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Time, Risk and Options
Chapter 16
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2
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Time and Risk
According to the sticker this model will use
$45 of electricity per year at today’s average
national price of $10.6 cents per kilowatthour. Sears sells another model (Energy
Guide not shown) that retails for $849 and
consumes $65 of power a year. Assuming
their lifespans are the same, which should
you buy to minimize the cost of keeping your
food cool? It depends on your alternatives.
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4
What’s Next?
There may be more proverbs and sayings about time than any
other single topic. Think about “time is money,” “no time like
the present,” “time will tell,” “can’t turn back time,” and
“living on borrowed time.” They reflect the many different
roles time plays in our lives and our economic decisions. In
this chapter we add time and chance to our models.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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duplicated, or posted to a publicly accessible website, in whole or in part.
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Positive Time Preference
Most humans have an innate preference for the present over
the future. Offered the choice between a good that is available
now and an identical amount of it at some future date almost
everyone prefers immediate availability. You will voluntarily
delay consuming it only if you are compensated for doing so.
Your positive time preference is generally rational. Borrowing
and lending are exchanges between the present and the future
that people evaluate according to the same principles.
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Positive Time Preference
Let’s begin with a loan to change your time pattern of
consumption. Assume that you are fresh out of college and have
a firm promise of a high-income job that starts next
year. To improve your standard of living, you are willing to pay
a premium in the future to have a six-pack of some beverage
(actually, beer) in your hands today. Consider me as a possible
lender. I currently earn a high income, but I am near the end
of my working years. During retirement I must live on income
from previous investments. One such investment is to buy a
sixpack today and lend it to someone who is willing to pay me a
relatively large amount in the future in order to have it today.
That someone is you.
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8
Positive Time Preference
Our rates of time preference are both positive, but they
differ. Each of us prefers to consume now rather than later, but
our different future prospects make you more impatient than
me. Assume that you are willing to pay for up to nine cans of
the beverage a year from today if you can have six right now. I
have a lower rate of time preference and would accept seven or
more cans a year from now as compensation for parting with
the six-pack today. Assume that we strike a deal for eight.
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Positive Time Preference
What do we learn from this example?
1. Neither of us is acting irrationally.
2. Interest is not “the price of money,” as it is frequently
called. Interest is the price of earlier availability.
3. Interest exists independently of any risk that you might
default on the loan, and we assumed no such risk in the
example.
4. Interest does not exist because inflation might degrade
the purchasing power of your repayment.
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10
The Productivity of Capital, Present Value,
and the Rate of Return - Loans for
Investment in Capital Goods
This graph shows the amount that
$10 grows to over various lengths
of time. At 10 percent interest
with annual compounding it
becomes $25.94, and in 20 years
it becomes $61.92. At 20 percent
interest the $10 grows to $67.27
in 10 years and $383.38 in 20
years.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
11
The Productivity of Capital, Present
Value, and the Rate of Return - Loans
for Investment in Capital Goods
In general, let P be a payment made today (i.e., the
deposit), i be the interest rate expressed as a decimal, and At
be the amount today’s payment grows to in t years. We have:
At =P(1 + i)t
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12
The Productivity of Capital, Present
Value, and the Rate of Return - Loans
for Investment in Capital Goods
A bank can pay interest to depositors because it receives
interest on money that it lends. It is a financial intermediary
between savers and borrowers. Some loans will be made to
people who wish to change the time shapes of their
consumption, as in the six-pack example, but many will also
be made to finance buildings, capital goods, inventories, and
durable consumer goods, such as homes and vehicles.
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13
The Productivity of Capital, Present
Value, and the Rate of Return –
Present Values
Assume that someone offers you a deal that requires an immediate
payment. Specifically, she promises you $100 a year from today in return
for $95 now. If you are certain she will pay your decision to accept the offer
depends on your alternatives. If the best of these is 10 percent interest per
year on a bank account, refuse her. Instead of the $95 she charges, all you
need to deposit today to get $100 in a year is $90.91. At 10 percent interest
it grows to $100 in a year, that is, $90.91 × 1.10 = $100.00. Thus, $90.91 is
the present value (sometimes called the discounted value or capital value)
of $100 that will be paid a year from now.
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14
The Productivity of Capital, Present
Value, and the Rate of Return –
Present Values
This shows the decreasing
relationship between the
present value of a $100
payment a year from now and
the interest rate at which that
payment is discounted. In that
figure the lower curve shows
the present value of the same
payment if it instead arrives
two years in the future.
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15
The Productivity of Capital, Present
Value, and the Rate of Return Annuities and Bond Prices
You may want to know the present value of a series of
payments that are to come (or be made) at several dates in the
future. For example, the present value of income created by a
drill press is the sum of the present values of each year’s net
income over its life span. An annuity is a set of equal annual
payments. A longer annuity has a higher present value.
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16
The Productivity of Capital, Present
Value, and the Rate of Return Annuities and Bond Prices
The figure to the right
shows how the value
of a $100 annuity
varies with its
duration when
discounted at 10
percent and 20
percent.
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17
The Productivity of Capital, Present
Value, and the Rate of Return Annuities and Bond Prices
A bond is a legally enforceable promise to pay a defined
stream of payments over the future. The simplest kind of bond
is a pure discount bond, like a U.S. Treasury bill, often
called a “T-bill.” It obligates the government to pay $1,000 to
whomever holds it at maturity, which can be three months, six
months, or a year ahead of the issuance date. A T-bill’s
promise of payment may be ironclad, but the value of the bill
itself is not. Instead, its market price varies inversely with
interest rates.
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18
The Productivity of Capital, Present Value,
and the Rate of Return - Net Present Value
and the Analysis of Investment
Assume that you are considering whether to construct a
building, an activity we will call “Project A.” If the building
can be constructed instantly and lasts for two years, you give
up P dollars today in return for payments net of operating
costs of A1 a year from today and A2 two years from today. At
discount rate i the net present value of the project is:
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19
The Productivity of Capital, Present Value,
and the Rate of Return - Net Present Value
and the Analysis of Investment
The alternative is construction Project B, which also entails
spending P dollars today, but in return it gives you three years
of net revenue in installments B1, B2, and B3. Project B’s net
present value is:
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20
The Productivity of Capital, Present Value,
and the Rate of Return - Net Present Value
and the Analysis of Investment
P is $100.00; A1 and A2 are
$100 and $45; B1,B2, and B3
are $50, $40, and $70; and
the interest rate is 10 percent.
we get NPV1 = $28.10 and
NPV2 = $31.10. The graph
shows the NPVs of the
projects for interest rates
between 0 and 0.5.
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21
The Productivity of Capital, Present Value,
and the Rate of Return - Net Present Value
and the Analysis of Investment
The NPVs of the two projects become equal at an interest rate
of 0.134, i.e., 13.4 percent. If you can only undertake one of the
projects, you will be wealthier with Project A if the interest
rate is below 13.4 percent, and with project B if it is above that
amount.
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22
Inflation, Default Risk, and Rational
Behavior - Inflation
Expectations of an inflation that will increase all prices will
also affect interest rates on loans. Assume that both you and
your lender (me) confidently expect that a dollar will be worth
some percentage of its original value when the loan is due. I
will insist on a repayment that compensates me for delayed
consumption and the opportunity cost of investing elsewhere.
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23
Inflation, Default Risk, and Rational
Behavior - Default Risk and Credit Rationing
In a market with people whose risk of repayment differs, you
will only lend to bad risks if your expected (i.e., mean) return
on loans to them is at least as much as you can earn with
certainty by lending to good risks. In a competitive market the
realized returns on loans of different riskiness will tend to
equality. A risky borrower who can increase the likelihood of
repayment (e.g., by finding a cosigner or supplying collateral)
will pay a lower interest rate.
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24
Inflation, Default Risk, and Rational
Behavior - How Do People Discount and
Anticipate the Future?
Whatever the behavior of individuals, businesses of every kind
now apply the insights of financial theory to investment choice
and risk management. Virtually all major corporations now
have a chief financial officer, and an increasing number have a
chief risk officer. Businesses increasingly apply the economics
of finance and uncertainty in their operations in matters that
include risk assessment, the valuation of information, and the
pricing of options.
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25
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26
Some Basics of Probability Frequencies and Probabilities
Economists often model situations in which risks can
be summarized by probabilities that various events will
occur. Probabilities are long-run frequencies. We cannot
know whether the next toss of a fair coin will show heads or
tails, but on the basis of past experience we can confidently
expect heads half the time. Heads and tails are the only two
elements (we call them events) in a sample space that
contains all possible outcomes of this rather simple exercise.
Because either heads or tails must occur, we set the sum of
their probabilities to 1.
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27
Some Basics of Probability Frequencies and Probabilities
Events A and B are independent if the probability that A will
happen is the same regardless of whether or not B has
happened. For example, tosses of the coin are independent
because the probability that the second toss is a head is .5,
regardless of whether the first toss was heads or tails.
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28
Some Basics of Probability Frequencies and Probabilities
If A and B are disjoint (i.e., have no events in common),
then:
The probability of either A or B occurring is:
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29
Some Basics of Probability Frequencies and Probabilities
We are often interested in the probability of event A,
knowing that B has occurred or will occur. This is the
conditional probability of A given B, denoted Pr[A|B]:
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30
Some Basics of Probability - Random
Variables and Distributions
A random variable is a function
that takes on a defined value for
every point in the sample space.
For example, in the figure at the
right, random variable X1 might be
the number of heads that come up
in two tosses.
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31
Some Basics of Probability - Mean and
Variance
The expectation of a random variable X that can take on any
of N possible values, Xi, is denoted E[X] and defined as:
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32
Some Basics of Probability - Mean and
Variance
Important aspects of risk are summarized in the range of values
that X can take and on the likelihood of extreme values. The
variance of X is defined as:
A distribution with a larger percentage of its observations
beyond a certain distance from the mean will have a higher
variance.
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33
Attitudes Toward Risk - Expected
Utility
Assume your house (i.e. your wealth W) is worth $100,000
and a fire would destroy all of its value. If the probability of a
fire is .01 and full insurance is $1,000 your expected wealth is
the same with or without insurance. Buying it gives you:
If you remain uninsured your expected wealth is:
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34
Attitudes Toward Risk - Expected
Utility
Whether or not you insure, your expected wealth is the same.
If there are costs of writing the insurance policy and handling
your claim you will actually pay more for it than your expected
loss.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
35
Attitudes Toward Risk - Expected
Utility
What you really care about is
the change in your well-being
(i.e., utility) if there is a
fire. If you are a risk-averse
person, your utility is related to
your wealth by a curve like the
one in the figure to the right.
The wealthier you are, the
higher your utility, but the
increase in utility (“marginal
utility”) from an extra dollar of
wealth falls as your wealth rises.
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36
Attitudes Toward Risk – Insurance and
Gambling
This shows why a riskaverse person will buy
insurance. Insurance
makes your level of
wealth a sure thing,
which will provide a
higher level of expected
utility than the gamble
you take by being
uninsured.
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37
The Risk–Return Trade-off Diversification
Taken at face value, the proverb that you should not put all
your eggs into one basket is not very helpful as a guide to how
you should act in risky situations. If you stumble while
carrying the basket you are indeed likely to break all of the
eggs. Assume that the probability of a stumble on any given
journey between the henhouse and your destination is .25.
Putting all of a dozen eggs in one basket and making one trip
leaves you with an expected nine eggs intact. If you choose to
carry two eggs on six trips you will end up with exactly the
same expected number of unbroken eggs as if you carried
them in one basket.
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38
The Risk–Return Trade-off Diversification
The real benefit of repeated trips with only a few eggs in each
basket is not the expected number of eggs that arrive
unbroken. It is the range of outcomes you can expect. If you
make six trips, the probability that you will stumble on every
one of them is 0.256 = .000244. The probability that you will
stumble on five trips is .004394, and on four it is
.032959.22 If you must eat at least six eggs to survive, taking
six trips with two eggs apiece lowers your probability of death
to .0376, the sum of these three probabilities. If you carry
the entire dozen in one trip, your probability of not surviving
is .25, nearly seven times higher.
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39
The Risk–Return Trade-off Diversification
Now instead of carrying eggs, think of a basket as an
investment. Instead of growing like an investment, however,
the eggs can at best stay constant in value by remaining
unbroken. The six-basket strategy deals with the risk of losing
too many. You have diversified your holdings (your portfolio)
by putting them into several baskets, and in the process you
have reduced the variance of your returns.
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40
The Risk–Return Trade-off Diversification
Suppose this figure
show hypothetical
annual returns in
various years on stock
shares in an oil
producer (X) and a
natural gas producer
(Y). We say these
returns are positively
correlated.
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41
The Risk–Return Trade-off Diversification
X and Y could also be negatively
correlated, as in The figure to the
right, with high values of one
variable (gasoline prices)
associated with low values of the
other (tire sales).
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42
The Risk–Return Trade-off Diversification
Finally, X and Y might be
uncorrelated, as shown in this
figure.
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43
The Risk–Return Trade-off Diversification
To summarize the association between X and Y, we introduce
the correlation coefficient between them, ρXY (ρ is the Greek
letter “rho”). It measures the closeness of that association
and tells us whether it is positive or negative.
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44
The Risk–Return Trade-off - The
Investor’s Preferences
If two investments, X and Y, have equal risk, you will rationally
prefer the one with the higher expected return. If they have the
same expected return you will prefer the one with lower risk.
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45
The Risk–Return Trade-off - The
Investor’s Preferences
The horizontal axis of the figure
measures the expected (mean) returns μx
and μy (expressed as decimals) on two
possible investments, X and Y. The
vertical axis shows the variance of those
returns. First look at X, whose mean
returns are 5 percent (.05) a year with a
variance of .07. Investment Y has both a
higher expected return (.09) and a higher
variance (.13) than X. Market forces will
see to it that the mean and variance of the
two stand in this relationship—a higher
expected return will only be available to
those willing to take on more risk.
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46
The Risk–Return Trade-off - The
Investor’s Preferences
I0 is known as an indifference
curve. For a risk-averse individual
it shows all levels of expected
returns and variance of returns
that provide an investor with a
given level of utility. I1 and I2 are
also indifference curves but they
represent successively lower levels
of utility.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
47
The Risk–Return Trade-off: Market
Possibilities
This is called the risk-return
frontier. The leftmost point
shows the mean and variance
of a portfolio that consists of X
alone, and the rightmost point
shows them for Y alone. The
variance of any mix of the two
is given by the curved line
between X and Y. The exact
shape depends on the
correlation between X and Y.
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48
The Risk–Return Trade-off - Market
Possibilities
Here is an illustration of how
the risk-return frontier
changes as correlation between
X and Y varies.
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49
The Risk–Return Trade-off – The
Investor’s Choice
The investor will choose
the combination of X & Y
(portfolio) that
maximizes their utility.
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50
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duplicated, or posted to a publicly accessible website, in whole or in part.
51
Risk and Uncertainty
Our models of risky choice thus far have assumed that people
accurately knew all of the probabilities they face. In some
situations they even knew the means and variances of
distributions of random variables. If all such knowledge were
there for the taking business decisions would be no more than
algebraic exercises, only slightly harder to evaluate than
probabilities of outcomes in a dice throw. Now we consider
what happens if decision makers are ignorant, but we need to
realize that like knowledge, ignorance is a matter of degree.
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52
Risk and Uncertainty
Unfortunately, all too often in business (and personal)
situations data that might improve the quality of our
decisions simply do not exist. At the extreme, you are
operating under pure uncertainty. In our terminology,
uncertainty differs from risk. In risky situations you have at
least some information about probabilities and the underlying
distributions of the possible outcomes of your choice. In
situations of pure uncertainty, you have absolutely none.
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53
Risk and Uncertainty
As a practical matter, such extreme uncertainty seldom exists,
and you will probably start your decision-making process with
educated guesses about the probabilities of various events.
You will start with a probability distribution of possible
outcomes in mind, called a prior distribution, or simply a
prior.
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54
Risk and Uncertainty
Whatever your prior, you want a method that will help you
reduce whatever uncertainty remains. The job has two
aspects:
1. You must try to devise tests whose results will allow
you to reduce the zone of uncertainty.
2. If such research is possible you must decide whether
the reduction in uncertainty is worth the cost of
undertaking it.
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55
The Cost and Value of Information The Parable of the Tycoon
Decision makers often engage in research whose outcomes
will help them update their priors. A famous tycoon has been
said to do just this when he acquires control of a company he
suspects is underperforming due to poor management. The
tycoon’s first action is to change its top management. If the
firm’s performance fails to improve he hires a new
management. If the second management also fails, he sells the
company.
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56
The Cost and Value of Information Determining the Value of a Test
This shows the possible outcomes of
investing $100 today. Tomorrow you will
get either $200 with probability .7 or $0
with probability .3. Assuming that you are
risk neutral over these amounts, the
payoff’s expected value is −100 + (0.7 ×
200) + (0.3 × 0) = $40.
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57
The Cost and Value of Information Determining the Value of a Test
A reliable person offers to sell you a test for $20 that will
determine the outcome with certainty. Buying the test means
you will only commit the $100 if it says you will gain $200.
Your payoff will be −20 − 100 + 200 = $80 if the test is
positive and −$20 if it is negative. Because the test is
perfectly accurate, the probability that it will give a result of
$200 is .7. Whatever the test result, if you buy it you lose
$20 with certainty. Your expected payoff if you buy the test
is −20 + 0.7 × (200 − 100) + (.3 × 0) = $50. The expected
payoff if you do not test is $40, but if you test, the expected
payoff, net of the cost of the test, is $50. Thus, it pays you to
buy the test as long as it costs less than $30.
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58
Bayesian Reasoning
Our approaches to the previous problems exemplify a more
general process called “Bayesian reasoning.” If A and B are
two events in the sample space, the definition of conditional
probability says that:
Substituting, we obtain Bayes’ theorem
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59
Bayesian Reasoning – An Example
A bag contains two coins. You know one is an ordinary
coin with a head and a tail, but the other has two heads. You
close your eyes, grab a coin, set it down without looking at the
other side, and see a head. What is the probability that you
chose the two-headed coin? Your prior is that the probability of
the two-headed coin is .5, because you picked it at random. But
you also know that three of the four sides that could have
appeared are heads, and that the ordinary coin has only one of
those three. Because heads are more likely on the two-headed
coin, you might want to revise your prior probability of the
two-headed coin upward.
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60
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duplicated, or posted to a publicly accessible website, in whole or in part.
61
Economics and Options
We often encounter a tension between keeping our options
open and making risky commitments. Economics is about
choices at the margin, and here too there is a margin. What if
instead of having to decide today you could buy the right to
delay that choice? During the extra time you might uncover
information that helps you better predict success or failure.
But time is also valuable—if the information is favorable the
delay in committing your funds reduces the present value of
your returns relative to investing earlier. Whether to buy the
time requires comparison of the costs of delay and the benefits
of additional information.
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62
Puts and Calls
A derivative is any financial instrument whose value depends on the
value of some underlying asset (also called the “underlying” or
“underlier”), for example a natural gas futures contract whose underlier
is the gas. An option is a derivative whose holder has the right, but not
the obligation, to buy or sell a certain quantity of an underlying asset
before a fixed expiration date.
A call option (usually just referred to as a call) allows but does not
require its holder to buy the underlying for a fixed amount known as its
strike price (or strike) at any date on or before its expiration. A put
option (or put) allows but does not require its holder to sell the
underlying at a preset strike price prior to expiration.
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63
Puts and Calls
Options cannot eliminate risk—nature still determines
whether this year’s corn crop will be large or small—but
options give people choices about which risks they will hold
and help to price those risks they might wish to assign to
others. The use of options to reduce uncertainty, like the use of
futures contracts discussed in Chapter 3, is another
instance of hedging.
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64
Puts and Calls - The Values of Puts and
Calls
Consider a call option with a
strike price of $20. The
Diagram to the right shows the
Value of such an option just
Before it’s expiration. If the
Value of the stock is less than
$20, than the call option is
Worthless. As the value of the
Stock rises above $20, the
Value of the call option will be
the difference between the
stock price and $20.
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65
Puts and Calls - The Values of Puts and
Calls
What about when the expiration date is further in the future?
Two factors influence the value of a long-term option:
1. The volatility of the companies stock in the past. The
more volatile the stock the greater the value of the
option.
2. The amount of time until expiration. The greater the
amount of time until expiration, the greater the value of
the option.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
66
Puts and Calls - The Black-Scholes
Formula
The Black-Scholes formula shows that the value of an
option depends on its strike price, the current price of the
underlying asset, the volatility of the underlying’s price, the
time to expiration, and the interest rate.
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67
Puts and Calls - The Diversity of
Options
Just a few of the many so-called exotic options include the
following:
•Compound options
•Barrier options
•Lookback options
•Average options
•Contingent-premium options
•Basket and rainbow options
•Leveraged options
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68
Irreversible Decisions and Real Options
- Real Options
This right to postpone is an example of a real option, as
opposed to a financial option like a call on a stock. For
example, oil in a storage tank and oil underground both have
option values because you have a choice of when to use or
extract them. The right to prepay a home mortgage
has option value. A plant that can be cheaply shut down for
short periods can be worth more than one that must run
continuously.
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69
Irreversible Decisions and Real Options The Value of Postponing a Commitment
Assume that you can build a plant instantly, and that your costs
and revenues all come due at the end of each year. Assume
that the decision to build the plant is completely irreversible—it
lasts forever, cannot produce any other good, and has no value
as scrap. The value of the real option to delay construction
depends upon several factors:
1. As a general principle, the greater the range of
uncertainty that can be resolved, the higher the value of
an option to delay.
2. Higher interest rates reduce the value of the option.
(c) 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.