11. Still the same: 0.35. Indeed, this is the case for any MVE portfolio now.

12. If you sketch it, the intuition is that when the risk-free rate is lower, the tangency portfolio should have less

risk and less reward. With rF = 3%, the weight on portfolio E1 is now 81.606%. The tangency portfolio has

a mean rate of return of 12.759% (excess return, 9.759%), and a standard deviation of 22.344%. The Sharpe

ratio is about 0.4368.

13. If you sketch it, the intuition is that when the risk-free rate is higher, the tangency portfolio should have more

risk and more reward. With rF = 3%, the weight on portfolio E1 is now ’23.949%. The tangency portfolio has

a mean rate of return of 28.592% (excess return, 19.592%), and a standard deviation of 82.607%. The Sharpe

ratio is about 2371.

14. needs to be changed from 17% G to 17.5% H

Cov(˜S&P500 , rE3 ) = Cov(˜S&P500 , 10.7%·˜S&P500 + +

r r r r 24.8%·˜Sony )

r

˜ 64.5%·˜IBM

= 10.7%·Cov(˜S&P500 , rS&P500 ) + 64.5%·Cov(˜S&P500 , rIBM ) + 24.8%·Cov(˜S&P500 , rSony )

r r r

˜ ˜ ˜

= 10.7%·Cov(˜S&P500 , rS&P500 ) + 64.5%·Cov(˜S&P500 , rIBM ) + 24.8%·Cov(˜S&P500 , rSony )

r r r

˜ ˜ ˜

= + +

10.7%·0.0362 64.5%·0.0330 24.8%·0.0477

≈ 0.0370

(16.57)

so the beta of S&P500 with respect to E3 is 0.0370/0.35652 ≈ 0.29, which con¬rms the computations in

Formula 16.23. The other betas are correct, too.

15.

(16.58)

8% = ± + (14% ’ ±) · 0.5 ⇐’ ± = 2% .

Therefore,

(16.59)

E (˜i ) = 2% + (14% ’ 2%) · 2 = 26% .

r

16.

(a) βE5,E2 = Cov(˜E5 , rE2 )/V (˜E2 ) = 17.974%/12.707% = 1.414.

r ar r

˜

(b) Yes, stock E5 does lie exactly on the line. This means that portfolio E2 is still MVE: you cannot do better.

¬le=optimalp¬o-g.tex: RP

419

Section A. Advanced Appendix: Excessive Proofs.

(c) The mean rate of return of this combination portfolio is E (˜) = 99%·17% + 1%·21.029% = 17.040%.

r

For E2 to remain MVE, the combination portfolio must not be more risky. It is V (˜) = (99%)2 ·12.707%+

ar r

(1%)2 ·41.978% + 2·1%·99% = 12.814%. This is higher than the variance of

rE2

˜

alone. Therefore, this combination portfolio is not superior to

rE2

˜

.

(d) The mean rate of return of this E (˜) = 101%·17% + (’1%)·21.029% = 16.960%. This portfolio already

r

has lower mean, so√ cannot dominate portfolio E2. You do not even have to compute the risk. (The

it

Sdv happens to be 0.126.)

17.

(a) Beta has not changed. βE5,E2 = Cov(˜E5 , rE2 )/V (˜E2 ) = 17.974%/12.707% = 1.414.

r ar r

˜

(b) No, stock E5 does not lie on the line. This means that portfolio E2 is no longer MVE: you can do better.

(c) The mean rate of return of this combination portfolio is E (˜) = 99%·17% + 1%·10.0% = 16.93%. This

r

portfolio already has lower mean, so it cannot dominate portfolio E2 .

(d) The mean rate of return of this combination portfolio is E (˜) = 101%·17%+(’1%)·10.0% = 17.07%. The

r

variance of the combination portfolio is V (˜) = (101%) ·12.707%+(’1%)2 ·41.978%+2·(’1%)·99% =

2

ar r

12.603%. Thus, this combination portfolio has both higher mean and lower risk than portfolio E2 .

Therefore, portfolio E2 is no longer MVE!.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)

¬le=optimalp¬o-g.tex: LP

420 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

CHAPTER 17

The CAPM: A Cookbook Recipe Approach

How much expected return should investments o¬er?

last ¬le change: Feb 5, 2006 (18:22h)

last major edit: Sep 2004

This chapter appears in the Survey text only.

The previous chapters explored your best investment choices. This chapter explores the

economy-wide consequences if all important investors in the economy were to engage in such

good investment choices”a fair relationship between expected rates of return and market beta,

called the CAPM.

This chapter o¬ers the “recipe version” of the CAPM. That is, it will show you how to use the

model even if you do not know why it is the right model or where it comes from. Chapter 18

will explain and critically evaluate the theory behind the CAPM. Chapters 21 and 22 will show

how to use the CAPM in a capital budgeting context.

Warning”this chapter is long and covers a lot of ground.

421

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422 Chapter 17. The CAPM: A Cookbook Recipe Approach.

17·1. The Opportunity Cost of Capital

Why is it important what portfolios other investors are buying? It is not important if you just

Knowing what (all) other

investors do is less want to invest and buy your own portfolio. Regardless of what other investors are holding, you

important to an

can make your own investment decisions. But it is very important for corporations who want

investor, than it is to a

to sell investments. They need to know what investors in the aggregate will want.

¬rm that wants to sell to

investors.

So put yourself into the shoes of a company that wants to act in the best possible way on

The ¬rm must know the

opportunities available behalf of its owner. The capital that investors have given the ¬rm has an opportunity cost”

to investors to know

investors could put their money into alternative projects elsewhere. Thus, a manager acting on

what they like and

behalf of the investors needs to determine what projects investors ¬nd worthwhile and what

dislike.

projects investors do not ¬nd worthwhile”which projects investors would want to buy and

which projects they would want to pass up. If the ¬rm™s projects are not good enough, it is

better to not adopt these projects and instead return the money to investors.

So, thinking about opportunity costs is how managers should determine the price at which their

This is why the

appropriate expected investors”representative investors in the economy”are willing to purchase projects. This is

rate of return is called

how managers should make the decision of what projects to take and what projects to avoid.

the cost of capital.

This task of “¬nding a fair price” is the process of determining an appropriate “cost of capital”

(expected rate of return) for a project. (See also Section 8·3.) The Capital Asset Pricing Model

(or CAPM) provides this number.

Important: The CAPM stipulates an opportunity cost of capital. This opportu-

nity cost helps corporate managers determine whether a particular project is ben-

e¬cial or detrimental for the corporate investors”whether the corporation should

take the project or instead return cash to its investors for better uses elsewhere.

¬le=capmrecipe-g.tex: RP

423

Section 17·2. The CAPM.

17·2. The CAPM

For the most important CAPM insight, you do not need a formal model. After the principle Higher Risk can mean

higher reward...but not

that diversi¬cation reduces risk, the second most important principle of investments is that it

always.

takes extra compensation to get investors to accept extra risk. This does not mean that, as an

investor, you can expect to receive extra compensation for risks that you do not need to take.

You can easily avoid betting in Las Vegas, so you are unlikely to receive extra compensation”a

high expected rate of return”when betting there. But it does mean that when a ¬rm wants

investors to accept risks that they cannot easily diversify away, the ¬rm must o¬er a higher

expected rate of return in order to get investors to accept this risk. The CAPM is a model that

translates this principle into speci¬c costs of capital.

17·2.A. The Premise and Formula

The basic premise of the Capital Asset Pricing Model is that investors”to whom the ¬rm wants The CAPM Formula is

similar to the Formula

to sell its projects”are currently holding a well-diversi¬ed market portfolio. The CAPM further

from Section 16·5,

stipulates that this market portfolio is also mean-variance e¬cient. For a new project to be except the ef¬cient

desired by investors and become part of their mean-variance e¬cient portfolio, it must o¬er a portfolio is M.

fair risk/reward trade-o¬. You already know from Section 16·5 what fair risk/rewards trade-

o¬s mean in speci¬c terms”a relationship between the project™s market-beta and the project™s

expected rate of return. Recall that the project™s risk contribution is measured by beta with

respect to the investors™ portfolio, here assumed to be the stock market M,

Cov(˜i , rM )

r˜

βi,M ≡ (17.1)

.

Var(˜M )

r

It is common to call this the beta, rather than the market-beta. Now, this beta is a parameter

which you are assumed to know”though in real life, you can only estimate it. Stocks that

strongly move together with the market (high beta) are relatively risky, because they do not help

investors”already holding the widely diversi¬ed market portfolio”to diversify. The project™s

reward contribution is measured by its expected rate of return, E( ri ). In the CAPM, securities

˜

that contribute more risk have to o¬er a higher expected rate of return.

Important: The CAPM states that the relationship between risk contribution

(βi,M ) and reward (E(˜i )) is

r

(17.2)

E (˜i ) = rF + E (˜M ) ’ rF · βi,M ,

r r

where rF is the risk-free rate, and E(˜M ) is the expected rate of return on the

r

stock market. Through its choice of beta, this formula assumes that the relevant

portfolio held by investors is the overall market portfolio.

E(˜M ) ’ rF is also called the Equity Premium or Market Risk Premium.

r

The CAPM speci¬cally places little importance on the standard deviation of an individual in- The CAPM Formula is

about the suitable

vestment (a stock, project, fund, etc.). A stock™s own standard deviation is only a measure of

risk/reward trade-off for

how risky it is by itself, which would only be of relevance to an investor who holds just this an investor holding the

one stock and nothing else. But our CAPM investors are smarter. They care about their overall market portfolio.

portfolio risk, and keep it low by holding the widely diversi¬ed overall stock market portfolio.

To them, the one new stock matters only to the extent that it alters their market portfolio™s

risk”which is best measured by the stock™s beta.

¬le=capmrecipe-g.tex: LP

424 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Market-beta has an interesting use outside the CAPM. For example, if you believe IBM will

Side Note:

outperform the stock market, but you do not know whether the stock market will do well or do poorly, you

may want to hedge out the market risk. If β is 2, for example, it would mean that for each 10% rate of return

in the stock market, IBM would move 20%. If you purchase $100 in IBM and go short β · $100 = $200 in the

stock market, then you have neutralized the systematic stock market risk and therefore are left only with the

idiosyncratic risk of IBM.

Solve Now!

Q 17.1 If the stock market has gone up by 16%, and the risk-free rate of return is 4%, then what

would you expect a ¬rm i with a stock-market beta of 1.5 to have returned?

So, to estimate an appropriate CAPM expected rate of return for a project or ¬rm, i.e., the cost

The CAPM has three

inputs. of capital, you need three inputs:

1. The risk-free rate of return, rF .

2. The expected rate of return on the market, E(˜M ).

r

3. A ¬rm™s or project™s beta with respect to the market, βi,M .

As always, you are really interested in the future expected rate of return on the market, and

the future beta of a ¬rm/project with respect to the market, and not in the past average rates

of return or beta. And, as always, you usually have no choice but to rely on estimates based

on historical data. In Section 17·4, we shall discuss in more detail how to best estimate each

CAPM input. But ¬rst we will explore the model itself, assuming we already know the inputs.

17·2.B. The Security Markets Line (SML)

Let us apply the CAPM in a speci¬c example. Assume that the risk-free rate is 3% per year,

An example of what rate

of returns individual and that the stock market o¬ers an expected rate of return of 8% per year. The CAPM formula

securities should offer.

then states that a stock with a beta of 1 should o¬er an expected rate of return of 3% + (8% ’

3%)·1 = 8% per year; that a stock with a beta of 0 should o¬er an expected rate of return of

3% + (8% ’ 3%)·0 = 3% per year; that a stock with a beta of 1/2 should o¬er an expected rate

of return of 3% + (8% ’ 3%)·0.5 = 5.5% per year; that a stock with a beta of 2 should o¬er an

expected rate of return of 3% + (8% ’ 3%)·2 = 13% per year; and so on.

The CAPM equation is often graphed as the security markets line, which shows the relationship

The Security Markets

Line, or SML, is just the between the expected rate of return of a project and its beta. Figure 17.1 draws a ¬rst security

CAPM formula.

markets line, using stocks named A through F. Each stock (or project) is a point in this coor-

dinate system. Because all securities properly follow the CAPM formula in our example, they

must lie on a straight line. In other words, the SML line is just a graphical representation of the

CAPM Formula 17.2 on Page 423. The slope of this line is the equity premium, E(˜M ) ’ rF , the

r

intercept is the risk-free rate, rF .

Alas, in the real world, even if the CAPM holds, you would not have the data to draw Figure 17.1.

The “Security Market

Line” in an Ideal CAPM The reason is that you do not know true expected returns and true market-betas. So, Figure 17.2

World

plots two graphs in a perfect CAPM world. Graph (A) repeats Figure 17.1 and presumes you

know CAPM inputs”the true market-betas and true expected rates of return”although in truth

you really cannot observe them. This line is perfectly straight. In Graph (B), presume you

know only observables”estimates of expected returns and betas, presumably based mostly

on historical data averages. Now, you can only plot an “estimated security market line,” not

the “true security market line.” Of course, you hope that our historical averages are good,

unbiased estimates of true market-beta and true expected rates of return (and this is a big if),

so the line will look at least approximately straight. (Section 13·1 already discussed some of

the pitfalls.) A workable version of the CAPM thus can only state that there should roughly be

a linear relationship between the data-estimated market beta and the data-estimated expected

rate of return, just as drawn here.

¬le=capmrecipe-g.tex: RP

425

Section 17·2. The CAPM.

Figure 17.1. The Security Market Line For Securities A“F

*

0.02 0.04 0.06 0.08 0.10

Project Expected Rate of Return (E(Ri))

D

˜

*M

*C

Slope is the 5% Equity Premium

F

Intercept is the 3%

*

risk free interest rate

*B

A

’0.02

*

’1.0 ’0.5 0.0 0.5 1.0 1.5

Project Beta (bi )

E (˜i ) E (˜i )

βi,M r βi,M r

Stock Stock

A ’1.0 ’2.0% C 0.5 5.5%

B ’0.5 M

0.5% 1.0 8.0%

F D

0.0 3.0% 1.5 10.5%

This graph plots the CAPM relation E (˜i ) = rF + [E (˜M ) ’ rF ] · βi,M = 3% + (8% ’ 3%) · βi,M . That is, we assume

r r

that the risk-free rate is 3%, and the equity premium is 5%. M can be the market, or any security with a βi,M = 1. F

can be the risk-free rate or any security with a βi,M = 0.

Solve Now!

Q 17.2 The risk-free rate is 4%. The expected rate of return on the stock market is 7%. What is

the appropriate cost of capital for a project that has a beta of 3?

Q 17.3 The risk-free rate is 4%. The expected rate of return on the stock market is 12%. What is

the appropriate cost of capital for a project that has a beta of 3?

Q 17.4 The risk-free rate is 4%. The expected rate of return on the stock market is 7%. A

corporation intends to issue publicly traded bonds which promise a rate of return of 6%, and

o¬er an expected rate of return of 5%. What is the implicit beta of the bonds?

Q 17.5 Draw the security market line if the risk-free rate is 5% and the equity premium is 4%.

Q 17.6 What is the equity premium, both mathematically and intuitively?

¬le=capmrecipe-g.tex: LP

426 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Figure 17.2. The Security Market Line in an Ideal CAPM World

(A) The Relationship Among Unobservable Variables

*

10

Known Expected Rate of Return

8 *

Market M

6

*

4

*

Risk’Free Treasury F

2

*

0

’2

*

’1.0 ’0.5 0.0 0.5 1.0 1.5

Known Beta With The Stock Market

(B) The Relationship Among Observable Variables

****

*

15

* *

****

* *

Historical Average Rate of Return

* *

*

***

10

*

**

*

*

** *

*

*M * *

**

**

** * * *

* *

*

**

5

* **

* *

* F ***

**

*

**

*

0

**

*

*

* *

*

**

’5

*

*

’1 0 1 2

Historical Beta With The Stock Market

Historical average returns and historical betas are estimated from the data”and hopefully representative of the

true underlying mean returns and true betas, which in turn means that they are also indicative of the future mean

returns and betas.

¬le=capmrecipe-joint.tex: RP

427

Section 17·3. Using the CAPM Cost of Capital in the NPV Context:Revisiting The Default Premium and Risk Premium.

17·3. Using the CAPM Cost of Capital in the NPV Context:

Revisiting The Default Premium and Risk Premium

An important reason why you worked through the CAPM in the ¬rst place was to obtain the We usually use the CAPM

expected rate of return

quantities that you need in the denominator of the NPV formula,

in the NPV denominator.

˜ ˜

E (C1 ) E (C2 )

NPV = CF0 + + + ... (17.3)

1 + E (˜0,1 ) 1 + E (˜0,2 )

r r

The CAPM tells you that cash ¬‚ows that correlate more with the overall market are of less value

to your investors, and therefore require a higher expected rate of return (E(˜)) in order to pass

r

muster (well, the hurdle rate).

Although mentioned before in Chapter 5, it is important to reiterate that the CAPM expected Do not lose the forest:

the CAPM has nothing to

rate of return (based on beta) does not take default risk into account. In the NPV formula, the

do with default risk.

default risk enters the valuation in the expected cash ¬‚ow numerator, not in the expected rate

of return denominator. So, recall the important box on Page 92, which decomposed rates of

return into three parts:

Promised Rate of Return = Time Premium + + Risk Premium .

Default Premium

(17.4)

= Time Premium + + Risk Premium .

Actual Earned Rate Default Realization

Expected Rate of Return = Time Premium + Expected Risk Premium .

The CAPM gives you the expected rate of return, which consists of the time premium and the

expected risk premium. It does not give you any default premium. This is important enough

to put in a box:

Important: The CAPM provides an expected rate of return. It does not include

a default premium. The probability of default must be handled in the NPV numer-

ator through the expected cash ¬‚ow, and not in the NPV denominator through the

expected rate of return.

How do you put the default risk and CAPM risk into one valuation? Here is an example. Say A speci¬c example.

you want to determine the PV of a corporate zero bond that has a beta of 0.25, and promises

to deliver $200 next year. This bond pays o¬ 95% of the time, and 5% of the time it totally

defaults. Assume that the risk-free rate of return is 6% per annum, and the expected rate of

return on the market is 10%. Therefore, the CAPM states that the expected rate of return on

your bond must be

(17.5)

E (˜Bond,t=0,1 ) = rF + E (˜M ’ rF ) · βBond,M = 6% + 4% · 0.25 = 7% .

r r

Of course, this has not yet taken the bond™s default risk into account. You must still adjust the

numerator (promised payments) for the probability of default”you expect to receive not $200,

but

˜

E (CBond,t=1 ) = 95% · $200 + 5% · 0 = $190.

(17.6)

= Prob(No Default) · Promise + Prob(Default) · Nothing .

Therefore, the CAPM states that the value of the bond is

˜

E (CBond,t=1 ) $190

PVBond,t=0 = = ≈ $177.57 . (17.7)

1 + E (˜Bond,t=0,1 ) 1 + 7%

r

¬le=capmrecipe-g.tex: LP

428 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Given this price, you can also compute the promised (or quoted) rate of return on this bond,

$200 ’ $177.57

Promised r0,1 = ≈ 12.6%

$177.57

(17.8)

Promised CF1 ’ CF0

= .

CF0

Although you rarely need to decompose quoted interest rates in practice, quantifying the three

components in this example helps to better conceptualize the magnitudes of the components

of quoted rates. For your bond, the time-premium of money is 6% per annum”it is the rate of

return that an equivalent-term Treasury bond o¬ers. The time-premium plus the risk-premium

is provided by the CAPM, and it is 7% per annum. Therefore, 1% per annum is your “average”

compensation for your willingness to hold this risky bond, rather than the risk-free Treasury

bond. The remaining 12.6% ’ 7% ≈ 5.6% per annum is the default premium: you do not expect

to earn money from this part “on average;” you only earn it if the bond does not default.

= + + .

12.6% 6% 5.6% 1%

(17.9)

Promised Interest Rate = Time Premium + Default Premium + Risk Premium .

As in the example, in the real world, most bonds have fairly small market betas and thus

risk premia. Instead, most of the premium that risky bonds quote above equivalent risk-free

Treasury rates is due to default risk.

Side Note: In the real world, corporate bonds also have important liquidity premia built-in, which compen-

sates investors for not being able to easily buy/sell these securities. The broker/market-makers tend to earn this

premium. The liquidity premium di¬ers across investors: retail investors are charged higher liquidity premia

than bond funds. As a retail investor, it is best not to purchase individual bonds.

Solve Now!

Q 17.7 A corporate bond with a beta of 0.2 will pay o¬ next year with 99% probability. The

risk-free rate is 3% per annum, the risk-premium is 5% per annum. What is the price of this bond,

and its promised rate of return?

Q 17.8 Continue: Decompose the bond™s quoted rate of return into its components.

Q 17.9 Going to your school has total additional and opportunity costs of $30,000 this year

and up-front. With 90% probability, you are likely to graduate from your school. If you do not

graduate, you have lost the entire sum. Graduating from the school will increase your 40-year

lifetime annual salary by roughly $5,000 per year, but more so if the stock market rate of return

is high than when it is low. For argument™s sake, assume that your extra-income beta is 1.5.

Assume the risk-free rate is 3%, the equity premium is 5%. What is the value of your education?

¬le=capmrecipe-joint.tex: RP

429

Section 17·4. Estimating CAPM Inputs.

17·4. Estimating CAPM Inputs

Let us now discuss how we can obtain reasonable estimates of the three inputs into the CAPM

formula,

(17.10)

E (˜i ) = rF + E (˜M ) ’ rF · βi,M .

r r

17·4.A. The Equity Premium E(˜M ) ’ rF

r

The most-di¬cult-to-estimate input in the CAPM is the equity premium. It measures the extra The equity premium

must be provided as a

expected rate of return that risky projects are o¬ering above and beyond what risk-free projects

CAPM input. Estimates

are o¬ering. The value you choose for the equity premium can have a tremendous in¬‚uence are all over the map;

on your estimated costs of capital. Of course, the CAPM model assumes that you know the reasonable ones can

range from 2% to 8%

expected rate of return on the market, not that you have to estimate it. Yet, in real life, the

per year.

equity premium is not posted anywhere, and no one really knows the correct number. There are

a number of methods to guesstimate it”but they unfortunately do not tend to agree with one

another. This leaves me with two choices: I can either throw you one estimate and pretend it

is the only one, or I can tell you the di¬erent methods that lead to di¬ering estimates. I prefer

the latter, if only because the former would eventually leave you startled to ¬nd that someone

else has used another number and has come up with another cost of capital estimate. We will

discuss the intuition behind each method and the speci¬c estimate the intuition would suggest.

In this way, you can make up your own mind as to what you deem to be an appropriate equity

premium estimate.

Historical Averages I: The ¬rst estimation method simply relies on historical average equity Method 1: Historical

Averages.

premia as good indicators of future risk premia. As of 2003, the arithmetic average

equity premium since 1926 was about 8.4% per annum. However, if you start computing

the average in 1869, the equity premium estimate drops to around 6.0%. Maybe you

should start in 1771? Or 1971? Which is the best estimation period? No one really knows

what the right start date should be. If you choose too few years, your sample average

could be unreliable. For example, what happened over the last 20 or 30 years might just

have been happen-stance and not representative of the statistical process driving returns.

Your estimate of the mean would carry a lot of uncertainty. The more years you use, the

lower would be your uncertainty (standard error about the mean). However, if you choose

too many years, the data in the earlier part of your sample period may be so di¬erent from

those today that they are no longer relevant”that is, you may incorrectly believe that the

experience of 1880 still has relevance today.

Historical Averages II: The second estimation method looks at historical equity premia in the Method 2: Inverse

Historical Averages.

opposite light. You can draw on an analogy about bonds”if stocks become more desirable,

perhaps because investors have become less risk-averse, then more investors compete to

own them, drive up the price, and thereby lower the future expected rates of return. High

historical rates of return are therefore indicative of low future expected rates of returns.

An extreme version thereof may even that high past equity premia are not indicative of

high future equity premia, but rather indicative of bubbles in the stock market. The

proponents of the bubble view usually cannot quantify the appropriate equity premium,

except to argue that it is lower after recent market run ups”exactly the opposite of what

proponents of the Historical Averages I argue.

A bubble is a run-away market, in which rationality has temporarily disappeared. There

Side Note:

is a lot of debate as to whether bubbles in the stock market ever occurred. A strong case can be made

that technology stocks experienced a bubble from around 1998 to 2000. No one has yet come up with

a rational story based on fundamentals that can explain both why the Nasdaq Index had climbed from

2,280 in March 1999 to 5,000 on March 27, 2000, and why it then dropped back to 1,640 on April 4,

2001.

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430 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Current Predictive Ratios: The third method tries to actively predict the stock market rate of

Method 3: Dividend or

Earnings Yields. return with historical dividend yields (i.e., the dividend payments received by stockhold-

ers). Higher dividend yields should make stocks more attractive and therefore predict

higher future equity premia. The equity premium estimation is usually done in two steps:

¬rst, you must estimate a statistical regression that predicts next year™s equity premium

with this year™s dividend yield; then, you substitute the currently prevailing dividend yield

into your estimated regression to get a prediction. Unfortunately, as of 2003, current div-

idend yields are so low that the predicted equity premia are negative”which is not a

sensible number. Variations of this method have used interest rates of earnings yields,

typically with similar results. In any case, the evidence suggests that this method has

yielded poor predictions”for example, it had predicted low equity premia in the 1990s,

which was a period of superb stock market performance.

Philosophical Prediction: The fourth method wonders how much rate of return is required

Method 4: What is

reasonable reward for to entice reasonable investors to switch from bonds into stocks. Even with an equity

risk?

premium as low as 3%, over 25 years, an equity investor would end up with more than

twice the money of a bond investor. Naturally, in an e¬cient market, nothing comes for

free, and the reward for risk-taking should be just about fair. So, equity premia of 8% just

seem too high for the amount of risk observed in the stock market. This philosophical

method generally suggests equity premia of about 1% to 3%.

Consensus Survey: The ¬fth method just asks people or experts what they deem reasonable.

Method 5: Just ask.

The ranges can vary widely, and seem to correlate with very recent stock market returns.

For example, in late 2000, right after a huge run up in the stock market, surveys by

Fortune or Gallup/Paine-Webber had investors expect equity premia as high as 15%/year.

(They were acutely disappointed: the stock market dropped by as much as 30% over

the following two years. Maybe they just got the sign wrong?!) The consulting ¬rm,

McKinsey, uses a standard of around 5% to 6%, and the social security administration

uses a standard of around 4%. In a survey of ¬nance professors in August 2001, the

common equity premium estimate ranged between 3.5% for a 1-year estimate to 5.5%

for a 30-year estimate. A more recent joint poll by Graham and Harvey (from Duke) and

CFO magazine found that the 2005 average estimate of CFOs was around 3% per annum,

although there was quite some dispersion in this number.

What to choose? Welcome to the club! No one knows the true equity premium. On Monday,

Some recent estimates.

February 28, 2005, the C1 page of the WSJ reported the following average annual after-in¬‚ation

forecasts over the next 44 years:

Anecdote: The Power of Compounding

Assume you invested $1 in 1925. How much would you have in December 2001? If you had invested in

large-¬rm stocks, you would have ended up with $2,279 (10.7% compound average return). If you had

invested in long-term government bonds, you would have ended up with $51 (5.3%). If you had invested

in short-term Treasury bills, you would have ended up with $17 (3.8%). Of course, in¬‚ation was 3.1%, so

$1 in 2001 was more like $0.10 in real terms in 1926. Source: Ibbotson Associates, Chicago. U

¬le=capmrecipe-joint.tex: RP

431

Section 17·4. Estimating CAPM Inputs.

Corp. Equity

Gov.

Name Organization Stocks bonds bonds Premium

William Dudley Goldman Sachs 5.0% 2.0% 2.5% 3.0%

Jeremy Siegel Wharton 6.0% 1.8% 2.3% 4.2%

David Rosenberg Merrill Lynch 4.0% 3.0% 4.0% 1.0%

Ethan Harris Lehman Brothers 4.0% 3.5% 2.5% 0.5%

Robert Shiller Yale 4.6% 2.2% 2.7% 2.4%

Robert LaVorgna Deutsche Bank 6.5% 4.0% 5.0% 2.5%

Parul Jain Nomura 4.5% 3.5% 4.0% 1.0%

John Lonski Moody™s 4.0% 2.0% 3.0% 2.0%

David Malpass Bear Stearns 5.5% 3.5% 4.3% 2.0%

Jim Glassman J.P. Morgan 4.0% 2.5% 3.5% 1.5%

Average 2.0%

The equity premium is usually quoted with respect to a short-term interest rate, because these

are typically safer and therefore closer to the risk-free rate that is in the spirit of the CAPM. This

is why you may want to add another 1% to the equity premium estimate in this table”long-term

government bonds usually carry higher interest rates than their short-term counterparts. On

the other hand, if your project is longer term, you may want to adopt a risk-free rate that is

more similar to your project™s duration, and thus prefer the equity premium estimates in this

table.

You now know that noone can tell you one authoritative number for the equity premium. Ev- Pick a good estimate,

and use it for all

eryone is guessing, but there is no way around it”you have to take a stance on the equity

similar-horizon projects.

premium. I am only able to give you the arguments that you should contemplate when you are

picking your number. I can however also give you my own take: First, I have my doubts that

equity premia will return to the historical levels of 8% anytime soon. (The twentieth century

was the American Century for a good reason: there were a lot of positive surprises for American

investors.) So, I personally prefer equity premia estimates between 2% and 4%. Interestingly, it

appears that I am not the only one, as the above table shows. (It is my impression that there is

relatively less dispersion in equity premia forecasts today than there was just ¬ve to ten years

ago.) But realize that reasonable individuals can choose equity premia estimates as low as 1%

or as high as 8%”of course, I personally ¬nd such estimates less believable the further they are

from my own, personal range. And I ¬nd anything outside this 1% to 8% range just too tough

to swallow. Second, whatever equity premium you choose, be consistent. Do not use 3% for

investing in one asset (say, project A), and 8% for investing in another (say, project B). Being

consistent will often reduce your relative mistakes in choosing one project over another.

¬le=capmrecipe-g.tex: LP

432 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Unfortunately, you cannot allow our limited knowledge of the equity premium stop you from

The CAPM is about

relative pricing, not using the CAPM”in fact, you cannot allow it to stop you from making investment choices. Yes,

absolute pricing.

the equity premium may be di¬cult to estimate, but there is really no way around taking a

stance. Indeed, you can think of the CAPM as telling you the relative expected rate of return

for projects, not their absolute expected rate of return. Given an estimate of how much risky

projects should earn relative to non-risky projects, the CAPM can tell you the right costs of

capital for projects of riskiness “beta.” But the basic judgment of the appropriate spread

between risky and non-risky projects is left up to you.

The need to judge the appropriate reward for risky projects relative to risk-free projects is not

No way around it: the

equity premium is the even just exclusive to the CAPM and corporations. It also matters for your personal investments:

most important number

if you believe that the equity premium is high, you should allocate a lot of your personal assets

in ¬nance, and we need

to purchasing stocks rather than bonds. So, it is not only because of the CAPM formula that

to pull it out of our hats.

the equity premium may be the single most interesting number in ¬nance.

Finally, I have been deliberately vague about the “market.” In CAPM theory, the market should

We shall use the S&P500

as our market. be all investable assets in the economy. In practice, we typically use only a stock market index.

And among stock market indexes, it often does not matter too much which index is used”be

it the value-weighted stock market index, the S&P 500, or the Dow-Jones 30. The S&P500 is

perhaps the most often used standin for the stock market, because its performance is posted

everywhere and historical data are readily available. In sum, using the S&P500 as the market is

a reasonable simpli¬cation from the perspective of a corporate executive.

Solve Now!

Q 17.10 What are appropriate equity premium estimates? What are not? What kind of reasoning

are you relying on?

Anecdote: The American Century?

Was this really the “American Century?”

The in¬‚ation-adjusted compound rate of return in the United States was about 6% per year from 1920 to 1995.

In contrast, an investor who would have invested in Romania in 1937 would have experienced not only the

German invasion and Soviet domination, but also a real annual capital appreciation of about ’27% per annum

over the 4 years of Hungarian stock market existence (1937“1941). Similar fates befell many other East European

countries”but even countries not experiencing political disasters often proved to be less stellar investments.

For example, Argentina had a stock market from 1947 to 1965, even though its only function seems to have

been to wipe out its investors. Peru tried three times: from 1941 to 1953, its stock market investors lost all their

money. From 1957 to 1977, its stock market investors again lost all their money. But three times is a charm:

From 1988 to 1995, its investors earned a whopping 63% real rate of return. India™s stock market started in

1940, and o¬ered its investors a real rate of return of just about ’1% per annum. Pakistan started in 1960, and

o¬ered about ’0.1% per annum.

Even European countries with long stock market histories and no political trouble did not perform as well as the

United States. For example, Switzerland and Denmark earned nominal rates of return of about 5% per annum

from 1921 to 1995, while the United States earned about 8% per annum.

The United States stock market was indeed an unusual above-average performer in the twentieth century. Will

the twenty-¬rst century be the Chinese century?

Source: Goetzmann and Jorion.

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433

Section 17·4. Estimating CAPM Inputs.

17·4.B. The Risk-Free Rate and Multi-Year Considerations (rF )

The second input of interest is the risk-free rate of return. The risk-free rate is relatively easily Which risk-free rate?

obtained from Treasury bonds. There is one small issue, though”which one? What if Treasury

bonds yield 2%/year over 1 year, 4%/year over 10 years, and 5%/year over 30 years? How would

you use the CAPM? Which interest rate should you pick in a multi-year context?

Actually, the CAPM o¬ers no guidance, because it has no concept of more than one single time- Advice: Pick the

closest-term interest

period. It therefore does not understand why there is a yield curve (di¬erent expected rates of

rate.

return over di¬erent horizons). However, from a practical perspective, it makes sense to use

the yield on a Treasury bond that is of similar length as a project™s approximate lifespan. So, a

good heuristic is to pick the risk-free rate closest in some economic sense (maturity or duration)

to our project. For example, to value a machine that produces for three years, it makes sense to

use an average of the 1-year, 2-year, and 3-year risk-free interest rates, perhaps 2.5% per annum.

On the other hand, if you have a 10-year project, you would probably use 4% as your risk-free

rate of return. This heuristic has an intuitive justi¬cation, too”think about the opportunity

cost of capital for a zero-beta investment. If you are willing to commit your money for 10 years,

you could earn the 10-year Treasury rate of return. It would be your opportunity cost of capital.

If you are willing to commit your money only for 3 months, you could only earn the 3-month

Treasury rate”a lower opportunity cost for your capital. One important sidenote, however, is

that you should use the same risk-free rate in the calculation of the equity premium”so, if you

use a higher risk-free rate because your project is longer-term, you would want to use a lower

equity premium where the risk-free rate enters negatively.

Solve Now!

Q 17.11 What is today™s risk-free rate for a 1-year project? For a 10-year project?

Q 17.12 Which risk-free rate should you be using for a project that will yield $5 million each

year for 10 years?

17·4.C. Investment Projects™ Market Betas (βi,M )

Finally, you must estimate your project™s market beta, which measures how your project rates Unlike the risk-free rate

and the equity premium,

of return ¬‚uctuate with the market. Unlike the previous two inputs, which are the same for every

beta is speci¬c to each

project/stock in the economy, the beta input depends on your speci¬c project characteristics: project.

di¬erent investments have di¬erent betas.

Let us gain some intuition on how the market-beta should relate the returns of individual Beta creates both a

mean rate of return, and

stocks to those of the market. The market-beta has both an in¬‚uence on the expected return of

an “ampli¬cation factor”

projects and on the range of observed returns. (This works through the part of a stock™s price of the market rate of

movement volatility that is caused by the market. There is also very important idiosyncratic return.

price movement, but we shall ignore it in this example.) Say the risk-free rate is 3%, the expected

rate of return on the market is 8%, and therefore the equity premium is 5%. A stock with a beta

of ’1 would therefore have an expected rate of return of ’2%, a stock with a beta of +2 would

have an expected rate of return of +13%. However, more than likely, the stock market rate

of return will not be exactly 8%. So, let us entertain one positive and one negative market

scenario as a standin for market volatility. If the stock market were to drop by 10% relative

to its mean of 8%”i.e., return an absolute ’2%”then our ¬rst stock would not earn ’2%, but

’2% + βi · (˜M ’ E(˜M )) = ’2% + (’1) · (’2% ’ 8%) = +8%, and our second stock would not

r r

earn +13%, but +13% + (+2) · (’10%) = ’7%. Conversely, if the stock market were to increase

by 10% relative to its mean”i.e. return an absolute +18%”we would expect our negative beta

stock to do really poorly (’2% + βi · (˜M ’ E(˜M )) = ’2% + (’1) · (+18% ’ 8%) = ’12%) and

r r

our positive beta stock to do really well (+33%).

¬le=capmrecipe-g.tex: LP

434 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Figure 17.3. The E¬ect of Market Beta on Stock Returns in Good and Bad Markets

T E(˜i )

r

t

+30%

t

+20%

t

t

d

If market returns ’2%

t d

t

+10%

s

t d

t

t d

t

u

t

qd

d

t t

' E

t

d t βi,M

’1.0 ’0.5 +0.5 +1.0 +1.5 +2.0

t t

Q

t

If market returns +8%

’10%

t

c

0

if beta (βi,M = Cov(˜i , rM )/V (˜M )) is

r˜ ar r

If market returns +18%

’1.0 +1.0 +2.0

0.0

’2% +3% +8% +13%

The stock™s expected rate of return

If market outperforms its mean of 8% by ’2% ’ 10% 3% ± 0% 8% + 10% 13% + 20%

10% (˜M = +18%), then our stock ri is

r ˜ = ’12% = 3% = 18% = 33%

expected to return

If market underperforms its mean of 8% ’2% + 10% 3% ± 0% 8% ’ 10% 13% ’ 20%

by 10% (˜M = ’2%), then our stock ri is

r ˜ = +8% = 3% = ’2% = ’7%

expected to return

Each line represents the range of return outcomes for one stock (with one particular market beta) if the market

rate of return were to be between ’10% and +10%. The black circle is the unconditional expected rate of return

(or conditional on the market turning in its expected performance of 8%)”i.e., points on the security markets line

E (˜i ) = rF + [E (˜M ) ’ rF ]·βi,M = 3% + 5%·βi,M . The red solid circles show the expected rate of return conditional

r r

on a market rate of return of ’10%. Stocks with negative beta are expected to perform well in this case. The blue

solid circles show the expected rate of return conditional on a market rate of return of +10%. Stocks with a negative

beta are expected to perform poorly in this case.

¬le=capmrecipe-joint.tex: RP

435

Section 17·4. Estimating CAPM Inputs.

In Figure 17.3, we repeat this computation for stocks with di¬erent market-betas. It shows how Beta can be thought of

as an ampli¬er of

they are expected to perform, conditional on whether the market beats its mean (of 8% by 10%,

market movements.

i.e., +18%), hits its mean (of 8%), or misses its mean (of 8% by 10%, i.e., ’2%). You can see how

beta determines both the stock™s expected rate of return”the mean given by the CAPM”and

how it dampens or ampli¬es the e¬ect of the stock market performance on our stock. The

latter is really just our de¬nition of market-beta”it measures how a project comoves with the

stock market. The sign of the market-beta determines whether the investment tends to move

with or against the stock market. And it is of course the CAPM that posits how the expected

rate of return should be increasing with the market-beta.

Depending on the project for which you need a beta, the estimate can be easy or di¬cult to Our plan: discuss

various methods to ¬nd

obtain. We now discuss the three most common sources for beta estimates.

betas

17·4.D. Betas For Publicly Traded Firms

For publicly trading stocks, ¬nding a market-beta is easy. There are many services (e.g., Beta is easy to get for

publicly traded stocks.

Yahoo!Finance) that publish betas. The average beta in the stock market is 1, and most stocks

have betas somewhere between about 0 and 3. Large, low-tech ¬rms tend to have lower betas

than small, high-tech ¬rms, but this is not always the case. In Chapter 13, we have already

talked about beta estimation, and even about shrinkage to improve on estimates. Some sample

market betas on asset classes, country portfolios, and Dow-Jones 30 stocks were in Tables 14.4“

14.7 (Page 346, Page 352, and Page 353). The published betas themselves are estimated from

historical time-series regressions, often monthly data, using our statistical technique that ¬ts

the best ± and β for the “regression” line ri = ±i + βi ·˜M + «. The regression estimator does

r

˜

exactly what we did in Section 13·4.B: it computes the covariance and divides it by the variance.

(Some more sophisticated data providers improve on this simple regression estimate with a

little bit of extra statistical wizardry called shrinkage, which we shall ignore.)

Solve Now!

Q 17.13 If you had representative historical project returns, how would you obtain the stock

market beta?

Q 17.14 Look up the beta for IBM at Yahoo!Finance. How does it compare to the beta of a young

upstart growth company? (Pick one!)

17·4.E. Betas From Comparables and Leverage Adjustments:

Equity Beta vs. Asset Beta

Individual betas are very noisy. For example, a pharmaceutical whose product happened to be Individual Betas can be

noisy”we often use

rejected by the FDA (usually causing a large negative return) in a month in which the market

similar company betas.

happened to go up (down) may end up having a negative (positive) market beta estimate”and

this would likely be totally unrepresentative of the future market beta. (This month would be

a “statistical outlier” or “in¬‚uential regression observation.”) In the long-run, such announce-

ments would appear randomly, so beta would still be the right estimate”but in the long-run,

we will all be dead. To reduce such noise in practice, it is common to estimate not just the beta

of the ¬rm, but to estimate the beta of a couple of similar ¬rms (comparables similar in size

and industry, perhaps), and then to use a beta that re¬‚ects some sort of average among them.

Indeed, if your project has no historical rate of return experience”perhaps because it is only a Using comparable

publicly traded stocks

division of a publicly traded company or because the company is not publicly traded (although

with unlevered Betas.

the CAPM is only meaningful to begin with if the owners need to hold most of their wealth

in the market portfolio)”you may have little choice other than to consider comparable ¬rms.

For example, if you believe your new soda company is similar to PepsiCo, you could adopt the

beta of PepsiCo and use it to compute the CAPM expected rate of return. Realizing that smaller

¬rms than PepsiCo tend to have higher betas, you might increase your beta estimate.

¬le=capmrecipe-g.tex: LP

436 Chapter 17. The CAPM: A Cookbook Recipe Approach.

It is however very important that you draw a clear distinction between equity betas and asset

Leverage Adjustments:

An Intuitive Example. betas. Usually, you have an intuition that your project beta (also called asset beta) is the same

as that of the publicly traded company”but all that you get to see is the comparable™s equity

beta. You must adjust the equity beta for the comparable™s leverage, because stocks that

are more levered have higher equity market-betas”they are riskier. Recall the example from

Chapter 5: when a project was split into debt and equity, the debt became less risky, while the

equity became more risky. This turns out to matter for betas, too.

Table 17.1. The E¬ect of Leverage on Beta

Choice of Capital Structure

(A) Unlevered (B) Split Project

Stock Market Project $150 Debt Equity

Value Today $10.0 trillion $200 $150 $50

if Good Times $13.0 trillion $230 $156 $74

if Bad Times $9.0 trillion $190 $156 $34

Expected Value $11.0 trillion $210 $156 $54

if Good Times +30% +15% +4% +48%

if Bad Times “10% “5% +4% “32%

Expected Rate of Return +10% +5% +4% +8%

Dollar Spread $4 trillion $40 $0 $40

Relative Spread 40% 20% 0% 80%

Market Beta 1.0 0.5 0.0 2.0

To determine how leverage changes beta, consider Table 17.1. In this example, the stock market,

Work one full example.

worth $10 trillion today, is expected to increase by 10% to $11 trillion next year. However,

relative to this expected value, the market can either underperform or overperform (by plus

or minus 20%). Now, your own unlevered project is worth $200 today and has a beta of 1/2.

Therefore, it is expected to return $210, but either 10% above or 10% below its mean of 5%,

and depending on the stock market rate of return. This makes sense: for a 40% di¬erence in

the rate of return on the stock market, your project would su¬er a 20% di¬erence in its rate of

return.

Now ¬nance your project di¬erently. Use an alternative capital structure that consists of $150

The beta of levered

equity scales with in debt and the rest in equity, i.e.,

leverage.

ValueProject = + ValueEquity ,

ValueDebt

ValueEquity

ValueDebt

= +

100%

ValueProject ValueProject (17.11)

= +

wDebt wEquity

100%

= + ,

100% 75% 25%

where the weight of each security in the capital structure is called w.

The debt is default-free, so it can command the risk-free rate, which we now assume to be 4%

The WACC remains the

same regardless of per annum. But being risk-free also means that the debt beta is 0. The value of the levered

capital structure.

equity must then be the remaining $50. Working through the remaining cash ¬‚ows, we ¬nd

that its expected rate of return is 8%, which is both above the risk-free rate and the unlevered

project™s expected rate of return. This higher expected rate of return is necessary to compensate

investors for risk. More importantly, note how your levered equity has a higher market-beta

than the original unlevered project. Instead of translating a market ¬‚uctuation of 40% into a

project ¬‚uctuation of ±20%, the levered equity translates the market ¬‚uctuation of 40% into a

rate of return ¬‚uctuation of ±80%! The beta is now 2, not 0.5.

¬le=capmrecipe-joint.tex: RP

437

Section 17·4. Estimating CAPM Inputs.

This example shows that the weighted expected rate of return and the weighted average beta The weighted average

beta is the overall beta.

add up to their overall project equivalents:

= 75% · 4% + 25% · 8%

5%

E (˜Project ) = wDebt · E (˜Debt ) + wEquity · E (˜Equity ) .

r r r

(17.12)

= 75% · 0 + 25% · 2

0.5

= wDebt · βDebt + wEquity · βEquity

βProject .

The ¬rst equation is called the ¬rm™s weighted-average cost of capital, abbreviated WACC, and

discussed in detail in Chapter 21. In our perfect world, the cost of capital remains invariant

to whatever capital structure you may choose. The latter equation is just a special version of

a general linear property of betas: as you learned earlier, you can take weighted averages of

betas. (On Page 333, we already discussed that the beta of a portfolio is the value-weighted

beta of its component securities.) Therefore, if you know how the ¬rm is ¬nanced, and if you

can guess the beta of the debt, it is easy to translate an equity beta into an asset beta.

βProject,mkt = wDebt ·βDebt,M + wEquity ·βEquity,M

(17.13)

= + = 0.5.

75%·0 25%·2.0

But, in general, where would you get the debt beta from? For large ¬rm stocks that are not

in ¬nancial distress, it is reasonable to presume that debt betas are reasonably close to zero.

This is because the debt is likely to be repaid”and, if not, repayment may not be contingent as

much on the stock market overall, as it may depend on the ¬rm™s circumstances. For small ¬rm

stocks or stocks in ¬nancial distress, bond betas can, however, become signi¬cantly positive.

Important: If project A consists of part B and part C,

(17.14)

A=B+C

then the overall market beta of the combined project A is the weighted average

market beta of its components,

(17.15)

βA,M = wB · βB,M + wC · βC,M ,

where w are weights according to value today, and add up to 1. The components

could be any type of investments, and in particular be the debt and equity of the

same ¬rm. Therefore,

= +

Project Debt Equity

(17.16)

’ βProject,M = wDebt · βDebt,M + wEquity · βEquity,M .

Solve Now!

Q 17.15 A project i is likely to go up by 20% if the stock market goes up by 10%. It is also likely

to go down by 20% if the stock market goes down by 5%. If the risk-free rate of return is 4%, what

would you expect the beta to be?

Q 17.16 A comparable ¬rm (in a comparable business) has an equity beta of 2.5 and a debt/asset

ratio of 2/3. The debt is almost risk-free. Estimate the beta for our ¬rm if projects have alike

betas, but our ¬rm will carry a debt/asset ratio of 1/3.

¬le=capmrecipe-g.tex: LP

438 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Q 17.17 (Continued.) If the risk-free rate is 3% and the equity premium is 2%, what is the expected

rate of return on the comparable ¬rm™s equity and on our own equity?

Q 17.18 A comparable ¬rm (in a comparable business) has an equity beta of 2.5 and a debt/equity

ratio of 2. The debt is almost risk-free. Estimate the beta for our ¬rm if projects have alike betas,

but our ¬rm will carry a debt/equity ratio of 1/2.

Q 17.19 (Continued.) If the risk-free rate is 3% and the equity premium is 2%, what is the expected

rate of return on the comparable ¬rm™s equity and on our own equity?

Q 17.20 You own a stock market portfolio that has a market beta of 2.4, but you are getting

married to someone who has a portfolio with 0.4. You are three times as wealthy as your future

signi¬cant other. What is the beta of your joint portfolio?

Q 17.21 Assume that you can short. If your portfolio has a market beta of 0.6 and you can short

a fund with a market beta of 1, what portfolio do you have to purchase to eliminate all market

risk?

17·4.F. Betas Based on Economic Intuition

Sometimes, there are projects for which there are no good publicly traded ¬rms from which

Intuitive Betas

Guestimating. you can extract a beta estimate. In such cases, you need to make a judgment: how will the

rate of return of your project covary with the stock market? To ¬nd out, rearrange the CAPM

Formula:

E (˜i ) ’ rF

r

E (˜i ) = rF + E (˜M ) ’ rF · βi,M ⇐’ βi,M =

r r . (17.17)

E (˜M ) ’ rF

r

The right side of this formula helps translate your intuition into a beta estimate. You can ask

such questions as “What rate of return (above the risk-free rate) will your project have if the

stock market were to have +10% or “10% rate of return (above the risk-free rate)?” Clearly, such

guess work is di¬cult and error-prone”but it can provide a beta estimate when no other is

available.

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439

Section 17·5. Value Creation and Destruction.

17·5. Value Creation and Destruction

Most of our CAPM applications will be explored in Chapters 21 and 22. Chapters 21 explains We delay applications

until later.

how to use the CAPM in a perfect world without taxes. Chapter 22 explains how to use the

CAPM in the presence of (corporate) income taxes. Because the primary use of the CAPM is to

determine appropriate costs of capital in corporations, it is only in these later chapters that

this book o¬ers enough examples to familiarize you with CAPM applications.

However, there are at least two important and basic concepts that were ¬rst raised in Chap- Important: How to add

value!

ter 7 that we can ¬nally discuss now, given that the CAPM illuminates the cost of capital. The

¬rst concept is almost trivial”it is the question of whether managers should seek to reduce id-

iosyncratic ¬rm risk. The second concept relates to the simplest of insights”that the total net

present value of two projects combined without project externalities is the sum of the project™s

net present value. As always, the concept is straightforward, but the devil is in the details.

17·5.A. Does Risk-Reducing Corporate Diversi¬cation (or Hedging) Create Value?

In the 1960s through 1970s, many ¬rms became conglomerates, that is, companies with widely Diversi¬cation reduces

risk, but does not create

diversi¬ed and often unrelated holdings. Can ¬rms add value through such diversi¬cation?

value.

The answer is “usually no.” Diversi¬cation indeed reduces the standard deviation of the rate

of return of the company”so diversi¬ed companies are less risky”but your investors can just

as well diversify risk for themselves. For example, if your $900 million ¬rm ABC (e.g., with

a beta of 2, and a risk of 20%) is planning to take over the $100 million ¬rm DEF (e.g., with

a beta of 1, and also risk of 20%), the resulting ¬rm is worth $1 billion dollars. ABC +DEF

has indeed an idiosyncratic risk lower than 20% if the two ¬rms are not perfectly correlated,

but your investors (or a mutual fund) could just purchase 90% of ABC and 10% of DEF and

thereby achieve the very same diversi¬cation bene¬ts. If anything, you have robbed investors

of a degree of freedom here: they no longer have the ability to purchase, say, 50% in ABC and

50% in DEF. (In a CAPM world, this does not matter.) The CAPM makes it explicit that the cost

of capital does not change unduly. Say both ¬rms follow the CAPM equation, and say that the

risk-free rate is 3% and the equity premium is 5%,

E (˜ABC ) = 3% + 5% · 2 = 13% ,

r

E (˜ABC ) = rF + E (˜M ) ’ rF · βABC,M

r r ,

(17.18)

E (˜DEF ) = 3% + 5% · 1 = 8%

r ,

E (˜DEF ) = rF + E (˜M ) ’ rF · βDEF,M

r r .

The newly formed company will have an expected rate of return”cost of capital”of

E (˜ABC +DEF ) = 90% · 13% + 10% · 8% = 12.5% ,

r

(17.19)

E (˜ABC +DEF ) = wABC · E (˜ABC ) + wDEF · E (˜DEF ) ,

r r r

and a market-beta of

βABC +DEF,M = 90% · 2 + 10% · 1 = 1.9

(17.20)

βABC +DEF,M = wABC · βABC,M + wDEF · βDEF,M .

The merged company will still follow the CAPM,

E (˜ABC+DEF ) = 3% + 5% · 1.9 = 12.5%

r

(17.21)

E (˜ABC+DEF ) = rF + E (˜M ) ’ rF · βABC+DEF,M

r r .

¬le=capmrecipe-g.tex: LP

440 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Its cost of capital has not unduly increased or declined. In an ideal CAPM world, no value has

been added or destroyed”even though ABC +DEF has a risk lower than the 20% per annum

that its two constituents had.

In the real world, diversi¬ed ¬rms often do not operate as e¬ciently as stand-alone ¬rms, e.g.,

Synergies or

Dis-synergies drive M&A due to limited attention span of management or due to more bureaucratization. Such mergers

value, not diversi¬cation.

destroy ¬rm value. Of course, other mergers can add value due to synergies, as we discussed

Managers also have

in Chapter 7. More often, however, the unspoken rationales for mergers are that managers

agency con¬‚icts in M&A

activity.

prefer the reduced idiosyncratic uncertainty and higher salaries guaranteed by larger ¬rms to

the higher risk and lower salaries in sharply focused, smaller ¬rms. In our context, to justify

a merger, managers will want to argue for a lower cost of capital any way they can”including

incorrectly using the acquirer™s cost of capital. (This is another example of an agency con¬‚ict,

which we have seen in Chapter 7 and which we will see again in our Chapter 28 on corporate

governance.)

Important: If there are no cash ¬‚ow synergies, combining ¬rms into conglom-

erates may reduce ¬rm risk, but does not create value for our investors. Investors

can diversify risk themselves.

Managers who want to create value through risk reduction should instead seek

to lower their ¬rms™ market betas”of course avoiding proportionally similar or

higher reductions in their ¬rms™ rewards.

Firms can also reduce their overall risk by hedging. The simplest example of a hedge would be

Hedging against stock

market risk. if the ¬rm itself shorted the stock market. For example, it could sell a contract that promises

to deliver the index level of the S&P500 multiplied by 1,000 in one year. Between now and

next year, whenever the stock market goes up, the value of this contract goes up. The contract

has a negative beta. Because the hedged ¬rm would consist of the unhedged ¬rm plus this

contract, the market-beta (or risk) of the hedged ¬rm would be lower than the market-beta of

the unhedged ¬rm. In fact, the ¬rm could sell the exact amount of contracts that make the ¬rm™s

market beta zero or even negative. But, this hedging contract would not create ¬rm value”the

¬rm™s expected rate of return would decline proportionally, too. If investors wanted to have

less exposure to the overall stock market, they could sell such hedging contracts themselves.

Firms do sometimes hedge against other risks. For example, oil companies often sell contracts

Hedging against other

risks. on oil that promise delivery in one year. This insulates them from the volatility of the price of

oil. In itself, in a perfect market, such fairly priced hedging contracts neither add nor subtract

value. But if the market is imperfect, as we noted in Chapter 6, a hedge may allow a ¬rm to

operate more e¬ciently (e.g., generating cash which avoids the need to borrow money), and

thereby add value. (Corporate hedging is further discussed in the web chapter on options.)

Anecdote: Risk and Conglomeration

In the 1970s, a lot of ¬rms diversi¬ed to become conglomerates. Management argued that conglomerates tended