<<

. 22
( 39)



>>



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
¬le=capmrecipe-g.tex: LP
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 )

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.
¬le=capmrecipe-joint.tex: RP
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

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