Chapter 2: Investors, Derivatives, and Risk ManagementChapter objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Section 2.1. Evaluating the risk and the return of individual securities and portfolios.. . .3

Section 2.1.1. The risk of investing in IBM.. . . . . . . . . . . . . . . . . . . . . . . . . . . .4Section 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. . . . . . . . .21Section 2.2.2. The risk premium for a security.. . . . . . . . . . . . . . . . . . . . . . . . .24

Section 2.3. Diversification and risk management.. . . . . . . . . . . . . . . . . . . . . . . . . . . .33Section 2.3.1. Risk management and shareholder wealth. . . . . . . . . . . . . . . . . .36Section 2.3.2. Risk management and shareholder clienteles. . . . . . . . . . . . . . . .41Section 2.3.3. The risk management irrelevance proposition. . . . . . . . . . . . . . .46

1) Diversifiable risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .462) Systematic risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463) Risks valued by investors differently than predicted by the CAPM. . .46Hedging irrelevance proposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Section 2.4. Risk management by investors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47Section 2.5. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50Literature Note. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52Key concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54Review questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57Figure 2.1. Cumulative probability function for IBM and for a stock with same return and

twice the volatility.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60Figure 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 thevolatility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61

Figure 2.3. Efficient frontier without a riskless asset. . . . . . . . . . . . . . . . . . . . . . . . . . .62Figure 2.4. The benefits from diversification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63Figure 2.5. Efficient frontier without a riskless asset.. . . . . . . . . . . . . . . . . . . . . . . . . .64Figure 2.6. Efficient frontier with a risk-free asset.. . . . . . . . . . . . . . . . . . . . . . . . . . . .65Figure 2.7. The CAPM.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66Box: T-bills.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67Box: The CAPM in practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68SUMMARY OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

Chapter 2: Investors, Derivatives, and Risk Management

December 1, 1999

Ren M. Stulz 1997, 1999

1Chapter 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.

2During 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

investors 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

3second 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 portfolios diversification is the

extent to which the funds invested are distributed across securities to reduce the dependence of the

portfolios 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 firms 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 firms 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

4initial 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 stocks

return probability distribution. A probability distribution provides a quantitative measure of the

5likelihood 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 IBMs 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%

6chance 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 IBMs return volatility is low, the absolute value of the difference between IBMs return

7and 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 IBMs 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

8less 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.

9 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 upthe 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

10

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.

11

By their actions, investors would bring pressure on IBMs 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,

12

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 investors 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:

(2.1.)w R Portfolio Returni ii 1

N

==

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

13

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 portfolios 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 +(1-

p)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:

(2.2.)w E(R ) Portfolio Expected Returni ii 1

N

= =

Applying this formula to our problem, we find that the expected return of the investors 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

14

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 portfolios return is the

expected value of the square of the difference between the portfolios 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)

15

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

16

w Var(R) w w Cov(R,R ) Variance of Portfolio Returni2

i i j i jj i

N

i 1

NN

+ ===

i 1

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) + w2

2Var(R2) +2w1w2Cov(R1,R2). More generally, therefore, the formula for the variance

of the return of a portfolio is:

(2.3.)

Applying the formula to our portfolio of IBM and XYZ, we need to know the covariance between

the return of the two securities. Lets 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

17

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.

18

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 portfolios 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

19

( )

Volatility of portfolio (1/ N) *0.3

N*(1/ N) *0.3 ((1/ N)*0.3)

2 2

i 1

N 0.5

2 2 0.5 2 0.5

=

= =

=

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 ones 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:

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,

20

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

21

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 well-

diversified 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. Lets 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

22

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 minimum-

volatility portfolio have positive investments in the risk-free asset. To the right of the minimum-

volatility 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%

23

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

24

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.

25

Cov(R R = Cov( w R R = w Cov(R Rp p i i pN

i i pi=1

N

, ) , ) , )i=

1

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 portfolios expected return resulting from a very small increase in the investors

holdings of the security must just compensate for the change in the portfolios 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:

(2.4.)

26

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.

Lets 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, lets 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. Lets 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

27

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.

28

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 securitys 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 securitys 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

29

hCov(R,R ) (1 h)Cov(R,R ) Cov(R,R )i m z m j m+ - =

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:

(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:

(2.6.)hCov(R R h)Cov(R R h* k*Cov(R Ri m z m j m, ) ( , ) , )+ - =1

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

30

hE(R ) (1 h)R E(R )i f j+ - =

security 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:

(2.8.)[ ]E(R R k E(R Ri f j f) )- = -

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:

(2.9.)[ ]E(R R Cov(R RCov(R R E(R Ri fi m

j mj f)

, ), )

)- = -

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. Lets choose security j to be the market portfolio in our equation. Remember that

31

[ ] [ ]E(R R = w E(R R w E(R Rpf f i i fi=1

N

i m f) ) )- - = - b i

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 is beta, or $i. In this case, our equation becomes:

Capital asset pricing model

(2.10.)[ ]E(R - R = E(R - Ri f i m f) )b

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 markets risk premium. Importantly,

with the CAPM diversifiable risk has no impact on a securitys 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 portfolios 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:

The portfolio share weighted sum of the securities in the portfolio is equal to the beta of the portfolio

32

wCov(R R

Var(R=

Cov(wR RVar(R

=Cov( w R R

Var(R=

Cov(R R

Var(R=i

i m

mi=1

Ni i m

mi=1

N i i

N

m

m

p m

mp

, ))

, ))

, )

)

, )

)

=i 1 b

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:

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

33

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, lets 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 risk-

34

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.

Lets 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

35

[ ]E(C)

1 R E(R ) RV

f M F+ + -=

b

E(C)1+R E(R R

M1+0.05+0.5(0.06)

Mf M f+ -

= =b[ ) ]

$350$324.074

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 firms beta

times the risk premium on the market. Consequently, firm value must be such that:

(2.12.)[ ]( )E(C) V 1 R E(R ) Rf m f= + + -b

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:

(2.13.)

Using this formula, we can value our gold mining firm. Lets 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:

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

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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? Lets 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

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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 firms operations to reduce risk.

1) Financial risk management policy that decreases the firms 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 firms 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 firms 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 firms 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

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expected cash flow by the cost of risk management. Managers will therefore never be rewarded by

shareholders for decreasing the firms diversifiable risk at a cost because shareholders can eliminate

the firms 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 firms systematic risk. We now

evaluate whether it is worthwhile for management to incur costs to decrease the firms 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 IBMs 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

39

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

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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 IBMs 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. Lets 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 investments 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 firms 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.

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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, Homestakes

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

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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 portfolios

expected return and volatility. They do not care about their portfolios 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 CAPMs 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, lets 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, lets consider explicitly the case where the firm hedges its gold price risk

with a forward contract on gold. We denote by Q the firms production and by G the price of gold

in one year. There is no uncertainty about Q.

Lets start with the case where there is no clientele effect to have a benchmark. In this case,

the CAPM applies