FINANCE I
FINANCIAL DECISION MAKING
Valuing decisions
The rst step in decision making is to identify the costs and bene ts of a decision. The
next step is to quantify these costs and bene ts. In order to compare the costs and
bene ts we need to evaluate them in the same terms - cash today.
A competitive market is a market in which goods can be bought and sold at the same
price - that price determines the cash value of the said goods. By evaluating costs and
bene ts using competitive market prices we can determine whether a decision will make
the rm and its investors wealthier. In extension, the valuation principle means that the
value of an asset to the rm or its investors is determined by its competitive market price.
The bene ts and costs of a decision should be evaluated using these market prices, and
when the value of the bene ts exceeds the value of the costs the decision will increase
the market value of the rm.
Interest rates and the time value of money
The time value of money is the di erence in value between money today and money in
the future. The rate at which we can exchange money today for money in the future is
determined by the current interest rate.
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r = interest rate
Present value (PV) = the value in terms of SEK today
Future value (FV) = the value in terms of dollars in the future
Interest rate factor = 1+r
Discount factor = 1/1+r one year discount factor
Discount rate = the risk free interest rate is also referred to as the discount rate for a risk
free investment
Present value and the NPV rule
One should always accept all NPV with value of 0 or higher when the possibility arises.
Net Present Value (NPV) = PV (bene ts) - PV (costs)
NPV Decision Rule = when making an investment decision you shall always choose the
alternative with the highest NPV due to choosing this alternative being equivalent to
receiving its NPV in cash today
Regardless of our preferences for cash today versus cash in the future, one should always
maximise NPV rst. We can then borrow or lend to shift cash ows through time and nd
our most preferred pattern of cash ow.
Arbitrage and the Law of One Price
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The practice of buying and selling equivalent goods in di erent markets to take
advantage of a price di erence is known as arbitrage. An arbitrage opportunity has a
positive NPV. Once they’re spotted they will quickly disappear, thus the normal state of
a airs in markets should be that no arbitrage opportunity exist.
We call a competitive market in which there are no arbitrage opportunities a normal
market. If the prices in two markets di er, investors will pro t immediately by buying in the
market where it is cheap and selling in the market where it is expensive. In doing so, they
will equalise the prices. As a result, prices will not di er. This important property is the
Law of One Price which means that if equivalent investment opportunities trade
simultaneously in di erent competitive markets then they must trade for the same price in
both markets.
No-arbitrage and the price of nancial assets
In nancial markets it is possible to sell an asset you do not own by doing a short sale. In
a short-sale, the person who intents to sell an asset rst borrows it from someone who
already owns it, then the person must either return the asset by buying it back or pay the
owner the cash ows he or she would have received. By executing a short-sale it is
possible to exploit the arbitrage opportunity when the bond is overpriced even if you do
not own it.
Asset/security = an investment opportunity that trades in a nancial market (bonds,
stocks, real estate etc)
No-arbitrage price = the price in all markets when there’s no arbitrage opportunity
Important to note is that the risk free interest rate equals the percentage gain that you
earn from investing in the bond, which is called a bond return. When assets trade at noarbitrage prices the cost and bene t are equal in a normal market and so the NPV of
buying an asset is zero.
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In a competitive market, if a trade o ers a positive NPV to one party it must give a
negative NPV to the other party, because all trades are voluntary they must occur at
prices at which neither party is losing value and therefor for which the trade of zero NPV
(no value created). Instead, value is created by real investment projects in which a rm
engages in developing new products, opening news stores or creating more e cient
production methods.
The separation principle = asset transactions in a normal market neither create nor
destroy value on their own, therefore we can evaluate the NPV of an investment decision
separately from the decision the rm makes regarding how to nance the investment
(borrowing or add stocks) or any other asset transactions the rm is considering
Portfolio = collection of assets/securities. Because asset C is equivalent to the portfolio
of A and B, by the Law of One Price, they must have the same price. This idea leads to
the relationship known as value additivity
Value additivity = the price of C must equal the price of the portfolio, which is the
combined price of A and B. The cash ows of the rm are equal to the total cash ows of
all projects and investments within the rm, therefore, value addivity, the price or value of
the entire rm is equal to the sum of the values of all projects and investments within it. To
maximize the value of the entire rm managers should make decisions that maximize NPV
Risk premium = the risk premium of an asset represents the additional return that
investors expect to earn to compensate for the asset/security risk. Because investors are
risk averse the price of a risky asset cannot be calculated by simply discounting its
expected cash ow at the risk-free interest rate, they want their risk premium. Rather,
when a cash ow is risky, to compute its present value we must discount the cash ow
we expect on average at a rate that equals the risk free interest rate plus an appropriate
risk premium. To eliminate any arbitrage opportunity, the highest bid-price should be
lower than the lowest ask price
THE TIME VALUE OF MONEY
There is a general assumption that money today is of more value than money tomorrow.
This is due to interest rates, possible nancial uncertainties and in ation
The timeline and the three rules of time travel
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An investor shall always draw a timeline with a stream of cash ows. There are 3 rules for
time travel:
- it is only possible to compare or combine values at the same point in time
- rule 1 can be applied with compounding (moving a value or cash ow forward in time)
by Cx(1+r)^n where n is the number of years in the future. This may lead to the e ect of
earning interest on interest which is known as compound interest
- the process of moving a value or cash ow backward in time is known as discounting,
and it’s done with PV=C/(1+r)^n
Calculating the NPV
After establishing the rules of time travel and determining the computation of present and
future values, comes the case of addressing our central goal - calculating the NPV of
future cash ows to evaluate an investment decision
NPV = PV(bene ts) - PV(costs)
The NPV of an investment opportunity is also the present value of the stream of cash
ows.
Perpetuities, annuities and other special cases
Perpetuity = a stream of equal cash ows that occur at regular intervals and last forever
Growing perpetuity = a stream of cash ows that occur at regular intervals and grow at a
constant rate forever (the rst payment does not grow)
Annuity = a stream of N equal cash ows paid at regular intervals. The di erence
between an annuity and a perpetuity is that an annuity ends after some xed number of
payments. To calculate the present value of an annuity we need all cash ows (with the
exception of the initial outlay) to be of equal value while also needing it to be equally long
between each cash ow
Growing annuity = a stream of N growing cash ows paid at regular intervals. It is a
growing perpetuity that eventually comes to an end
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Sometimes, we know the present value or future value but do not know one of the
variables we have previously been given as an input. In such situations. We use the
present and/or future values as inputs then solve for the variable we are interested in.
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The present value of a cash ow stream is the summation of the discounted values:
Internal rate of return (IRR) = the interest rate that assets the net present value of the
cash ows equal to zero. In a few cases it is impossible to solve the IRR directly. One
other approach is to use trial and error to nd the IRR or other variables than the IRR
INVESTMENT DECISION RULES
One interpretation of the internal rate of return is the average return earned by taking on
the investment opportunity. The IRR investment rule is based on this idea.
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IRR Investment Rule = an investor skall take any opportunity where the IRR exceeds the
opportunity cost of capital. Turn also down any opportunity where IRR is less than the
opportunity cost of capital
NPV Decision Rule = when making an investment decision you shall always choose the
alternative with the highest NPV due to choosing this alternative being equivalent to
receiving its NPV in cash today
NPV pro le = a graph of the project’s NPV over a range of discount rates
Pitfalls with the IRR rule
There are several situations in which the IRR fails:
- during delayed investments
- when there are multiple IRRs present
- when there exists no IRR
Delayed investments
Imagine someone o ers you a million SEK upfront to write a book that takes 4 years to
write which would mean having to stop working in order to write the book, which would in
turn mean you forgo a certain amount of money as alternative income. For most
investment opportunities the expenses occur initially and cash is received later, but in this
case you receive cash upfront which incurs the costs of producing the book later. It is as
if you borrowed money, receiving money today for a future liability (and when you borrow
money you prefer as low a rate as possible. In this case IRR better represents the rate
that you are paying rather than earning - the optimal choice is then to borrow money so
long as this rate is less than the cost of capital.
Multiple IRRs
Imagine that you would like the publisher to sweeten the deal before accepting to write
the book. In response, the publisher o ers to give you a royalty payment when the book
is published in exchange for taking a smaller upfront payment. This deal would entail that
you get one million SEK in 4 years after publishing the book and only 500K SEK as
upfront payment. In this case there exist multiple IRRs, which leads us to relying on the
NPV rule instead.
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Nonexistent IRR
Imagine after your latest negotiations that the publisher is willing to increase your initial
payment to 750K SEK in addition to the 1 million SEK royalty when the book is published
in 4 years. With these cash ows there exists no IRR which means that there is no
discount rate that makes NPV equal to zero, thus the IRR rule provides no guidance
whatsoever.
Equivalent Annual Annuity (EAA) = is a method of evaluating projects with di erent life
durations used as a metric for determining how nancially e cient projects are. It
smoothes out all cash ows and generates a single average cash ow for all periods that
(when discounted) equal the project’s NPV which means that a high EEA is more
attractive
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The result is that Project A has a higher EAA which implies that it also has a higher NPV,
therefore should the investor choose Project A.
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The Payback Rule = states that you should only accept a project if its cash ow pay
back its initial investment within a pre-speci ed time period. To apply the rule we must
rst calculate the amount of time it takes to pay back the initial investment (called the
payback period) and then accept the project if the payback period is less than a prespeci ed length of time (usually a few years)
INTEREST RATES
Interest rates are generally stated a an annual rate, thought the payments themselves
may occur at di erent intervals such as monthly or quarterly. When evaluating cash ows
however, we must use a discount rate that matches the time period of our cash ows this discount rate should re ect the annual return we could earn over that time period.
The E ective Annual Rate (EAR)
EAR is the actual amount of interest that will be earned at the end of one year. For
example, with an EAR of 5% and an investment of 100K SEK, our investment would grow
to 105K SEK at the end of the year.
At the end of the second year our investment grows to:
Adjusting the discount rate to di erent periods is important since we want to know
what the total interest earnings for these 2 years look like. With this, we could also
calculate the interest earnings for periods shorter than a year, for example half year.
By raising the interest rate factor (1+3) to the appropriate power, we can compute an
equivalent interest rate for a desired time period, in this case half a year:
The general equation for discount rate period conversion is:
Annual Percentage Rates (APR)
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Banks also quote annual percentage rate, which indicates the amount of simple interest
earned in one year - aka the amount of interest earned without the e ect of
compounding. Since it does not include the e ect of compounding, the APR is typically
less than the actual amount of interest that we will earn. To compute the actual amount
that we will earn in one year, we need to convert the APR to EAR.
For example, if we have a savings account with the interest rate of 6% APR with monthly
compounding, we will learn 6%/12=0,5% monthly interest which means that APR with
monthly compounding is actually a way of quoting a monthly interest rate rather than an
annual interest rate. Since the interest compounds each month we will (with 1 SEK
investment) earn:
Since the APR doesn't itself re ect the true amount that we earn over one year it is
important to remember that we then cannot use APR itself as a discount rate, instead the
APR with a k amount of compounding periods is a way of quoting actual interest earned
each compounding period.
Once we computed the interest earned per compounding period we can then compute
the EAR by compounding using the rst formula on this page. The e ective annual rate
corresponding to an APR with k compounding periods per year is given by the following
conversion formula:
This means that when working with APR we must rst divide the APR by the number of
compounding periods per year to the determine the actual interest rate per compounding
period, and then compute the appropriate discount rate by compounding the rst formula
on this page. Once these step are completed, we can then use the discount rate to
evaluate the present or future values of a set cash ows.
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The link between EAR and APR is:
Interest rates are split into nominal (the ones we and banks use for discounting cash
ows) and real.
Nominal interest rate = indicates the rate at which your money will grow if invested for a
certain period of time
Real interest rate = indicates the growth of your purchasing power after adjusting for
in ation
Growth in purchasing power is therefore calculated by dividing the growth of money with
the grow of prices!
Taxes and interest rates
If you earn interest on savings, or pay interest on borrowing, taxes reduce the amount
that you can actually keep or have to pay. The after-tax interest rate formula is (r is still
interest rate, while t is tax rate):
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Real and Nominal interest rates
In some cases, the interest on loans is tax deductible, which means that the cost of
paying interest on the loan is o set by the bene t of de tax deduction. The net e ect is
that when interest on a loan is tax deductible, the e ective after-tax interest rate is r(1-t).
In other words, the ability to deduct the interest expense lowers the e ective after-tax
interest rate paid on the loan.
Risk and interest rates
Interest rates are a ected by default risk, which means that the possibility that the
borrower will pay the loan. The general consensus is that if you for example buy a bond
which promises 1000SEK in the future, the price of the bond depends on various factors.
If a party whose interest rate on borrowed money is very low, that means that the party is
very reliable and thus the bond has a high price which in turn means that the bond has
lower potential upside - lower yield. The rule goes:
Price UP - Yield DOWN
Price DOWN - Yield UP
This in turn means that the greater the risk, the greater the possibility of return. The
potential gap between what you're willing to pay for the bond and what you're going to
get determines the amount of risk involved with the trade.
The Opportunity Cost of Capital
The opportunity cost of capital is the best available return o ered in the market on an
investment of comparable risk and retur. The opportunity cost of capital provides a
benchmark against which the cash ows of a new investment should be evaluated.
BONDS
A bond is a security sold by governments and corporations to raise money from investors
today in exchange for promised future payments. A bond certi cate shows the terms of
the bond, for example:
- maturity date (the time remaining until the repayment date, also known as the term of
the bond)
- coupons (the promised interest payments of a bond)
- coupon rate (the amount of each payment)
- face value (the notional amount used to compute interest payments)
Zero-Coupon Bonds
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This is the simplest type of bond, which does not make coupon payments. The only cash
payment the investor receives is the face value of the bond on the maturity date. ZeroCoupon bonds generally trade at a discount, aka a price lower than the face value, which
is why they're also called pure discount bonds.
YTM = yield to maturity, which is the IRR of an investment in a bond
Coupon Bonds
Coupon bonds also pay investors their face value at maturity, but in addition to that they
also make regular coupon interest payments. These can be treasury notes (original
maturities from 1 to 10 years) and treasury bonds (original maturities of more than 10
years).
The YTM of a coupon bond can also be computed. Remember than the YTM of a bond is
the IRR of investing in it and holding it to maturity - it is the single discount rate that
equates the present value of a bond's remaining cash ows to its current price, as below:
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Because the coupon payments represent an annuity, the YTM is the interest rate y that
solves the formula:
Here is an example for the price of a coupon bond:
The risk-free interest rate for an investment until date n equals the YTM of a risk-free
zero-coupon bond that matures on date n. A plot of these rates against maturity is called
the zero-coupon yield curve. The YTM of a coupon bond is the discount rate y, which
equates the present value of the bond's future cash ows with its price:
When a bond issuer does not make a bond payment in full, the issuer has defaulted (the
risk that default can occur is called default/credit risk).
Constant Dividend Growth
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The simplest forecast of the rm's future dividends states that they will grow at a constant
rate, g, forever. The case yields the following timeline for the cash ows for an investor
who buys a stock today and holds it:
Because the expected dividends are a constant growth perpetuity, we can use the
following formula to calculate the stock price (with the help of rst calculating the present
values rst):
Constant dividend growth model = states that the value of the rm depends on the
dividend level for the coming year divided by the equity cost of capital adjusted by the
expected growth rate of dividends
Dividend = a distribution of cash or stock to a class of shareholders in a company
(dividends are drawn from a company's retained earnings and therefore based on them
for the most part)
STOCK VALUATION
The Law of One Price implies that to value any security we must rst determine the
expected cash ows an investor will receive from owning it, thus we begin our analysis of
stock valuation by considering the cash ows for an investor with a one-year investment
horizon. We then consider the perspective of investors with longer investment horizons.
The Dividend-Discount Model
By using this model we can nd the value of a stock by analysing the cash ows coming
to it - the dividends. The model says that the value (or price) of a stock today is equal to
the present value of all future dividend payments!
Stock = common stocks represent a part of ownership in a company (usually includes
voting rights and therefore a small control of the company - if a large enough sum is
bought)
Dividend yield = the return you get from the dividend (dividend/current price)
Capital gain = represents the increase in price and is calculated using (p1-p0)/p0
Total return = the amount of money we made on this investment
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Imagine buying a stock for 50SEK. One year later you sell the stock for 50,50SEK, but
before that you managed to get a dividend as well, worth 1,50SEK. In this case, your total
return is worth 2SEK, which is a 4% return on the investment. Of these 4%, the dividend
yield stands for 3% and the increase in stock price stands for 1%. You invest now (period
0) into a stock with the current price p0, only to sell it a year later (period 1) ,where your
total return amounts to dividend1+p1 (where p1 represents the price of the stock in
period 1).
These are the results of someone investing one year into the future. For a multiyear
investor the calculations are di erent. For example, if we planned to hold the stock for 2
years, we would receive dividends in both year 1 and year 2 before selling the stock, as
shown below:
Of course, the future dividend payment and stock price in the timeline above are not
known with certainty, rather these values are based on the investor’s expectations at the
time the stock is purchased. Given these expectations, the investor will be willing to buy
the stock at today’s price as long as the NPV of the transaction is not negative - aka as
long as the current price does not exceed the present value of the expected future
dividend and sale price. Because these cash ows are risky, we can’t compute their
present value using the risk-free interest rate, instead we must discount them based on
the equity cost of capital, rE, for the stock, which is the expected return of other
investments available on the market with equivalent risk to the rm’s shares. The investor
is willing to then buy the stock under these conditions:
For an investor to be willing to sell the stock, she must receive at least as much today as
the present value she would receive if she waited to sell next year:
But because for every buyer of a stock there must a seller, both equations must then
hold, therefore the stock price should satisfy:
Setting the stock price equal to the present value of the future cash ows in this case
implies:
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As a two-year investor, in this case, we care about the dividend and stock price in year 2,
which would imply that if you’re a one-year investor you would then not care about these
numbers for year 2. While a one-year investor doesn’t really directly care about the
dividend and stock price in year 2, she has to care about them indirectly since they will
a ect the price for which the investor can sell the stock for at the end of year 1. This
means that the formula for the stock price for a two-year investor is the same as the one
for a sequence of two one-year investors!
This process can be continued for any number of years by replacing the nal stock price
with the value that the next holder of the stock would be willing to pay. Doing so leads to
the general dividend-discount model for the stock price where the horizon N is arbitrary
and dependent on the speci c exercise done:
In the end, investors do this to earn money, therefore what we’re looking for is the
maximum amount of total return, calculated by:
Because the expected dividends are a constant growth perpetuity, we can calculate their
present value and then obtain the following formula for the stock price:
Constant dividend growth model = a model that shows that the value of the rm
depends on the dividend level for the coming year divided by the equity cost of capital
adjusted by the expected growth rate of dividends
Total Payout Model
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This model allows us to ignore the rm’s choice between dividends and share
repurchases. While using the dividend-discount model we assumed that any cash paid
out by the rm to shareholders takes the form of a dividend, however in recent years an
increasing number of rms have replaced dividend payouts with share repurchases.
Share repurchase = the rm uses excess cash to buy back its own stock
Share repurchases have two consequences on the dividend-discount model:
- the more cash the rm uses to repurchase shares, the less it has available to pay
dividends
- by repurchasing shares, the rm decreases its share count which increases its earnings
and dividends on a per-share basis
The total payout model instead values all of the rm’s equity rather than a single share. To
do so we discount the total payouts that the rm makes to shareholders, which is the
total amount spent on both dividends and share repurchases, then we divide by the
current number of shares outstanding to determine the share price.
Discounted FCF Model
This model focuses on the cash ows to all of the rm’s investors, both debt and equity
holders, allowing us to avoid estimating the impact of the rm’s borrowing decisions on
earnings. In the total payout model we rst value the rm’s equity rather than just a single
share, but the discounted free cash ow model goes one step further and begins by
determining the total value of the rm to all investors - both equity and debt holders. That
is, we begin by estimating the rm’s enterprise value with:
Enterprise value = the value of the rm’s underlying business, unencumbered by debt
and separate from any cash or marketable securities. We can interpret the enterprise
value as the net cost of acquiring the rm’s equity, taking its cash, paying o all debt and
thus owning the unlevered business
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The advantage of the discounted FCF model is that it allows us to value a rm without
explicitly forecasting its dividends, share repurchases or its use of debt. To value an
enterprise we compute the present value of the free cash ow that the rm has available
to pay all investors, both debt and equity holders. FCF measures the cash generated by
the rm before any payments to debt or equity holders are considered. Just as we can
determine the value of a project by calculating the NPV of its free cash ow, so can we
estimate a rm’s current enterprise value V0 by computing the present value of the rm’s
free cash ow:
When we are looking at the entire rm, it is natural to de ne the rm’s net investment as
its capital expenditures in excess of depreciation:
Net investment = investment intended to suppor the rm’s growth, above and beyond
the level needed to maintain the rm’s existing capital
Depreciation = a way to calculate the reduction in value of an asset due to use, wear and
tear or obsolescence (used for intangible assets)
Amortization = paying o a debt over time (used for tangible assets)
Net working capital = how much cash we have “stuck” in for example inventory (an
increase here would a ect cash ow negatively)
Capital expenditures = money that the rm outlays in order to expand or maintain its
operations, upgrade/replace its machines/property etc
WACC = stands for weighted average of cost of capital and is the average cost of capital
the rm must pay to all its investors, both debt and equity holders (if the rm has no debt
then rwacc=rE but if the farm has debt then rwacc is an average of the rm’s debt and equity
cost of capital). WACC is then interpreted as re ecting the average risk of all of the rm’s
investments
FCF = stands for free cash ow and is calculated by EBIT(1- taxes) - depreciation capital expenditure - the increase in working capital)
Firms have several ways of measuring pro t:
EBIT = earnings before interest and taxes (can also be calculated with EBITDA depreciation and amortization)
EBITDA = EBIT + depreciation and amortization
E = earnings, which can be calculated by EBIT(1- taxes)
RISK AND RETURN
Every investor has to be aware of the risk involved and the likely return an investment will
earn, before the purchase of a security. When an investment is risky, there are di erent
returns it may earn, each with their own likelihood of occurring. To make investments
comparable, we will now summarise all information by calculating a probability
distribution. This will assign a probability pR that each possible return R will occur.
Expected Return
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Given the probability distribution of returns, we can now compute the expected return.
We calculate the expected return as a weighted average of the possible returns where the
wrights correspond to the probabilities.
Expected Value
The expected value of an investment can also be called “the mean of a random variable
x” where the variable x represents in this case a random stock or any other kind of
investment. This is our expectation of x. The expected value corresponds to the average
outcome if we invest in x many times over, therefore if we invest in x once then the
expected value cannot be the outcome.
Variance and Standard Deviation
Two common measures of the risk of a probability distribution are its variance and
standard deviation. The variance is the expected squared deviation from the mean, and
the standard deviation is the square root of the variance.
If the return is risk-free and never deviates from its mean then the variance is zero.
Otherwise, the variance increases with the magnitude of the deviations from the mean,
therefore the variance is a measure of how “spread out” the distribution of the return is.
In nance, we refer to the standard deviation of a return as its volatility aka how much an
investment deviates from its expected returns.
In nance, we would also like to diversify our investments, so as to not put all of our eggs
in the same basket. For that we have to compute the covariance of investments,
meaning how similarly investments move (if two stocks move almost the same then
investing in both does not achieve diversi cation). Covariance stands for co-movement.
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Since the covariance between X and Y can take any value, it is hard to interpret the
resulting number. Instead, we may use a correlation coe cient for X and Y. The coe cient
Pxy is always between 1 and -1.
The formula for covariance is:
And an example of calculating correlation is:
Historical Returns
Out of all possible returns, the realised return is the return that actually occurs over a
particular time period.
Suppose that you invest in a stock on date t for price Pt. If the stock pays dividend Divt
on the date of t+1 and you sell the stock at that time for the price of Pt+1 then the
realised return from your investment in the stock from t to t+1 is:
Average Annual Returns
The average annual return of an investment during some historical period is simply the
average of the realised returns for each year. That is, if Rt is the realised return of a
security in year t then the average annual return for years 1 through T is:
PORTFOLIO THEORY
To nd an optimal portfolio, which is a group of investments, we need a method to de ne
a portfolio and analyse its return. We can describe a portfolio by its portfolio weights,
which are the fraction of the total investment in the portfolio held in each individual
investment in the portfolio.
The portfolio weights add up to 1, so that they represent the way we divided our money
between the di erent individual investments in the portfolio. Think of it as percentage of
the total value of portfolio, where 1 represents 100% and likewise 0,2 represents 20%
etc. Consider a portfolio with 200 shares of the stock D and 100 shares of the stock C.
The corresponding portfolio weights are as follows:
Given the portfolio weights we can calculate the return on the portfolio. Suppose x1, x2
etc are the portfolio weights of the n investments in a portfolio, and these investments
have returns R1, R2 etc, then the return on the portfolio RP is the weighted average of the
returns on the investments in the portfolio where the weights correspond to portfolio
weights:
Covariance and Correlation
In order to nd the risk of a portfolio, we need to know more than the risk and return of
the component stocks - we need to know the degree to which the stocks face common
risks and their returns move together. Covariance is the expected return of the deviations
of two returns from their means. The covariance between returns Ri and Rj is:
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While estimating the covariance from historical data we instead use:
While the sign of the covariance is easy to interpret, its magnitude is not. It will be larger if
the stocks are more volatile and it will be larger the more closely the stocks move in
relation to each other. In order to control for the volatility of each stock we quantify the
strength of the relationship between them - we can calculate the correlation between
two stock returns, de ned as the covariance of the returns divided by the standard
deviation of each return:
After understanding how to calculate expected return and volatility of a portfolio, we can
now try and create an e cient portfolio. Generally, an e cient portfolio will have
maximised return and minimised volatility.
Short Sales
So far we have only considered portfolios in which we invest a positive amount in each
stock, thus we refer to a positive investment in a security as a long position in the
security. It is though also possible to invest a negative amount in a stock, called a short
position, by engaging in short sales (a transaction in which you sell a stock today that
you do not own, with the obligation of buying it back in the future).
Risk vs Return
Variations in the risk associated with or the price of an individual stock are mainly of two
types, rm-speci c (risks and variations speci c to the rm) and market-speci c (risks
and variations speci c to all rms in a market). Important to note is that rm-speci c risk
does not lead to higher expected return, thus the term “high risk, high return” should
instead be “high systematic risk, high expected return”!
You can also borrow money to invest. Borrowing money to invest in stocks is referred to
as buying stocks on margin or using leverage. A portfolio that consists of a short
position in the risk-free investment is known as a levered portfolio.
Tangent and Market Portfolios
To earn the highest possible expected return for any level of volatility we must nd the
portfolio that generates the steepest possible line when combined with a risk-free
investment. The slope of the line through a given portfolio P is often referred to as the
Sharpe ratio of the portfolio.
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The Sharpe ratio measures the ratio of reward-to-volatility provided by a portfolio where
the optimal portfolio to combine with the risk-free asset will be the one with the highest
Sharpe ratio, where the line with the risk-free investment just touches (and so is tangent
to) the e cient frontier of risky investments. The portfolio that generates this tangent line
is known as the tangent portfolio.
Another type of portfolio is the market portfolio, which is a portfolio created out of all
available risky nancial assets in an economy. The market portfolio uses weights as
relative market value of each one of these nancial assets. According to CAPM, the
forces of nancial marktes will make the market portfolio and tangent portfolio equal.
The Capital Allocation Line (CAL)
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The CAL is used when investing in a portfolio of a risk-free asset and a risky asset. The
weight of the risky one is x, while the weight of the risk-free one is 1-x (since they have to
add upp to 1). The weighted sum of the expected return of its two assets is:
This leaves us with the capital allocation line:
Beta and the Required Return
So far we know how to conclude if a portfolio is e cient or not based on which one has
the highest Sharpe ratio and therefore the best risk-return trade-o . What we want to do
now is to determine whether we can improve a portfolio by adding more of a given
security and use it to calculate an investor’s required return for holding an investment.
The required return is the expected return that is necessary to compensate for the risk
that our investment will contribute to the portfolio. The required return for an investment is
equal to the risk-free interest rate plus the risk premium of the current portfolio, scaled by
the investment’s sensitivity to the portfolio.
The beta of investment i with portfolio P is de ned with the help of combining the
volatility and correlation terms of the following equation:
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Which gives the equation for the beta of our investment:
Expected Return
If a security’s expected return exceeds its required return, then we can improve the
portfolio by adding more of that speci c security, but how much should we add? As we
buy shares of security I, its correlation (and therefore its beta) with our portfolio will
increase, ultimately raising its required return. This works vice versa as well (reducing the
holding in a security who’s expected return is lower than the required return). Important to
remember - a portfolio is e cient if and only the expected return of every available
security equals its required return!
The Capital Asset Pricing Model (CAPM)
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The CAPM is a model which allows us to identify the e cient portfolio of risky assets
without having any knowledge of the expected return of each security. Instead, the CAPM
uses the optimal choices investors make ot identify the e cient portfolio as the market
portfolio, the portfolio of all stocks and securities in the market.
To obtain this result we need to make 3 assumptions regarding the behaviour of investors:
- Investors can buy and sell all securities at competitive market prices and can borrow
and lend at the risk-free interest rate
- Investors hold only e cient portfolios of traded securities, portfolios that yield the
maximum expected return for a given level of volatility
- Investors have homogenous expectations regarding the volatilities, correlations and
expected returns of securities
When the CAPM assumptions hold, the market portfolio is e cient, so the tangent
portfolio then becomes the market portfolio. When the tangent line goes through the
market portfolio it is called the capital market line (CML). According to CALM, all
investors should choose a portfolio on the capital market line, by holding some
combination of risk-free security and the market portfolio.
Since the market portfolio in this case is the e cient portfolio, if we don’t know the
expected return of a security or the cost of capital of an investment, we can use the
CAPM to nd it by using the market portfolio as a benchmark.
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The beta of the security is then de ned as:
The beta of the security measures its volatility due to market risk relative to the market as
a whole, and thus captures the security’s sensitivity to market risk. Following the Law of
One Price, in a competitive market, investments with similar risk should have the same
expected return. Because investors can eliminate rm-speci c risk by diversifying their
portfolios, the right measure of risk is the investment’s beta with the market portfolio. As
the CAPM model states, the investment’s expected return should therefore match the
expected return of the capital market line portfolio with the same level of market risk.
Knowing that there is a linear relationship between a stock’s beta and its expected return,
we can see (in the picture below) that there is a line going through the risk-free investment
with a beta of 0 and the market with a beta of 1 - that is called the security market line
(SML). Under the CAPM assumptions, the security market line is the line along which all
individual securities should lie when plotted according to their expected return and beta.
Because the security market line applies to all tradable investment opportunities, we can
apply it to portfolios as well. Consequently, the expected return of a portfolio depends on
the portfolio’s beta, which we can calculate as follows:
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In other words, the beta of a portfolio is the weighted average beta of the securities in the
portfolio!
OPTIONS
A nancial option contract gives its owner the right to purchase or sell an asset at a xed
price at some future date. These can be split into 2 types of contracts, call options and
put options. A call option gives the owner the right to buy the asset, while a put option
gives the owner the right to sell the asset. Since an option is a contract between two
parties, there is also an option writer aka the person who takes the other side of the
contract.
When a holder of an option enforces the agreement and buys or sells a share of stock at
the agreed-upon price, he is exercising the option. The price at which the holder buys or
sells the share of stock when the option is exercises is called a strike/exercise price.
There are 2 kinds of options - American options and European options. American
options, the most common kind, allow their holders to exercise the option on any date up
to and including a nal date called the expiration date. European options though only
allow their holders to exercise the option on the expiration date exactly. The option buyer,
also called the option holder, has then a long position in the contract, while the option
seller, also called the option writer, has a short position in the contract.
The reason selling options is lucrative is because options always have positive prices,
thus if you sell an option you get paid for it. The market price of the option is also called
the option premium. This upfront payment compensates the seller for the risk of loss in
the event that the option holder chooses to exercise the option.
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The total number of outstanding contracts of a speci c option is called open interest.
When the exercise price of an option is equal to the current price of the stock, the option
is said to be at-the-money. If the payo from exercising an option immediately is
positive, the option is said to be in-the-money. Call options with strike prices below the
current stock price are in-the-money. Conversely, if the payo from exercising the option
immediately is negative, the option is out-of-the-money.
Options on other nancial securities
Although the most commonly traded options are written on stocks, options on other
nancial assets do exist. For example, a stock index put option can be used to o set the
losses on an investor’s portfolio in a market downturn. Using an option to reduce risk in
this way is called hedging. Options also allow investors to speculate, or place a bet on
the direction in which they believe the market is likely to move.
Option Payo s at Expiration
Assume you own an option with a strike price of 20SEK. If on expiration date the stock
price is greater than the strike price, say 30SEK, you can make money by exercising the
call by buying the stock for 20SEK and immediately selling it in the open market for
30SEK. The 10SEK di erence is then what the option is worth. If S is the stock price at
expiration, K is the exercise price and C if the value of the vall option, the value of the call
at expiration is:
The holder of a put option will exercise the option if the stock price S is below the strike
price of K. Because the holder receives K when the stock is worth S, the holder’s gain is
equal to K-S, thus the value of a put at expiration is:
An investor holding a short position in an option has an obligation - this investor takes the
opposite side of the contract to the investor who is long, thus the short position’s cash
ows are the negative of the long position’s cash ows. Because the investor who is long
an option can only receive money at expiration, aka the investor will not exercise an
option that is out-of-the-money since a short investor can only pay money.
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Assume you have a short position in a call option with an exercise pice of 20SEK. If the
stock price is greater than the strike price of the call, say 25SEK, the holder will exercise
the option, which means that you then have the obligation to sell the stock for the strike
price of 20SEK. Since you must purchase the stock at the market price of 25SEK you
then lose the di erence of 5SEK. However, if the stock price is less than the strike price at
the expiration date then the holder will not exercise the option. In this case, you lose
nothing, you have no obligation.
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Options where the strike price and the stock price are far apart as referred to as deep-inthe-money or deep-out-of-the-money. Important to remember is that stock option
contracts are always written on 100 shares of stock.
Combination of Options
Sometimes investors combine option positions by holding a portfolio of options. In this
case, we describe the most common combinations. Imagine at expiration you have both
a put option and a call option at the same strike price - that is a straddle.
By combining a call option and a put option you will receive cash so long as the options
do not expire at-the-money. The farther away from the money the options are, the more
money you will make. However, to construct the combination requires purchasing both
options, so the pro ts after deducting this cost are negative for stock prices close to the
strike price and positive elsewhere. This combination is known as a straddle. This
strategy is used by investors who expect the stock to be very volatile and move up and
down a large amount but who do not necessarily have a view on which direction the stock
will move.
Put-Call Parity
If you imagine K as the strike price of the option, C as the call price, P as the put price
and S as the stock price, then if both positions have the same price we get:
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In the following equation, the left side of the equation is the cost of buying the stuck and
a put while the right side is the cost of buying a zero-coupon bond with face value K and
a call option. Recall that the price of a zero-coupon bond is just the present value of its
face value. Rearranging terms gives an expression for the price of a European call option
for a non dividend paying stock as:
This relationship between the value of the stock, the bond and call and put options is
known as a put-call parity. It says that the price of a European call equals the price of
the stock plus an otherwise identical put minus the price of a bond that matures on the
exercise date of the option.
But what happens if the stock pays a dividend? In that case, the two di erent ways to
construct portfolio insurance do not have the same payout because the stock will pay a
dividend while the zero-coupon bond will not, thus the two strategies will cost the same
to implement only if we add the PV of future dividends to the combination of the bond
and the call:
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The left side of the following equation is the value of the stock and a put while the right
side is the value of a zero-coupon bond, a call option, and the future dividends paid by
the stock during the life of the option, denoted by Div. Rearranging terms gives the
general put-call parity fortmula: