overwhelms market imperfections so that purchasing power parity holds fairly
well. A good example of departures from purchasing power parity is given by the article from The
Economist reproduced in Box 8.1. The Big Mac Index.
The real exchange rate is the exchange rate adjusted for purchasing power. Generally, the real
Chapter 8, page 21
exchange rate is computed against a base year. Using our notation, if AUS(t) and AF(t) are price
indices,
the
real
exchange
rate
at
date
t+1
using
date
t
as
the
base
is
(S(t+1)AF(t+1)/AUS(t+1))/(S(t)AF(t)/AUS(t)). If purchasing power parity holds in years t and t+1, the
real exchange rate does not change. In this case, equation (8.2.) implies that the real exchange rate
is equal to one. Suppose that purchasing power parity does not hold and that the real exchange rate
increases. With our example using the UK and the US, the dollar price of the pound could increase
without changes in the price indices. This would be a real appreciation of the pound. In this case, a
pound buys more US goods at t+1 than it did at t and a dollar buys fewer UK goods at t+1 than it
did at t. This means that a car that sells for £10,000 at t+1 before the increase is suddenly worth more
goods in the US at t+1 than at t. Hence, a US consumer who wants to buy that automobile would
have to give up more US goods to get it. A real appreciation therefore makes the export goods of
a country less competitive at their domestic currency price. In contrast, an appreciation of the pound
that does not correspond to a change in the real exchange rate has no impact on a firm’s competitive
position if all prices adjust to inflation. If the pound appreciates to offset US inflation, the price of the
car in the US can be increased to match inflation so that the proceeds in pounds of selling the car in
the US are unchanged - the producer gets more dollars that have less value and the consumers do not
care because the price of the car has increased by the rate of inflation like everything else. The fact
that purchasing power parity works poorly means that, in the short run, almost all exchange rate
changes are real exchange rate changes when inflation is low.
It is important to remember, however, that changes in exchange rates that do not correspond
to changes in real exchange rates affect the domestic currency value of foreign currency exposures
that are fixed in nominal terms. For instance, for Export Inc., a depreciation of the SFR is costly
Chapter 8, page 22
because its SFR exposure is fixed even if the SFR depreciates to reflect inflation in Germany. This
is because it is too late for Export Inc. to adjust prices to reflect inflation.
Section 8.2.3. Competitive exposures
Let’s consider the determinants of Motor Inc.’s competitive exposure to the dollar exchange
rate. If a change in the exchange rate purely offsets a change in inflation, it has no impact on
competitive exposures in the long run because all prices adjust. Consequently, to study competitive
exposures, we consider changes in real exchange rates. There is a large literature that investigates
competitive exposures. It generally emphasizes the importance of the type of competition the firm
faces.1 To understand this, let’s consider two polar cases. At one extreme, Motor Inc. could have no
competitors. In this case, the key determinant of its exposure to the dollar would be the pricesensitivity of the demand for its cars (i.e., the percentage change in demand for a percent change in
the price). Figure 8.1. presents the case where the demand is not very price-sensitive. Figure 8.1.A.
shows the demand curve in pounds and Figure 8.1.B. shows it in dollars. Motor Inc. sells cars up to
the point where the marginal revenue (the impact on total return of selling one more car) equals the
marginal cost (the impact on total cost of selling one more car). As the dollar depreciates, there is no
effect on the demand curve in dollars. However, for each quantity sold, the demand curve in pounds
falls by the extent of the depreciation of the dollar. Hence, a depreciation of the dollar shifts the
demand curve in pounds downwards in Figure 8.1.A. but leaves the demand curve in dollars
unchanged in Figure 8.1.B. If costs are exclusively denominated in pounds, the marginal cost curve
1
See Flood and Lessard (1984), Levi (1994), and Marston (1996) for papers that deal
with the issues we address here.
Chapter 8, page 23
in pounds is unaffected by the depreciation of the dollar. The net effect of the depreciation of the
dollar is to reduce the quantity sold in the US and increase the dollar price of the cars sold in the US.
The dollar price of cars sold in the US does not, however, increase by the full amount of the
depreciation. In the case of the monopolist, the exchange rate exposure is roughly equal to the dollar
revenue before the depreciation.
The other polar case is the one where Motor Inc. sells cars in a highly competitive market.
Suppose first that the competitors of Motor Inc. are American firms with dollar costs. In this case,
as the dollar depreciates, nothing changes for Motor Inc.’s competition. Figure 8.2. shows this case
where the demand for Motor Inc.’s cars is highly price-sensitive. If Motor Inc. offers cars at a slightly
higher price than its competitors, its demand vanishes. Motor Inc. sells cars up to the point where the
marginal cost of producing cars equals the marginal revenue from selling them. In this case, however,
the marginal revenue curve is equal to the demand curve. Looking at Figure 8.2.A., where the
demand curve is in pounds, we see that a depreciation of the dollar moves the demand curve
downwards by the extent of the depreciation. In the example shown in Figure 8.2.A., the
depreciation has the effect of pushing the demand curve completely below the marginal cost curve,
so that it is no longer profitable for the firm to sell in the US. In this case, the exposure of the firm
depends on its ability to shift sales away from the US. If the firm can shift sales rapidly and at low
cost, it may have very little exposure to the exchange rate.
What if Motor Inc. is a small player in a highly competitive US market dominated by German
car makers? Suppose that the demand for the market as a whole is not very price-sensitive, but the
demand for Motor Inc. cars is extremely price-sensitive. In other words, Motor Inc. has very little
ability to set its prices. In this case, the key question in understanding the exposure of Motor Inc. to
Chapter 8, page 24
the dollar exchange rate has to do with how the SFR and the pound prices of the dollar move
together. Suppose that Motor Inc. expects the two exchange rates to move closely together. In this
case, a depreciation of the dollar with respect to the pound also means a depreciation of the dollar
with respect to the Euro. Consequently, the German producers increase their dollar prices and Motor
Inc. can do the same. This means that the impact of the depreciation on cash flow should be fairly
small. Alternatively, suppose that the pound price of the dollar and the Euro price of the dollar move
independently. In this case, an appreciation of the pound with respect to the dollar that is not
accompanied by an appreciation of the Euro with respect to the dollar has an extremely adverse effect
on Motor Inc.’s income from dollar sales and may force it to exit the US market. An appreciation of
the Euro against the dollar that is not accompanied by an appreciation of the pound with respect to
the dollar is good news for Motor Inc. because its competitors in the US increase their dollar price.
In this case, Motor Inc. has an exposure to the Euro even though it does not export to Germany. The
reason for this is that its competitors are German.
In our discussion, we have assumed that everything else is maintained constant. In other
words, the only source of exchange rate exposure for Motor Inc. is its revenue from the US. In
general, however, some of the costs of Motor Inc. are likely to also depend on the exchange rate,
which complicates the analysis. For instance, it could be that Motor Inc. incurs some costs in the US
to sell cars there. In this case, the pound costs of Motor Inc. cars sold in the US fall since the dollar
costs are now less in pounds. This shifts downwards the pound marginal cost curve in our examples
and hence reduces the loss to Motor Inc. from a depreciation of the pound. In the extreme case where
Motor Inc. has only dollar costs for its sales in the US, the exposure is limited to the repatriated
profits. Hence, instead of having all the dollar revenue exposed to changes in the exchange rate,
Chapter 8, page 25
Motor Inc. would only have its dollar profits exposed to the exchange rate.
Section 8.3. Using the pro forma statement to evaluate exposures.
Using a firm’s income statement, one can obtain a cash flow statement. Each item in that cash
flow statement is risky and its exposure to a risk factor can be evaluated by estimating by how much
that item would change if that risk factor changed unexpectedly. Adding up the impact of a change
in the risk factor across all items of the cash flow statement, one can compute the exposure of cash
flow with respect to that risk factor for a given change in the risk factor. We call this the pro forma
approach to evaluating exposures.
Let’s go back to the cash flow statement of Motor Inc. and the base case assumptions
described in section 8.2. We can forecast the cash flow statement for a particular year based on
assumptions about the variables that affect cash flow. Suppose that the only risk factor that affects
Motor Inc.’s cash flow is the dollar/pound exchange rate. The cash flow exposure of Motor Inc. to
the exchange rate is the sum of the exposures of the components of the cash flow statement. To see
this, consider our simple cash flow statement for Motor Inc.:
Cash flow = Sales - Costs of goods sold Taxes - Investment
The exposure of cash flow to the exchange rate as defined by equation (8.2.) is the impact of a unit
change in the exchange rate on cash flow. Consequently, the impact on cash flow of a £0.01 change
in the pound price of the dollar is:
Chapter 8, page 26
Cash flow exposure*0.01
However, using the right-hand side of the cash flow equation, it follows that the impact of a £0.01
exchange rate change on cash flow must be equal to its impact on sales minus its impact on costs of
goods sold, minus its impact on taxes, and minus its impact on investment.
We can look at the cash flow of Motor Inc. for 2002 assuming 10% growth in sales, costs,
and investment:
Cash flow = Sales - Costs of goods sold - Taxes - Investment
£27.5M = £440M - £330M - 0.25*£110M - £55M
A change in the exchange rate can affect each component of cash flow. Suppose for now that it
changes only the pound value of US sales. In this case, the exposure of cash flow is simply the
amount of sales in the US times 1 minus the tax rate, which we assumed to be 25%. The tax rate is
generally ignored in discussions of exposure, but it is important. If the dollar depreciates, this reduces
the firm’s cash flow, but it also reduces its taxable income. Each pound lost because of the decrease
in the pound value of dollar sales means that taxes paid are reduced by a quarter of a pound. We can
obtain this exposure by noticing that the pound value of US sales is:
Dollar price per car * Number of cars sold * Pound price of the dollar
If the dollar price per car and the number of cars sold are constant, the dollar revenue is constant. The
Chapter 8, page 27
dollar exposure of the cash flow of Motor Inc. is then simply its dollar revenue times one minus the
tax rate (which we assumed to be 25%). In this case:
Cash flow exposure of Motor Inc. to the dollar = Dollar revenue of Motor Inc.*(1 - 0.25)
Using the pro forma cash flow statement, this corresponds to $440M* (1 - 0.25) = $330M. The cash
flow exposure can then be used to compute cash flow under different assumptions about exchange
rate changes. First, suppose that one believes that the worst possible exchange rate move is a ten
percent depreciation. In this case, one can use the exposure measure to compute the cash flow
shortfall if that ten percent depreciation takes place. Note that a ten percent depreciation means that
cash flow has a shortfall relative to no depreciation of:
Cash flow exposure of Motor Inc. to the dollar x Pound value of 10% depreciation of the dollar
Remember that we assume the pound to be worth two dollars in the base case. Hence, this means that
a 10% depreciation of the dollar brings the dollar from £0.50 to £0.45. Consequently, with our
assumptions, this shortfall is for £16.5M, namely 330M*0.05.
Alternatively, one can use the exposure to compute the volatility of cash flow. Remember that
with our assumptions, the dollar exchange rate is the only risk affecting cash flow. Consequently:
Volatility of cash flow in pounds = Exposure*Volatility of exchange rate
Chapter 8, page 28
Suppose that the volatility of the exchange rate is 10% p.a. In this case, the pound volatility of 2002
cash flow viewed from the beginning of 2001 depends on the volatility of the exchange rate over a
two-year period. Using the square root rule, this volatility is the square root of two times 10%, or
14.14%. This gives us a cash flow volatility of 14.14% of £330m, or £46.66M. Using the volatility
of cash flow, we can compute CaR. There is one chance in twenty that the cash flow shortfall will be
at least £76.99M below projections assuming no change in the exchange rate. Remember that if the
cash flow is distributed normally, the fifth percentile of the distribution of cash flow is 1.65 times the
volatility of cash flow. Consequently, £46.66M*1.65 gives us £76.99M.
The analysis becomes more complicated if the dollar price of cars and/or the quantity of cars
sold in the US also change with the dollar exchange rate. We ignore taxes and investment for
simplicity, so that the cash flow is the revenue from selling cars minus the cost of producing the cars
sold. It is still possible to compute the worst outcome if one believes that the dollar will at most
depreciate by 10%. In this case, however, one has to make assumptions about the demand curve and
the marginal cost curve. We consider the case corresponding to Figure 8.2., but for simplicity we
assume that Motor Inc. only sells in the US and does not sell in the UK. Let’s further assume that the
marginal revenue of selling in the US for Motor Inc. is fixed and does not depend on the actions of
Motor Inc. because Motor Inc. sells in a highly competitive market. In this case, the dollar marginal
revenue is equal to the price of a car, that is $20,000. The marginal cost in pounds for Motor Inc.
depends on the cost function of Motor Inc. We assume that the total cost for Motor Inc. is given by
the following function:
Cost = 10M + 0.25*(Quantity produced)2
Chapter 8, page 29
Consider the numbers from Table 8.1. but ignore taxes for simplicity. We assumed that Motor Inc.
sold cars in the US at $20,000 a piece and sold 20,000 cars there. Using our cost function, the
marginal cost of a car produced is:
Marginal cost of a car produced = 0.5*(Quantity produced)
If Motor Inc. produces 20,000 cars, the marginal cost of a car is 0.5*20,000, which amounts to
£10,000 or, when the dollar costs £0.5, $20,000. Consequently, when Motor Inc. sells 20,000 cars
in the US, marginal cost equals marginal revenue. In this case, Motor Inc.’s cash flow is £90M.
Figure 8.3. shows that the cash flow in pounds is an increasing convex function of the pound
price of the dollar. The fact that cash flow is not a linear function of the exchange rate has extremely
important implications for exposure calculations. It implies that the change in cash flow is not
proportional to the size of the change in the exchange rate: the change in cash flow if the exchange
rate increases by 2x is not twice the change in cash flow if the exchange rate increases by x. The
exposure now depends on the exchange rate change and the level of the exchange rate. To see this,
suppose we start from an exchange rate of £0.5 and it falls to £0.45. In this case, the quantity of cars
sold in the US becomes 18,000 and cash flow falls to £71M. Consequently, the change in cash flow
per unit change in the exchange rate is £19m/0.05, or $390. However, if we consider a change in the
dollar so that it depreciates by 50%, the profit falls by £75M, so that for such a change the exposure
per unit change is £75M/0.25, or $300M. Had the initial exchange rate been £0.80 instead, a drop
of £0.05 to £0.80 would have led to a loss of £31M since income would have fallen from £246M to
£215M. Consequently, at £0.80, the exposure of cash flow per unit change is $620M. This means that
Chapter 8, page 30
to compute the loss resulting from a depreciation of the dollar we have to compute the cash flow for
the exchange rate after the depreciation and compare it to the cash flow before the depreciation.
Instead of having one exposure measure that applies irrespective of the magnitude of the exchange
rate change, we now have an exposure measure for each exchange rate change! There are two
approaches to deal with this problem. One approach uses simulations and we discuss it in section 8.4.
The alternative approach uses the delta exposure measure which we discuss next.
An approach to computing multiple exposures is to treat exposure as if it were linear and
approximate this linear exposure by the exposure that the firm has at the current exchange rate for
a small change in the exchange rate. The exposure is then given by the exchange rate delta of the cash
flow which is defined as the change in cash flow for a small change in the exchange rate. More
generally, the delta measure of exposure with respect to a risk factor is given by:
Risk factor delta of cash flow
This measure of exposure is defined as follows:2
Cash flow risk factor delta = Change in cash flow for a small change in the risk factor
from its current value
(8.6.)
The exchange rate delta exposure of cash flow is given in Figure 8.3. by the slope of the cash flow
function since that slope measures the change in cash flow for an infinitesimal change in the exchange
2
If cash flow is a function that can be differentiated, the cash flow risk factor delta is
technically the partial derivative of cash flow with respect to the risk factor.
Chapter 8, page 31
rate. Expressed in terms of a unit change in the exchange rate, the exposure approximated this way
is equal to $400M, which corresponds to the US sales revenue in dollars at that exchange rate. Using
this approximation, we find that a £0.05 increase in the value of the dollar increases pound profits by
£20M, since we have -0.05*400M. Computing the loss by comparing profits when the exchange rate
is £0.50 to profits when the exchange rate is £0.45, we saw above that the exchange rate depreciation
reduces profits by £19M. The two numbers are sufficiently close that measuring the exposure using
the cash flow exchange rate delta works well. Note now that this approximation implies that we need
a hedge where we gain £20M if the pound price of the pound decreases by £0.05. We therefore want
to be short $400M.
The delta approximation can often become very imprecise for larger changes in the risk factor
and can even lead to absurd results. For instance, suppose that the pound price of the dollar falls by
£0.25. In this case, the delta measure implies a loss of £0.25*400M, or £100M. We have seen that
the actual loss when we compute the profits explicitly at that new exchange rate is £75M, or threequarters of the predicted loss. The reason for why the approximation does not work well for large
changes here is that for large changes the firm adjusts production and sales, whereas the delta
approximation assumes that everything stays unchanged.
The mistake made with the delta measure for large changes is a problem because it increases
the risk of hedged cash flow. If we use the delta measure, we are short $400M dollars, so that we
gain £100M as the price of the dollar falls in half. Yet, our cash flow falls by £75M, so that we make
an unexpected gain in our hedged position of £25M. One might be tempted to say that there is no
reason to complain here since we make more money than expected. However, in terms of risk
management, what matters is that we make an unexpected gain, so that our cash flow is volatile.
Chapter 8, page 32
What we want to do through hedging is minimize the volatility of hedged cash flow. Here we do not
succeed because the cash flow is not a linear function of the exchange rate. As a result of this feature
of cash flow, a static hedge (a hedge that is unchanged over the hedging period) cannot work equally
well for all changes in the exchange rate. A hedge that works well for small changes may not work
well for large changes. Importantly, all we can say with the use of delta exposure is that it works
poorly as a hedge for large changes. We cannot say whether the hedged position has gains or losses
for large changes because this depends on the precise nature of the cash flow function. In chapter 10,
we will see that options can help to solve such hedging problems.
One issue that arises from investigating the approximation we used is whether the result that
the exposure to a small exchange rate change is equal to sales in that currency at the current exchange
rate is a general result. In a recent working paper, Marston (1996) argues that this result holds in a
large number of cases. For instance, it holds both in the monopolistic case and the duopoly case
where the firms take into account the impact of their actions on their competitor. Note that we have
ignored taxes. Had we taken into account taxes, we would have needed to use dollar sales times one
minus the tax rate.
In the example we just discussed, Motor Inc. had to have a fair amount of information to
compute its exposure to a large exchange rate change and ended up with an exposure that itself
depended on the level of the exchange rate. Despite being complicated, the example we looked at was
heavily simplified. For instance, there was only a single source of risk. This raises the question of
whether there are alternative approaches that can be used when the analytical approach we described
cannot be used in practical situations or can be used to complement this analytical approach. We
discuss such alternative approaches in the next two sections. First, in section 8.4, we show how to
Chapter 8, page 33
use a firm’s past history to get a measure of exposure. Then, in section 8.5, we discuss simulation
approaches.
Section 8.4. Using past history to measure exposure.
We saw in section 8.3. that measuring a firm’s exposure can be quite complicated even in
extremely simplified examples. If we are concerned about how exchange rate changes affect the value
of a large firm with lots of different activities, an analytic approach similar to the one described earlier
would seem rather difficult to implement. Even if one were able to implement such an approach, one
would want to have some way to check whether its results are sensible. An important tool to evaluate
a firm’s exposure is simply to look at the past history of that firm. Such an approach is quite valuable
when one believes that the future will not be too different from the past. In other words, such an
approach works well when one believes that past exposures are similar to future exposures.
A firm’s exposure to a market risk makes its value sensitive to this risk. Consequently, we can
evaluate this sensitivity using regression analysis building on the results of the last two chapters. This
approach decomposes the random return of the firm into one part that depends on the market risk and
one part that does not. Consider the situation where one would like to estimate the sensitivity of the
firm’s equity to some risk factor x. A risk factor need not be the price of an asset and, therefore, it
does not necessarily have a return. We can obtain this sensitivity by regressing the return of the firm’s
equity on the percentage change or return of that factor:
Rit ' " % $i Rxt % ,it
(8.7.)
In this regression, Rit is the firms cash flow or return on its securities, " is the constant, Rxt is the
Chapter 8, page 34
return of the risk factor, $i measures the exposure of the firm to the particular financial risk, and ,it
is the error term. The beta measure of the sensitivity of the value of the firm to the risk factor of
interest is equally valid if this factor is the interest rate, foreign exchange rate, and/or commodity
prices or some macroeconomic variable of interest. Note, however, that we assumed that the i.i.d.
returns model holds for all the variables in the regression. This model need not be the right one, as
discussed in chapter 7.
There is an important difference between the analytical approach used in the previous section
and the approach estimating a firm’s sensitivity to a market risk with regression analysis. In the
previous section, we obtained the sensitivity of Motor Inc. to the dollar/pound exchange rate. If the
cash flow of Motor Inc. is sensitive to other exchange rates because Motor Inc. also exports to these
countries, this would not affect the measure of exposure derived in the previous section. However,
with the statistical approach used here, all sources of correlation of firm value with the US
dollar/pound exchange rate affect the beta coefficient. For instance, suppose Motor Inc. sells to
Canada instead of selling to the US. We do not know this and estimate a regression of Motor Inc.’s
equity return on the change in the US dollar/pound exchange rate. In this case, Motor Inc.’s sales to
Canada would affect the regression beta of Motor Inc. on the US dollar/pound exchange rate because
the Canadian dollar/pound exchange rate is correlated with the US dollar/pound exchange rate. As
a result, we might conclude that the firm is exposed to a US dollar/pound exchange rate when it is
not because it is exposed to a risk factor that is correlated with the currency used in the regression.
If the correlation between the US dollar and the Canadian dollar exchange rates are stable, then using
the US dollar to hedge works, but probably not as well as using the Canadian dollar. However, if the
correlation is not stable, then changes in the correlation between the two currencies can create serious
Chapter 8, page 35
problems. In that case, using the US dollar as a hedge for the true risk factor may no longer work if
the correlation has changed, but since we don’t know that the US dollar proxies for the true risk
factor, the Canadian dollar, we don’t know that it no longer works! This means that one should
always be cautious when using regressions and evaluate the results using our understanding of the
firm’s economic situation.
To estimate the sensitivity of the equity of Motor Inc. to the US dollar/pound exchange rate
only, one would therefore have to estimate the regression controlling for other sources of risk that
affect the equity of Motor Inc. and are correlated with the US dollar/pound exchange rate. This
means that if there are two sources of risk for the equity of Motor Inc., namely the US dollar/pound
exchange rate and the Canadian dollar/pound exchange rate, one would have to regress the return on
the equity of Motor Inc. on the two sources of risk. However, if one is interested in Motor Inc.’s
sensitivity with respect to exchange rates for the purpose of risk management, one would have to
regress Motor Inc. only on the sources of risk for which hedging instruments are available and will
be used. If the only hedging instrument one intends to use is a forward contract on the US dollar, one
can use this contract to partly hedge the firm’s exposure to the Canadian dollar since that currency’s
changes are positively correlated with those of the US dollar.
If one wants to estimate the exposure of the firm to several risk factors, one has to use a
multivariate regression. With such a regression, one regresses the return on the firm’s equity on all
the sources of risk one is interested in. For instance, if one is concerned about the firm’s exposure to
the stock market as well as to one financial risk factor, one estimates the following regression:
Chapter 8, page 36
Rit ' " % $i Rmt % (i Rxt % ,it
(8.8.)
where Rmt is the return on the market portfolio, $i is the exposure of the firm to the stock market,
and (i is its exposure to the financial risk factor. In the case of exchange rates, Jorion (1990) uses
the regression in equation (8.8.) to evaluate the exposure of US multinational firms to the dollar price
of a trade- weighted basket of foreign currencies. He shows that the exposure to the price of a basket
of foreign currencies systematically increases as a firm has more foreign operations.
The multivariate regression analysis could be used to evaluate all of a firm’s exposures to
financial risks. A firm may be exposed to exchange rates from more than one country as well as to
interest rates and to commodity prices it uses as inputs to production. The implementation of the
multivariate approach then uses many different financial prices as independent variables. The problem
that arises with this approach, though, is that market risks are correlated. This makes it hard to
estimate the exposure of a firm to many market risks precisely using the regression approach.
Let’s now look at an example. General Motors is a large automobile manufacturer with global
sales and competition. The Opel subsidiary of the firm is located in Europe and accounts for a large
portion of the firm’s international vehicle sales. A very high percentage of these sales are made in
the German market. Also, it faces major competition from Japanese firms in both the North American
and European market. Finally, a high percentage of the firm’s vehicle sales are financed, so the
vehicle sales should be exposed to the level of interest rates. With these factors in mind, one would
expect that the firm has exposure to the value of the yen and Euro against the dollar and to the level
of US interest rates. For the purpose of our analysis, we will assume that the exposure of GM against
the Euro is well approximated by its exposure against the German mark, so that we can use a long
Chapter 8, page 37
time-series of GM returns in our regression analysis.
If we first wanted to know the exposure of GM to the value of the German mark, while
controlling for the general market factors, we would consider the following regression:
RGM,t ' " % $ Rmt % ( SDM .
(8.9.)
where the dependent variable is the monthly return on GM, Rmt is the return on the market portfolio
(here the S&P500), SDM is the percentage change in the price of the DM, and $ and ( represent the
exposures of GM to the market and the mark/dollar exchange rate, respectively. The resulting
regression for monthly exchange rate and returns data from 1973 - 1995 is the following:
RGM,t ' 0.001 % 0.924 Rmt % 0.335SDM
(0.368) (11.157)
(2.642)
(8.10.)
where the number in parentheses represent the t-statistic of the regression coefficient. The
interpretation of the above regression is that a 10% appreciation of the DM leads to a 3.35% increase
in the value of the firm’s equity keeping the level of the stock market constant. This positive
coefficient on the percentage change in the dollar price of the DM reflects the increased value of the
German operations as the DM appreciates.
Next, as stated above, the firm is also exposed to the value of the yen because of the
competition from the Japanese firms in the North American market. Thus, an increase in sales of the
Japanese firms should result in a decrease in sales for GM. We define S¥ as the percentage change in
the dollar price of the yen. Also, since cars are durable goods, we would expect that the firm would
be exposed to the level of interest rates. We use as an explanatory variable RB, which is the
Chapter 8, page 38
percentage change in interest rates for corporate bonds of maturities from 3 to 10 years. To test the
firm’s exposure to these factors, we estimate the following regression:
RGM,t ' " % $ Rmt % ( SDM % * S¥ % 8 RB.
(8.11.)
Where " is a constant, $ is the firm systematic risk, (, *, and 8 are the sensitivities to the rates
of change of the German mark, Japanese yen and US interest rates, respectively. The estimates (tstatistics in parentheses) are:
RGM,t ' 0.0007 % 0.965 Rmt % 0.472SDM & 0.353 S¥ & 0.198RB
(0.181)
(11.300) (2.997)
(&2.092) (&1.902).
(8.12.)
So, GM has significant exposures to the mark, to the yen, and to the level of US interest rates. A
depreciation of the yen relative to the US dollar decreases firm value. Also, an increase in the US
interest rates decreases firm value. Numerically, a 10% appreciation in the dollar relative to the yen
and the DM along with a 10% increase in interest rates leads to a decrease of 3.53% and 1.98% from
the yen and the interest rate, respectively, and an increase of 4.72% as a result of DM increase. This
results in a 0.61% decrease in the value of the firm’s equity.
The example of General Motors shows how we can analyze the exposure of a firm to various
financial prices by including these prices in a simple linear regression. A similar analysis could be
done with the inclusion of the firm’s input prices or any other factor that the analyst feels may be
important after evaluation of the firm and industry characteristics along with a pro forma analysis.
Careful examination of the results of the regression can assist the financial manager in determining
the proper risk management strategy for the firm.
Chapter 8, page 39
There are a few caveats about the use of the regression approach that must be kept in mind
before its implementation. First, the regression coefficients are based on past information and may
not hold for the firm in the future. For example, the world automotive industry went from a period
of relatively benign competition in the 1970s to a period of more intense competition beginning in the
early 1980s. This competitive change probably had some effect on the exposure of cash flows of GM
to the changes in the yen. Also, the sales in Europe became more significant over this period. This
means that this approach might have done a poor job of estimating exposures for the 1980s using
data from the 1970s. More recently, the DM has been replaced by the Euro. It may well be that GM
exposure to the Euro is different from its exposure to the DM.
Secondly, the firm may be exposed to more market risks than those used in the regressions.
To understand how much of the volatility in equity is explained by the market risks used in the
regression, we already know that we can use the R2 of the regression. The R2 of the second regression
is 35%. This shows that our risk factors explain slightly more than 1/3 of the variance of General
Motors’ stock. Further examination of the characteristics of the firm, including its financial position,
sourcing, marketing, and the competitive nature of the industry would be helpful in identifying
additional risk factors.
Finally, the above regressions assume a simple linear relation between the cash flows of the
firm and the financial risk factors. This may not be appropriate. The relation can also be nonlinear.
For instance, the interaction between two market risks can matter for firm value. Such a nonlinear
structure can be incorporated in the regression, but to know that it exists, one has to already have a
good understanding of the firm’s exposures.
Despite all these caveats, the regression approach provides a way to check one’s
Chapter 8, page 40
understanding of a firm’s exposures. If our analytical approach tells us that an increase in the price
of the SFR affects a firm adversely and the regression approach tells us the opposite, it is reasonable
to ask some tough questions about the analytical approach. Going further, however, we can also use
the exposures from the regression approach as hedge ratios for the present value of the cash flows
since firm value is the present value of the firm’s cash flows.
Section 8.5. Simulation approaches.
Consider a firm whose current situation is quite different from the situation it faced in its
recent history. Consequently, historical exposures estimated through a multiple regression do not
correctly describe the exposures that will apply in the future. If future cash flows can be modeled in
the way that we modeled the cash flow of Motor Inc., the cash flow exposures can be obtained
analytically for each exchange rate. However, the problem with that approach is that generally one
cannot obtain a measure of the volatility of cash flow analytically because one generally does not
know the distribution of cash flow even though one knows the distribution of the exchange rate
changes. As a result, one cannot compute CaR analytically. To obtain CaR, one then has to use
simulations.
Let’s consider the example of Motor Inc. studied in Section 8.3. where sales depend on the
exchange rate. In this case, we can solve for the quantity produced that maximizes cash flow using
the relation that the marginal cost of a car produced must equal its marginal revenue when production
is chosen optimally. With our assumptions, marginal revenue is $20,000y, where y is the pound price
of the dollar. This must equal 0.5*Quantity produced. Consequently, the quantity produced is
20,000y/0.5. We can then compute the cash flow as:
Chapter 8, page 41
Cash flow in pounds = Price*Quantity - Cost
= (20,000y)(20,000y/0.5) - 10M - 0.25(20,000y)2
To compute CaR, we therefore need the fifth percentile of the distribution of cash flow. Note,
however, that cash flow depends on the square of the exchange rate. We may not even know what
the distribution of the square of the exchange rate is, so that an analytical solution may not be
possible. The simplest way to get CaR is therefore to perform a Monte Carlo analysis. Such an
analysis generates realizations of a function of random variables from draws of these random variables
from their joint distribution. Consequently, if we know the distribution of the exchange rate, we can
generate squared values of the exchange rate without difficulty even if the squared exchange rate does
not have a statistical distribution to which we can give a name. As a result, instead of having a
historical sample, we have a simulated sample. These realizations can then be analyzed in the same
way that we would analyze an historical sample. To understand this approach, note that here cash
flow depends on a random variable, the exchange rate which we denote by y. We therefore first have
to identify the distribution of y. Having done this, we use a random number generator to draw values
of y from this distribution. For each drawing, we compute cash flow. Suppose we repeat this process
10,000 times. We end up with 10,000 possible cash flows. We can treat these cash flows as
realizations from the distribution of cash flow. This is like having a sample. If we repeat the process
more often, we have a larger sample. We can then use the sample to estimate the mean, volatility, and
the fifth percentile of cash flow. As we increase the number of drawings, we are able to estimate the
distribution of cash flow more precisely. However, an increase in the number of drawings means that
the computer has to work longer to create our sample. In the example considered here, this is not an
Chapter 8, page 42
issue, but this may be an important issue for more complicated cash flow functions.
In Section 8.3., we computed the volatility of cash flow assuming a volatility of the exchange
rate of 10% p.a. Let’s assume that the current exchange rate is £0.5, that the return is distributed
normally, that its percentage volatility for two years is 14.14%, and that its mean is zero. We saw
earlier that if sales are constant, the CaR is £76.99M. Let’s now see what it is if sales change so that
the cash flow function is the one we derived in the previous paragraph. In this case, we know the
distribution of y, since the exchange rate in two years is the current exchange rate plus a normally
distributed return with mean zero and volatility of 10%. We therefore simulate 0.5 + x*0.5, where
x is a normally distributed variable with mean zero and volatility of 0.10.
We perform a Monte Carlo simulation with 10,000 drawings. Doing this, we find that the
expected cash flow is £90.91M, that the volatility is £20.05M, and that the fifth percentile shortfall
is £30.11M. The distribution of the cash flows is given in Figure 8.4. Note first that the expected cash
flow is not £90M, which is the cash flow if the dollar costs half a pound, which is its expected value.
There are two reasons for the discrepancy. First, with the Monte Carlo simulation, we draw a sample.
Different samples lead to different results. The results are expected to differ less for larger samples.
Second, because the cash flow is a nonlinear function of the exchange rate, its expectation is not equal
to the cash flow evaluated at the expected exchange rate. We can compare these results to the case
discussed earlier where sales are constant. In that case, the volatility was £46.66M and the fifth
percentile was £76.99M. The fact that the firm adjusts sales as the exchange rate changes reduces
cash flow volatility and CaR sharply relative to the case where it cannot do so. This shows that
flexibility in production can itself be a risk management tool, in that the firm that can adjust
production has less risk than the one that cannot.
Chapter 8, page 43
The above example allowed us to show how to use the Monte Carlo method. This method
can be applied when there are multiple sources of risk as well and can handle extremely complex
situations where one would be unable to compute exposures analytically. Box 8.1.: Monte Carlo
Analysis Example: Motor Inc. B describes a Monte Carlo analysis in detail for a variant of Motor
Inc. This analysis involves exchange rates for a number of years in the future, cash flows that depend
on past and current exchange rates as well as on whether a competitor enters the market. After having
computed cash flows for each year in each simulation trial, we have the distribution of the cash flows
that obtains given the distribution of the risk factors. We can then relate cash flows to risk factors.
In particular, we can measure the covariance between a given year’s cash flows and the exchange rate
in that year. This allows us to obtain a hedge coefficient for that year’s cash flow like we did when
we used the regression approach. The difference between the regression approach and the approach
discussed here is that the regression approach assumes that the future is like the past. The Monte
Carlo approach assumes that the distribution of the risk factors in the future is the same as in the past.
It does not assume that the distribution of cash flows is the same as in the past. Consequently, the
Monte Carlo approach could be used for a firm that has no history. This would not be possible with
the regression approach.
In the example Monte Carlo analysis discussed earlier in this section as well as the one in the
box, we make a number of assumptions that limit how the firm can change its operations in response
to changes in risk factors. For instance, the firm cannot switch methods of production or cannot start
producing in another country. It is important to stress that the Monte Carlo approach is generally the
best method to deal with situations where the firm has more flexibility than we assume here in
adjusting its operations in response to changes in risk factors. For instance, the Monte Carlo approach
Chapter 8, page 44
could be used to study a situation where a firm produces in one country if the exchange rate is lower
than some value and in another country if it has a higher value.
Section 8.6. Hedging competitive exposures.
We have now seen that foreign exchange changes affect firm value through a variety of
different channels. Firms often choose to focus on some of these channels, but not others. For
instance, they might be concerned about transaction exposure but not about competitive exposure.
Whether a firm concerned about transaction exposure only is right in doing so depends on why it is
concerned about risk. However, focusing on the wrong exposure can have dramatic implications for
a corporation. An oft-cited example is the case of a European airline concerned about the volatility
of its cash flow.3 It therefore decides to hedge its transaction exposure. Its most important transaction
is an order of planes from Boeing. The payment will have to be made in dollars. The company
interprets this to mean that the firm is short dollars and hedges by buying dollars forward. During the
period of time that the hedge is maintained, the dollar depreciates and the airline gets in financial
trouble despite being hedged.
What went wrong? The prices that the airline charges are fixed in dollars because of the
structure of the airline market. Hence, if the dollar falls, the airline loses income in its home currency.
This does not correspond to booked transactions when the firm decides on its risk management
policy, so that focusing on transaction exposure, the airline forgot about the exposure inherent in its
operations. Because of the airline’s blindness to its competitive exposure, its risk management policy
was inadequate. Had the firm taken into account its competitive exposure, it could have hedged its
3
This example is cited in a recent controversial article on foreign exchange risk
management, Copeland and Joshi (1996).
Chapter 8, page 45
cash flow effectively.
The airline company example makes it clear that hedging transaction exposure does not hedge
cash flow. For this company, the impact of exchange rate changes on the firm’s competitive position
is immediate. The firm cannot raise ticket prices denominated in dollars, so that it receives less
income. The firm could have hedged so that it can keep selling tickets at constant dollar prices and
keep its hedged income unaffected by the change in the exchange rate. However, hedging to keep
activities unchanged as exchange rates change does not make sense and raises important questions
about hedging competitive exposure. Suppose that the airline hedged its competitive exposure. The
exchange rate changes, so that at the new exchange rate, the airline is not as profitable before the
income from hedging as it was before. Consequently, some flights that were profitable before may
no longer be so. Even though the airline is hedged, it does not make sense for the airline to continue
flights that are no longer profitable. This is because each unprofitable flight makes a loss. If the airline
stops the flight, it eliminates the loss. The fact that the airline makes money on its hedges is gravy it does not have to waste it on losing flights. Hence, the fact that the airline is hedged does not have
implications for its operations except insofar as the absence of hedging would prevent it from
maintaining flights that it views as profitable. For instance, it could be that if it had not hedged, it
would not have enough working capital and would be unable to fly profitable routes.
In the case
of the airline discussed above, the competitive effects of changes in exchange rates manifest
themselves very quickly. In general, competitive effects take more time to affect a company and are
more subtle. Does it make sense to hedge competitive exposure and if yes, how? Suppose that a firm
has long-term debt and needs a minimum level of cash flow to make debt repayments. Adverse
developments in the firm’s competitive position due to exchange rate changes could make it
Chapter 8, page 46
impossible for the firm to make these debt repayments if it is not hedged and hence might force the
firm to incur bankruptcy and distress costs. Therefore, the arguments for hedging obviously provide
reasons to hedge competitive exposure. However, these arguments are not arguments for leaving
operations unaffected. A firm with low cash flow because of its unexpectedly poor competitive
position is a firm that has low internal funds to spend on new projects. One of these projects might
be to keep investing in the business with a poor competitive position because the situation is
temporary. However, it might be that the firm is better off to close the business with a poor
competitive position and invest in other operations.
One reason the firm might believe that the poor competitive position is temporary is that real
exchange rates are volatile and tend to move toward their purchasing power parity level in the long
run. Hence, the firm’s competitive position might change. Often, a firm’s activities in a country are
expensive to stop and start up later. The firm might therefore be better off to suffer short-term losses
in a country rather than close its activities. If that is the case, however, it has to have the resources
to pursue such a strategy. It may not have these resources if it is not hedged because capital markets
might be reluctant to provide funds. This is because it might be quite difficult for capital markets
participants to understand whether a strategy of financing losing operations is profitable from a longrun perspective. At the same time, though, it is also easy for the firm to convince itself that financing
these operations is profitable when it is not.
Despite the importance of competitive exposure for the future cash flows of firms, a lot of
firms are content to hedge only transaction and contractual exposures with financial instruments.
When it comes to hedging economic exposure, if they do it at all, they turn to “real” hedges. A real
hedge means that one is using the firm’s operating strategy to reduce the impact of real exchange rate
Chapter 8, page 47
changes. For our car manufacturing company, a real hedge would mean having production in the US
as well as in the UK. This way it would still be able to sell profitably in the US market if the pound
appreciates because it could sell cars produced in the US. Such a strategy sometimes makes sense.
Suppose that by having facilities in the UK and the US the firm can quickly increase production where
it is cheapest to produce and decrease it where it is most expensive. This might be profitable
regardless of hedging in that the firm’s marginal production cost over time could be lower. However,
in general having production in different countries can be quite expensive since it involves large setup
costs, reduces economies of scale, and leads to higher management costs. In contrast, the transaction
and management costs of hedging with financial instruments are fairly trivial.
Section 8.7. Summary and conclusions.
In this section, we investigated the impact of quantity risks on the firm’s risk management
policies. We saw that quantity risks do not alter the formula for the optimal hedge with futures and
forwards, but the hedge differs when quantity is random from when it is not. Quantity risks make it
harder to figure out a firm’s exposure to a risk factor. We explored this issue extensively for foreign
exchange rate exposure. We saw in this case that exchange rate changes have a complicated impact
on firm value because they change revenue per unit as well as the number of units sold. We developed
three approaches that make it possible to figure out exposures when there are quantity risks. First,
the pro forma approach works from the cash flow statement and identifies directly how changes in
the exchange rate affect the components of cash flow. The problem with this approach is that it does
works well when the impact of exchange rate changes is straightforward. Otherwise, the exposure
cannot be identified and it is not possible to compute CaR. Second, we showed how multivariate
Chapter 8, page 48
regressions can help identify exposures in the presence of quantity risks. Third, we introduced the
Monte Carlo method as a way to estimate the impact of changes in risk factors and to compute CaR.
We saw that this method is quite versatile and makes it possible to estimate exposures when
alternative approaches do not work. We concluded with a discussion of the separation between
hedging and operating policies. The fact that a losing operation is hedged does not mean that one has
to keep it.
Chapter 8, page 49
Literature note
This chapter uses material in Stulz and Williamson (1997). The Stulz and Williamson (1997) paper
provides more details on some of the issues as well as more references. The problem of hedging
quantity and price risks is a long-standing problem in the hedging literature with futures. As a result,
it is treated in detail in most books on futures contracts. The formula we present is derived and
implemented in Rolfo (1980). The problem of hedging foreign currency cash flows is at the center
of most books on international corporate finance. An early but still convincing treatment of the issues
as well as a collection of papers is Lessard (1979). Shapiro ( ) provides a state-of-the-art discussion.
However, international financial management books generally do not focus on the quantitative issues
as much as we do in this chapter. Levi (
) has the best textbook analysis in terms of demand and
supply curves and is the inspiration for our discussion. Marston (1997) provides a careful analysis.
The discussion of exchange rate dynamics is basic material in international finance textbooks. A good
treatment can be found in Krugman and Obstfeld ( ), for instance. The regression approach has led
to a large literature. Adler and Dumas (1984) develop this approach as an exposure measurement
tool. Jorion (1990) is a classic reference. For references to more recent work and new results, see
Griffin and Stulz (2000). Williamson (2000) has a detailed analysis of exposures in the automotive
industry. His work underlies our analysis of GM. The role of real hedges has led to a number of
papers. Mello, Parsons, and Triantis (1995) have an interesting paper where they analyze hedging in
the presence of flexible production across countries. Logue (1995) has an article where he questions
hedging competitive exposures with financial instruments and Copeland and Joshi (
) argue that
firms often hedge too much against changes in foreign exchange rates because of the diversification
effect discussed in chapter 7.
Chapter 8, page 50
Key concepts
Quantity risks, transaction exposure, contractual exposure, competitive exposure, cash flow delta to
a risk factor.
Chapter 8, page 51
Questions and exercises
1. An estimate by Goldman Sachs is that a 10% depreciation of the Japanese yen translates in a 2.5%
increase in the cost of Chinese exports in Chinese currency. How could that be?
2. Suppose that McDonalds hedges its current purchases of potatoes against an unancipated increase
in the price of potatoes, so that if the price of potatoes increases, it receives a payment equal to the
current purchases of potatoes times the change in the price of potatoes. How should an increase in
the price of potatoes affect the price of fries at McDonalds?
3. Suppose that purchasing power parity holds exactly. What would a 10% depreciation of the
Japanese yen that increases the cost of Chinese exports by 2.5% imply for prices in Japan and China?
4. How could a US car dealer in Iowa City that sells cars produced in the US by a US car
manufacturer have a competitive exposure to the Euro?
5. Consider a farmer who wants to hedge his crop. This farmer produces corn in Iowa. If his crop is
smaller than expected, then so is the crop of most farmers in the U.S. Should the farmer sell more or
less than his expected crop on the futures market? Why?
6. A wine wholesaler buys wine in France. He proceeds as follows: He goes on a buying trip in
France, buys the wines, and ships them to the US. When the wines arrive in the US, he calls up
Chapter 8, page 52
retailers and offers the wine for sale. There is a shipping lag between the time the wine is bought in
France and the time it arrives in the U.S. The wholesaler is concerned about his foreign exchange
exposure. His concern is that he has to pay for his purchases in French Franc when the wine arrives
in the US. Between the time the wine is bought in France and the time it is sold in the US, the French
Franc could appreciate, so that the wine becomes more costly in dollars. What are the determinants
of the cash flow exposure of the wholesaler to the dollar price of the French Franc?
7. Suppose you are a German manufacturer that competes with foreign firms. You find that your cash
flow is normally distributed and that for each decrease in the Euro price of the dollar by 1/100th of
a Euro, your cash flow falls by Euro1M. Yet, when the Euro appreciates, you sell fewer cars. How
do you hedge your cash flow? How does quantity risk affect your hedge?
8. Consider the analysis of Motor Inc. in Section 2. Suppose that Motor Inc. exports instead 11,000
cars to the US and 11,000 cars to Sweden. Let’s assume that the price of a Swedish Krona in pounds
is the same as the price of a US dollar and the price of a car sold in Sweden is Kr20,000. We assume
now that the jointly normal increments model holds. The volatility of each currency is 0.10 pounds
per year. The two prices have a correlation of zero. What is the CaR in this case?
9. Let’s go back to the base case discussed in the text where Motor Inc. exports 22,000 cars to the
US. Let’s assume that half the costs of producing a car are incurred in dollars six months before the
sale and that all sales take place at the end of the calendar year. The US interest rate is 10% p.a. What
is Motor Inc.’s exposure to the dollar?
Chapter 8, page 53
10. Assume that the volatility of the pound price of the dollar is 0.10 pound per year. What is the CaR
corresponding to the situation of question 9?
Chapter 8, page 54
Figure 8.1.A.
Pounds
DD’
P
P’
MC
DD
MR’ MR
0
Q’ Q
Quantity
Before the depreciation of the dollar, the pound price of cars sold in the U.S. is P and is given by the
point on the demand curve DD corresponding to the quantity where the marginal revenue curve MR
intersects the marginal cost curve MC. As the dollar depreciates, the demand shifts to DD’, the marginal
revenue curve shifts to MR’. The price in pounds falls to P’ and the quantity exported falls to Q’.
Figure 8.1.B.
Dollars
P’
P
MC’
MR
0
MC
DD
Q’ Q
Quantity
Before the depreciation of the dollar, the dollar price of cars sold in the U.S. is P and is given by the
point on the dollar demand curve DD corresponding to the quantity where the dollar marginal revenue
curve MR intersects the marginal cost curve MC. As the dollar depreciates, the marginal cost curve
moves to MC’. The dollar price increases to P’ and the quantity exported falls to Q’.
Chapter 8, page 55
Figure 8.2.A
Pounds
MC
P
DD=MR
DD’=MR’
0
Q
Quantity
Before the depreciation of the dollar, the pound price of cars sold in the U.S. is P and is given by the
point on the demand curve DD corresponding to the quantity where the marginal revenue curve MR
intersects the marginal cost curve MC. As the dollar depreciates, the demand shifts to DD’, the marginal
revenue curve shifts to MR’. Sales in the U.S. are no longer profitable.
Figure 8.2.B.
Dollars
MC’
MC
P
DD=MR
0
Q
Quantity
Before the depreciation of the dollar, the dollar price of cars sold in the U.S. is P and is given by the
point on the dollar demand curve DD corresponding to the quantity where the dollar marginal revenue
curve MR intersects the marginal cost curve MC. As the dollar depreciates, the dollar marginal cost
curve shifts upward to MC’. Sales in the U.S. are no longer profitable.
Chapter 8, page 56
Figure 8.3. Cash flow and exchange rate.
This figure gives the cash flow for Motor Inc. as a function of the pound price of the dollar. Cash
flow is a nonlinear function of the exchange rate. The delta foreign exchange rate exposure
measures the impact of a small increase in the exchange rate by using the slope of the cash flow
function. This approach is accurate for small changes in the exchange rate, but here for larger
increases it predicts a bigger fall in cash flow than actually occurs. If the exchange rate falls from
£0.50 to £0.25, the loss predicted by the delta exposure of 400m is £100m, while the actual loss
predicted by the cash flow function is £75m.
Chapter 8, page 57
Figure 8.4. Distribution of Motor Inc.’s cash flow.
The exchange rate is £0.5 for the dollar with a return volatility of 10% p.a. and zero mean.
Marginal revenue is $20,000 per car. The cash flows are obtained through a Monte Carlo
simulation with 10,000 draws using @Risk.
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
2
43 8
.0
47
58 3
.0
94
7
73
.1
88 42
.1
89
10 4
3.
23
11 7
8.
28
13 4
3.
33
14 1
8.
37
16 9
3.
42
6
PROBABILITY
Distribution of Motor Inc.'s cash flow
Cash flow
Chapter 8, page 58
Table 8.1.
Cash flow statement of Motor Inc. for 2000 (in million)
The dollar is assumed to be worth half a pound. The tax rate is
assumed to be 25%.
Sales in US
20,000 units at $20,000 each
£200
Sales in UK
20,000 units at £10,000 each
200
Cost of sales
(300)
Investment
(50)
Taxes
0.25x(400-300)
(25)
Net cash flow
25
Chapter 8, page 59
Box 8.1.: Monte Carlo Analysis Example: Motor Inc. B.
Motor Inc. B is a car producer in the UK. It has a vehicle in a market segment that it now
dominates in the UK and wants to expand its success to the US market. The firm is worried about
the impact of changes in the exchange rate on the net present value of the project. An analyst is
therefore asked to evaluate the exposure of the net present value of the project as well as the
exposure of each annual cash flow to the dollar/pound exchange rate. To evaluate the exposures the
analyst uses her knowledge of the costs of Motor Inc. B and of its competitive position in the US
market.
The demand function for the cars in the US is assumed to be:
Number of cars customers are willing to buy = 40,000 - Dollar price of a car
The marginal cost of a car is £7,500 per unit and it is assumed to be constant over time. At an
exchange rate of $2 per pound, it is optimal for the firm to sell 25,000 cars at $15,000. The analyst
realizes that there is a strong possibility that a US producer will enter the market segment with a
comparable vehicle. This will have an effect on the pricing and thus the sales volume of Motor Inc.
B cars. The demand function with an entrant becomes:
Demand = 40,000 - 1.25*Dollar price of a car
Consequently, the entrant makes the demand for Motor Inc. B cars more sensitive to price. The
analyst believes that the probability that Motor Inc. B will face a competitor is 75% if the exchange
Chapter 8, page 60
rate exceeds $1.95 per pound and 50% otherwise. To sell in the US, Motor Inc. B has to spend
£100m in 2001. It can then sell in the US starting in 2002. Depreciation is straight line starting in
2001 over five years. The tax rate on profits is 45% if profits are positive and zero otherwise. Motor
Inc. B fixes the dollar price of its cars at the end of a calendar year for all of the following calendar
year. The sales proceeds in dollars are brought back to the UK at the end of each calendar year at the
prevailing exchange rate.
This example includes nine random variables: whether an entrant comes in or not, and the
exchange rate for each calendar year from 2001 to 2009. Whether an entrant comes in or not is
distributed binomially with a probability of 0.75 if the exchange rate in 2001 exceeds 1.95. The
percentage change of the exchange rate from one year to the next is distributed normally with mean
of 2.2% and standard deviation of 14.35%. These random variables do not influence the present value
in a straightforward way. We argue in the text that the Monte Carlo analysis is particularly useful in
the presence of path-dependencies. In this case, there are two important path dependencies: first, the
cash flows depend on whether there is an entrant in 2001 which itself depends on the exchange rate;
second, the cash flow for one year depends on that year’s exchange rate as well as on the exchange
rate the year before.
We performed a Monte Carlo analysis using 400 trials. The output of the Monte Carlo
analysis can be used to understand the exposure of Motor Inc. B to the dollar/pound exchange rate
in many different ways. Here, we show two of these ways. First, the figure shows the relation
between the 2003 cash flow and the exchange rate in that year. At high exchange rates, it becomes
likely that the cash flow will be negative and there is a decreasing nonlinear relation between pound
cash flow and the dollar/pound exchange rate. Because of the path dependencies we emphasized, the
Chapter 8, page 61
Figure Box 8.1. Motor Inc. B Corporation
This figure shows the relation between the pound cash flow in 2003 and the dollar price of the pound
in 2003 for the Motor Inc. B simulation. We draw four hundred different exchange rate series from
2001 to 2009. For each of these exchange rate series, we use the binomial distribution to obtain the
decision of whether a competitor enters the market or not. Based on the realization of the random
variables, we compute the cash flows from 2001 to 2009. The 2003 cash flows and their associated
2003 exchange rates are then plotted on the figure.
2003 cash flow and exchange rate
400.00
350.00
250.00
200.00
150.00
100.00
50.00
Dollar price of pound
Chapter 8, page 62
$2.96
$2.60
$2.40
$2.24
$2.10
$2.00
$1.91
$1.79
$1.73
$1.66
$1.57
$1.46
$1.37
-50.00
$1.22
0.00
$0.85
Pound cash flow
300.00
cash flow depends on the exchange rate in earlier years also.
An alternative way to use the output of the Monte Carlo analysis is to regress the pound net
present value of the project (NPV) of selling in the US on the future exchange rates. This shows the
sensitivity of the net present value to future exchange rates. The NPV is measured in million pounds.
We get:
NPV =
1285 - 16.123X(2001) - 144.980X(2002) - 61.34X(2003) - 39.270X(2004)
(28.95)
(-0.500)
(-4.382)
(-1.951)
(-1.245)
-109.892X(2005) - 42.596X(2006) - 9.843X(2007) - 82.941X(2008) - 14.318X(2009)
(-3.225)
(-1.230)
(-0.307)
(-2.509)
(-0.612)
where X(j) is the exchange rate in year j and t-statistics are in parentheses below the regression
coefficients. The R-squared of this regression is 78%. Whereas the NPV is negatively related to all
exchange rates, not all future exchange rates are equally important determinants of the net present
value. This is not surprising. First, the net present value calculation puts less weight on the cash flows
that are received farther in the future. Second, our example has complicated path dependencies. Note,
however, that some exchange rates have an extremely large effect on the NPV. For instance, if the
2002 dollar price of the pound is unexpectedly higher by 10 cents, the NPV is lower by 14.5 million
pounds. The NPV obtained by averaging across 400 trials is £345 million. A 10 cents deviation in the
2002 exchange rate (about 4%) correspond to a change in the NPV of slightly more than 4%.
Chapter 8, page 63
Chapter 9: Measuring and managing interest rate risks
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section 9.1. Debt service and interest rate risks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Section 9.1.1. Optimal floating and fixed rate debt mix. . . . . . . . . . . . . . . . . . . . . . . 3
Section 9.1.2. Hedging debt service with the Eurodollar futures contract. . . . . . . . . . . 5
Section 9.1.3. Forward rate agreements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Section 9.2. The interest rate exposure of financial institutions. . . . . . . . . . . . . . . . . . . . . . . . 12
Section 9.2.1. The case of banks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Section 9.2.2. Asset-liability management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Section 9.3. Measuring and hedging interest rate exposures. . . . . . . . . . . . . . . . . . . . . . . . 22
Section 9.3.1. Yield exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Section 9.3.2. Improving on traditional duration. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Section 9.4. Measuring and managing interest rate risk without duration. . . . . . . . . . . . . . . . 41
Section 9.4.1. Using the zero coupon bond prices as risk factors. . . . . . . . . . . . . . . 44
Section 9.4.2. Reducing the number of sources of risk: Factor models. . . . . . . . . . . 46
Section 9.4.3. Forward curve models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Section 9.5. Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Review questions and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Box 9.1. The tailing factor with the Euro-dollar futures contract . . . . . . . . . . . . . . . . . . . . . . 59
Box 9.2. Orange County and VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Box 9.3. Riskmetrics™ and bond prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Table Box 9.1. Riskmetrics™ mapping of cash flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Table 9.1. Eurodollar futures prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Table 9.2. Interest Rate Sensitivity Table for Chase Manhattan . . . . . . . . . . . . . . . . . . . . . . 67
Table 9.3. Example from Litterman and Scheinkman(1991) . . . . . . . . . . . . . . . . . . . . . . . . . 68
Figure 9.1. Example of a Federal Reserve monetary policy tightening . . . . . . . . . . . . . . . . . . 69
Figure 9.2. Bond price as a function of yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure 9.3. The mistake made using delta exposure or duration for large yield changes . . . . . 71
Figure 9.4. Value of hedged bank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Figure 9.5.Term Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Table 9.1. Factor loadings of Swiss interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Chapter 9: Measuring and managing interest rate risks
September 22, 2000
© René M. Stulz 1998, 2000
Chapter objectives
At the end of this chapter, you will:
1. Understand tradeoffs between fixed and floating rate debt for firms.
2. Be able to measure interest rate exposures.
3. Have tools to measure interest rate risks with VaR and CaR.
4. Have reviewed interest rate models and seen how they can be used to hedge and measure interest
rate risks.
Chapter 9, page 1
In 1994, interest rate increases caused a loss of $1.6 billion for Orange County and ultimately led
the county to declare bankruptcy. If Orange County had measured risk properly, this loss would most likely
never have happened. In this chapter, we introduce tools to measure and hedge interest rate risk. In one of
our applications, we show how these tools could have been used by Orange County. Much of our analysis
deals with how interest rate changes affect the value of fixed income portfolios, but we also discuss how
financial institutions as well as non-financial corporations can hedge interest rate risks associated with their
funding. Firms can alter their interest rate risks by changing the mix of fixed and floating rate debt they have.
We investigate the determinants of the optimal mix of floating and fixed rate debt for a firm and show how
a firm can switch from floating rate debt to fixed rate debt using a futures contract.
Financial institutions are naturally sensitive to interest rate risks. Remember that we use exposure
to quantify the sensitivity to a risk factor. Exposure of a security price to a risk factor is our estimate of the
change in the security price per unit change in the risk factor. We therefore review the determinants of the
exposure of financial institutions to interest rate changes and why these institutions care about this exposure.
Interest rate risks affect a firm’s cash flow as well as its value. As a result, some firms focus on the effect
of interest rate risks on cash flow whereas others are concerned about their effect on value. The techniques
to measure and manage the cash flow impact and the value impact of interest rate changes differ and are
generally used by different types of institutions. After reviewing these issues, we focus more directly on the
quantitative issues associated with the measurement and management of interest rate risks.
When measuring interest rate risk, a popular approach is to use duration. We will review how
duration is used, when it is and is not appropriate, and how one can do better than using duration. We also
Chapter 9, page 2
present approaches to measuring interest rate risk that do not rely on duration. In Chapter 4, we showed
how to estimate value-at-risk (VaR) in general. In this chapter, we present approaches to estimate VaR for
fixed income securities. However, fixed income securities differ widely in complexity. We will restrict our
attention mostly to fixed income securities without embedded options. Options as they relate to fixed income
securities will be studied in Chapter 14.
Section 9.1. Debt service and interest rate risks.
In this section, we first discuss the optimal mix of floating and fixed rate debt for a firm. We then turn
to using the Eurodollar futures contract to hedge the interest rate risks of floating rate debt. An alternative
financial instrument to hedge interest rate risks is a forward rate agreement (FRA). We explain this
instrument in the last part of the section.
Section 9.1.1. Optimal floating and fixed rate debt mix.
As we have already discussed, a firm’s choice of funding is part of its risk management strategy. An
all-equity firm may be able to avoid risk management altogether. However, if equity is expensive relative to
debt, firms choose to have some debt and have to decide the mix of floating and fixed rate debt that is
optimal. Fixed rate debt does not contribute to the volatility of cash flow since it has constant payments. In
contrast, floating rate debt has variable payments indexed to an interest rate and may increase or decrease
the volatility of cash flow. Consider a firm with revenues that are higher when interest rates are high. Such
a firm has more variable cash flow if it finances itself with fixed rate debt than with floating rate debt. With
floating rate debt, the firm has high interest rate payments when its revenues are high. In contrast, a firm
Chapter 9, page 3
whose revenues fall as interest rates increase is in a situation where floating rate debt could lead to substantial
cash shortfalls since the firm might not be able to make interest payments when interest rates are high. Such
a firm might want to finance itself so that its interest rate payments are inversely related to the level of interest
rates.
It is important to understand that the debt maturity of a firm affects the interest rate sensitivity of its
cash flows. Fixed rate debt that matures in one year means that in one year the firm has to raise funds at
market rates. Hence, its cash flows in the future will depend on the rates it has to pay on this new debt. An
important issue with debt maturity is that the credit spread a firm has to pay can change over time as well.
A credit spread is the difference between the interest payment a firm has to promise and the payment it
would have to make if its debt were risk-free. This means that short-term debt increases future cash flow
volatility because the firm might have to promise greater coupon payments in the future to compensate for
greater credit risk.
The tradeoffs we have just discussed imply that there is an optimal mix of fixed and floating rate
financing for a firm. Once a firm has financed itself, there is no reason for the mix to stay constant. If a firm
suddenly has too much floating rate debt, for instance, it can proceed in one of two ways. First, it can buy
back floating rate debt and issue fixed rate debt. Doing this involves floatation costs that can be substantial.
Second, the firm can hedge the interest rate risks of the floating rate debt. Having done this, the firm has
transformed the floating rate debt into fixed rate debt since the interest payments of the hedged debt do not
fluctuate. Often, hedging the interest rate risks of floating rate debt is dramatically cheaper than buying back
fixed rate issuing new floating rate debt.
A firm might want to use financial instruments to add interest rate risk to its debt service. For
Chapter 9, page 4
instance, a firm with fixed rate debt whose operating income was insensitive to interest rates finds that it has
become positively correlated with interest rates. That firm would have a more stable cash flow if it moved
from fixed-rate debt to floating-rate debt since debt payments would then be positively correlated with
operating income. Rather than buying back fixed rate debt and issuing floating rate debt, the firm might be
better off using derivatives to change its debt service so that it looks more like the debt service of floating
rate debt than of fixed rate debt. Alternatively, a firm may be in a situation where some sources of funding
are attractive but using them would force the firm to take on interest rate risk it does not want. For instance,
non-US investors find some tax advantages in buying dollar debt issued outside the US. In particular, such
debt is bearer debt, so that the owners of the debt are generally hidden from the fiscal authorities. As a result
of these tax advantages, American companies have been able to issue debt offshore at lower rates than
within the US. In extreme cases, American companies were even able to raise funds offshore paying lower
debt service than the US government on debt of comparable maturity. Access to offshore markets was
generally available only for well-known companies. These companies therefore benefitted from taking
advantage of the lower cost financing. Yet, a company might have wanted to use floating-rate financing
which would not have been typically available on the offshore market for long maturity debt. Consequently,
after raising fixed rate funds abroad, the corporation would have to transform fixed-rated debt into floating
rate debt.
Section 9.1.2. Hedging debt service with the Eurodollar futures contract.
Let’s now look at how a firm would change the interest rate risks of its debt service. Consider a firm
that wishes to hedge the interest rate risks resulting from $100M of face value of floating rate debt. The
Chapter 9, page 5
interest rate is reset every three months at the LIBOR rate prevailing on the reset date. LIBOR stands for
the London InterBank Offer Rate. It is the rate at which a London bank is willing to lend Eurodollars to
another London bank. Eurodollars are dollars deposited outside the US and therefore not subject to US
banking regulations. US banking regulations are onerous to banks. In contrast, lending and borrowing in
Eurodollars is much less regulated. As a result, the Eurodollar market offers more advantageous lending and
borrowing rates than the domestic dollar market because the cost of doing business for London banks is
lower than for US domestic banks. In the 1960s and 1970s, though, the advantage of the London banks
was much higher than it is now. The LIBOR rate is provided by the British Bankers Association (
http://www.bba.org.uk/ ) through a designated informationvendor. The designated information vendor polls
at least eight banks from a list reviewed annually. Each bank “will contribute the rate at which it could
borrow funds, were it to do so by asking for and then accepting inter-bank offers in reasonable market size
just prior to 1100.” for various maturities. The British Bankers Association averages the rates of the two
middle quartiles of the reporting banks for each maturity. The arithmetic average becomes the LIBOR rate
for that day published at noon, London time, on more than 300,000 screens globally.
Suppose that today, date t, the interest rate has just been reset, so that the interest rate is known
for the next three months. Three months from now, at date t+0.25, a new interest rate will be set for the
period from date t+0.25 to date t+0.5. As of today, the firm does not know which interest rate will prevail
at date t+0.25. LIBOR is an add-on rate paid on the principal at the end of the payment period. The interest
payment for the period from date t+0.25 to date t+0.5 is 0.25 times the 3-month LIBOR rate determined
on the reset date, date t+0.25, and it is paid at the end of the payment period, date t+0.5. If 3-month
LIBOR at the reset date is 6%, the payment at t+0.5 is 0.25* 6%*100M, which is $1.5M. The general
Chapter 9, page 6
formula for the LIBOR interest payment is (Fraction of year)*LIBOR*Principal. LIBOR computations use
a 360-day year. Typically, the payment period starts two business days after the reset date, so that the end
of the payment period with quarterly interest payments would be three months and two business days after
the reset date. The interest rate payment for the period from t+0.25 to t+0.5 made at date t+0.5 is equal
to the principal amount, $100M, times 0.25 because it corresponds to a quarterly payment, times LIBOR
at t+0.25, RL(t+0.25). To obtain the value of that payment at the beginning of the payment period, we have
to discount it at RL(t+0.25) for three months:
Value of interest payment at beginning of payment period =
100M *0.25RL(t +0.25)
(1 + 0.25RL(t +0.25))
Using our numerical example with LIBOR at t+0.25 at 6%, we have:
$100M *0.25*0.06
= $1.415M
1 + 0.06
To hedge the interest payment for the period from t+0.25 to t+0.5 at date t, we could think of using
a futures contract. The appropriate futures contract in this case is the Eurodollar contract. The Eurodollar
futures contract is traded on the International Money Market (IMM) of the Chicago Mercantile Exchange.
It is for a Eurodollar time deposit with three-month maturity and a $1 million principal value. The futures
price is quoted in terms of the IMM index for three-month Eurodollar time deposits. The index is such that
its value at maturity of the contract is 100 minus the yield on Eurodollar deposits. Cash settlement is used.
Chapter 9, page 7
The Eurodollar offered yield used for settlement is obtained by the IMM through a poll of banks. These
banks are not the same as those used by the British Bankers Association to obtain LIBOR, so that the
Eurodollar offered yield used for settlement is not exactly LIBOR. In our discussion, though, we ignore this
issue since it is of minor importance. Each basis point increase in the index is worth $25 to the long. Table
9.1. shows the futures prices on a particular day from the freely available 10 minute lagged updates on the
IMM web site (http://www.cme.com/cgi-bin/prices.cgi?prices/r_ed.html). An important characteristic of
Eurodollar futures is that they are traded to maturities of up to ten years.
To see how the contract works, suppose that we have a short position of $100M in the contract
expiring in four months quoted today at 95. If the contract matured today, the yield on Eurodollar deposits
offered would be 5% (100-95 as a percentage of 100). The yield that would obtain if the contract matured
today is called the implied futures yield. In this case, the implied futures yield is 5%. Four months from now,
Eurodollar deposits are offered at 6%. In this case, the index at maturity of the contract is at 94. Since the
index fell and we are short, the settlement variation (the cash flows from marking the contract to market)
over the four months we held on to the position is 100 basis points annualized interest (6% - 5%) for three
months applied to $100M. Hence, we receive 0.25*0.01*$100M, which amounts to $250,000. Now,
suppose that we have to borrow $100M at the market rate in four months for a period of three months. We
would have to pay interest for three months of 0.25*0.06*100M, or $1.5M. The interest expense net of
the gain from the futures position will be $1.5M - $0.25M, which is $1.25M and corresponds to an interest
rate of 5%. This means that the futures position allowed us to lock in the implied futures yield of 5%.
The only difficulty with the hedge we just described is that the futures settlement variation has been
completely paid by the time the futures contract matures. In contrast, the interest to be paid on the loan has
Chapter 9, page 8
to be paid at the end of the borrowing period of three months. Hence, as of the maturity of the futures
contract, the value of the interest payment is $1.5M discounted at LIBOR for three months whereas the
settlement variation on the futures contract is $0.25M. To obtain a more exact hedge, therefore, the futures
hedge should be tailed. The tailing factor discussed in Chapter 6 was a discount bond maturing at the date
of the futures contract. Here, however, because interest paid on the loan is paid three months after the
maturity of the futures contract, this means that we get to invest the settlement variation of the futures
contract for three more months. To account for this, the tailing factor should be the present value of a
discount bond that matures three months after the maturity of the futures contract. The details of the
computation of the tailing factor are explained in Box 9.1. The tailing factor with the Eurodollar
contract.
In our example, we were able to take floating rate debt and eliminate its interest rate risks through
a hedge. Consequently, the hedged floating rate debt becomes equivalent to fixed rate debt. For the
corporation, as long as there are no risks with the hedge, it is a matter of indifference whether it issued fixed
rate debt or it issued floating rate debt that it hedges completely against interest rate risks. The Eurodollar
futures contract enables the corporation to take fixed rate debt and make it floating as well. By taking a short
position in the Eurodollar futures contract, one has to make a payment that corresponds to the increase in
the implied futures yield over the life of the contract. Adding this payment to an interest payment from fixed
rate debt makes the payment a floating rate payment whose value depends on the interest rate. It follows
therefore that we can use the Eurodollar futures contract to transform fixed rate debt into floating rate debt
or vice versa. With this contract, therefore, the interest rate risk of the firm’s funding is no longer tied to the
debt the firm issues. For given debt, the firm can obtain any interest rate risk exposure it thinks is optimal.
Chapter 9, page 9
To understand how the futures contract is priced, we consider a portfolio strategy where we
borrow on the Euro-market for six months and invest the proceeds on the Euro-market for three months
and roll over at the end of three months into another three-month investment. With this strategy, we bear
interest rate risk since the payoff from the strategy in six months increases with the interest rate in three
months which is unknown today. We can hedge that interest rate risk with the Euro-dollar contract. If we
hedge the interest rate risk, our strategy has no risk (we ignore possible credit risk) and therefore its payoff
should be zero since we invested no money of our own.
To implement the strategy, we have to remember that the futures contract is for $1M notional
amount. Therefore, we can eliminate the interest rate risk on an investment of $1M in three months. The
present value of $1M available in three months is $1M discounted at the three-month rate. Suppose that the
three-month rate is 8% annually and the six-month rate is 10% annually. We can borrow for six months at
10% and invest the proceeds for three months at 8%. In three months, we can re-invest the principal and
interest for three months at the prevailing rate. With the 8% rate, the present value of $1M available in three
months is $980,392. We therefore borrow $980,392 for six months and invest that amount for three
months. In six months, we have to repay $980,392 plus 5%, or $1,029,412. To hedge, we short one Eurodollar contract. Ignoring marking to market, the only way that we do not make a profit on this transaction
is if the Eurodollar futures contract allows us to lock in a rate such that investing $1M for three months at
that rate yields us $1,029,412. This corresponds to an annual rate of 11.765%. With this calculation, the
futures price should therefore be 100 - 11.765, or 88.235.
The price of the futures contract obtained in the last paragraph ignores marking-to-market, so that
departures from the price we have computed do not represent a pure arbitrage opportunity. Nevertheless,
Chapter 9, page 10
the example shows the determinants of the futures price clearly. Suppose that in three months the Eurodollar
rate is 15%. In that case, we lose money on our futures position. In fact, we lose 3.235/4 per $100 (15% 11.765% for three months). We therefore invest in three months $1M - 32,350/4/1.0375 for three months
at 3.75%. Doing that, we end up with proceeds of $1,029,412, which is what we would have gotten had
interest rates not changed. Our hedge works out exactly as planned.
Section 9.1.3. Forward rate agreements.
In contrast to the Eurodollar contract, forward rate agreements (FRAs) are traded over the counter.
A forward rate agreement is an agreement whereby the buyer commits to pay the fixed contract rate on
a given amount over a period of time and the seller pays the reference rate (generally LIBOR) at maturity
of the contract. The amount used to compute the interest payment plays the role of principal amount, but
it is not paid by anybody. Amounts used solely for computations are usually called notional amounts.
Consider the following example of a FRA. The contract rate is 10% on $10M notional starting in
two months for three months. With this contract, the buyer would pay the 10% and the seller would pay
three-month LIBOR set in two months on $10M. The notional amount of $10M would not be exchanged.
With the FRA, the floating rate buyer locks in the borrowing rate. This is because in two months the buyer
pays 10% and receives LIBOR. Consequently, he can take the LIBOR payment and pass it on to the
lender. The payment from the FRA is computed so that in two months the required payment for the buyer
is the 10% annual payment in three months minus the payment at the reference rate discounted at the
reference rate for three months. This insures that the FRA is a perfect hedge. In other words, if the reference
rate is 8% in two months in our example, the payment in two months is 10% annual for three months (say
Chapter 9, page 11
91 days) minus 8% annual for the same period discounted at 8% annual for that period. This amounts to
(91/360)*0.02*10M/(1+(91/360)*0.08), or $49,553. Note that we divide by 360 days because the day
count for Eurodollars is based on 360 days. If the buyer has to make a floating rate payment based on the
rate in two months, he will have to pay 8% annual in five months or $202,222 (which is
0.08*(91/360)*10M). If the buyer borrows $49,553, he will have to pay back $50,556 in three months.
Hence, at that time, the buyer’s total payment will be $252,778. The payment on a $10M loan at 10% for
91 days is exactly the same amount. If it were not, there would be an arbitrage opportunity and the FRA
would be mispriced.
Section 9.2. The interest rate exposure of financial institutions.
Financial institutions can focus on measuring and hedging the impact of interest rate changes on their
earnings or on their portfolio value. We discuss these two approaches in this section. We then focus on
issues of implementation in the following two sections.
Section 9.2.1. The case of banks.
Let’s look at a typical bank. It has assets and liabilities. Most of its liabilities are deposits from
customers. Its assets are commercial and personal loans, construction loans, mortgages and securities. This
bank faces interest rate risks as well as other risks. If interest rate risks are uncorrelated with other risks,
they can be analyzed separately from other risks. However, if they are correlated with other risks, then the
bank cannot analyze its interest rate risks separately from its total risk. Even if the bank has to consider its
total risk, it is helpful for it to understand its interest rate risks separately. Doing so can help it hedge those
Chapter 9, page 12
risks and understand the implications for the bank of changes in interest rates.
Suppose that interest rates increase. A measure of the impact of the increase in interest rates is how
it affects the bank’s income. Banks have generally used various exposure measures that tell them how their
net interest income (NII) is affected by changes in rates. If a bank keeps its holdings of assets and liabilities
unchanged, the only impact on NII has to do with changes in the interest payments of various assets. As
interest rates change, some assets and liabilities are affected and others are not. An asset or liability whose
interest payment changes during a period as a result of a change in interest rates either because it has a
floating rate or because it matures and is replaced by a new asset or liability with a market rate is said to
reprice during that period of time. The impact of the increase in interest rates on NII comes through the
repricing of assets and liabilities. The period of time over which the impact of the increase in interest rates
is computed determines which assets and liabilities are repriced. For instance, banks have fixed rate
mortgages outstanding whose interest rate payments are not affected by interest rate changes. They also
have deposits outstanding whose interest rate payments increase as interest rates increase. A bank with a
large portfolio of fixed-rate mortgages financed by deposits with short maturities therefore experiences a fall
in its interest income as interest rates increase until its mortgage portfolio reflects the new interest rates. If
we look at the impact of an interest rate increase in the long run, it is much less than in the short run, because
new mortgages are written at the new rates. A bank whose liabilities reprice faster than the assets is called
liability sensitive. This is because the increase in interest rates increases the payments made to depositors
more than it increases payments received. Alternatively, it could be that the bank’s liabilities have a lot of
time deposits for one year and more, whereas its loans are floating-rate loans. In this case, the inflows might
Chapter 9, page 13
increase more than the outflows and the bank would be asset sensitive.
Why would the bank care about net interest income in this way? If none of the assets and liabilities
of the bank are marked to market, the impact of interest rate changes on net interest income is the only effect
of interest rate changes on accounting income and on the balance sheet assuming that the bank’s portfolio
stays constant. If the bank wants its earnings to be unaffected by interest rate changes and starts from a
position of being liability sensitive, it can do so through financial instruments. First, it could rearrange its
portfolio so that its assets are more interest-rate sensitive. For instance, it could sell some of the fixed rate
mortgages it holds and buy floating rate assets or short-term assets. Alternatively, it could take futures
positions that benefit from increases in interest rates. This would mean shorting interest rate futures contracts.
Remember from our discussion of the T-bond futures contract that interest rate futures contracts are similar
to forward purchases of bonds. An increase in interest rates decreases bond prices, so that the short position
benefits.
There exist a number of different approaches to measure the exposure of net interest income to
interest rate changes. The simplest and best-known approach is gap measurement. The first step in gap
measurement is to choose a repricing period of interest. Consider a bank’s assets and liabilities. A new
thirty-year mortgage does not reprice for thirty years if held to maturity in that the rate on that mortgage is
fixed for that period. In contrast, a one-month CD reprices after one month. At any time, therefore, a bank
has assets and liabilities that reprice over different time horizons. Suppose we want to find out how the
bank’s income next month is affected by a change in interest rates over the current month. The only
payments affected by the change in rates are the payments on assets and liabilities that reprice within the
month. A deposit that reprices in two months has the same interest payments next month irrespective of how
Chapter 9, page 14
interest rates change over the current month. This means that to evaluate the interest-rate sensitivity of the
bank’s interest income over the next month, we have to know only about the assets and liabilities that reprice
before the next month. Consider a bank with $100B in assets for which $5B assets and $10B liabilities
reprice before next month. $5B is a measure of the bank’s asset exposure to the change in rates and $10B
is a measure of the bank’s liability exposure to the change in rates. This bank has a net exposure to changes
in rates of -$5B. The difference between a bank’s asset exposure and its liability exposure measure over
a repricing interval is called the bank’s dollar maturity gap. Alternatively, the gap can be expressed as a
percentage of assets. In this case, the bank has a percentage maturity gap of 5%.
Table 9.2. shows how Chase Manhattan reported gap information in its annual report for 1998. The
first row of the table gives the gap measured directly from the bank’s balance sheet. The gap from 1 to 3
months is $(37,879) million. The parentheses mean that between 1 and 3 months the bank’s liabilities that
reprice exceed the assets that reprice by $37,879M. Derivatives contracts that are not on the balance sheet
of the bank affect its interest rate exposure. Here, the derivatives increase the bank’s interest rate gap over
the next three months. As a result, the bank’s gap including off-balance sheet derivatives for that period is
$42,801M. Though the balance-sheet gap at a short maturity is negative, it becomes positive for longer
maturities. The relevant gap measure if one looks at one year is the cumulative interest rate sensitivity gap
reported in the second row from the bottom. The cumulative interest rate sensitivity gap for 7-12 months
is $(37,506) million. This gap tells us that including all assets and liabilities that reprice within one year, the
bank has an excess of liabilities over assets of $37,506M.
The one-year gap is an extremely popular measure. It measures assets and liabilities that will be
repriced within the year. It is important to understand the assumptions made when using such a gap measure.
Chapter 9, page 15
The gap measure is a static measure. It takes into account the assets and liabilities as they currently are and
assumes that they will not change. This can make the interest rate exposure measure extremely misleading.
To see this, let’s go back to the example discussed above. Let’s assume that $50B of the assets correspond
to fixed rate mortgages. The bank has a -$5B gap. On an annual basis, therefore, a 100 basis points
decrease in rates would increase interest income by $50M since the bank’s income from assets would fall
by $50M and its interest payments on deposits would decrease by $100M. Now, however, new mortgage
rates are more attractive for existing borrowers. This makes it advantageous for borrowers to refinance their
mortgages. Suppose then that half of the mortgages are refinanced at the new lower rate which is 100 basis
points lower than the rate on existing mortgages. In this case, $25B of the mortgages are refinanced and
hence repriced. As a result, $25B of assets are repriced in addition to the $5B used in the gap measure.
Instead of having an exposure of -$5B, the bank has an exposure of $20B. Using the gap measure would
therefore lead to a dramatic mistake in assessing the firm’s exposure to interest rates. If the bank hedged
its gap, hedging would add interest rate risk. To see this, note that to hedge the gap the bank would have
to go short interest rate futures so that it benefits from an increase in rates. The true exposure is such that
the bank makes a large loss when interest rates fall, so that it would have to be long in futures. By being
short, the bank adds to the loss it makes when interest rates fall. The additional complication resulting from
mortgages is, however, that the refinancing effect is asymmetric: the bank loses income if rates fall but does
not gain income if the interest rates increase. We will get back to the impact of such asymmetries on hedging
repeatedly.
Gap measures are static measures. They are therefore more appropriate for banks that have sticky
portfolios. For instance, suppose that the bank we discussed has a portfolio of 8% mortgages when market
Chapter 9, page 16
rates are at 12%. A 100-basis points decrease in interest rates does not make it advantageous for
borrowers to refinance their mortgages. Some borrowers refinance because they are moving, but this is not
that important in this context. Consequently, in this case, a static measure may be a good indicator of interest
rate exposure. However, if instead market rates are 7.5%, a 100 basis point decrease may have a large
impact on refinancings. Though we used mortgages as our example of portfolio changes due to interest rate
changes, other assets and liabilities can be affected. For instance, consider an individual with a two-year CD.
One month after buying the CD interest rates increase sharply. Depending on the early withdrawal provisions
of the CD, the individual may decide to withdraw his money and incur the penalty to reinvest at the new rate.
When using gap measures, it is therefore important to check whether using a static measure is really
appropriate to measure a bank’s exposure. Another important issue with the gap measure is that it presumes
that the change in interest rates is the same for all assets and liabilities that reprice within an interval. This
need not be the case. There can be caps and floors on interest payments that limit the impact of interest rate
changes. It can also be the case that some interest payments are more sensitive to rate changes than others.
For instance, the prime rate tends to be sticky and some floating rate payments can be based on sticky
indexes or lagging indexes.
There is a direct connection between a gap measure and a CaR measure. Consider our bank with
a gap of -$5B. Suppose that the standard deviation of the interest rate is fifty basis points. If the rate is 5%
now and its changes are normally distributed, there is a 5% chance that the rate in one year will be greater
than 5% + 1.65*0.5%, or 5.825%. This means that there is a 0.05 probability of a shortfall in interest
income of 0.825%*5B for next year, or $41.25M. We can therefore go in a straightforward way from a
gap measure to a CaR based on gap that takes into account the distribution of the interest rate changes. This
Chapter 9, page 17
CaR makes all the same assumptions that gap does plus the assumption that interest rate changes are
normally distributed. We know how to compute CaR for other distributions using the simulation method, so
that we can compute the gap CaR for alternate distributions of interest rates changes.
A natural improvement over the gap approach is to look at the bank’s balance sheet in a more
disaggregated way and model explicitly the cash flows of the various assets and liabilities as a function of
interest rates. In this way, one can take into account the dependence of the repayment of mortgages on
interest rates as well as limits on interest payment changes embedded in many floating-rate mortgage
products. Rates for different assets and liabilities can be allowed to respond differently to interest rate
shocks. To follow this approach, we therefore need to model the assets and liabilities of the bank carefully
so that we understand how their cash flows are affected by interest rate changes. Once we have done this,
we can figure out how the bank’s NII changes with an interest rate shock. A standard approach is to use
the model to simulate the impact of changes on NII over a period of time for a given change in interest rates,
say 100 basis points or 300 basis points.
At the end of 1999, Bank One reported that an immediate increase in rates of 100 bp would
decrease its pretax earnings by 3.4% and that an immediate decrease in rates of 100 bp would increase its
earnings by 3.7%. Bank One is fairly explicit about how it measures the impact of interest rate changes on
earnings. It examines the impact of a parallel shock of the term structure, so that all interest rates change by
100 bp for a positive shock. It then makes a number of assumptions about how changes in interest rates
affect prepayments. In the process of computing the impact of the interest rate change on earnings, it takes
into account the limits on interest payments incorporated in adjustable rate products. As part of its evaluation
of a change in interest rates on earnings, it estimates the impact of the change in interest rates on fee income
Chapter 9, page 18
as well as on deposits of various kinds.
The approach of Bank One assumes a parallel shift of the term structure. Events that create
difficulties for banks often involve changes in the level as well as in the shape of the term structure. For
instance, if the Federal Reserve increases interest rates, the impact will generally be stronger at the short end
of the curve than at the long end. Figure 9.1. shows an example of a dramatic increase in rates, namely the
one that took place in 1979. Further, spreads between interest rates of same maturity can change. For
instance, some loans might be pegged to LIBOR while other loans might be pegged to T-bill rates. It could
be that interest rates increase but that the spread between the T-bill rates and LIBOR falls. One approach
to deal with changes in the shape of the term structure and in spreads is to consider the impact on earnings
of past changes in rates corresponding to specific historical events. Estimating the impact of a specific
scenario on earnings is called a stress test. Chase reports results of stress tests in its 1999 annual report. It
measures the impact on earnings of a number of scenarios. Some of these scenarios are hypothetical,
corresponding to possible changes in the term structure that are judged to be relevant. Other scenarios are
historical. In 1994, the Federal Reserve increased interest rates sharply. One of the historical scenarios that
Chase uses corresponds to the changes in rates in 1994. For such a historical scenario, Chase uses the
changes in rates of various maturities as well as the changes in rates with different credit risks that happened
in 1994. In its report, Chase concludes that the “largest potential NII stress test loss was estimated to be
approximately 8% of projected net income for full-year 2000.” These stress tests represent an extreme
outcome, in that they assume an instantaneous change in rates that is followed by no management response
for one year.
A natural approach to estimating the exposure of earnings to interest rates would be to simulate
Chapter 9, page 19
earnings using a forecast of the joint distribution of interest rates. With this simulation, we could compute a
CaR measure that would take only interest rates as risk factors. The advantage of such an approach is that
it could take into account the probabilistic nature of interest rate changes. Though the scenario approach
discussed above is useful, it does not take into account the probability that the various scenarios will occur.
The difficulty with computing a CaR measure is that one has to be able to forecast the joint distribution of
interest rates. We will consider various approaches to do that later in this chapter.
In general, a bank is likely to want to compute a CaR or EaR measure that takes into account other
risk factors besides interest rates. If it simulates earnings or cash flows, then it can measure its exposure to
interest rates of earnings or cash flows by estimating their covariance with relevant interest rates in the way
we measured foreign exchange exposures in Chapter 8.
Section 9.2.2. Asset-liability management.
A financial institution or a pension fund manager might be more concerned about the value of their
portfolio of assets and liabilities than with its cash flow. This will always be the case when assets and
liabilities are marked to market. For an institution with assets and liabilities marked to market, a drop in the
value of the assets marked to market can create a regulatory capital deficiency, an income loss, and possibly
greater problems. It is therefore important for the institution to understand the distribution of possible gains
and losses. Alternatively, a pension fund manager has to make certain that the fund can fulfill its
commitments. Let’s consider how an institution can measure the interest rate risks of its portfolio of assets
and liabilities. It wants to compute how the market value of assets and liabilities changes for a given change
in interest rates. This problem is one that we have already discussed in principle. When we compute VaR,
Chapter 9, page 20
we estimate the distribution of value given the distribution of risk factors. When the only risk factors are
interest rates, then the VaR provides an estimate of the interest rate sensitivity of a firm. The advantage of
a VaR measure is that it takes into account the probability of changes in risk factors as well as the level of
those changes. To estimate VaR, we have to be able to forecast the distribution of interest rate changes. This
is a topic that we will discuss later in this chapter.
Banks often use more simple approaches than VaR to evaluate interest rate risks. The simplest way
to measure the value impact of a change in rates is to compute the exposure of the market value of assets
and liabilities to a change in rates. The exposure is the change in market value of the portfolio for a change
in the reference interest rate. Say that we compute the exposure for a 100 basis point change. Since there
are multiple interest rates, the solution is generally to assume a parallel shift in the term structure, so that all
rates change by the same amount. Once we have the exposure, we can then compute the impact of a
specific rate change on the value of the portfolio. Suppose that we find an exposure of -$500M. This means
that a 100-basis points change decreases portfolio value by $500M. Assuming a fixed exposure, we can
then evaluate the impact of a 50-basis points change or a 200-basis point change since these impacts are
just scaled impacts of the 100 basis points change. Using the exposure, we can compute a VaR. Using the
same volatility as before, we would find that the VaR is $412.5M (1.65*0.5*500M).
The fixed exposure approach is one that is widely used. We will develop techniques to estimate this
exposure in the next section. There is, however, an obvious alternative. Instead of measuring the exposure
and then computing the VaR from the exposure, we can also compute the distribution of the asset and
liability values and then directly estimate the VaR using that distribution. In effect, we can compute the VaR
of a portfolio in the way we learned to do it earlier for a portfolio of securities. We saw that with a portfolio
Chapter 9, page 21
of securities we can compute the VaR analytically if the securities have normally distributed returns.
Alternatively, we can simulate changes in risk factors and compute the value of assets and liabilities for each
realization of changes in risk factors. This approach therefore requires the use of pricing models. We will
consider these approaches in the last section of this chapter.
Section 9.3. Measuring and hedging interest rate exposures.
In this section, we examine how to measure the interest rate exposure of securities and portfolios
of securities. The tools we develop that would have allowed Orange county in early 1994 to answer
questions like: Given our portfolio of fixed income securities, what is the expected impact of a 100 basis
increase in interest rates? What is the maximum loss for the 95% confidence interval? Once we know how
to measure the impact of interest rate changes on a portfolio, we can figure out how to hedge the portfolio
against interest rate changes.
There is a fundamentaldifficulty in evaluating interest rate exposures of fixed income securities. When
yields are low, a small increase in yield leads to a large fall in the bond price, but when yields are extremely
large, the impact of a small increase in yield on the bond price is trivial. This makes the bond price a convex
function of the yield as shown in Figure 9.2.1 For a given increase in yield, the sensitivity of the bond price
to yield changes falls as the yield rises, so that the bond’s yield exposure depends on the yield. The
economic reason for this is that the yield discounts future payments, so that the greater the yield, the smaller
the current value of future payments. As the yield increases, payments far in the future become less important
1
Remember that a convex function is such that a straight line connecting two points on the
function is above the function. A concave function is the opposite.
Chapter 9, page 22
and contribute less to the bond price. Payments near in the future contribute more to the bond price when
the yield increases, but because they are made in the near future, they are less affected by changes in the
yield.
Ideally, we would like to find an estimate of exposure so that exposure times the change in yield
gives us the change in the bond value irrespective of the current yield or the size of the change in yield. Since
the exposure of a fixed income security changes with the yield, this cannot be done. However, for small
changes in yield, the exposure is not very sensitive to the size of the yield change. As a result, it is reasonable
to use the same measure of exposure for small changes in yield. This is the approach we present first in this
section. We then discuss the limitations of this approach and show how it can be improved.
Section 9.3.1. Yield exposure.
Consider a coupon bond with price B that pays a coupon c once a year for N years and repays the
principal M in N years. The price of the bond is equal to the bond’s cash flows discounted at the bond yield:
B=
∑
i= N
c
i= 1
(1 + y )
i
+
M
(1 + y ) N
(9.1.)
In the bond price formula, each cash flow is discounted to today at the bond yield. A cash flow that occurs
farther in the future is discounted more, so that an increase in the yield decreases the present value of that
cash flow more than it decreases the value of cash flows accruing sooner. Consequently, a bond whose cash
flows are more spread out over time is more sensitive to yield changes than a bond of equal value whose
cash flows are received sooner.
Chapter 9, page 23
Let’s consider the exposure of a bond to a small change in yield evaluated at the current yield. We
called exposure evaluated for a very small change in a risk factor the delta exposure to the risk factor. The
bond delta yield exposure is the change in the price of the bond for a unit change in yield measured for an
very small(infinitesimal) change in the yield. This exposure provides an exact measure of the impact of the
yield change on the bond price for a very small change in the yield. The dollar impact of a yield change )y
on the bond price evaluated using the delta exposure is equal to )y times the bond yield delta exposure.
There is an explicit formula for the bond yield delta exposure that can be derived from the bond price
formula by computing the change in the bond price for a very small change in the yield: 2
N* M 
 i= N i *c
+
∑
i
 i= 1
(1+ y) (1 + y) N 
−B 
Bonddelta yield exposure =

B
(1+ y) 



B
= − D*
= − MD * B
(1+ y)
(9.2.)
Note that this expression depends on the current yield y, so that a change in the yield will change the bond
delta yield exposure. The term in square brackets, written D in the second line, is the bond’s duration. The
bond’s duration is the time-weighted average of the bond cash flows. It measures how the cash flows of the
bond are spread over time. A bond with a greater duration than another is a bond whose cash flows, on
average, are received later. We already discussed why such a bond is more sensitive to the yield. Duration
2
Technical argument. The impact of the yield change is obtained by taking the derivative of
equation (9.1) with respect to the yield. For instance, taking the derivative of the present value of the ith coupon with respect to yield gives us -i*c/(1+y) i+1.
Chapter 9, page 24
divided by one plus the yield, D/(1+y), is called the modified duration, which we write MD. The change
and the percentage change in the bond price resulting from a yield change )y are respectively:
∆B = − M D * B * ∆y
(9.3.A.)
∆B
= − M D * ∆y
B
(9.3.B.)
When coupon payments are made once a year, duration is automatically expressed in years. The
convention with duration is to use duration expressed in years. If there are n coupon payments per year, the
discount rate for one payment period is (1+y/n) rather than (1+y). Consequently, we replace (1+y) by
(1+y/n) in equations (9.1.) and (9.2.). The resulting duration is one in terms of payment periods. To get a
duration in terms of years, we divide the duration in terms of payment periods by n.
Though the delta yield exposure or the duration are exact measures of exposure for very small
changes, they are generally used as approximate measures for larger changes. Using the bond’s delta yield
exposure or duration to estimate the impact of a change in the yield, we use a linear approximation to
approximate a nonlinear function. More precisely, we use the slope of the tangent at the point of
approximation as shown in Figure 9.3.3 If we move up or down this tangent line, we are very close to the
bond price function for small changes in yields, so that using the approximation works well. For larger
3
Technical point. The delta yield exposure of a bond is obtained by taking a first-order Taylorseries expansion of the bond pricing function around the current value of the yield. The tangent line
corresponds to the straight line given by the first-order Taylor-series expansion as we vary the yield.
Let B(y) be the bond price as a function of the yield and y* be the current yield. A first-order Taylorseries expansion is B(y) = B(y*) + By(y*)(y - y*) + Remainder. Ignoring the remainder, the first-order
Taylor-series expansion of the bond price is given by a straight line with slope By(y*). By(y*) is the
bond delta yield exposure.
Chapter 9, page 25
changes, the point on the tangent line can be far off the bond pricing curve, so that the approximation is poor
as can be seen in Figure 9.3. For a large increase in yields, the bond price obtained from the approximation
is lower than the actual bond price. The same is true for a large decrease in yields. This is because the line
that is tangent to the bond price at the point of approximation is always below the function that yields the
bond price.
Let’s look at an example. Suppose that we have a 25-year bond paying a 6% coupon selling at
70.357. The yield is 9%. The modified duration of that bond is 10.62. The dollar price impact of a 10-basis
point change using equation(9.3.A) is -10.62*70.357*0.001 or -$.0747. Using equation (9.3.B), we can
obtain the percentage price impact of a 10 basis point change as -10.62*0.001, or 1.06%. Had we
computed that price impact directly, we would have gotten -1.05%. Duration works well for small yield
changes. If we had a 200-basis point change, however, duration would imply a fall of 21.24% whereas the
true fall in the bond price is 18.03%.
We have now seen that we can use duration to compute the impact of a change in yield on a bond
price. If we know the duration of the assets and liabilities, we can then compute the impact of a change in
interest rates on the value of a portfolio. To do that, though, we have to deal with an obvious problem. If
we look at one bond, it is perfectly sensible to look at the impact on the bond price of a change in its yield.
It is less clear what we should do when we have many different bonds. With a flat term structure, all bonds
have the same yield. In this case, we can change all yields in the same way and the problem is solved. If the
term structure is not flat, we can change all yields by the same amount. This is called a parallel shift in the
term structure.
With a flat term structure, all future payments are discounted at the same interest rate. In this case,
Chapter 9, page 26
the yield on a bond must equal that interest rate and all bonds have the same yield. Consequently, we can
compute the impact of a parallel shift of the term structure using duration, i.e., a shift where all yields change
by the same amount. Denote by A the value of the assets and by L the value of the liabilities, so that the
value of the portfolio is A - L. Using modified duration, the value of the assets changes by -A*MD(A))y
and the value of the liabilities changes by -L*MD(L)))y. Consequently, the change in the value of the
portfolio is:
Change in value of portfolio = [-A*MD(A) - (-L*MD(L)) ])y
(9.4.)
where MD(A) is the modified duration of assets, MD(L) is the modified duration of liabilities, and )y is the
parallel shift in yields. Consider a bank with assets of $100M with a modified duration of five years and
liabilities of $80M with a modified duration of one year. The bank’s modified duration is therefore
5*(100/20) - 1*(80/20), or 21. The market value of the bank falls by 20*21*0.01 = $4.2M if the interest
rate increases by 100 basis points.
To hedge the value of the bank’s balance sheet with futures contracts using the duration approach,
we would have to take a futures position that pays off $4.2M if the interest rate increases by 100 basis
points. The duration of a futures contract is measured by the duration of the underlying bond discounted at
the yield that equates the value of the underlying bond to the futures price. If there is a futures contract with
a modified duration of five years, we would have to go short that futures contract by $84M. In this case, the
impact of a 100-basis point increase on the balance sheet would yield a futures gain of 84M*5*0.01 =
$4.2M. Hence, the 100 basis points increase in interest rates would leave the value of equity unchanged.
Chapter 9, page 27
We can obtain a general formula for hedging interest rate risk with duration. Suppose that we have
a portfolio with value W and we have a security with price S that we want to use to hedge. The portfolio
has modified duration MD(W) and the security has modified duration MD(S). We want to invest in n units
of the security with price S so that the hedged portfolio has no duration. If we do that, we invest W + nS.
To satisfy the constraint that we do not invest more than what we have, we need to be able to borrow or
lend in the form of a security that has no duration. Let this security be equivalent to a money market account
that pays every day the market interest rate for that day and let’s assume that this security can be shorted.
We denote by K the value of the investment in the money market account. In this case, we must have W
+ nS + K = W. If S is a futures price, we do not need the money market account since we do not use any
cash to take the futures position. Let n be the number of units of the security with price S we purchase to
hedge. We want to choose n and K so that:
W
nS




Modifieddurationof hedgedportfolio = 
MD (W)+ 
MD (S)=0

 W+ nS+ K 
 W+ nS+ K 
(9.5.)
Value of portfolio = W+ nS+ K = W
(9.6.)
Solving for n, we get the volatility-minimizing duration hedge:
Volatility-minimizing duration hedge
The volatility minimizing hedge of a portfolio with value W and modified duration MD(W) using a security
with price S and modified duration MD(S) involves taking a position of n units in the security:
Chapter 9, page 28
Volatility - minimizing hedge = n = -
W* MD (W)
S* MD (S)
(9.7.)
Let’s go back to our example. We have $100M with modified duration of five years and $80M with
modified duration of one year. The portfolio has value of $20M. Its modified duration is (100/20)*5 (80/20)*1 = 25 - 4 = 21. A yield change of one basis point decreases the portfolio value by 21*20M*0.01,
or $4.2M. W*MD(W) is equal to 20M*21, or 420M. Let’s first look at the case where we have a security
S with price $92 and modified duration of 5 years. Using our formula, we get:
n=-
W * M D (W )
$20 M * 21
=−
= − 0.913043M
S * M D (S)
$92 * 5
To construct this hedge, we go short 913,043 units of the security with price of $92, for proceeds of
$83.999956M. We invest these proceeds in a money market account that has no duration to insure that our
hedge portfolio has no duration.
Let’s now consider the case where we hedge with futures. The futures contract we use has a price
of 92 and modified duration of 5 years. The size of the contract is $10,000. In this case, a change in yield
of )y changes the value of a futures position of one contract by -0.92M)y, or -$4.6M*5)y. We want
a position of n contracts so that -n*0.92M*5)y - 20M*21)y = 0. In this case, S in equation (9.7.) is the
futures price times the size of the contract and is therefore equal to $920,000. S*MD(S) is equal to
0.92M*5, or 4.6M. With this computation, n gives us the size of our futures position expressed in number
Chapter 9, page 29
of contracts. Dividing -W*MD(W) by S*MD(S) gives us -420M/4.6M, or a short position of 91.3043
contracts.
It is useful to note that the duration hedge formula is nothing else than the minimum-volatility hedge
formula derived in Chapter 6 when the risk factor is the interest rate and its impact on a security is given by
the duration formula. To see this, note that the change in portfolio value, if the interest rate risk is the only
source of risk, is equal to the yield delta exposure times the change in the interest rate, -W*MD(W))y, and
the change in the value of the security is equal to the yield delta exposure of the security times the change
in the interest rate, -S*MD(S))y. If, instead of taking the change in the interest rate to be given, we let it be
a random variable, then we can use our minimum-volatility hedge ratio formula. This formula gives us a
hedge ratio which is Cov[-W*MD(W))y,-S*MD(S))y]/Var[-S*MD(S))y]. This hedge ratio is equal to
W*MD(W)/S*MD(S). Using our result from Chapter 6, we would go short W*MD(W)/S*MD(S) units of
asset with price S to hedge.
The duration approximation makes it possible to obtain a VaR measure based on duration. Since
the change in the portfolio value is -W*MD(W))y, we can interpret )y as the random change in the interest
rate. If that random change is distributed normally, the volatility of the change in portfolio value using the
duration approximation is W*MD(W)*Vol[)y]. Using the formula for VaR, we have that there is a 5%
chance that the portfolio value will be below its mean by more than 1.65*W*MD(W)*Vol[)y]. Applying
this to our example, we have 1.65*420M*0.005, or $3.465M. Box 9.2. Orange County and VaR shows
how the duration-based VaR discussed in this section would have provided useful information to the officials
of Orange County.
Chapter 9, page 30
Section 9.3.2. Improving on traditional duration.
Let’s look at the reasons our duration hedge might not work well and the duration VaR might
provide a poor assessment of the risk of a portfolio. We assumed that all yields on assets and liabilities
change by exactly the same number of basis points, )y. This corresponds to a parallel shift in the term
structure. It also implies that the yield spreads of the securities in the portfolio relative to the term structure
of government bond yields remain unchanged. A balance sheet that is hedged using duration as in our
example is not hedged against changes in the term structure which are not parallel shifts or against changes
in yield spreads. Does that mean that the duration strategy is worthless? The answer is no. If we relate
changes in bond yields to changes in one yield, one generally explains a large fraction of the bond yield
changes - at least 60% across countries. In the U.S., the duration model generally explains more than 75%
for U.S. government bonds, but about 10% less for corporate bonds. In other words, this means that we
can eliminate a substantial fraction of the volatility of a bond portfolio through a duration hedge. At the same
time, though, there is always the potential for changes to take place that imply large losses or large gains for
a hedged balance sheet when duration is used to hedge.
One way to hedge against shifts in the yield curve using duration is as follows. Suppose that we have
assets and liabilities and that our objective is to keep assets and liabilities in balance. If all changes in yields
for the assets and liabilities are brought about by parallel shifts in the yield curve, all we have to do is keep
the duration of assets equal to the duration of liabilities. Suppose, however, that changes in yields are not
perfectly correlated. In this case, we want to put our assets and liabilities in duration bins. Say one bin is for
duration between one and two years. Rather than having the duration of all our assets match the duration
of all our liabilities, we want the value of assets for a duration bin to equal the value of liabilities. If the term
Chapter 9, page 31
structure changes, assets and liabilities that have similar durations change similarly. Hence, if yields increase
for one duration bin and fall for another duration bin, the assets and liabilities for each duration bin should
remain in balance.
With the asset/liability management technique we just described, we still have not eliminated interest
rate risks, however. Remember that a bond with a given duration is a portfolio of discount bonds with
different maturities. Two different bonds could have a five-year duration for very different reasons. For
instance, one bond could be a five-year discount bond and the other bond could be a fifteen-year bond with
declining coupon payments. The fifteen year bond’s value depends on yields that may change without
affecting the price of the five-year discount bond at all. For instance, the yield to maturity of a fifteen-year
discount bond could change in such a way that all yields to maturity for maturities of five year or less would
be unchanged. In this case, even though both bonds have identical durations, the price of one bond would
change but the price of the other bond would not.
Taking into account, the slope of the term structure explicitly makes the duration approximation more
precise. To see this, remember that a coupon bond is a portfolio of zero-coupon bonds. Hence, the duration
of a coupon bond must be the duration of a portfolio of zero-coupon bonds. Each coupon’s duration has
a weight given by the present value of the coupon as a fraction of the present value of the coupon bond.
When we look at a coupon bond this way, it is possible to understand why the duration method using yields
runs in trouble. When we use duration with a yield, we do not use the present value of the coupon at market
rates. Rather, we use the present value of the coupon computed using the yield as a discount rate. We
therefore discount all coupons at the same rate. Obviously, if the term structure is flat, this is not a problem
since the market discount rate is also the yield of the bond. When the term structure is not flat, however, this
Chapter 9, page 32
presents a difficulty. If we look at the bond as a portfolio of zero coupon bonds, the duration approximation
using the yield takes the approximation at the same yield for all bonds rather than at the discount rate
specified by the term structure. Since the duration approximation depends on the slope of the bond price
function, this means that using the duration approximation at the same yield for all zero coupon bonds used
to price a coupon bond introduces an additional approximation error. This approximation error depends on
the slope of the term structure and its location.
To avoid this approximation error, the best approach is to treat each cash flow separately and
discount it at the appropriate rate from the term structure. Let r(t+i) be the continuously-compounded rate
at which a discount bond maturing at t+i is discounted, where today is t. The value of a coupon payment c
paid at t+i is consequently:
Current value of coupon paid at t+i = Exp[-r(t+i)*i]*c
(9.8.)
The impact on the value of the coupon of a small change in the discount rate is:
Change in current value of coupon paid at t + i for interest rate change ∆ r(t +i)
= − i * Exp[-r(t +i) *i]* c * ∆ r(t + i)
= - i *Current value of coupon paid at t +1* ∆ r(t +i)
(9.9.)
The proportional change is the change given in equation (9.9) divided by the current value coupon, which
amounts to -i*)r(t+1). When using continuous compounding, there is no difference between duration and
modified duration. Consequently, the duration of a zero-coupon bond using a continuously-compounded
Chapter 9, page 33
discount rate is the time to maturity of the discount bond. Using equation (9.9), we can estimate the change
in the bond price from a change in the whole term structure by adding up the changes in the present value
of the coupons. If )r(t+i) is the change in the rate for maturity t+i, the change in the bond price is:
N
Change in bond price = ∑ − i *e − r(t + i)*i c * ∆ r(t + i) − N*e − r(t + N)*N M* ∆ r(t + N)
(9.10.)
i =1
A parallel shift of the yield curve is the special case where all rate changes are equal to the same amount.
The approximation in equation (9.10.) is exact for each coupon payment when the change in the interest rate
is very small (infinitesimal). It does not eliminate the approximation error resulting from the convexity of bond
prices for larger changes in interest rates. Expressing this result in terms of duration for a shift in the term
structure of x for all rates, we have:
 N

− i * P(t + i) * c − (N) * P(t + N) * M
∑

∆B
 *x
=  i= 1
B 
B





= − DF * x
(9.11.)
The term in square brackets, written DF, is called the Fisher-Weil duration. It is actually what Macaulay, the
father of duration, had in mind when he talked about duration. With this duration, the duration of each zerocoupon bond used to discount the coupon and principal payments is weighted by the portfolio weight of the
current value of the payment in the bond portfolio. For instance, the i-th coupon payment’s duration has
weight P(t+i)*c/B. This formula can be used for any term structure and it gives the exact solution for a very
small parallel shift of the term structure.
Chapter 9, page 34
The Fisher-Weil duration provides an exact approximation for a very small shift in the term structure.
The obvious difficulty is that when one hedges, shifts are not always very small. For larger shifts, we know
that the duration approximation is not exact. Let’s now evaluate the magnitude of the mistake we make for
large changes and see whether we can find ways to make this mistake smaller. Let P(t+i) be the price of a
discount bond that matures at i. We can approximate the impact of a change of the interest rate on this bond
using the Fisher-Weil duration. Duration tells us that the proportional change in the bond price due to a
change in the interest rate of )r(t+i) is i*)r(t+i), where r(t+i) is the continuously-compounded rate for the
discount bond maturing at i. Hence, for a thirty-year zero, a one basis point change in the yield leads to a
proportional fall in price of 0.0001*30 or 0.003, namely three-tenth of one percent.
We can get the change in value of the bond directly using the delta exposure, which is the bond price
times the Fisher-Weil duration. Consider the case where the interest rate is 10%. In this case, the bond price
is $0.0497871. The delta exposure is -30*$0.0497871. The impact of a one basis point change on the price
using the delta exposure is -30*$0.0497871*0.0001, whichis -$0.000149361. Computing the price change
directly, we find that the new price of $0.0496379 is below the old price by $0.000149137. In this case,
the duration mistake is trivial. Suppose now that the change in the rate is 100 basis points. The new price
is $0.0368832. The price predicted using duration is $0.0348509. Using duration in this case, the predicted
price is 5.5% lower than the new price of the bond, which is an economically significant difference. Note
that the difference will be less for shorter maturity bonds and higher for longer maturity bonds.
Since the duration approximation is a first-order or linear approximation of the bond price function,
we can obtain a more precise approximation by taking into account the curvature of the bond price function.
Gamma exposure is the change in delta exposure for a unit change in the risk factor evaluated for an
Chapter 9, page 35
infinitesimal change in the risk factor. It is therefore a measure of the convexity of the bond price function.
Instead of using the delta exposure measure, we use a delta-gamma exposure measure. If the value of a
security is a linear function of a risk factor, its gamma exposure is equal to zero. With a bond, the delta
exposure falls as the interest rate increases, so that the delta exposure is negative. The delta-gamma
exposure measure is obtained by taking into account the second-order effect of a change in the risk factor
as well as the first-order effect. We therefore use the following approximation:4
)P(t+i) = Delta exposure*)r(t+i) + 0.5*Gamma exposure*[)r(t+i)]2
(9.12.)
The gamma exposure is the second-order impact of the change in the interest rate on the bond price:
Gamma exposure =
Change in delta exposure per unit change in r(t+i) evaluated for infinitesimal change in r(t+i)
= i2*P(t+i)
= Duration squared*P(t+i)
(9.13.)
The change in duration for an infinitesimal change in the interest rate is called the convexity. It is expressed
in the same units as the duration. The duration squared is the convexity of a zero-coupon bond. We can
4
Technical note: This approximation is a second-order Taylor-series expansion around the
current rate assuming that the remainder is negligible. A second-order Taylor-series expansion of a
function f(x) around X is f(x) = f’(X)(x - X) + 0.5f”(X) (x-X)2 + Remainder.
Chapter 9, page 36
collect these results to get the proportional change in the bond price using both delta and gamma exposures,
or equivalently duration and convexity. Consequently, we have:
Approximation of bond price change using duration and convexity
Consider a zero-coupon bond maturing at t+i with price P(t+i) and continuously-compounded discount rate
r(t+i). The duration of that bond is i and its convexity is i2. The delta exposure of the bond is -i*P(t+i) and
its gamma exposure is i2*P(t+i). The approximation for the bond price and the proportional bond price
changes are:
)P(t+i) = -i*P(t+i)*)r(t+i) + 0.5*i2*P(t+i)*()r(t+i))2
(9.14.)
)P(t+i)/P(t+i) = -i* )r(t+i) + 0.5*i2*()r(t+i))2
(9.15.)
The conventional approach to duration discussed above uses discrete compounding and yields. With that
conventional approach, the percentage change in the bond price using duration and convexity is:
Percentage price change = - Modified duration* )y + 0.5*Convexity*()y)2
If we measure duration in periods corresponding to coupon payments and coupon payments take place
every six months, we can get duration in years by dividing modified duration by the number of payment
periods per year. For instance, if duration in coupon payment periods is 20, duration in years is 20/2, or 10.
The convexity measure corresponding to duration in coupon payment periods has to be divided by the
square of the number of payment periods to get convexity in years. So, if convexity is 100 when it is
Chapter 9, page 37
computed using coupon payment periods, it becomes 25 in years.
Using convexity and duration, we add positive terms to the expression for the change or percentage
bond price using duration only when convexity is positive, so that we increase the bond price following a
change in the interest rate. We saw that the convexity of a zero-coupon bond is positive. A plain vanilla
coupon bond is a bond with no options attached. A callable bond is not a plain vanilla bond because the
issuing firm has attached to the bond the option to call the bond. The convexity of a plain vanilla coupon
bond is always positive also. Let’s use equations (9.14.) and (9.15.) for the 30-year discount bond for a
100-basis point change. In this case, i is equal to 30, i squared is equal to 900, and the bond price is
$0.0497871. Consequently, the approximation for the change using equation (9.14.) is:
)P(t+i) = -i*P(t+i)*)r(t+i) + 0.5*i2*P(t+i)*()r(t+i))2
= -30*0.0497871*0.01 + 0.5*900*0.0497871*0.0001
= 0.0126957
We saw earlier that the true price change is $0.0129039. The mistake made using the delta-gamma
approximation is therefore of the order of 0.5% of the bond price. Here, therefore, using the more precise
approximation reduces the mistake as a percentage of the true price by a factor of 10. Note that with the
delta approximation, we overstated the price fall. Now, with the delta-gamma approximation, we understate
the price fall slightly.
When constructing a hedge using duration, we took a short position in a security that had the same
duration than the portfolio we were trying to hedge. Duration captures the first-order effect of interest rate
Chapter 9, page 38
changes on bond prices. The duration hedge eliminates this first-order effect. Convexity captures the
second-order effect of interest rate changes. Setting the convexity of the hedged portfolio equal to zero
eliminates this second-order effect. To eliminate both the first-order and the second-order effects of interest
rate changes, we therefore want to take a hedge position that has the same duration and the same convexity
as the portfolio we are trying to hedge.
To understand how convexity can affect the success of a hedge, let’s go back to our bank example,
but now we assume that the bank has a duration of 21 years and a convexity of 500 using the Fisher-Weil
duration instead of the modified duration. Suppose that we hedge with the 30-year discount bond. This
discount bond has a duration of 30 and a convexity of 900. The bank has net value of $20M. So, if we use
only the 30-year bond, we have to go short 0.7*20M face value of the discount bond to get the hedged
bank to have a duration of zero. Unfortunately, this hedged bank has a convexity of -130 (which is 500 0.7*900). In this case, the hedged bank has negative convexity and its value is a concave function of the
interest rate. Figure 9.4. shows the value of the hedged bank as a function of the current interest rate. This
value reaches a maximum at the current interest rate which we take to be 5%. At the current interest rate,
the slope of this function is zero because the hedged bank has no duration. The current value of the hedged
bank is the present value of its hedged payoff. Since the bank value is highest if rates do not change, this
means that if rates do not change, the bank value is higher than expected and if rates change by much in
either direction the bank value is lower than expected. Negative convexity of the bank hedged using duration
therefore means that the value of the bank falls for large changes of either sign.
To improve the hedge, we can construct a hedge so that the hedged bank has neither duration nor
convexity. This requires us to use an additional hedging instrument so that we can modify the convexity of
Chapter 9, page 39
the bank hedged to have zero duration. Let a be the short position as a principal amount in a discount bond
with maturity i and b be the short position in a discount bond with maturity j. We have a money market
instrument in which we invest the proceeds of the short positions so that the value of the bank is unaffected
by the hedge. In this case, the hedge we need is:
Duration of unhedged bank = (a/Value of bank)*i + (b/Value of bank)*j
Convexity of unhedged bank = (a/Value of bank)*i2 + (b/Value of bank)*j 2
Let’s use a discount bond of 30 years and one of 5 years. We then need to solve:
21 = (a/20)*30 + (b/20)*5
500 = (a/20)*900 + (b/20)*25
The solution is to have a equal to 10.5333 and b equal to 20.8002. In this case, the duration is
-(10.5333/20)*30 + 21 - (20.8002/20)*5, which is zero. The convexity is -(10.5333/20)*900 + 500 (20.8002/20)*25, which is also zero.
We used the delta approximation to compute a VaR measure. We can do the same with the deltagamma approximation. We can view equation (9.14.) as an equation that gives us the random change in the
discount bond price as a function of the random change in the interest rate. Any fixed income portfolio of
plain-vanilla bonds can be decomposed into a portfolio of investments in zero coupon bonds. This allows
us to model the random change in the value of the portfolio in a straightforward way. Let the portfolio be
Chapter 9, page 40
such that W(i) represents the dollar principal amount invested in the discount bond with maturity i. For
instance, if the portfolio holds one coupon bond which makes a coupon payment at date i equal to c, W(i)
is equal to c. In this case, the value of the portfolio W is given by:
N
W = ∑ P(i) *W(i)
i= 1
If we make discount bond prices depend on one interest rate only, r, then we have:
N
N
i=1
i= 1
∆W = ∑ W(i)∆ P(i) = ∑ W(i) *(i * ∆r − 0.5* i 2 * ( ∆ r) 2 )
(9.15.)
Using this equation, we can then simulate the portfolio return and use the fifthpercentile as our VaR measure.
Section 9.4. Measuring and managing interest rate risk without duration.
The analysis of Section 9.3. uses the duration approximation. This approximation makes it possible
to evaluate the impact of an infinitesimal parallel shift in the term structure. When we use the yield
formulation, the approximation is right starting from a flat term structure. When we use the Fisher-Weil
formulation, the approximation is right irrespective of the initial term structure. In all cases, though, the
approximation requires a parallel shift and degenerates when the shift is not infinitesimal. We saw that we
can improve on duration by using convexity as well. Whether we use duration only or duration and
convexity, we still use an approximation. As we construct a VaR measure for an institution or a portfolio,
we use a distribution of interest rate changes. When interest rate changes are random, there is generally
Chapter 9, page 41
some probability that large changes for which the approximation is poor can take place. Further, the duration
approach does not make it possible to use interest rate distributions where changes in interest rates for
different maturities are not perfectly correlated. Consequently, using the duration approximation for VaR
measurement means ignoring potentially important sources of risk, namely the risks associated with changes
in the shape of the term structure. Further, we ignored so far changes in spreads among rates. As rates
change, spreads can change also. For instance, the AAA rate for a 10-year discount bond can fall relative
to the 10-year default-free discount bond as the interest rate for the 10-year default-free discount bond
increases. Whereas an institutionwithonly plain-vanilla high grade fixed income positions may not miss much
if it uses the duration approximation, this is definitely not the case for institutions that hold positions with
embedded options or bonds with more significant default risks. For positions with embedded options, small
changes in the shape of the term structure can have dramatic impacts and the approximation errors
associated with the duration approximation can be large. We will see examples as we proceed to more
complicated derivatives. We will discuss the determinants of credit spreads in Chapter 18.
The alternative to the duration approximation is to consider the impact of interest rate changes on
prices directly. In this case, if one wants to consider how an interest rate change affects the value of a fixed
income position, one simply computes the value of the position for the interest rates that hold after the
change. For instance, one can evaluate the impact on the portfolio of a one basis point change in rates by
recomputing the value of the portfolio for that change. This approach is becoming increasingly common.
A good example of how the impact on the portfolio of a one basis point change is used can be found
in the practices of Chase. In 1999, Chase stopped reporting the gap measures that we discussed earlier in
this Chapter and started reporting the basis point value for the portfolio (BPV). This basis point value gives
Chapter 9, page 42
the change in the market value of its portfolio for a one basis point change in rates. Before 1999, Chase used
BPV for its trading portfolio only. Starting with 1999, it uses it more broadly to include other assets and
liabilities. It considers two different BPVs. First, it estimates the impact of a one basis point change in interest
rates. It reports that at the end of December 31, the BPV for a one basis point change in rates is -$6.4M.
Second, it computes the impact of a one basis point change in spreads between liabilities and assets (basis
risk). It reports that the BPV of a one basis point change in spreads is -$10.7M.
By recomputing the basis point value for the portfolio value for changes in interest rates, one can
compute the portfolio’s VaR as well as construct minimum-volatility hedges. For instance, one can estimate
a joint distribution for the interest rates one is interested in, and use Monte Carlo simulation to obtain draws
of interest rates. For each draw of the interest rates, we compute the value of the position. After a large
number of draws, we end up with a simulated distribution of the value of the position given the joint
distribution of the interest rate changes. We can then use the fifth percentile of that simulated distribution as
our VaR and use the distribution to construct minimum-volatility hedges. Alternatively, we could use
historical changes in interest rates to obtain simulated values of the position. While such approaches would
have been ludicrous when Macaulay introduced his duration measure because repricing a portfolio 100,000
times would have taken years, it can now be implemented within minutes or hours.
The way we used the duration approximation to compute VaR, we assumed that there were random
level shifts in the term structure. In other words, the term structure changed because of only one source of
risk. A model where all rates depend on only one random variable is called a one-factor model of the term
structure. The duration model is an extreme case of a one factor model of the term structure in that it allows
only for a flat term structure. Other one-factor models allow for term structures that are not flat and can
Chapter 9, page 43
change shape as the level changes. Yet, with a one-factor model of the term structure, all rates depend on
a single random variable and are therefore perfectly correlated. To allow for rates to be imperfectly
correlated, one must allow for more sources of variation in rates. For instance, one can make the rates
depend on two random variables or sources of risk. Such a model would be called a two-factor model of
the term structure. With more sources of risk affecting interest rates, a wider range of changes in the shape
of the term structure becomes possible. We discuss three alternate approaches to modeling interest rate risk
for purpose of risk management. The first approach considers the distribution of the zero-coupon bond
prices. The second approach models spot interest rates as functions of factors. The third approach uses the
distribution of forward rates.
Section 9.4.1. Using the zero coupon bond prices as risk factors.
The simplest way to model the risk of a fixed income portfolio without using duration would be to
simply use the joint distribution of the returns of the securities in the portfolio. If these returns are normally
distributed, we know how to compute a VaR analytically. Knowing the joint distribution of the returns of
the securities, we could construct minimum-volatility hedges using the covariances and variances of the
returns of these securities. This approach is a natural use of modern portfolio theory, but it runs into some
serious difficulties. The first difficulty is that if we proceed this way, we are likely to have arbitrage
opportunities among our fixed-income securities. To see this, our portfolio might have many different bonds.
Each one of these bonds has its own return. We know that to avoid arbitrage opportunities the law of one
price must hold: there can be only one price today for a dollar delivered for sure at a future date. This means
that if 20 different bonds pay a risk-free cash flow in 30 years exactly, all 20 bonds must discount that riskChapter 9, page 44
free cash flow at the same rate. Such an outcome does not occur if we allow all bonds to have imperfectly
correlated returns. The second difficulty is that there are too many securities if one treats each security
separately.
To make sure that there are no arbitrage opportunities, one can start by generating zero-coupon
bond prices and use these bonds to price other bonds. Doing this insures that all risk-free cash flows have
the same prices across bonds. Taking zero-coupon bonds as primitive instruments is the approach chosen
by Riskmetrics™. Riskmetrics™ makes available correlations and volatilities of the returns of selected zerocoupon bonds. Since any plain-vanilla security is a portfolio of zero-coupon bonds, it is a portfolio of
securities for which the distribution of returns is available using Riskmetrics™. With this distribution of
returns, a portfolio of fixed-income securities is not treated differently from a portfolio of stocks. A portfolio
of stocks has a return that is a weighted average of the returns of the stocks that compose it. We know how
to compute the analytical VaR of portfolios of securities and, consequently, we can compute an analytical
VaR of portfolios of zero-coupon bonds. In addition, since we know the distribution of the return of the
portfolio, we can also manage the risk of the portfolio by constructing a minimum-volatility hedge.
There are too many potential zero-coupon bonds for Riskmetrics™ to make the distribution of the
returns of all these bonds available. Consequently, Riskmetrics™ makes the distribution of the returns of
a subset of zero-coupon bonds available. In implementing the Riskmetrics™ approach, this creates a
difficulty. It might be that in a country a five-year zero and a seven-year zero are available, but no six-year
zero. Yet, if we have a coupon bond in that country, that coupon bond will have a coupon payment six years
from now. To use the Riskmetrics™ approach, we have to transform the coupon bond into a portfolio of
zero-coupon bonds. We therefore have to find a solution where we can know the distribution of the return
Chapter 9, page 45
of the six–year zero. The approach recommended by Riskmetrics™ involves an interpolation method that
is described in Box 9.3. Riskmetrics™ and bond prices.
Section 9.4.2. Reducing the number of sources of risk: Factor models.
Any plain vanilla coupon bond is a portfolio of discount bond. This means that to measure the risk
of a coupon bond or to hedge a coupon bond, we can use discount bonds. Unfortunately, as discussed
above, too many such bonds are required to price a long maturity coupon bond. This makes it impractical
to simulate prices for all required zero-coupon bonds. Riskmetrics™ resolves this problem by selecting a
subset of bonds. A different approach is to assume that a few risk factors explain changes in all bond prices.
The simplest such model is to assume that returns on bonds depend on two factors, a short rate and a longer
maturity rate or a spread between long maturity and short maturity. The idea motivating such a model is that
the short rate captures the level of the term structure and the spread or the long maturity rate capture the
steepness of the curve. With such a model, two risk factors are used to explain the term structure. It is
therefore called a two-factor model. Such a model explains returns on bonds better than the duration model.
The duration model is a one factor model, in that returns on bonds depend on one risk factor. It turns out
that the two-factor model explains about 10 to 20% more of the returns of government bonds than the one
factor model.
Litterman and Scheinkman (1991) use a different approach. They use a statistical technique called
factor analysis to identify common factors that influences returns of zero coupon bonds. These factors are
extracted from the data rather than pre-specified as in the approach discussed in the previous section.
Consequently, they do not correspond directly to the return of individual bonds. Nevertheless, Litterman
Chapter 9, page 46
and Scheinkman (1991) find that these three factors roughly correspond to a level effect, a steepness effect,
and a curvature effect. With such an approach, the return of each zero coupon bond depends on the three
factors. If one has a time series of factor returns, one can estimate the exposure of each zero coupon bond
to the factors using a regression of zero coupon bond returns on factor returns. They find that the three
factors explain at least 95% of the variance of the return of zeroes with the level or duration effect explaining
at least 79.5%.
A three-factor model like the one of Litterman and Scheinkman can be used to hedge. To use their
example, they consider a duration-balanced portfolio given in Table 9.3. That portfolio is bought on
February 5, 1986, and sold on March 5, 1986. Over that month, it loses $650,000. Using their three
factors, they find that the portfolio loses $84,010 for a one-standard-deviation (monthly) shock to the level
factor, gains $6,500 for a one-standard deviation shock to the steepness factor, and loses $444,950 for a
one-standard deviation in the curvature factor. During the month, the level factor change by -1.482 standard
deviations, the steepness factor by -2.487 standard deviations, and the curvature factor by 1.453 standard
deviations. To compute the total impact of these changes on the value of the portfolio, we add up their
effects:
Total change in portfolio value = -1.482*(-$84,010) + -2.487*$6,500 + 1.453*(-$444,950) = -$538,175
Since the portfolio is duration matched, one would predict no change in its value with a change in rates.
However, duration ignores the large effect on the value of the portfolio of changes in the curvature of the
term structure. This effect is captured by the three factor model. To hedge the portfolio with the three factor
Chapter 9, page 47
model, one would have to go short portfolios that mimic the factors.
For instance, one would take a position in a portfolio that mimics the curvature factor so that we gain
$444,950 for a one standard deviation move in the curvature factor. To hedge a portfolio that is exposed
to the three factors, we would therefore have to take positions in three factor mimicking portfolios to set the
exposure of the hedged portfolio to each one of the factors equal to zero.
In a factor approach, knowing the exposure of a position to the factors is useful for three reasons.
First, if we know the joint distribution of the factors, we can use this joint distribution to compute a VaR.
If the factor returns are normally distributed, we can take a fixed-income position and compute an analytical
VaR. For instance, if the factors are normally distributed returns and we have three factors, the return on
a fixed-income position is equivalent to the return of a three-security portfolio, where each security
corresponds to a factor. We already know how to compute the VaR analytically in this case. Second,
knowing the exposure of a position to the factors allows us to construct a hedge against this exposure. All
we have to do is take a position in existing securities or futures contracts that has factor loadings that cancel
out the factor loadings of our fixed income position. Finally, exposure to factors can be an active
management tool. We may feel that exposure to one of the factors is rewarding while exposure to the other
factors is not. In this case, our understanding of the factor exposures of our position and of individual
securities allows us to make sure that we are exposed to the right factor.
Section 9.4.3. Forward curve models.
The approaches discussed so far do not generally guarantee that the distribution of discount rates
is such that there cannot be arbitrage opportunities. This is unfortunate because one does not expect to find
Chapter 9, page 48
pure arbitrage opportunities in the markets. If a term structure allows for arbitrage opportunities, one ends
up with prices that do not correspond to prices that one could find in actual markets, and hence the value
of the position one obtains is not right. A key requirement for a term structure to exhibit no arbitrage
opportunities is that if we have two discount bonds, P(t+i) and P(t+j), it must be the case that P(t+j) < P(t+i)
if j > i. The reasoning for this is straightforward. If P(t+i) is the less expensive bond, we can buy that bond
and sell the other bond short. We then make a profit today of P(t+j) - P(t+i) that we get to keep. At date
t+i, we keep the proceeds from the bond that matures in the risk-free asset. At date t+j, we use these
proceeds to reverse the short sale. The problem is that if one is not careful in generating draws of the term
structure of interest rates, one can end up with a situation where a discount bond is more expensive than
a discount bond that matures sooner. For instance, if the one-day returns of two discount bond prices are
normally distributed and not perfectly correlated, there is always some probability that the bond with a short
maturity ends up being less expensive since one bond could increase in price and the other could fall. To
avoid this problem, one must therefore impose some structure on how term structures are generated.
A straightforward way to proceed to avoid arbitrage opportunities is to use the forward curve. Let
f(t+i) be the forward rate at t for delivery of a one-year zero at t+i. Using continuous compounding, f(t+i)
is equal to the logarithm of P(t+i)/P(t+i+1). The forward rates f(t+i) define the forward curve. This curve
differs from the term structure obtained using yields of zero coupon bonds or using coupon bonds. The
traditional term structure curve is given by the yields of coupon bonds. It is generally called the par curve
since it gives the coupon yield of bonds issued at par. Figure 9.5. shows the par curve for an upward-sloping
term structure. For such a term structure, the zero-coupon yield curve, usually called the zero curve, is above
the par curve. The reason for this is that a coupon bond is a portfolio of zero coupon bonds. Consequently,
Chapter 9, page 49
the yield of a par bond is a weighted average of the yields of the zero coupon bonds that compose the
portfolio of zero coupon bonds. The yield of the zero coupon bond with the same maturity as the coupon
bond is the zero coupon bond in that portfolio that has the highest yield when the term structure is upward
sloping. The forward rate corresponding to the maturity of the zero coupon bond is the six-month rate
implied by that zero coupon bond and the one maturing six months later. With an upward-sloping term
structure, the forward rate is therefore higher than the yield of the zero coupon bond because the yield of
the zero coupon bond increases as the maturity lengthens, which can only be achieved by having a higher
implied yield for the next period of time. Figure 9.5. shows how the forward curve, the zero curve, and the
par curve relate to each other for an upward-sloping term structure.
If we have the forward curve, we can use it to compute the value of any bond. To see this, note that
we can buy at t the equivalent of a coupon payment to be made at t+j for forward delivery at t+j-1. The
value of the coupon payment to be made at t+j as of t+j-1 is the coupon payment discounted for one period.
We can enter c forward contracts today where we agree to pay the forward price at t+j-1 for a discount
bond whose value at t+j is $1. If F(t+j-1) is the forward price today for delivery of a discount bond at t+j-1
that pays $1 at t+j, we pay F(t+j-1)c at t+j-1. Remember that F(t+j-1) = P(t+j)/P(t+j-1), where P(t+j) is
the price at t of a discount bond that pays $1 at t+j. We can enter F(t+j-1)c forward contracts to purchase
discount bonds at t+j-2 that pay each $1 at t+j-1. By paying F(t+j-2)*F(t+j-1)c at t+j-2, we therefore pay
for a portfolio of discount bonds and forward contracts that gives us c at t+j. We can then go backwards
one period, so that we enter F(t+j-2)*F(t+j-1)c forward contracts with forward price F(t+j-3) at t to
purchase discount bonds at t+j-3 that pays $1 at t+j-2. Therefore, by paying F(t+j-3)*F(t+j-2)*F(t+j-1)c
at t+j-3, we acquire a portfolio of discount bonds and forward contracts that pays c at t+j. We can then
Chapter 9, page 50
work backwards from t+j-3 up to today using the forward contracts, so that we end up with the price of
the coupon today. Suppose that j = 4. Then, F(t+j-3)*F(t+j-2)*F(t+j-1)c = F(t+1)*F(t+2)*F(t+3)c. We
can buy F(t+1)*F(t+2)*F(t+3)c discount bonds at t for P(t+1) that each pay $1 at t+1. Consequently, the
current value of the coupon paid at t+4 is P(t+1)*F(t+1)*F(t+2)*F(t+3)c. Note that P(t+1) is also the
forward price at t for immediate delivery with price F(t), so that we can write the present value of the
coupon as F(t)*F(t+1)*F(t+2)*F(t+3)c. More generally, using our reasoning, we have:
Value of coupon today
= F(t) * F(t + 1) * F(t + 2)... F(t + j - 2) * F(t + j − 1)* c
P(t + 1) P(t + j -1) P(t + j)
=
...
*c
P(t)
P(t + j − 2) P(t + j -1)
= P(t + j) * c
(9.16.)
P(t) is the price of a discount bond that matures at t and is therefore 1. Using this expression, we can use
the forward curve to compute the value of future cash flows. We can therefore also compute the risk of the
value of the coupon today by using the distribution of the forward prices or the forward rates since:
Present value of coupon payment at t + j =
P(t +1) P(t + j- 1) P(t + j)
...
*c
P(t)
P(t + j − 2) P(t + j -1)
= e − r(t) * e− f(t +1) ...e− f(t + j− 3) * e− f(t + j− 2) * e− f(t + j-1) *c
= e −[r(t) + f(t +1)...+ f(t + j− 3) + f(t + j− 2)+ f(t + j-1)] *c
(9.17.)
Note that the discount rate in the last line is the sum of the one-period forward rates plus the current spot
rate. If we use the forward rates, we start from the current forward rates. We can then estimate the joint
Chapter 9, page 51
distribution of the forward rates. Having this joint distribution, we can simulate forward curves and value the
portfolio for each simulated forward curve.
Conceptually, there is an important difference between the forward curve approach and the
approaches discussed previously that use zero-coupon bond prices or rates for various maturities. With
continuous compounding, the discount rate that applies to a bond maturing at t+i is the sum of the forward
rates as we just saw. Hence, any change in a forward rate affects the discounting rate for a zero coupon
bond with a maturity that is later than the period for which the forward rate is computed. This means that
the zero prices we use take into account the term structure we generate by simulating the zero curve. A
shock to the forward rate for maturity t+j has a one-for-one effect on the discount rate of all zeroes that
mature later. In contrast, if we simulate zeroes using the joint distribution of zero returns, a realization of the
price of a zero for one maturity could be high without affecting the price of the zeroes maturing later on
because of idiosyncratic risk. The same could be the case if one uses discount rates instead of zero prices.
The forward rate approach therefore insures that the no-arbitrage condition holds for the term structure.
One can obtain simulated forward curves in essentially two ways. One way is to use the historical
changes in the forward curve. To do that, one computes either absolute changes or proportional changes
of the forward curve over past periods. For instance, if we want to estimate a one-month VaR, we use
monthly changes. Whether we use absolute or proportional changes depends on whether we believe that
the distribution of absolute or proportional changes is more stable. Though there are disagreements on which
changes to use, there are good arguments to use proportional changes since, as rates become small, one
would think that a one hundred basis point change in rates becomes less likely. Computing changes using
past history allows us to construct a database of changes. We can then apply these changes to the current
Chapter 9, page 52
term structure to obtain simulated term structures for which we compute portfolio values. The fifth percentile
of these portfolio values gives us the VaR. Alternatively, we can estimate the joint statistical distribution of
changes in the forward curve using past data. For instance, we could assume that proportional changes are
jointly normally distributed and then estimate the parameters for this joint distribution. Having done this, we
could use that distribution for a Monte Carlo analysis. With this analysis, we would draw forward curves
and price the portfolio for these forward curves to get a VaR. We could also assume that forward rate
changes follow a factor model, so that the whole forward curve changes depend on relatively few sources
of risk.
The models presented in this section give us points along a term structure. For instance, the forward
rate models give us forward rates for various maturities. To get a complete term structure, one then has to
join those various points using term structure smoothing techniques that insure that the resulting term structure
does not have arbitrage opportunities.
Section 9.5. Summary and conclusion
In this chapter, we started by considering the optimal mix of floating and fixed rate debt of a firm.
A firm’s mix of floating and fixed rate debt is a tool of risk management. If the firm at a point in time has the
wrong mix, it can change that mix by refinancing or by using derivatives. We saw how to transform floating
rate debt into fixed rate debt using the Eurodollar futures contract and vice versa. We then turned to the
interest rate exposure of financial institutions and portfolios in general. The interest rate exposure of a
financial institution can be evaluated in terms of the effect of interest rate changes on income or on value. We
showed how one can compute a CaR measure for a financial institution. To compute a VaR measure for
Chapter 9, page 53
a financial institution, one has to understand how changes in interest rates affect the value of the securities
held by that institution. Traditionally, the duration approximation has been the most popular tool for such an
assessment. We discussed various ways to implement the duration approximation and showed how one can
compute a duration-based VaR. In the last section of the chapter, we consider ways of computing VaR and
evaluating interest rate risk that do not rely on the duration approximation but rather evaluate the impact of
interest rate changes on the value of the securities directly. One such approach, proposed by Riskmetrics™,
treats zero-coupon bond prices as risk factors. Another approach treats interest rates as risk factors. The
last approach we discussed treats forward rates as risk factors.
Chapter 9, page 54
Literature note
There are a large number of useful books on fixed income markets. Fabozzi (1996) provides the ideal
background. More advanced books include Ho (1990) and Van Deventer and Imai (1997). Van Deventer
and Imai (1997) have a more formal treatment of some of the arguments presented here. Their book also
covers some of the topics discussed in Chapter 14. That book also has a lot of material on how to construct
smooth term structures. Duffie (1994) reviews the issues involved in interest rate risk management and
presents the forward rate approach that we discuss in this chapter. The paper by Litterman and Schenkman
(1991) addresses the key issues involved in factor models. Ilmanen (1992) presents an up-to-date empirical
assessment of duration that supports our discussion. Bierwarg, Kaufman, and Toevs (1983) provide a nice
collection of papers evaluating duration in a scientific way. Ingersoll, Skelton, and Weil provide the FisherWeil duration measure discussed in the text and relate it to the work of Macaulay. The Riskmetrics™
manual provides complete information on how Riskmetrics™ deals with interest rate risks.
Chapter 9, page 55
Key concepts
Forward rate agreements, LIBOR, asset sensitive, liability sensitive, modified duration, Fischer-Weil
duration, convexity, interest rate delta exposure, factor models, forward curve.
Chapter 9, page 56
Review questions and problems
1. A firm has fixed income of $10m per year and debt with face value of $100m and twenty-year maturity.
The debt has a coupon reset every six months. The rate which determines the coupon is the 6-month
LIBOR. The total coupon payment is half the 6-month LIBOR on the reset date in decimal form times the
face value. Today is the reset date. The next coupon has just been set and is to be paid in six months. The
6-month LIBOR at the reset date is 5.5% annually. The volatility of 6-month LIBOR is assumed to be 100
basis points annually and the term structure is flat. What is the one-year CaR for this firm assuming that the
only risk is due to the coupon payments on the debt?
2. What is the value of the debt at the reset date if the debt has no credit risk? What is the duration of the
debt at the reset date if the debt has no credit risk?
3. A position has modified duration of 25 years and is worth $100m. The term structure is flat. By how much
does the value of the position change if interest rates change by 25 basis points?
4. Suppose that you are told that the position has convexity of 200 years. How does your answer to
question 3 change?
5. Consider a coupon bond of $100m that pays coupon of $4m and has no credit risk. The coupon
payments are in 3, 9, and 15 months. The principal is paid in 15 months as well. The zero-coupon bond
Chapter 9, page 57
prices are P(t+0.25) = 0.97531, P(t+0.75) = 0.92427, and P(t+1.25) = 0.82484. What is the current bond
price? What is the forward price today for a six-month zero-coupon bond delivered in 9 months? What is
the forward rate associated with that forward price?
6. Using the Fisher-Weil duration, compute the impact of a parallel shift in the yield curve of 50-basis points?
How does your answer compare to the change in the bond price you obtain by computing the new bond
price directly and subtracting it from the bond price before the parallel shift?
7. Compute the yield of the bond and then compute the modified duration using the data in question 5. What
is the impact of a 50 basis points change in the yield using the modified duration? How does it compare to
your answer in question 6? How does your answer compare to the change computed directly from the new
and the old bond prices?
8. How does your answer to question 7 change if you also use convexity?
9. Assume that the volatility of the zero-coupon bond prices expressed per year in question 5 is 1% for the
3-month bond, 3% for the 9-month bond, and 5% for the 15-month bond. The correlation coefficients of
the bonds are 0.9 between all bonds. Ignoring expected changes in value of the bonds, what is the volatility
of the bond price? Assuming that the returns of the zero-coupon bond prices are normally distributed, what
is the one-day VaR of the bond?
Chapter 9, page 58
10. Show how the par curve, the zero curve, and the forward curve relate to each other if the term structure
is downward sloping.
Box 9.1. The tailing factor with the Euro-dollar futures contract
We are at date t and want to hedge an interest rate payment to be made at date t+0.5 based on an
interest rate determined at date t+0.25, the reset date. We therefore want to use a Euro-dollar futures
contract that matures at date t+0.25. Let’s define F to be the implied futures yield at date t for the
contract that matures at t. Settlement variation on the contract at t+0.25 is 0.25*(RL(t+0.25) - F) times
the size of the short position. At t+0.25, we get to invest the settlement variation for three months at the
rate of RL(t+0.25). This means that at t+0.5 we have (1+0.25*RL(t+0.25))*(0.25*(RL(t+0.25)-F))
times the short position. We therefore want to choose a short position such that at t+0.5 we have no
interest rate risk. The net cash flow at t+0.5 is equal to minus the interest payment plus the proceeds of
the short position:
- 0.25*RL(t+0.25)*100m +
Short position* (1+0.25*RL(t+0.25))*0.25*(RL(t+0.25)-F)
If we choose the short position so that at t+0.25 it is equal to 100m/ (1+0.25*RL(t+0.25)), we end up
with:
- 0.25*RL(t+0.25)*100m + (100m/(1+0.25*RL(t+0.25))*
(1+0.25*RL(t+0.25))*(0.25*(RL(t+0.25)-F))
= -0.25*F*100m
Consequently, if our short position in the futures contract is such that at date t+0.25 it is equal to
100m/(1+0.25*RL(t+0.25)), we eliminate the interest rate risk and our net cash flow at date t+0.5 is
equal to an interest payment based on the futures yield known when we enter the futures contract.
To have a futures position of 100m/(1+0.25*RL(t+0.25)) at date t+0.25, note that
1/(1+0.25*RL(t+0.25)) at date t+0.25 is the present value of one dollar paid at t+0.5 discounted at the
LIBOR rate. This must be equivalent to the value at t of a discount bond maturing at t+0.5 which has a
risk equivalent to LIBOR time-deposits. Therefore, if we use as the tailing factor such a discount bond
that matures at t+0.5, we will have the appropriate futures position at t+0.25. At date t, we therefore take
a short position equal to 100m times the price of such a discount bond. At date t, the price of such a
bond is the price of a dollar to be paid at t+0.5 discounted at six-month LIBOR.
Chapter 9, page 59
Box 9.2. Orange County and VaR
To see that a VaR computation using duration can provide extremely useful information, let’s look at the
case of Orange County. In December 1994, Orange County announced that it had lost $1.6 billion in an
investment pool. The investment pool collected funds from municipalities and agencies and invested them
on their behalf. The pool kept monies to be used to finance the public activities of the County. The amount
in the pool had been $7.5 billion and was managed by the County Treasurer, Bob Citron. As of April 30,
1994, the portfolio had an investment in securities of $19.860b with reverse repo borrowings of $12.529b.
The securities were mostly agency fixed-rate and floating rate notes with an average maturity of about 4
years. The fund was therefore heavily levered. The investment strategy was to borrow short-term using the
repo market discussed in Chapter 2 to invest in securities with longer duration to take advantage of the slope
of the term structure. The problem with such a strategy is that as interest rates increase, the value of the
assets falls while the liabilities do not. Valuing the securities held on April 30 at cost, we have $19,879b
worth of assets. The duration of the pool was estimated at 7.4 in a subsequent analysis. The yield of the fiveyear note went from 5.22% in December 1993 to 7.83% in December 1994. Let’s assume that 7.4 is the
appropriate duration for December 1993 and that the portfolio given in the table is the one that applies at
that time. Assuming further that this duration measure is modified duration and that the yield of the 5-year
note is the appropriate yield given the duration of the pool, we have the loss to the pool for a 261 basis
points change in rates:
7.5b*7.4*0.0261 = $1.406b
Chapter 9, page 60
This is quite close to the actual loss reported of $1.6b! Let’s assume that percentage changes in yields are
normally distributed. Using monthly data from January 1984 to December 1993, the volatility for the
percentage change in the yield of the 5-year note is 4.8% per month. Applying the square-root rule, this
gives us a yearly proportional volatility of 16.63%. The 5-year note had a yield of 5.22% in December
1993, so that 16.63% of 5.22% is 0.868%. There was therefore a 5% chance that the 5-year yield would
increase by more than 1.65*0.868% over the coming year, or more than 1.432%. The loss corresponding
to an increase of 1.432% is given by the duration formula:
One-year VaR = 1.65*7.5b*7.4*0.01432 = $2.106b
In other words, as of December 1993, the one-year VaR of the Orange County pool was $2.106b. There
was therefore a five percent chance Orange County would lose at least $2.106b given its investment strategy
based on the assumptions of this calculation. Obviously, many refinements could be made to this calculation.
However, it shows that the duration VaR tool can be sufficient to understand risk well enough to avoid big
mistakes. It is hard to believe that the officials of Orange County would have used the investment strategy
they chose had they known the VaR estimate we just constructed.
Sources: The Orange County debacle is described in Philippe Jorion, Big bets gone bad: Derivatives and
bankruptcy in Orange County, Academic Press, 1995. Professor Jorion also maintains a Web Site that
includes a case on Orange County involving various VaR computations for December 1994 and useful other
materials: http://www.gsm.uci.edu/~jorion/oc/case.html.
Chapter 9, page 61
Box 9.3. Riskmetrics™ and bond prices
The Riskmetrics™ approach to evaluating interest rate risk involves two steps. The first step consists
in mapping cash flows from fixed income securities to the cash flows available from Riskmetrics™. These
cash flows are called vertices in the language of Riskmetrics™. In our language, this operation amounts to
matching the cash flows of default-free fixed income securities to the cash flows of zero-coupon bonds.
Through this operation, the portfolio of fixed income securities is transformed into an equivalent portfolio of
zero-coupon bonds. Its return then becomes the return of a portfolio of zero-coupon bonds. Riskmetrics™
makes available the volatilities and the correlations of the returns of these zero-coupon bonds. We can
therefore compute an analytical VaR using this information and using as portfolio weights the investments
in the zero-coupon bonds of the transformed portfolio.
The complication with the Riskmetrics™ approach is that there are at most 14 vertices available 1m, 3m, 6m, 12m, 2yr, 3yr, 4yr, 5yr, 7yr, 9yr, 10yr, 15yr, 20yr, 30yr. Obviously, a portfolio will have cash
flows with maturities that do not correspond to the vertices. For instance, a portfolio might have a cash flow
that matures in 6 years. The solution is then to attribute that cash flow to the two adjoining vertices, 5 year
and 7 year, in a way that does not affect the risk of the portfolio compared to what it would be if we had
the 6 year vertex. This means that (a) the total value of the cash flow must not be affected by the split, (b)
the market risk must be preserved, and (c) the sign is preserved.
The implementation of the approach works by first interpolating the yields linearly. In our example,
the 6-year yield is half the 5-year and half the 7-year. With the interpolated 6-year yield, we can compute
the present value of the 6-year cash flow. The volatility of the 6-year return is computed by taking a linear
interpolation of the 5-year and 7-year volatilities. We then solve for weights on the 5-year and the 7-year
Chapter 9, page 62
zeroes so that the variance of this portfolio of two zeroes has the same variance as the 6-year return and the
same value. These weights are then used to split the 6-year cash flow into a 5-year and a 7-year cash flow.
The Table Box 9.1. reproduces an example from the Riskmetrics™ manual.
Table Box 9.1. Riskmetrics™ mapping of cash flows
Problem:
On July 31, 1996, we expect a cash flow occurring in 6 years of USD 100.
Data from
Riskmetrics™:
r(t+5)
r(t+7)
1.65std(r(t+5))
5-year yield
7-year yield
Riskmetrics™ volatility on the 5-year
bond price return
Riskmetrics™ volatility on the 7-year
bond price return
Correlation between the 5-year and
the 7-year bond return
6.605%
6.745%
0.5770%
Interpolation of
yields
0.5*6.605% + 0.5*6.745%
6.675%
Standard
deviation of
r(t+6)
0.5*(0.5770%/1.65)
+ 0.5*(0.8095%/1.65)
0.4202%
1.65std(r(t+7))
D(r(t+5),r(t+7))
Variance of
r(t+6)
r(t+5)
r(t+7)
0.8095%
0.9975
1.765*10-3%
1.223*10-3%
2.406*10-3%
Chapter 9, page 63
Computation of
weight " of 5year bond and
(1- " ) of 7-year
bond
1.765= " 2*1.223 + (1- " )2*2.406
+ 2*" *(1- " )*(0.5770%/1.65)
*(0.8095%/1.65)
Chapter 9, page 64
Solution: One solution of this equation is
" = 5.999 and the other is " = 0.489.
The first solution does not preserve the
sign. The second solution is chosen.
The 6-year cash flow is worth USD
93.74. 48.9% of that is invested in 5year bond.
Table 9.1. Eurodollar futures prices.
Eurodollar
Daily Prices As of :- Wednesday, 10 May
Date
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
Open
High
Low
Last
Chge
125
Prev.
Volume
1520
Prev.
Open_Int
35446
M a y 932700
00
Jun 00 931400
Jul 00 930200
A u g 929350
00
S e p 928050
00
Oct 00 926000
D e c 925300
00
M a r 924500
01
Jun 01 923650
S e p 923400
01
D e c 922950
01
M a r 923450
02
Jun 02 923550
S e p 923600
02
D e c 923150
02
M a r 923650
03
Jun 03 923350
S e p 923150
03
D e c 922550
03
M a r 922950
04
Jun 04 922550
S e p 922250
04
D e c 921550
932850
932650
932775
931650
930300
929400
931350
930200
929100
931500
930250
929150
50
100
300
50676
451
41
503450
7331
2540
928500
928000
928100
150
82742
563283
926400
925950
926000
925050
926400
925450
500
350
0
74909
35
458937
925200
924250
924750
450
60037
359860
924400
924100
923550
923300
924000
923750
500
500
34372
21612
247951
186444
923700
922900
923300
450
10536
137124
924000
923300
923800
500
10597
126764
924000
924000
923500
923450
923900
923900
550
550
5784
4897
88815
87919
923550
922950
923450
550
4286
72646
924050
923350
923950
550
4950
72575
923750
923550
922950
922750
923700
923500
600
600
2148
4117
49702
49648
922950
922100
922900
600
1837
37798
923350
922500
923300
600
2019
32982
922950
922650
922100
921800
922900
922600
600
600
1912
2320
29965
25471
921950
921100
921900
600
2557
26416
Chapter 9, page 65
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
5/10/00
Composite
5/9/00
04
M a r 921900
05
Jun 05 921300
S e p 920950
05
D e c 920250
05
M a r 920500
06
Jun 06 920350
S e p 920000
06
D e c 919300
06
M a r 919650
07
Jun 07 919000
S e p 918700
07
D e c 918000
07
M a r 918250
08
Jun 08 917900
S e p 917550
08
D e c 916850
08
M a r 917550
09
Jun 09 916550
S e p 916200
09
D e c 915500
09
M a r 916400
10
V o l u Open_
me
Int
3 8 9 5 2 331989
6
0
922300
921450
922250
600
2135
18824
921900
921550
921100
920750
921900
921550
600
600
288
276
11228
9930
920800
920050
920800
550
201
7055
921050
920300
921050
550
201
7478
920700
920350
919950
919600
920700
920350
550
550
170
145
6494
6108
919650
918900
919600
500
145
6085
919900
919150
919850
500
145
4740
919550
919250
918800
918500
919500
919150
500
450
97
72
3798
3647
918550
917800
918400
400
72
5130
918800
918050
918650
400
97
4289
918450
918100
917700
917350
918300
917950
400
400
243
217
4593
4152
917400
916650
917200
350
241
3180
917650
916900
917450
350
240
2647
917300
916950
916550
916200
917100
916750
350
350
55
55
2554
2438
916250
915500
916050
350
55
1724
916500
915750
916300
350
55
692
Chapter 9, page 66
Table 9.2. Interest Rate Sensitivity Table for Chase Manhattan
At December 31, 1998 ( in
millions)
1-3 Months
4-6 Months
7-12 Months
1-5 years
Over 5
years
$(37,879)
$480
$6,800
$43,395
$(12,796)
(4,922)
803
(2,788)
2,542
4,365
Interest Rate Sensitivity Gap
(42, 801)
1,283
4,012
45,937
(8,431)
Cumulative Interest Rate
Sensitivity Gap
$(42,801)
$(41,518)
$(37,506)
$8,431
$--
(12)%
(11)%
(10)%
2%
–%
Balance Sheet
Derivative Instruments Affecting
Interest Rate Sensitivitye
% of Total Assets
e
Represents net repricing effect of derivative positions, which include interest rate swaps,
futures, forward rate agreements and options, that are used as part of Chase’s overall ALM.
Chapter 9, page 67
Table 9.3. Example from Litterman and Scheinkman(1991).
Face value ($)
Coupon
Maturity
Price on February
5, 1986
Price on March 5,
1986
-118,000,000
12 3/8
8/15/87
106 5/32
106 9/32
100,000,000
11 5/8
1/15/92
112 12/32
115 20/32
-32,700,000
13 3/8
8/15/01
130 16/32
141 14/32
Chapter 9, page 68
Figure 9.1. Example of a Federal Reserve monetary policy tightening.
16%
16%
14%
14%
12%
12%
10%
10%
8%
8%
6%
6%
4%
4%
2%
2%
0%
0
5
10
15
20
Time-to-maturity (Years)
25
30
0%
0
A. Term structure in March 1979.
5
10
15
20
Time-to-maturity
(Years)
25
B. Term structure in March 1980.
Chapter 9, page 69
30
Figure 9.2. Bond price as a function of yield.
This figure shows the price of a bond paying coupon annually of $5 for 30 years with principal value of $100.
Chapter 9, page 70
Figure 9.3. The mistake made using delta exposure or duration for large yield changes.
Chapter 9, page 71
Figure 9.4. Value of hedged bank.
This figure shows the value of the hedged bank when the hedged bank has a duration of 21 years and a
convexity of 500. The bank is hedged with a discount bond with maturity of 30 years. This discount bond has
a duration of 30 years and a convexity of 900. The hedged bank has no duration at the current rate but has
negative convexity. Note that the present value of the hedged bank is the present value of the payoff
represented in this figure. Hence, the bank will be worth more than expected if rates do not change and will
be worth less if they change by much.
Chapter 9, page 72
Figure 9.5.Term Structures.
Yield
Forward rate
Zero-coupon yield
Coupon-bearing
bond yield
0
Time to maturity
Chapter 9, page 73
Table 9.1. Factor loadings of Swiss interest rates
This table is extracted from Alfred Buhler and Heinz Zimmerman, A statistical analysis of the term structure of interest rates in Switzerland Germany,
Journal of Fixed Income, December 1996, 55-67. The numbers in parentheses are the percent of the variance of interest rate changes explained by
the factor.
Maturity
Factor 1 (%)
Factor 2 (%)
Factor 3 (%)
1-month
0.162 (48.30%)
-0.139 (35.97%)
0.079 (11.46%)
3-month
0.142 (62.56%)
-0.099 (30.50%)
0.016 (0.80%)
6-month
0.138 (70.09%)
-0.075 (20.83%)
-0.020 (1.41%)
12-month
0.122 (70.25%)
-0.056 (14.57%)
-0.041 (7.84%)
2-year
0.130 (90.10%)
0.011 (0.69%)
-0.022 (2.47%)
3-year
0.125 (89.51%)
0.030 (5.04%)
-0.013 (1.00%)
4-year
0.116 (87.30%)
0.040 (8.38%)
-0.004 (0.10%)
5-year
0.108 (84.95%)
0.039 (10.93%)
0.002 (0.03%)
7-year
0.097 (77.73%)
0.043 (15.19%)
0.016 (2.14%)
10-year
0.092 (73.79%)
0.043 (16.14%)
0.020 (3.33%)
Chapter 9, page 74
Chapter 10: Options, downside risk, and nonlinear exposures
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section 10.1. Using options to create static hedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Section 10.1.1. Options as insurance contracts: The case of an investor. . . . . . . . . . . 6
Section 10.1.2. Options as insurance contracts: The case of Export Inc. . . . . . . . . . . 9
Section 10.1.3. Options to hedge nonlinear exposures. . . . . . . . . . . . . . . . . . . . . . . 14
Section 10.1.4. How we can hedge (almost) anything with options. . . . . . . . . . . . . . 17
Section 10.2. Some properties of options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Section 10.2.1. Upper and lower bounds on option prices. . . . . . . . . . . . . . . . . . . . 23
Section 10.2.2. Exercise price and option values . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Section 10.2.3. The value of options and time to maturity . . . . . . . . . . . . . . . . . . . . 29
Section 10.2.4. Put-call parity theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Section 10.2.5. Option values and cash payouts . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Section 10.2.6. Cash payouts, American options, and put-call parity . . . . . . . . . . . . 37
Section 10.3. A brief history of option markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Section 10.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Figure 10.1. Wealth of Ms.Chen on January 19, 2001 as a function of Amazon.com stock
price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 10.2.Payoffs of cash, forward, and put positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Figure 10.3. Difference in Firm Income Between Put Option and Forward . . . . . . . . . . . . . 53
Figure 10.4. Payoffs to exporter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 10.5.Arbitrary payoff function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 10.6.Approximating an arbitrary payoff function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Table 10.1. Puts on Amazon.com maturing in January 2001. . . . . . . . . . . . . . . . . . . . . . . . . 57
Box 10.1. Hedging at Hasbro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Box 10.2. Cephalon Inc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 10: Options, downside risk, and nonlinear
exposures
September 25, 2000
© René M. Stulz 1999, 2000
Chapter objectives
At the end of this chapter, you will know how to:
1. Choose between forwards and options as hedging instruments.
2. Use options to hedge complicated nonlinear payoffs.
3. Decide whether a firm should buy or sell options on itself.
4. Set bounds on option prices.
Chapter 10, page 1
Suppose that you invested in a common stock and the value of your investment has increased
dramatically. You are worried that you might make large losses because of an increase in volatility. You
could sell the stock, but you would have to pay capital gains taxes and would have to give up upside
potential you believe is significant. Hedging all the exposure with forward and futures contracts would allow
you to eliminate all downside risk, but it would also eliminate the upside potential. Buying put options would
eliminate the downside risk without at the same time eliminating the possibility of making large gains if the
stock price increases sharply. Options offer enormous flexibility in tailoring hedges. In this chapter, we
show why options are so useful in hedging programs and discuss some properties of options. In the next
two chapters, we will study the pricing of options in detail before returning to risk management with options
in Chapter 13.
We saw in Chapter 8 that quantity risks make the value of the cash position a nonlinear function
of the cash price. For instance, an exporter who becomes more successful when the domestic currency
depreciates has a currency exposure that increases as foreign currencies become more valuable. When
exposure is a nonlinear function of the price of the cash position, the optimal hedge using forward and
futures contracts changes as the price of the cash position changes. For instance, in our example in Chapter
8 of a British exporter who sells cars in the U.S., the exporter sells more cars as the pound depreciates
relative to the dollar, so that the cash flow of the exporter in dollars increases as the pound depreciates and
hence the dollar exposure of the exporter increases as the pound depreciates. A static hedge is a hedge
position you put in place at one point in time and leave unchanged until maturity of the exposure - in other
words, you can put on the hedge and can go to the beach. In the case of a nonlinear exposure such as the
Chapter 10, page 2
one of the British exporter, a static hedge with forward and futures contract may work poorly.1 This is
because such a hedge eliminates a fixed exposure but the size of a nonlinear exposure depends on the cash
price at maturity of the hedge. Therefore, there can be a large hedging mistake whenever the price at
maturity of the hedge is very different from its expected value.
There are two ways to improve on a static hedge with forward and futures contracts when the
exposure is nonlinear. These two approaches are closely related. One approach is to change the hedge
over the hedging period. Remember that the static hedge does not work well for nonlinear exposures when
the price changes by a large amount. Typically, prices do not change by large amounts overnight. Whenever
the price changes over the hedging period, one can therefore adjust the hedge to reflect the newly acquired
knowledge about the distribution of the price at maturity of the hedge. This is called a dynamic hedging
strategy. The important question withsuch a strategy is how to compute the initial hedge and how to adjust
it over time as we learn more about the price of the cash position. The alternative approach is to use
additional financial instruments to construct the static hedge. Starting with this chapter, we investigate
financial instruments that make it possible to construct static hedging strategies that achieve the same
objective as dynamic hedging strategies with forwards and futures more efficiently. Since these dynamic
hedging strategies are used to hedge risks that are nonlinear in the price of the cash position, these financial
instruments have payoffs that are nonlinear in that price. We will find out that knowing how to price these
financial instruments also allows us to design efficient dynamic hedging strategies.
1
Remember that futures hedges have to be tailed. We consider hedges before tailing, so that a
fixed hedge ratio before tailing is a static hedge.
Chapter 10, page 3
What is the benefit of a static hedge relative to a dynamic hedge? A dynamic hedge requires
frequent trading. Consequently, it has transaction costs that a static hedge does not have. Further, a
dynamic hedge requires monitoring. Somebody has to follow prices and make sure that the right trades are
implemented. Finally, a dynamic hedge can fail because of model risk and/or trading risk. With a dynamic
hedge, we have to have the right model to adjust hedge ratios over time. If the model turns out to be
inaccurate, we are less well hedged than we thought. We will see precisely what this means with derivatives
later on. In addition, since we have to trade, there has to be a market for us to trade in when we want to.
Sometimes, however, markets become very illiquid precisely when we would like to trade. An example
of this is what happened during the European currency crisis in 1991. As uncertainty increased, spreads
widened and trading became difficult. The same difficulties occurred in the bond markets following the
default by Russia in August 1998. As we progress, we will discuss these problems in many different
contexts. However, it may also turn out that a dynamic hedge is cheaper. We will examine the tradeoff
between static and dynamic hedging strategies in greater detail later when we better understand how to
implement these strategies.
In this chapter, we introduce options. We will see that options are building blocks of any financial
instrument that has a payoff that is nonlinear in the price. We therefore will use options throughout the
remainder of this book. In the first section, we focus on the payoffs of options and on how options can be
used to reduce risk. We conclude the first section showing that almost any payoff can be hedged using
options. In the second section, we discuss some fundamental properties of options that do not depend on
the distribution of the price of the assets on which options are written. In the third section, we survey the
Chapter 10, page 4
markets for options and their evolution. The chapter is briefly summarized in the last section.
Section 10.1. Using options to create static hedges.
Forward and futures contracts obligate the long to buy at maturity and the short to sell at maturity
at a fixed price. A call option gives the right to the holder of the option to buy at a fixed price. The fixed
price is called the exercise or strike price. A put option gives the right to the holder to sell at a fixed
price. The seller of the option is called the option writer. Options differ with respect to when the holder
can exercise his right. An option is a European option if the holder can only exercise his right at maturity.
An option is an American option if the holder can exercise his right at any time until maturity.
We saw in Chapter 5 that a forward contract has no value when we enter the contract. Options
are different. With a put option, we have a right which we exercise only if it is profitable for us to do so.
Consequently, we will not exercise our right to sell SFRs at maturity if the spot price of the SFR exceeds
the exercise price. If we exercised in this case, we would have to give money to the option writer that we
could otherwise keep. Since, as an option holder, we never lose money when we exercise optimally, the
option must have value. Otherwise, an option would be a free lunch: the option holder would get something
for nothing. The price paid for an option is called the option premium.
In this section, we focus on the use of European options. We will see in Section 10.3. that using
American options complicates matters because it is sometimes advantageous to exercise American options
early. In this section, we examine first the purchase of puts to hedge a stock portfolio. We then turn to
using puts to hedge a fixed exposure going back to the case of Export Inc. Before considering the case
Chapter 10, page 5
where options can help hedge exposures that involve quantity risks, we examine the impact on the firm’s
risk of a forward hedge versus a put hedge.
Section 10.1.1. Options as insurance contracts: The case of an investor.
Consider Ms.Chen, an investor who invested all her wealth in Amazon.com except for $150,000
which she holds in cash. As she sees the volatility of the stock increase, she becomes concerned that she
could lose a large fraction of her net worth. She would like to purchase insurance that pays her the loss in
net worth she incurs if the stock price falls below some threshold level. Say Ms.Chen has 10,000 shares
of Amazon.com and the price of a share is $58 1/2, which is what it was on May 5, 2000. On May 5, her
net worth is $735,000. She is acutely aware that five months before her net worth was about double what
it is now. To make sure that she does not lose much more, she could sell her shares, but she believes that
Amazon.com has substantial upside potential. To avoid giving up this upside potential, she decides to buy
puts on her shares as discussed in Chapter 2. To investigate protecting her wealth with puts, she chooses
to go the Chicago Board Options Exchange web site to investigate. Using the free delayed quote
information (http://quote.cboe.com/QuoteTableDownload.htm) she immediately finds that a large number
of different options are traded on Amazon.com. 2 In particular, there are options maturing in four different
months during 2000 as well as options maturing in 2001.
Ms.Chen decides to investigate options maturing in January 2001. She finds four different quotes
for puts maturing in January 2001 with an exercise price of $55. The quotes are shown on Table 10.1. If
2
See the book web site for the Excel file she could have downloaded.
Chapter 10, page 6
she buys a put contract on Amazon.com with exercise price of $55 and maturity in January 2001, she gets
to sell 100 shares of Amazon.com at $55 per share on or before the maturity in January 2001. CBOE
options expire 11:59 pm Eastern Time on the Saturday immediately following the third Friday of the
maturity month. The expiration calendar available on the CBOE web site tells us that the last trading day
of the put is January 19, 2001. Each quote is for a different exchange identified by the last letter identifying
the option. Let’s consider the first quote. The quoted put is ZCM ML-E. The E stands for Chicago Board
Options Exchange (A is for AMEX, P is for the Philadelphia Stock Exchange and X is for the Pacific Stock
Exchange). The next column gives us the last price traded, which is $14 1/4. We then get the bid and the
ask. Since our investor wants to buy, she needs to know the ask, which is $13 7/8. To protect all the
shares in her portfolio, Ms.Chen would have to buy 100 contracts for a total cost (remember that a
contract allows you to sell 100 shares at the exercise price) of 100*100*$13 7/8, or $138,750. In addition
to this, Ms.Chen would have to pay a commission to her broker. Ignoring commissions, Ms.Chen could
therefore guarantee that her wealth in January would be at least $550,000 plus the cash left over after she
pays for the options, $150,000 minus $138,750 or $11,250, plus the interest on that cash.
To see that Ms.Chen could guarantee that her wealth is at least $561,250 on January 19, 2001,
suppose that on that day Amazon.com were to trade at $40. In this case, she could sell her puts on that
day or exercise. To exercise, Ms.Chen has to tender her options to the exchange before 4:30 pm on
January 19 (her broker may impose an earlier time in the day for the tendering of the options). If she
exercises, she knows that she will get $55 per share of Amazon.com. Hence, when she gets the $55 per
share, her wealth will be $550,000 plus $11,250 plus the interest from May 5 to January 19 on $11,250.
Chapter 10, page 7
Figure 10.1. shows graphically how Ms.Chen protects her wealth through the purchase of puts.
One might conclude that the insurance that Ms.Chen buys this way is extremely expensive.
However, before one reaches this conclusion, it might be useful to compute the one-day VaR of Amazon
as of May 5, 2000. To do that, Ms.Chen goes to the web site of e-trade, where she can get volatilities for
free ( http://www.etrade.com/cgi-bin/gx.cgi/AppLogic+ResearchStock ). She finds that the annualvolatility
of Amazon.com is 110%. The percentage one-day VaR is 11.5%. This means that if we are willing to
believe that Amazon.com’s return is normally distributed, Ms.Chen has a 5% chance of losing about
$67,000 the next day!
Ms. Chen might find that the insurance is worth it. However, she would want to understand the
quote tables better before doing so. She would notice that there was no trade in the put option she wants
to buy on May 5. Further, she would also notice that the existing open interest is 4,273 contracts, so that
investors have the right to sell 427,300 shares of Amazon.com for $55 a share on or before January 19,
2001. The absence of trades might concern her. This would suggest that this put contract is not very liquid.
This is consistent with a bigger bid-ask spread for January options than May or June options. For instance,
a June option with about the same price has a spread of $0.75 rather than $1 for the January option. She
also notices that the stock itself had a much smaller bid-ask spread of 1/8. Looking at this data, she realizes
that she could get exposure to the upside potential of Amazon.com by selling all her shares and using the
proceeds to buy call options and to invest in a money market account. She therefore looks at call options
with an exercise price of $55. As seen on Table 10.1, she finds that she can buy a call at $19. If she buys
100 call contracts, she would have $548,750 (1,000*58 7/8 + 150,000 - 100*100*19) that she could
Chapter 10, page 8
invest in the money market account ignoring commissions. This is less than she would have buying the put.
However, she would earn interest on $548,750 until January 19, 2001, and she gets the right to exercise
the calls until expiration. If she exercises the calls, she receives the shares in exchange of $55 per share
three business days after she tenders her exercise notice.
For Ms.Chen to make the right decision as to whether she should buy calls or puts, she has to
know how the price of a put should relate to the price of a call. We will see that there is an exact relation
between call prices and put prices in Section 10.2.4. She would also like to know whether she can
compute the price at which the puts and calls should trade. We will study the pricing of puts and calls in
this and the next two chapters.
Section 10.1.2. Options as insurance contracts: The case of Export Inc.
In Chapter 6, we considered the problem of Export Inc. whichexpects to receive SFR 1M on June
1. We saw that Export Inc. can make its cash flowriskless by selling SFR 1M forward. Figure 10.2. shows
the payoff of the unhedged position, the payoff of a short SFR 1M forward position, and the payoff of the
put option as a function of the spot exchange rate at maturity. If we hedge with a forward contract, the
payoff is the same irrespective of the spot exchange rate at maturity.
Suppose Export Inc. believes that the forward price of the SFR is too low, so that its expected
cash flow would be higher absent hedging. If the only hedging instrument available to Export Inc. is the
forward contract, it can only take advantage of its beliefs about the SFR by hedging less and hence having
more risk. For the SFRs it sells forward, Export Inc. does not benefit from an increase in the exchange rate.
Chapter 10, page 9
Consequently, it makes sense for Export Inc. to try to figure out ways to hedge that protect it against
downside risk but enable it to benefit from its beliefs about the future price of the SFR if it is right. Let’s see
how it can do that.
Consider a static hedge where Export Inc. buys a European put option on the SFR. Such a put
option gives it the right to sell the SFR at a fixed price. Figure 10.2. shows the payoff of the put option at
maturity. If Export Inc. buys a put that gives it the right to sell SFR 1M at the forward price, for instance,
it will earn at maturity an amount equal to the forward price minus the spot price times one million if that
amount is positive. Export Inc. must pay the option premium when it buys the put. Since we want to
compare the payoff of the option strategy with the payoff of the forward hedge strategy, we have to
consider the value of the option premium as of the maturity of the hedge. To do that, we assume that Export
Inc. borrows at the risk-free rate to pay the option premium and repays the loan at maturity of the hedge.
Let S be the price of one SFR and p(S, F, June 1, March 1) be the option premium paid on March 1 for
a put option on one SFR maturing on June 1 with an exercise price equal to the current forward exchange
rate for the SFR for delivery on June 1, F. Let’s assume that F is equal to $0.75, which is also the spot
exchange rate, and that the premium paid by Export Inc. is $0.01111 per SFR, or slightly more than one
cent. Using our notation, if Export Inc. buys a put option on SFR1M today, borrows the option premium
and repays the loan at maturity of the hedge, it will have to pay 1M*p(S, F, June 1,March 1)/P(June 1)
at maturity of the hedge. Using a continuously-compounded interest rate of 5%, the price of a discount
bond that matures on June 1 is $0.987578. Consequently, the loan reimbursement on June 1 is equal to
1M*0.01111/0.987578, or $11,249.7.
Chapter 10, page 10
Figure 10.3. compares the payoffs of three strategies: no hedge, the forward hedge studied in
Chapter 6, and the put hedge. Let’s consider the payoffs of these strategies. With the strategy of no
hedging, Export Inc. has SFR 1M on June 1. With the forward hedge strategy, Export Inc. receives
$1M*0.75 at maturity. With the option strategy where it buys a put with an exercise price equal to the
forward rate, the payoff at maturity depends on whether the spot exchange rate is higher or lower than the
exercise price of the option. Consider first the case where the spot exchange rate is lower than the exercise
price of the option. Since Export Inc. has the right to sell SFRs at the forward price and the forward price
exceeds the spot price, it decides to exercise that right. Therefore, denoting the spot exchange rate on June
1 by S(June 1), it receives at maturity:
Cash position + Payoff from put option - Repayment of loan used to pay put premium
1M*S(June 1) + (1M*F - 1M*S(June 1)) - 1M*p(S, F, June 1,March 1)/P(June 1)
1M*F - 1M*p(S, F, June 1,March 1)/P(June 1)
= $750,000 - $11,249.7
= $738,750.30
This is less than it would get with the forward hedge because it has to reimburse the loan it took out to pay
for the option premium.
Consider now the case where at maturity the spot exchange rate exceeds the forward exchange
rate. In this case, Export Inc. chooses not to exercise the option at maturity. If it made the mistake of
Chapter 10, page 11
exercising the option, it would be selling SFR1M for 1M*F when it could get 1M*S(June 1) for them.
Consequently, its payoff at maturity if it exercises optimally is:
Cash position - Repayment of loan used to pay put premium
1M*S(June 1) - 1M*p(S, F, June 1,March 1)/P(June 1)
If the spot exchange rate is $0.80, Export Inc. receives $800,000 - $11,249.7, or $788,750.30. This
payoff is less than if it had not hedged by $11,249.7 because it purchased insurance against an unexpected
fall in the SFR exchange rate that it ended up not needing. The payoff of the insurance at maturity is the
maximum of 0 or F - S(June 1), which we denote by Max(F - S(June 1), 0). Putting together the payoff
of the position when Export Inc. exercises the put and when it does not results in the payoff of the hedged
position:
Payoff of hedged position = Cash position + Payoff of put - Repayment of loan used to pay the premium =
1M*S(June 1) + 1M*Max(F - S(June 1),0) - 1M*p(S, F, June 1, March 1)/P(June 1)
In summary, the firm earns more with the put strategy than with the forward strategy if the SFR appreciates
sufficiently to offset the cost of the put and earns less otherwise since it has to pay for the put. In deciding
which strategy to use, Export Inc. would have to trade off the benefit from the put strategy if the SFR
appreciates sufficiently to cover the cost of the put against the cost of the put if it does not.
Chapter 10, page 12
There could be other reasons for Export Inc. to use the put strategy instead of the forward strategy.
In Chapter 3, we saw that if a firm has investment opportunities correlated with a risk factor, the firm might
want to hedge so that it has more resources available when it has good investment opportunities. Here,
Export Inc. sells goods in Switzerland. As the SFR appreciates, its profits from selling in Switzerland at
constant SFR prices increase. This means that an unexpected appreciation of the SFR corresponds to an
unexpected improvement in Export Inc.’s investment opportunities. Hence, Export Inc. might want to be
in a situation where it has more resources when the SFR appreciates. If, however, Export Inc. incurs costs
with low cash flow, it wants to protect itself against downside risk and get some benefit from an unexpected
SFR appreciation. The forward hedge strategy protects Export Inc. against downside risk but does not
provide it with more resources to invest if the SFR appreciates. In contrast, the put strategy protects Export
Inc. against downside risk and provides it with more resources when its investment opportunities improve
unexpectedly. Section 8.7. of Chapter 8 on hedging at Merck showed that the considerations discussed
here play a role in Merck’s decision to hedge with options. Box 10.1. Hedging at Hasbro shows how
accounting rules affect the choice of hedging instrument for some firms.
We just saw how Export Inc. could design an option strategy so that it has more resources
available when it has good investment opportunities. If the market learns that a firm discovered good
investment opportunities, the stock price of that firm increases. Consequently, a firm that buys calls on itself
has more resources when it has good investment opportunities. Cephalon Inc. implemented such a strategy.
The Box 10.2. Cephalon Inc. describes how it went about it. Firms often take option positions on their own
stock. For instance, the Wall Street Journal reported on May 22, 1997 that Intel, Microsoft, Boeing,
Chapter 10, page 13
and IBM had all sold puts.3 The article explained that more than 100 firms had sold puts on their stock. In
the case of IBM, it sold puts to hedge a convertible debt issue. At that time, Intel had puts on $1B worth
of its stock and had received $423M in proceeds from selling puts. Microsoft had puts on $2B worth of
its stock and had made $300M after tax on puts. Selling puts for a firm makes sense if it has valuable
information about itself that lead it to believe that the puts are overpriced. This could be the case if the firm
believes that the probability of a drop in the stock price is smaller than the market thinks it is, perhaps
because the stock price is too low or because the market’s estimate of the volatility is too high. The concern
with having a firm sell puts on itself is that it amounts to increasing the risk of the firm. If the firm experiences
an adverse shock, its resources are further depleted by having to pay the holders of the puts.
Section 10.1.3. Options to hedge nonlinear exposures.
Let’s look at one more example. Consider a situation where a firm, Nonlinear Exports Inc., exports
goods to Switzerland. In three months, it will have completed production of one million widgets that it can
sell in the U.S. for $1 a piece or in Switzerland for SFR2 a piece. Ignoring transportation costs, it will sell
in Switzerland if the SFR exchange rate is more than $0.5. Hence, its SFR exposure depends on the
exchange rate. If the SFR is less than $0.5, Nonlinear Exports Inc. receives no SFRs; if the SFR is more
than $0.5, it receives SFR2M in 90 days. Its payoff in 90 days if it does not hedge is:
$1M + SFR2M*Max(S(t+0.25) - 0.5,0)
3
See “More firms use options to gamble on their own stock,” E.S. Browning and Aaron
Lucchetti, Wall Street Journal, May 22, 1997, C1.
Chapter 10, page 14
To understand this payoff, note that if the spot exchange rate at t+0.25 is less than $0.5, it does not export
and it gets $1M. If the spot exchange rate is greater than 0.5, it exports and gets 2M*S(t+0.25). With the
above formula for the payoff, it gets $1M plus 2M*(S(t+0.25) - 0.5) in that case, which is 2M*S(t+0.25).
Figure 10.4. shows the payoff if the firm does not hedge.
Let’s see how Nonlinear Export Inc. can hedge this payoff. A static forward hedge will not work
very well. This is because the exposure is nonlinear in the exchange rate. At maturity, the exposure is a
SFR2M exposure if the spot exchange rate exceeds $0.5 and no SFR exposure otherwise. Suppose the
firm sells SFR2M forward. In this case, its hedged payoff is:
Cash market payoff + Forward payoff
$1M + SFR2M*Max(S(t+0.25) - 0.5,0) + SFR2M(F - S(t+0.25))
We show this payoff on Figure 10.4. This hedge works very well if the spot exchange rate exceeds $0.5.
In this case, the firm exports to Switzerland and has an exposure of SFR2M which is completely hedged
through a forward position. Unfortunately, if the spot exchange rate is below $0.5, this hedge gives the firm
an exposure to the SFR when it has none from its sales. Look at the case where the spot rate at maturity
happens to be $0.4 and the forward rate is $0.6. In this case, the firm does not export to Switzerland. Yet,
it has an exposure to the SFR that arises from its forward position. This is because Nonlinear Export Inc.
is obligated to sell SFR2M at maturity, but it receives no SFRs from its clients in Switzerland because it
did not sell to them. As a result, if the SFR depreciates, Nonlinear Export Inc. makes a gain on its forward
Chapter 10, page 15
contract even though it does not export. To fulfill its forward obligation, it buys SFRs at $0.4 and sells them
at $0.6, making a net gain of $400,000. Through its forward hedge, Nonlinear Export Inc. eliminated the
exchange rate risk when the exchange rate is above $0.5 per SFR but at the expense of adding exchange
rate risk when it is below $0.5. Whenever the exchange rate is less than $0.5, its income depends on the
exchange rate even though it does not export to Switzerland, so that the firm still incurs risk - its CaR is not
zero despite the fact it hedges with forward contracts! Having the forward position does not result in
unexpected losses on the forward position at maturity when the firm does not export, but it creates
uncertainty about the firm’s cash flow which is bad if that uncertainty is costly.
Let’s see how Nonlinear Export Inc. could construct an exact hedge. Remember that a call option
gives the right to buy SFRs if the spot price exceeds the exercise price. By writing a call instead of buying
one, we take on the obligation to sell when the call option buyer exercises. This means that we sell when
the spot price exceeds the exercise price. Consider a SFR call option with an exercise price of $0.5. If the
firm writes this option, it sells SFRs for the exercise price if the spot exchange rate at maturity exceeds
$0.5. The option expires unexercised if the spot exchange rate is below $0.5. The payoff of the option at
maturity for the option holder is Max(S(t+0.25) - 0.5,0) and it is -Max(S(t+0.25) - 0.5,0) for the seller.
Suppose the firm sells 2M call options on the SFR with exercise price of $0.5 per SFR. It receives the
option premium now, c(S,0.5,t+0.25,t) per SFR, and can invest it until maturity. Its payoff at maturity is:
Payoff of unhedged position + payoff of option position
$1M + 2M*Max(S(t+0.25) - 0.5,0) + 2M*c(S,0.5,t+0.25,t)/P(t+0.25) - 2M*Max(S(t+0.25) - 0.5,0)
Chapter 10, page 16
= $1M + 2M*c(S,0.5,t+0.25,t)/P(t+0.25)
In this case, the option hedge creates a perfect hedge. Using options, the firm obtains a hedged position
which has lower volatility than the hedged position that minimizes volatility with a static forward hedge. The
payoff from the position hedged by writing call options is shown on Figure 10.5.
Note that in this example, the corporation hedges by writing an option. Some corporations have
policies that do not permit the writing of options by the corporation because of the fear that writing options
could expose the firm to large losses. If the corporation in this example had such a policy, it would not have
been able to implement a hedge that completely eliminates its exposure!
Section 10.1.4. How we can hedge (almost) anything with options.
For starters, we can construct a static hedge of a forward contract with options but we cannot
construct a static hedge of an option with a forward contract. Let’s see why. Suppose we have a forward
contract to buy 100,000 shares of GM stock at date T for F and we want to hedge it with options that
mature at T. With the forward contract, we have one of two outcomes. First, the stock price at T, S(T),
exceeds the forward price so that S(T) - F is positive. This payoff increases linearly with the price of GM
stock as long as the price exceeds the forward price. Hence, this payoff is Max(S(T) - F,0). This is the
payoff of a call option with exercise price equal to F. Second, the stock price is lower than the forward
price. In this case, S(T) - F is negative. This is equivalent to the payment we would have to make had we
written a put on GM stock with exercise price F, -Max(F - S(T),0). Consequently, buying a call with
Chapter 10, page 17
exercise price F and selling a put with exercise price F has the same payoff as a forward contract.
Remember though that a forward contract has no value when entered into. Consequently, the portfolio long
one call and short one put with exercise price equal to the forward price has no value if bought at the
initiation date of the forward contract. Let c(S,F,T,t) be the call price and p(S,F,T,t) be the put price,
where S is the underlying of the options (for instance, the price of the common stock in an option on a
common stock), F is the forward price of the underlying, T is the maturity date of the options, and t is the
current date. With this notation, the forward equivalence result is:
Forward equivalence result
A forward contract is equivalent to a long position in a call and a short position in a put. The put and the
call have the same maturity as the forward contract and a strike price equal to the forward price. This result
implies that at inception of the forward contract:
c(S,F,T,t) = p(S,F,T,t)
(10.1.)
Note that equation (10.1.) does not hold for a forward contract after inception because then the
forward contract has value. It is still the case, however, that a long forward position is equivalent to a long
call position and a short put position. Consequently, the value of a long forward position is given by
c(S,F,T,t) - p(S,F,T,t). Since one can hedge a long forward position with a short forward position, it
follows that going short the call and long the put hedges a long forward position. We already saw with our
Chapter 10, page 18
discussion of the put as an insurance contract that we cannot hedge a nonlinear payoff like the payoff of
a put option with a static forward position.
Consider now an arbitrary payoff function like the one represented in Figure 10.5. This is a payoff
to be received at date T and it is a function of the price of an underlying asset. Let G(S(T)) be the payoff
at time T when the underlying asset has price S(T). This payoff function is chosen arbitrarily to make a
point, namely that even a payoff function as unusual as this one can be hedged with options. This function
is not linear, but piecewise linear. It has straight lines, but they are connected at angles. It is not possible
to use a static hedge composed of futures or forward positions to hedge this payoff. For instance, a short
forward position could be used to hedge the part of the payoff that increases with the price of the underlying
asset. However, from Sa to Sb, the payoff falls with the price of the underlying asset, so that over that price
range, a long position is needed to hedge the payoff. Hence, a short forward position aggravates the
exposure of the hedged payoff to the price of the underlying asset over the price range from Sa to Sb. Over
that price range, an increase in the price of the underlying asset decreases the payoff and decreases the
value of the short forward position.
Using a portfolio of options, it is possible to replicate the payoff corresponding to the straight line
segment from a to b on Figure 10.5. exactly. To see this, note that the straight line segment represents a
payoff that falls as the price of the underlying asset increases. To hedge this payoff, we need a financial
position that has a positive payoff when the price exceeds Sa. The assumed slope of the straight line in the
figure is such that every dollar increase in the price of the underlying asset results in a decrease in the payoff
of two dollars. A long position of two units of the underlying asset pays off two dollars for each dollar
Chapter 10, page 19
increase in the price of the underlying asset. However, this long position pays off irrespective of the price
of the underlying asset. In contrast, a portfolio of two call options on the underlying asset with exercise price
Sa pays off two dollars per dollar increase in the price of the underlying asset when the price equals or
exceeds Sa since its payoff is:
2*Max(S - Sa, 0)
The problem with the call is that it pays off even if the price exceeds Sb. Suppose, however, that we now
sell a call on two units of the underlying asset with exercise price Sb. In this case, the payoff of the long call
position and the short call position is:
2*Max(S - Sa, 0) - 2*Max(S - Sb, 0)
This portfolio of calls pays two dollars per dollar increase in the price of the underlying asset as long as the
price of the underlying asset is higher than Sa and lower than Sb. We already saw that the calls with exercise
price Sa pay two dollars per dollar increase in the price of the underlying asset at or above the exercise
price. Suppose now that the price of the underlying asset is Sb plus one dollar. In this case, the calls with
exercise price Sa pay off 2*(Sb + 1 - Sa) and we have to pay 2*(Sb + 1 - Sb) to the holders of the calls we
sold. The net payoff of the call portfolio when the price of the underlying asset is Sb + 1 is therefore 2*(Sb
+ 1 - Sa) - 2*(Sb + 1 - Sb) = 2*(Sb - Sa).Therefore, for each dollar change of the price of the underlying
Chapter 10, page 20
asset between Sb and Sa the call portfolio has a gain that exactly offsets the loss in the cash position. For
price changes outside the range from Sa to Sb, the call portfolio pays off nothing. We therefore created a
perfect hedge for the payoff function between the points a and b on Figure 10.5.
Using the method that allowed us to create a perfect hedge of the payoff function between the
points a and b, we can create a perfect hedge for any straight line segment of the payoff function. To do
so, for each segment, we need to find out the slope of the payoff function. The slope of the payoff function
dictates the size of the option position. From a to b, we saw that the slope is minus two. Let’s look now
from b to c. There, the slope is 0.5. This means that the payoff function increases as the price of the
underlying asset increases. To hedge, we need to take a position that falls in value by 50 cents whenever
the price of the underlying asset increases by one dollar. Selling half an option on the underlying asset with
exercise price Sb provides this hedge. However, the written call has a negative payoff for any price greater
than Sb. Since for a price of the underlying asset equal to or greater than Sc the slope of the payoff function
of the cash position is different, we buy half a call option with exercise price Sc. This way we have created
a perfect hedge for payoffs between b and c. Continuing this way, we can hedge the whole payoff function.
What if the payoff function does not have straight lines? In this case, using calls and puts, we can
construct a static approximate hedge by using a linear approximation of the payoff function. Figure 10.6.
provides an example of a payoff function that does not have straight lines. We can compute the exposure
at each point using the delta exposure as discussed in Chapter 8. Using that approach, we can create a
piecewise linear function that is arbitrarily close to the payoff function. Figure 10.6. shows one such possible
approximation. One could construct a more precise approximation. Once we have a piecewise linear
Chapter 10, page 21
approximation, we can then proceed as we did with the piecewise linear function of Figure 10.5. A more
precise approximation involves more straight lines, but these straight lines are shorter. Since two options
are needed to replicate the payoff corresponding to a straight line segment, it follows that a more precise
approximation requires a portfolio composed of a larger number of options. An alternative approach that
we will discuss later is to use exotic options.
Section 10.2. Some properties of options.
We already saw in the previous section that options cannot have a negative price. Options are
limited liability instruments for those who hold them: the holder never has to exercise them if exercising them
would hurt her. Options are not limited liability instruments for those who write them. The liability of a call
writer is potentially infinite. Remember that if we write a call, we have to sell the underlying asset at a fixed
price. The higher the value of the underlying asset (for example, the common stock in a stock option), the
more we lose if the option is in the money at maturity.
In the remainder of this section, we discuss properties of options that hold irrespective of the
distribution of the underlying asset. These properties apply to options as long as the underlying asset is a
financial asset. By a financial asset, we mean a security such that all the payoffs received by the holder of
the asset are cash payoffs or payoffs that can be converted into cash by selling the security. In other words,
there is no convenience yield. Stocks and bonds are financial assets. Currency is not a financial asset: It
earns a convenience yield, which is the benefit from carrying cash in one’s wallet - it saves trips to the ATM
machine. A foreign currency discount bond that pays one unit of foreign currency at maturity is a financial
Chapter 10, page 22
asset. Further, until later in this section, we assume that the underlying asset has no cash payouts until
maturity of the option. This means that all its return accrues in the form of capital gains. Finally, we assume
that financial markets are perfect.
Section 10.2.1. Upper and lower bounds on option prices.
The upper bound on a call option price is the price of the underlying asset. Remember that the
option allows the holder to buy the underlying asset at a fixed price. Consequently, the most the holder can
ever get out of an option contract is the underlying asset. A put contract allows the holder to sell the
underlying asset at a fixed price. This means that the upper bound on the price of a put contract is the
exercise price.
A call option cannot have a negative price. Can we find a higher minimum price than zero? If the
call option is an American option, we know that its value cannot be less than S - K, where S is the price
of the underlying asset and K is the exercise price. If the price of the option is smaller than S - K, we can
buy the option, exercise it, and make a profit equal to S - K minus the call price. This result does not apply
to European options since they cannot be exercised early.
Consider a European call with three months to maturity on a stock that does not pay a dividend.
The price of the stock is $50, the exercise price K is $40, the price of a discount bond maturing in three
months is $0.95. Suppose the option premium is $10. In this case, we can buy the stock for delivery in
three months by paying $10 today and setting aside 0.95*$40 in discount bonds, or $48 in total. This is
because in three months we can exercise the option and get the asset by paying $40, which we will have
Chapter 10, page 23
from our investment of $38 in discount bonds. If we don’t exercise in three months because the option is
out-of-the-money, we can buy the asset for less than $40 and have money left over, so that our effective
cost of buying the asset is at most $48 today. If we want the stock in three months, we are therefore better
off buying it using a call option and setting aside the present value of the exercise price than buying it today.
In fact, we can make money through an arbitrage transaction. If we sell the stock short, buy the call, and
set aside the present value of the exercise price in discount bonds, we make $2 today. At maturity, if the
stock price is greater than $40, we get the stock with the call, use the proceeds from the discount bonds
to pay the exercise price, and use the stock to settle the short-sale. If the stock price is less than $40, the
call is worthless, and we use the proceeds from the discount bonds to buy the stock. In this case, we have
some money left over. With this arbitrage portfolio, we make money for sure today and possibly also at
maturity. The only way the arbitrage opportunity disappears is if the stock price does not exceed the call
price plus the present value of the stock price. In that case, it is not cheaper to buy the stock for deferred
delivery through the call.
Let’s consider the case where the stock price is less than the call price plus the present value of
the exercise price. Suppose that the call price is $15, so that a portfolio of the call and of discount bonds
maturing in three months with face value equal to the exercise price is more expensive than the stock. It
would seem that this is an arbitrage opportunity, but it is not. Buying the stock and selling the portfolio of
the call and of the discount bonds does not make money for sure. If the call is in the money, we get nothing
at maturity and make money today. If the call is out of the money, we have to repay K but the stock is
worth less than K so that we lose money at maturity. Potentially, we could lose K if the stock price is zero.
Chapter 10, page 24
Hence, we could make money today but lose much more at maturity. Let S be the price of the stock at date
t and date T be the maturity of the option. We have shown that there is an arbitrage opportunity if the call
price plus the present value of the exercise price is smaller than the stock price, which is if c(S,K,T,t) +
P(T)K < S, and that there is no arbitrage opportunity when c(S,K,T,t) + P(T)K $ S. This analysis
therefore implies that the call price must exceed the difference between the stock price and present value
of the exercise price:
c(S,K,T,t) $ S(t) - P(T)K
(10.2.)
In words, we have:
Property 1. The European call price cannot be less than the current price of the underlying
asset minus the present value of the exercise price.
What about put options? If a put option is an American put option, we can always exercise
immediately, so that it’s value must be at least K - S(T). If the put option is a European put option,
note that we can get K at maturity by investing the present value of K or at least K by buying the asset and
the put. With the approach of buying the asset and the put, we get more than K if the put is out-of-themoney because we have the asset and its price exceeds K. Consequently, a portfolio of the put and the
asset must be worth more than a portfolio whose value is equal to the present value of the exercise price.
Chapter 10, page 25
This implies that the following relation holds:
p(S,K,T,t) $ P(T)K - S(t)
(10.3.)
In words, we have:
Property 2. The European put price cannot be less than the present value of the exercise
price minus the current price of the underlying asset.
This property must hold to avoid the existence of arbitrage opportunities. If it does not hold, we make
money today buying the put, buying the asset, and borrowing the present value of K. At maturity, if the put
is out of the money, we have S(T) - K, which is positive. If the put is in the money, the value of the position
is zero. Hence, we make money today and never lose money later.
Section 10.2.2. Exercise price and option values.
Consider two calls on the same underlying asset that have the same maturity. One call has exercise
price of $40 and one has exercise price of $50. If the asset price is greater than $50 at maturity, the call
with the lower exercise price pays $10 more than the call with the higher exercise price. If the asset price
is between $40 and $50, the call with the lower exercise price pays something and the other nothing.
Finally, if the asset price is below $40, neither call is in-the-money. Consequently, the call with the lower
Chapter 10, page 26
exercise price never pays less than the call with the higher exercise price, so that its value cannot be less.
More generally, if we have a call with exercise price K and one with exercise price K’, where K’ > K, it
must be that:
c(S,K,T,t) $ c(S,K’,T,t)
(10.4.)
This result is our third property of option prices:
Property 3. Ceteris paribus, the price of a European call option is a decreasing function
of its exercise price.
If both options are in the money, the difference between the two option payoffs is K’ - K. This is
because one option pays S(T) - K and the other pays S(T) - K’, so that S(T) - K - (S(T) - K’) is equal
to K’ - K. If K’ > S(T) > K, one option pays nothing and the other pays S(T) - K < K’ - K. If S(T) < K,
each option pays zero. Therefore, the maximum difference between the payoff of the two options is K’ K. Therefore, by investing the prevent value of K’ - K, one always gets at least the payoff of being long
the option with exercise price K and short the option with exercise price K’. Consequently, it must be the
case that:
c(S,K,T,t) - c(S,K’,T,t) # P(T)[K’ - K]
Chapter 10, page 27
(10.5.)
With puts, everything is reversed since we receive the exercise price. Consequently, it must be that:
p(S,K’,T,t) $ p(S,K,T,t)
(10.6.)
p(S,K’,T,t) - p(S,K,T,t) # P(T)[K’ - K]
(10.7.)
In words, the first inequality tells us that:
Property 4. Ceteris paribus, the price of a European put option is an increasing function
of the exercise price.
Finally, we showed for both puts and calls that:
Property 5. The absolute value of the difference in price of two otherwise identical
European put options or call options cannot exceed the absolute value of the difference of the of
the present value of the exercise prices.
Let’s use the various properties we have presented on a numerical example. Let’s use our example
of two call options with exercise prices of $40 and $50. Let’s say that the price of the call with exercise
price of $40 satisfies Property 1. A satisfactory price would then be $20. Property 1 must also hold for
the call with exercise price of $50. Consequently, $5 would be an acceptable price for that call, but $1
Chapter 10, page 28
would not be. If the call with exercise price of $50 sells for $5 and the call with exercise price of $40 sells
for $20, the present value of the difference in the exercise prices of $9.5 is exceeded by the absolute value
of the difference in option prices, which is $15. These option prices violate Property 3. To take advantage
of this violation, we buy the call with exercise price of $50, sell the call with exercise price of $40, and
invest the present value of the difference in exercise prices. The cash flow associated with establishing this
portfolio is -$5 + $20 - 0.95*$10, or $5.50. At maturity, if the stock price is below $40, both options
expire unexercised and we have $10, which is the value of our investment in discount bonds. If the stock
price expires between $40 and $50, say at $45, we have to pay $5 on the option we wrote, get nothing
from the option we bought, and still have the $10, so on net we have $5. Finally, if the stock price is above
$50, say at $55, we have to pay $15 on the option we wrote, get $5 on the option we bought, and still
have the $10, so that on net the portfolio is worthless. Consequently, the worst that can happen with our
portfolio strategy is that we get $5.50 today and have a portfolio that has no value at maturity. This means
that we make money for sure and that failure of Property 3 to hold represents an arbitrage opportunity.
Section 10.2.3. The value of options and time to maturity.
Consider two American calls that are equivalent except one has a longer time to maturity. The call
with a longer time to maturity cannot be worth less than the one with the shorter time to maturity. If we hold
the call with the longer maturity, we can exercise it at the maturity of the call with the shorter time to
maturity. If we do that, the two calls are equivalent. If we choose not to do that, it is because we benefit
from not exercising and in that case our call has an advantage that the call with the shorter maturity does
Chapter 10, page 29
not have. The same reasoning applies to American puts. This reasoning does not apply to European calls
and puts because we cannot exercise them before maturity. In summary:
Property 6. The value of American options increases with time to maturity.
If the underlying asset does not pay a dividend, this result applies also to European call options, but not to
European put options. With a European put option, suppose that the stock price is equal to zero. In this
case, if we could exercise, we would get the exercise price. We cannot exercise now. Anything that
happens subsequently can only make us worse off. The most we can get at maturity is the exercise price,
so that the longer we have to wait to exercise, the smaller the present value of what we will get and hence
the lower the value of the option.
Section 10.2.4. Put-call parity theorem.
If there is a market for European calls, we can manufacture European puts on our own. Suppose
that we want to buy a put with exercise price of $40 that matures in three months, but no such puts can be
bought. However, we find that we can buy a call with exercise price of $40 that matures in three months.
Consider the strategy of buying the call, investing the present value of the exercise price in discount bonds,
and selling the asset short. If the call price is $15, this strategy costs us $15 plus $38 minus $50, or $3.
Let’s see what we have at maturity. First, if the stock price is below $40, the call is worthless. Hence, we
have $40 from the sale of the discount bonds and have to buy the asset to settle the short-sale. On net, the
Chapter 10, page 30
value of our portfolio is $40 - S(T). This is the payoff of a put with exercise price of $40 when it is in-themoney. Second, if the stock price is above $40, the call pays S(T) - $40. We use the discount bonds to
pay the exercise price, get the stock and deliver it to settle the short-sale. On net, the value of our portfolio
is zero. This is the payoff of a put with exercise price of $40 when it is out-of-the-money. It follows that
we manufactured the payoff of the put at a cost of $3 by buying the call, investing in discount bonds the
present value of the exercise price, and selling the stock short. One would expect $3 to be the price of the
put, since the production cost of the put to us is $3. This result is called the put-call parity theorem. It states
that the price of a European put is equal to the value of a portfolio long the call, long an investment in
discount bonds for the present value of the exercise price of the call and same maturity as the call, and short
the underlying asset:
Put-call parity theorem
p(S,K,T,t) = c(S,K,T,t) + P(T)K - S(t)
(10.8.)
The put-call parity theorem holds because otherwise we can make money for sure by creating an arbitrage
position. The theorem does not hold for American puts and calls, however. We will see why below.
Let’s consider a numerical example of how we can make money out of situations where the put-call
parity theorem does not hold. We consider a situation where the put and call options mature in 90 days.
The underlying asset is worth $50, the put is worth $10, the exercise price is $55, and a discount bond that
matures in 90 days is worth $0.97. This means that S(t) = 50, K = 55, p(50,55,t+0.25,t) = 10, P(t+0.25)
Chapter 10, page 31
= 0.97. Using the put-call parity theorem, the call price must be:
c(50,55,t+0.25,t) = 10 + 50 - 0.97*55 = 60 - 53.35 = 6.65
Suppose now that the call sells for $7 instead. We therefore want to sell the call because it is too expensive
relative to the put. Since we do not want to take the risk of a naked short call position, we hedge the short
call by buying a portfolio that does not include the call but has the same payoff. The put-call parity theorem
tells us that this portfolio is long the put, long the underlying asset, and short discount bonds that mature at
T and have face value $55. Buying this hedge portfolio creates a cash outflow of $6.65 ($10 plus $50
minus the present value of the exercise price). This cash outflow is financed by selling the call for $7, so that
we have $0.35 left. These thirty-five cents are our arbitrage profit, because no cash is needed at maturity.
This is shown on the arbitrage table below.
Suppose that instead the call is worth $5. In this case, the call is too cheap. We buy it and sell the
portfolio that has the payoff of the call. This means that we short the asset, write the put, and invest in
discount bonds with a face value of $55 and maturity in three months. This means that we have a cash flow
of -$5 - $53.35 + $50 + $10, or $1.65. At maturity, we get Max[S(T) - K,0] - S(T) + K - Max[K S(T),0]. If the call is in the money, the put is not. In this case, we get S(T) - K - S(T) + K, or zero. If the
put is in the money, we get -S(T) + K - (K -S(T)), or zero. Note that with this strategy we never make or
lose money at maturity. We therefore gain $1.65 today and pay nothing at maturity. Hence, there is an
arbitrage opportunity whenever we make money today. For us not to make money today, the call price
Chapter 10, page 32
has to be at least $6.65. Yet, if the call price exceeds $6.65, we are
Arbitrage table to exploit departures from put-call parity
In the example, c(50,55,t+0.25,t) = 7, p(50,55,t+0.25,t) = 10, S(t) = 50, K = 55, and P(t+0.25) =
0.97. Put-call parity does not hold since: 7 > 10 + 50 - 53.35 = 6.65.
Position at t
Cash outflow at t
Cash outflow at
t+0.25 if S(t+0.25) <
55
Cash outflow at
t+0.25 if S(t+0.25) >
55
Long put
-10
55 - S(t+0.25)
0
Long underlying asset
-50
S(t+0.25)
S(t+0.25)
Borrow K
53.35
-55
-55
Write call
7
0
-(S(t+0.25) - 55)
Total
0.35
0
0
in the situation where the call price is too high and make money by writing the call as already discussed.
Consequently, there is an arbitrage opportunity whenever put-call parity does not hold.
Put-call parity does not hold exactly in the presence of transaction costs. We saw with the example
of Ms.Chen in Section 10.1.1. that the bid-ask spread for options can be substantial. This raises the
question of whether we can still use put-call parity in markets with significant transaction costs. With
transaction costs, there is an arbitrage opportunity if either one of the strategies we talked about makes
money today net of transaction costs. If this happens, we make money today implementing the strategy and
owe nothing at maturity.4
4
This assumes that there are no transaction costs at maturity.
Chapter 10, page 33
Section 10.2.5. Option values and cash payouts.
Now, we allow the underlying asset to have cash payouts before the maturity of the option. For
concreteness, we assume that the underlying asset is a stock, so that the cash payouts are dividend
payments. The discussion is not restricted to options on stocks. It would apply equally well to an option
on a coupon bond that makes coupon payments before maturity of the option.
If we hold a European call option on a stock, we get to buy the stock at maturity for the exercise
price. The dividends that accrue to the stockholder between the time we buy the call and the time it matures
do not belong to us unless the option contract includes some provision increasing the stock price at maturity
by the amount of dividends paid. These dividends therefore reduce the value of our option. If the firm were
to pay a liquidating dividend the day after we buy a European call, the call would become worthless unless
there is some protection against dividend payouts. Consequently, in the absence of such protection, the
results discussed so far in this section hold for European options for the stock price minus the present value
of dividends to be paid between now and maturity of the option. This is similar to our discussion of forward
contracts on common stock where we subtracted the present value of the dividend payments from the
current stock price.
In the presence of dividends, it is easy to see why it could be the case that a European call option
that matures at t” > t’ is worth less than the option that matures at t’. If the firm pays a liquidating dividend
between t’ and t”, the call option that matures at t’ has value, but the option that matures at t” does not.
With an American call option, the call option that matures at t” is worth at least as much as the option that
matures at t’ because one can always exercise the option maturing at t” at t’. With an American call option,
Chapter 10, page 34
if we knew that tomorrow the stock will pay a liquidating dividend, we would exercise the option today.
This would give us the stock and thereby the right to collect the liquidating dividend. The American call
option therefore has a benefit that the European call option does not have. Consequently, the American call
option has to be worth at least as much as the European call option. The early exercise provision of
American options is a right, not an obligation. It therefore cannot hurt us.
Suppose we have a call option on a stock that never pays a dividend. Would we ever exercise an
American call option on that stock early? The answer is no for a simple reason. Exercising early has a cost
and a benefit. The benefit is that we get S(t) now. The cost is that we must pay the exercise price K now.
If the stock does not pay a dividend, we are not better off having the stock now than having it at maturity
of the option. What we get now by exercising is simply the present value of the stock at maturity. However,
we are worse off paying the exercise price now than at maturity. By paying now, we lose the opportunity
to invest our money until maturity. If we exercise at date t, we therefore lose the gain from investing the
exercise price from t to T, K/P(T) - K. We also lose the limited liability feature of the option. Note that by
exercising, we receive S(t) - K. If the stock price falls, we could end up losing money on the exercised
position that we would not have lost had we kept the option. This means that exercising early an American
call on a stock that does not pay a dividend is unambiguously a mistake in the absence of counterparty
risks. If exercising early an American call on a stock that does not pay a dividend is never worth it, then
the price of an American call on such a stock is equal to the price of a European call on such a stock. As
a result, a European call on a stock that does not pay a dividend increases in value with time to maturity.
Note that our reasoning does not hold if the stock pays a dividend. If the stock pays a dividend and
Chapter 10, page 35
we exercise early, we get the dividend. So, suppose that the stock pays a dividend just after we exercise
at t and that it does not pay a dividend again until maturity. Say that the dividend is D(t). If we exercise just
before the dividend payment, we get D(t) that we would not otherwise have but lose K/P(T) - K and the
limited liability feature of the option. Whether we exercise or not depends on the magnitude of D(t) relative
to the costs of exercising the option. If the dividend payment is large enough, we exercise.
Would we ever exercise at some time other than just before a dividend payment? No. By
postponing exercise until just before the dividend payment, we get the benefit of investing the exercise price
and there is no cost to us from not exercising since no payout takes place. Consequently, American call
options on stocks are exercised just before a dividend payment or not at all. 5
What about American put options? With a put option, a dividend payment increases the payment
we receive upon exercise since it decreases the stock price. There is therefore no reason to exercise a put
option before a dividend payment. When we exercise an American put option, we receive the exercise
price which we then get to invest. However, we lose the limited liability feature of the option position. This
limited liability feature protects us relative to a short position in the stock if the stock price increases to
exceed the exercise price. In that case, the short position will keep incurring losses if the stock price
5
A word of warning about currency options. These options are analyzed in detail in Chapter
12. We will see there that American currency options can be exercised early. Currencies do not pay a
dividend in cash, but a convenience yield (see Chapter 5) is equivalent to a non-cash dividend. The
convenience yield on currency is the advantage of not having to go to the ATM machine. By holding
cash, one gives up the interest rate one would earn if the money was in a money market fund. This
interest rate is the convenience yield of cash. By exercising early a currency call option, one gets cash
now rather than getting cash later which is a benefit, but one pays the exercise price now rather than
later, which is a cost. It is possible for the benefit of exercising early to be greater than the cost.
Chapter 10, page 36
increases further, but the put option will not. On the other hand, if the stock price is sufficiently low, this
limited liability feature does not have much value relative to the benefit of receiving the exercise price now
and investing it until maturity. Hence, when the stock price is low, it pays to exercise because there is
nothing to gain from waiting until maturity: the lost interest on investing the exercise price overwhelMs.the
possible gain from further decreases in the stock price.
Section 10.2.6. Cash payouts, American options, and put-call parity.
Let’s go back to put-call parity and see how it is affected by early exercise. Let C(S,K,T,t) and
P(S,K,T,t) be respectively the American call price and the American put price for an option on the
underlying asset with price today of S(t), maturity at T, and exercise price K. Now, suppose that put-call
parity as defined above does not hold. One possibility is that:
C(S,K,T,t) > P(S,K,T,t) + S(t) - P(t,T)K
(10.9.)
Can we make money? Relative to put-call parity, the call price is too high. Were we to believe that put-call
parity should hold, we would write a call, buy a put, buy the stock, and borrow the present value of the
exercise price in discount bonds. Consider our situation at time t* if the call is exercised at that point. Let
P(t*,T) be the price at t* of a discount bond that matures at T. Our portfolio then has value:
-(S(t*) - K) + P(S,K,T,t*) + S(t*) - P(t*,T)K = P(S,K,T,t*) + (1 - P(t*,T))K > 0 (10.10.)
Chapter 10, page 37
This is positive at any time until maturity, when it could be zero. Since we made money opening the
position, we have identified an arbitrage opportunity. Consequently, we have shown that there is an upperbound to the call price:
C(S,K,T,t) # P(S,K,T,t) + S(t) - P(t,T)K
(10.11.)
Is there a lower bound to the call price in terMs.of the put price, the stock price, and the discount
bonds? Think of a situation where the call price is extremely low. We buy the call, sell the put, sell the
stock, and buy discount bonds equal to P(t,T)K. We make money now. This strategy must be such that
we have some risk of losing money later; otherwise, it is an arbitrage opportunity. Let’s look at the value
of our portfolio if it becomes optimal for the put holder to exercise:
C(S,K,T,t*) - (K - S(t*)) - S(t*) + P(t*,T)K = C(S,K,T,t*) - (1 - P(t*,T))K (10.12.)
This could be negative if the value of the call is small. If, at inception of the strategy, we put K in discount
bonds instead of P(t,T)K, the value of the investment in discount bonds when the put is exercised at t* is
K/(P(t,t*). If the put is exercised, the value of the portfolio long the call, short the stock, short the put, and
long an investment of K in discount bonds is:
C(S,K,T,t*) - (K - S(t*)) - S(t*) + K/P(t,t*) = C(S,K,T,t*) - (1 - 1/P(t,t*))K
Chapter 10, page 38
(10.13.)
In contrast to the right-hand side of equation (10.12), the right-hand side of equation (10.13) cannot be
negative - if the call is worth zero, its lowest value would be zero as long as the interest rate is not negative
because P(t,t*) cannot be greater than one. This means that a portfolio long the call, short the put, short
the stock, and long an investment in discount bonds worth K never has a negative value (exercising the call
will never make us worse off). Consequently, the value of that portfolio must be positive - otherwise, we
make money for sure since we get something worth something and get paid for taking it. This implies that:
C(S,K,T,t) - P(S,K,T,t) - S(t) + K $ 0
(10.14.)
Putting both results together, we get:
Bounds on American call prices
P(S,K,T,t) + S(t) - K # C(S,K,T,t) # P(S,K,T,t) + S(t) - P(t,T)K (10.15.)
Let’s look at an example. Suppose that the stock price is $50, the exercise price is $40, the put price is
$5, and the price of a discount bond maturing in three months is $0.95 today. With this, the price of the call
must be less than $5 + $50 - 0.95*$40, or less than $17. It must also be more than $5 + $50 - $40, or
$15. Suppose the price of the call is $13. We can buy the call and hedge it.
To hedge the call, we would like to have a position that offsets the payoff of the call. Since the call
Chapter 10, page 39
makes money when the stock price exceeds the exercise price, we want a position that loses money in that
case. Being short the stock price will result in losses when the stock price increases. However, if the stock
price falls, a short position makes gains when we gain nothing from the call. Consequently, we have to write
a put to eliminate these gains. If we write an American put, however, it can be exercised at any time.
Consequently, if we want our hedge portfolio to have no cash flow requirements during its life, we have to
set aside the exercise price and invest it. This implies that to hedge the call, we sell the stock short, write
a put, and invest the exercise price. The proceeds from this hedge portfolio are $50 +$5 - $40, or $15.
Since we paid $13 for the call, we make two dollars out of buying the call and hedging it. Since we bought
the call, we only exercise at maturity. Consequently, suppose that the stock price goes to $55. In this case,
we have $15 from the call at maturity and our hedge portfolio is worth -$55 - $0 + $40/0.95, or $12.8947 since the put is worthless. On net, we have $2.1053. Alternatively, suppose the stock falls to
$35 by tomorrow and stays there. In this case, the call is worthless and the put has value. The issue that
we have to face is that the holder of the put could exercise tomorrow or at any time until maturity. If the
holder exercises at maturity, the hedge portfolio is then worth -$35 - $5 + $40/0.95, or $2.1053. If the
holder exercises tomorrow, the hedge portfolio is worth tomorrow -$35 -$5 + $40, or zero. The hedge
portfolio will never make a loss when the stock price falls below the exercise price of the put. Further, if
the stock price increases, the loss of the hedge portfolio will never exceed the value of the call. As a result,
we always make money buying the call and hedging it with the hedge portfolio.
Section 10.3. A brief history of option markets.
Chapter 10, page 40
Options have a long history. Not all of this history is illustrious. One can trace their use to the
Greeks and Romans. They were used extensively in the 17th century and in the early 18th century. A famous
financial event of the beginning of the 18th century, the South Sea Company bubble, involved extensive use
of options as an incentive device. The South Sea Company’s history is complicated, but to succeed the
Company had to obtain approval of Parliament. As a result, it gave options to various members of
Parliament. These options would have expired out of the money had Parliament not done what was
necessary for the Company to succeed. Eventually, share prices of the Company increased so much that
their value exceeded the present value of future cash flows six or seven times. In the aftermath of the
collapse of the South Sea bubble, England’s Parliament banned options with Sir John Barnard’s Act and
the ban stayed in place until the middle 19th century.
Options have been traded in the U.S. for a long time over the counter. For instance, newspapers
in the 1870s carried quotes for options offered by brokerage houses. The Securities Act of 1934
empowered the Securities and Exchange Commission to regulate options. Until 1973, options were traded
over the counter and there was little secondary trading. All this changed in the early 1970s. In 1973, the
Chicago Board Options vExchange, or CBOE, became the first registered exchange to trade options.
Trading grew quickly and it stopped being unusual that options on a stock would have higher volume on
a day than the stock itself. Though options had interested academics for a long time, until the beginning of
the 1970s, the formulas that could be used to price them required knowledge of the expected return of the
underlying asset, which made them of limited practical use. This changed with the publication in 1973 of
a paper by Fisher Black and Myron Scholes. In that paper, they provided a formula for the pricing of
Chapter 10, page 41
options that did not require knowledge of the expected return of the underlying asset and proved of great
practical use. In the next two chapters, we will derive this formula and explore its properties extensively.
This formula quickly became the pricing tool of traders on the CBOT. Texas Instruments even started
selling a calculator with the Black-Scholes option pricing formula already programmed. In the 1970s,
economic events also provided the impetus for a dramatic extension of option markets. In 1974, the
industrialized market economies switched from a regime of fixed exchange rates to a regime where
exchange rates were determined by market forces. As a result, exchange rate volatility became a concern
of corporations and investors. As discussed in Chapter 5, futures contracts on currencies became available.
Soon thereafter, option contracts on currencies became available also. These option contracts were traded
over the counter. Eventually, the Philadelphia Options Exchange started trading foreign currency options.
However, it has always been the case that most of the volume for the currency options market is traded
over the counter.
Another important event of the 1970s was the increase in the volatility of interest rates. This
development created a demand for instruments to hedge against interest rate changes. Some of these
instruments were futures contracts. Other instruments took the form of options of various kinds. Options
on bonds became available. Another type of option that became popular were options paying the difference
between a stated interest rate and the market interest rate at fixed dates over a period of time. These
options are called caps. We consider options on bonds and interest rates in Chapter 14.
Since the 1980s, innovation in the options markets has taken two directions. First, options on new
underlyings were introduced. This led to options on futures contracts, on stock indices, on commodities,
Chapter 10, page 42
and so on. Second, new types of options, so-called exotic options were introduced. Exotic options are
options which are not standard puts or calls. They can be options on one or many underlyings. Examples
of exotic options are Bermudan options, which can be exercised at fixed dates, barrier options which can
be exercised only if the underlying crosses a barrier (or does not cross a barrier for other types of barrier
options), rainbow options which give the right to buy one of several underlyings at a given price. Chapter
13 discusses strategies involving index options and Chapter 17 discusses exotic options.
Section 10.4. Conclusion.
In this chapter, we started by showing how options can be useful to an exporter to hedge its
exposure. It is well-known that options can be used to eliminate downside risk. Perhaps more importantly,
options can be used to hedge nonlinear payoffs that cannot be hedged with static positions in forwards or
futures. We saw that options could be used to hedge just about anything. A forward contract is equal to
a portfolio of a long position in a European call and a short position in a European put where both
options have the maturity of the forward contract and an exercise price equal to the forward price. A
piecewise linear payoff function can be hedged exactly with a portfolio of options, but not with a forward
contract. We then examined properties of options that do not depend on the distribution of the price of the
underlying asset. Call option prices fall with the exercise price; put option prices increase with the exercise
price. American options increase in value with time to maturity. A European call on a dividend-paying stock
and a European put can decrease in value with time to maturity. European put and call prices are related
to each other through the put-call parity theorem. American calls are exercised only immediately before
Chapter 10, page 43
a dividend payment, whereas American puts are exercised when the price of the underlying asset is low
enough. The put-call parity theorem does not apply to American options because they can be exercised
early. We concluded with a brief review of option markets.
Chapter 10, page 44
Literature note
The results on option prices that do not require assumptions about the distribution of the underlying are
classic results first published in Merton (1973). The put-call parity theorem is due to Stoll (1969). Cox and
Rubinstein (1985) discuss the use of options to hedge and also have some historical material on options.
The use of convertible debt to mitigate incentives on shareholders to take on risk is discussed in Green
(1984). The role of options in the South Sea bubble is explained in Chancellor (1999). Valerio and Kayris
(1997) discuss the option markets in the 1870s.
Chapter 10, page 45
Key concepts
Static hedge, dynamic hedge, American option, European option, forward equivalence result, put-call
parity, early exercise.
Chapter 10, page 46
Review questions
1. Where could you buy an option on a stock/
2. How does the cash flow of Export Inc. differ if it hedges by buying a put on SFR1M and if it hedges
by selling SFR1M forward?
3. Why is it possible to construct a static hedge of nonlinear payoff functions with options when it is not
possible to do so with forwards or futures contracts?
4. How does one replicate a forward contract with options?
5. Why could it make sense for a firm to buy calls on its stock?
6. What is the minimum price greater than zero that a European call price must exceed?
7. How is the value of an option related to its exercise price? Does it depend on whether the option is a
put or a call?
8. Why can time to maturity have a different impact on European and American options?
9. Why would you want to exercise an American call early?
10. Does put-call parity hold for American options?
Chapter 10, page 47
Problems.
1. You are the Treasurer of a large multinational corporation. The company’s competition is mostly
Japanese, but you do not produce in Japan and have no assets or liabilities denominated in yen. The Board
looks at your derivatives portfolio and finds that you have puts on the yen, so that you get to sell yen for
a fixed dollar price. You are accused of speculating. Is there a plausible reason why you would have these
puts if you are only hedging?
2. Consider a car manufacturer. In six months, the firm has to start producing 100,000 cars. It can produce
them in Europe or in the US. After the cars are produced, they are sold in the US at a fixed price. The cost
of producing a car in Europe is fixed in Euros and is equal to Euro30,000. This cost is to be paid when
production starts. The cost of producing a car in the US is fixed at $20,000. The current price of a Euro
is $0.6. The forward price of a Euro for delivery in six months is also $0.6. The firm can decide where it
will produce at any time between today and six months from now. You work for this company and the
CEO tells you that he wants to decide today because this way the firm can hedge its foreign exchange rate
risk. Looking at the forward rate, he finds out that producing in Europe is cheaper. Assuming that
production takes six months and cars are sold immediately, what is the net present value of producing
100,000 cars for the company after having hedged exchange rate risk assuming that the continuouslycompounded dollar risk-free interest rate is 10%?
Chapter 10, page 48
3. Suppose that you did what the CEO wanted you to do and six months from now the Euro is selling for
$1. The CEO comes to you and says that it no longer makes sense to produce in Europe. Since you
hedged, is the company better off to produce in Europe or in the US?
4. Can you show the CEO how the firm can eliminate foreign exchange rate risk but still choose to produce
in six months in the country where the cost per car evaluated in dollars is the lowest?
5. Suppose that you recommend to the CEO to decide in six months. He tells you that he does not
understand your recommendation because he does not see the value of waiting. He argues that all that can
happen by waiting is that the firm could lose the benefit of producing in Europe at current exchange rates.
Can you provide him with an estimate of the value of waiting by comparing the firm’s profits if it waits and
if it does not wait?
6. Many firms include options to their managers as part of their compensation. It is often argued that a sideeffect of this practice is that if management wants to pay out funds to shareholders, it prefers to do so in
the form of a share repurchase than an increase in dividends. What is the rationale for this argument?
7. Your broker recommends that you buy European calls on a non-dividend paying stock maturing in one
year. The calls are worth $50, the stock is worth $100, and the exercise price is $49. He tells you that
these calls are a great buy. His reasoning is that if they were American calls, you could exercise them, and
Chapter 10, page 49
immediately gain $51. He admits that these calls are European calls, but his argument is that since the stock
pays no dividends, there is no difference. Should you buy the calls?
8. Consider a European put and a European call on a stock worth $50. The put and the call mature in one
year, have the same exercise price of $50, and have the same price. Can you make money from a situation
like this one if interest rates are 0% p.a.? What would you do to make money if interest rates are 10% p.a.?
9. A European put is worth $5 and matures in one year. A European call on the same stock is worth $15
and matures in one year also. The put and the call have the same exercise price of $40. The stock price
is $50. The price of a discount bond that matures in one year is worth $0.90. How do you make risk-free
profits given these prices?
10. You hold an American call on a non-dividend paying stock. The call has a two-year maturity. The
exercise price is $50 and the stock is now worth $70. Your broker calls you and tells you that you should
exercise. He argues that the benefit of the call is the downward protection, but that you don’t need this
protection anymore because the stock price is so high relative to the exercise price. Should you exercise?
How do you justify your decision?
11. You wrote an American put with exercise price of $50. The stock price is now $1 and the option has
one year to go. Do you gain if the put holder does not exercise today?
Chapter 10, page 50
Figure 10.1. Wealth of Ms.Chen on January 19, 2001 as a function of Amazon.com stock price.
To insure her portfolio, Ms.Chen buys puts with exercise price of $55. Buying the puts has two effects on
her wealth on January 19, 2001. First, it reduces her wealth for each stock price since she has to write
a check of $138,750 on May 5. Second, it increases her wealth for every stock price below $55 since the
puts pay off for those stock prices.
Wealth on January 19
Uninsured portfolio
Insured portfolio
Stock portfolio
Wealth
on May 5
Payoff of put
Stock price
$55 $58.5
Chapter 10, page 51
Figure 10.2. Payoffs of cash, forward, and put positions
The forward is equal to the exercise price of the put option. The option premium is
subtracted from the option payoff at maturity. Both the forward and the exercise price are
equal to the spot exchange rate when the positions are taken. The cash position payoff is
defined as the gain resulting from entering a cash position at the current spot exchange rate.
Payoff
Put option
exercise price
Cash position payoff
Price of SFR
Put option payoff
Forward
price
Payoff of short forward position
Chapter 10, page 52
Figure 10.3. Difference in Firm Income Between Put Option and Forward
Hedges
Unhedged cash flow
Export Inc. cash flow
Value of option
premium at maturity:
$11,249.7
Gain with
option hedge
0.75
Loss with option hedge
Cash flow hedged
with forward
Price of DM
Chapter 10, page 53
Figure 10.4. Payoffs to exporter
This figure shows the payoff for an exporter who exports and gets SFR2M only if the SFR is
worth more than $0.5 and otherwise gets $1M.
Payoff in
dollars
Unhedged payoff
$1M + option
Position hedged through the sale of a call
premium
option on SFR2M with exercise price at $0.5.
invested for 90
days
Position hedged with a forward position when
$1M
the forward price is $0.5.
0
$0.5
Price of SFR
Chapter 10, page 54
Figure 10.5. Arbitrary payoff function
Arbitrary payoff function where G(S(T)) is the payoff function and S(T) is the price of the
underlying asset.
G(S(T))
a
c
b
0
Sa
Sb
Chapter 10, page 55
Sc
S(T)
Figure 10.6. Approximating an arbitrary payoff
function
Piecewise linear approximation of arbitrary payoff function where G(S(T)) is the payoff function and S(T)
is the price of the underlying asset.
G(S(T))
0
S(T)
Chapter 10, page 56
Table 10.1. Puts on Amazon.com maturing in January 2001.
The quotes are from the Chicago Board Options Exchange web site for May 5, 2000. The last sale is the
price at which the last sale took place. Net stands for the net change from the previous last sale. The bid
price is the price at which options can be sold and the ask price is the price at which options can be bought.
Volume is the number of contracts traded during the day. Open interest is the number of option contracts
outstanding.
Puts
01 Jan 55 (ZCR MK-E)
01 Jan 55 (ZCR MK-A)
01 Jan 55 (ZCR MK-P)
01 Jan 55 (ZCR MK-X)
Last Sale
14 1/4
17 1/2
11 1/8
13 1/4
Net
Calls
01 Jan 55 (ZCR AK-E)
01 Jan 55 (ZCR AK-A)
01 Jan 55 (ZCR AK-P)
01 Jan 55 (ZCR AK-X)
Last Sale
14
16 3/8
50 1/2
16 5/8
Net
Bid
pc 12 7/8
pc 12 3/8
pc 12 5/8
pc 12 1/4
Ask
Bid
pc 18
pc 18 1/8
pc 17 5/8
pc 18 1/8
Ask
Vol
13 7/8
13 1/8
14 7/8
13
0
0
0
0
Open Int
4273
4273
4273
4273
0
0
0
0
Open Int
1997
1997
1997
1997
Vol
19
18 7/8
19 7/8
19
Box 10.1. Hedging at Hasbro.
Hasbro is global toy company based with headquarters in the U.S. Its classic game is Monopoly, but its
toy lines include Teletubbies and Mr. Potato Head. It sources products in five currencies and sells toys in
30 different currencies. In 1997, it had revenues of $3.2B dollars. It estimated an adverse effect of foreign
Chapter 10, page 57
exchange fluctuations of $92M on its net revenues due to the strengthening of the dollar. Revenues for toy
manufacturers are highly seasonal - more than 60% of the sales take place in the second half of the year.
However, prices are set early in the year and then the firm loses its flexibility to deal with changes in costs
because of exchange rate changes. Faced with increasing concerns about foreign exchange exposure,
Hasbro hired a new treasurer in October 1997 who had experience in international treasury and foreign
exchange risk management. He immediately developed a hedging program to hedge transaction and
anticipated exposures. By the end of 1997, Hasbro had hedged a considerable fraction of its estimated
1998 foreign currency transactions using forwards and options. It had forwards in place for a notional
amount of $35M and options for $135m. In mid-April 1998, the corporation disclosed that a 10%
unfavorable movement in exchange rates would affect its operating revenue by only $10M.
Let’s consider now how Hasbro made the choice to use options rather than only forwards.
Accounting considerations play a key role in the firm’s practice. Forward contracts have to be marked-tomarket when a forward position does not meet the conditions for hedge accounting, which will usually be
the case for transactions that are uncertain. Option premiuMs.for options used to hedge anticipated
transactions can be amortized over time and cash flows from the options hits the income when they accrue.
As a result, if a firm enters a forward contract that expires in the next quarter because it expects to have
a transaction next quarter, its earnings this quarter will be affected by the change in value of the forward
contract. The forward hedge therefore affects earnings in a way that the option hedge does not. If the
transaction is one that the firm is committed to, this problem does not arise because the forward contract
qualifies for hedge accounting, so that the loss of the forward is directly matched with a gain on the
Chapter 10, page 58
transaction. Importantly, these accounting issues do not affect the firm’s economic cash flow. Yet, they
affect the reasoning of Hasbro’s treasurer substantially: “We are sensitive to mark-to-market risk as a
company, which is driven by the seasonality of our business cycle. We’ve historically had lower earnings
in the first quarter and higher earnings in the fourth quarter. But unfortunately, when we hedge, we are
typically hedging for the whole year. So the time we are asked to mark-to-market our forward hedges is
typically when our earnings are weakest on a cyclical basis. Purchased options are a way to deal with that
mark-to-market issue. We use options for further out commitments, and forwards for near-term
commitments that already qualify for hedge accounting.” Hasbro used options in eight critical currency
markets. Interestingly, Hasbro factored the premiuMs.into the budgeting process as a cost of doing
business.
The accounting treatment of derivatives changes over time and is often quite complex. The newest
FASB ruling on derivatives accounting, FASB 133, specifies that under some circumstances the option
premium has to be marked-to-market, so that the benefit that Hasbro finds in using options does not always
apply.
Hasbro’s foreign exchange hedging strategies are presented in “Managing risk in toyland,” by William
Falloon, Risk, May 1998, 40-43.
Chapter 10, page 59
Box 10.2. Cephalon Inc.
On May 7, 1997, Cephalon Inc. acquired derivatives on its own stock, paying for them with newly
issued common stock. The derivatives have the following payoff: If the share price on October 31, 1997,
was below $21.50, Cephalon would get nothing. If the share price was between $21.50 and $39.50, it
would get a standard call payoff, namely the difference between the stock price and $21.50. If the share
price was above $39.50, the firm would get $18, or the difference between $39.50 and $21.50. Such
derivatives are called capped calls. Suppose that the stock price at maturity is $50. In this case, we can
think of the payoff to Cephalon as receiving the payoff of a call with exercise price of $21.50, which would
be $28.50, and having to pay the payoff of a call with exercise price of $39.50, which would be $10.50,
so that on net it gets $28.50 minus $10.50, or $18. Hence, the payoff to Cephalon of the capped call is
the payoff of having a long position in a call with exercise price of $21.50 and writing a call with exercise
price of $39.50. Cephalon bought capped calls on 2,500,000 shares and paid for them a total premium
of 490,000 shares. The price of a share at that time was close to $20.
Why would Cephalon enter such a transaction? Cephalon entered this transaction just before the
FDA was supposed to decide whether a drug developed by the firm, Myotropin, was ready for commercial
distribution. If the drug was approved, the stock price would increase sharply. Taking the position would
therefore have been a good bet for Cephalon if it thought that approval was more likely than the market
thought it was. In that case, the stock price would have been too low and hence Cephalon could have
gotten the capped calls cheap. Straight calls would have generated more profits for Cephalon if it was right,
but would have required more shares to be issued since with capped calls, Cephalon sells some calls also.
Chapter 10, page 60
Cephalon justified its transaction on risk management grounds rather than speculative grounds. If
the drug was approved, Cephalon had a large need for cash. Cephalon had sold the rights to the drug to
a partnership, but had an option to buy these rights back. With approval, it would become profitable to buy
the rights back. This would require Cephalon to raise funds to pay for the repurchase of the rights. If
external financing is expensive for Cephalon, then it might face difficulties in raising funds externally. The
capped options, though, would pay off precisely when Cephalon would require the funds. Viewed this way,
one could argue that Cephalon’s sale of calls amounts to hedging so that it has funds available when it has
attractive investment opportunities. This motivation for hedging was discussed in Chapter 4.
Immediately after Cephalon entered this derivatives transaction, the FDA announced that the drug
was not ready for commercialization. The stock price fell and the capped calls expired out of the money.
Source: Tufano, Peter, Markus Mullarkey, and Geoffrey Verter, 1998, Cephalon Inc., Harvard Business
School Case Study, 298-116.
Chapter 10, page 61
Chapter 11: The binomial model and dynamic hedging strategies
Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section 11.1. Arbitrage pricing and the binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Section 11.1.1. The replicating portfolio approach to price an option. . . . . . . . . . . . . 6
Section 11.1.2. Pricing options by constructing a hedge . . . . . . . . . . . . . . . . . . . . . . . 8
Section 11.1.3. Why the stock’s expected return does not affect the option price
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Section 11.1.4. The binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Section 11.2. The binomial approach when there are many periods . . . . . . . . . . . . . . . . . . . 20
Section 11.2.1 Pricing a European option that matures in n periods . . . . . . . . . . . . . 30
Section 11.3. Implementing the binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Section 11.4. The binomial model and early exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Section 11.4.1.A. Using the binomial approach to price a European call on a dividend
paying stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Section 11.4.1.B. Using the binomial approach to price an American call on a dividend
paying stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Section 11.4.1.C. Pricing American puts using the binomial approach . . . . . . . . . . . 40
Section 11.5. Back to hedging everything and anything . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Section 11.6. Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Literature note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 11.1. Evolution of stock price and replicating portfolio for the one period case . . . . . 53
Figure 11.2. Evolution of stock price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 11.3. Two-period trees for the stock price and the option price . . . . . . . . . . . . . . . . 55
Figure 11.4. Stock price evolution and option payoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 11.5. Cash flows of arbitrage strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 11.6. Tree with dividend payment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 11.7. Payoff of a power option compared to payoff of a call option. . . . . . . . . . . . . . 59
Chapter 11: The binomial model and dynamic hedging
strategies
September 25, 2000
© René M. Stulz 1996, 2000
Chapter objectives
1. Present stock-price dynamics if returns follow the binomial distribution.
2. Demonstrate how to hedge options with the binomial model.
3. Provide an approach to price options with that model.
4. Show how early exercise can be handled.
5. Provide an approach to hedge complex payoffs using that model.
Chapter 11 page 1
In chapter 10, we introduced options and showed how options can be used to hedge complicated
payoffs. Options differ fundamentally from forward and futures contracts because their payoff is not a linear
function of the underlying. Financial instruments whose payoff is not linear in the underlying are called
contingent claims. Besides traditional call and put options, we will consider many different contingent claims
in the remainder of this book. For instance, we will discuss exotic options such as digital options. A cashor-nothing digital option pays a fixed amount of cash if the underlying exceeds the exercise price. Because
a portfolio of options allows us to create a forward contract (see chapter 10), any payoff that a firm might
want to hedge can be viewed as the payoff of a position in a risk-free asset and a contingent claim. For
instance, in chapter 10, we showed that we could approximate a complicated nonlinear payoff with a
portfolio of options. Instead of using this approach, we could hedge the nonlinear payoff by selling a
contingent claim with that nonlinear payoff. A contingent claim with such a payoff would have to be created
before it could be sold.
In this chapter and the next, we develop the basic approach to price options and other contingent
claims. Knowing how to price options turns out to be crucial to quantify the risk of options. As importantly,
however, the techniques used to price options are techniques that allow us to create new contingent claims.
The reason for this is that we will price options like we price forward contracts. Remember from Chapter
5 that to price forward contracts, we use a replicating portfolio. This portfolio has the same payoff at
maturity as a position in the forward contract, it makes no cash distributions, requires no cash infusions
between its purchase and maturity of the forward contract, and it has no position in the forward contract.
Once we have constructed the replicating portfolio, we can price the forward contract by arbitrage. Since
the replicating portfolio pays the same as the forward contract, its value must be the same. Consequently,
Chapter 11 page 2
by holding the replicating portfolio, we get the same payoff as if we held the forward contract. This allows
us to create the payoff of a derivative on our own by buying the replicating portfolio.
Though we take the replicating portfolio approach to price contingent claims, there is, however,
a key difference. When we price forward contracts, the replicating portfolio is a buy and hold portfolio.
We set up the portfolio when we price the contract and hold it unchanged until maturity of the contract.
With contingent claims, the replicating portfolio is a portfolio that has to be adjusted over time. It is still,
however, a portfolio that makes no payments and requires no cash inflows between the time it is purchased
and the maturity of the contingent claim. Since by buying the replicating portfolio and adjusting its
investments over time, we receive at maturity a payoff equal to the payoff of the contingent claim, it must
be that the value of the contingent claim is equal to the value of the replicating portfolio.
To understand why the replicating portfolio has to change over time, think of the case of a
European call option on IBM with exercise price of $100 and maturity in one year. At maturity, the option
is either worth one share of IBM minus $100 or nothing. Suppose that after some time in the life of the
option, the price of a share has increased so much that it is sure that the option will be in the money at
maturity irrespective of what happens to the share price subsequently. At that point, holding one share of
IBM and borrowing the present value of $100 forms the perfect replicating portfolio. Instead, if after some
time, it is sure that the option will not be in the money at maturity because the share price is low enough,
then the replicating portfolio has to be worth nothing irrespective of changes in the stock price. This can
only happen if the replicating portfolio holds no share of IBM at that point. With our example, at some point
the portfolio either ho