. 23
( 39)


to have lower risk, which created value for shareholders. This argument was, of course, total nonsense: Investors
could diversify for themselves. It was the managers who valued lower risk, with the lower chance to lose their
jobs and the higher compensation due to running a bigger company. Worse, because conglomerates often
operate less e¬ciently than individual stand-alone, focused companies, diversi¬cation actually often destroyed
¬rm value. In the 1980s, there were many “bust-up buyouts,” which created value by purchasing conglomerates
to sell o¬ the pieces.
A good example of such a conglomerate was Gulf and Western. It was simultaneously involved in oil, movies
(Paramount), recording (Stax), rocket engines, stereo components, ¬nance, publishing (Simon and Schuster),
auto parts, cigars, etc. It promptly crashed and split up in the 1980s. A more current example is Tyco, which
has over 260,000 employees in 50 (!) separate business lines, including electronics, undersea ¬ber optic cables,
health care, adhesives, plastics, and alarm systems. (Its former executive, Dennis Kozlowski, became famous
for his extravagant looting of Tyco™s assets. With so many business lines, no wonder no one noticed for years!)
The most interesting conglomerate, however, is General Electric. It has hundreds of business lines, but unlike
most other conglomerates, GE seems to have been running most of its divisions quite well.
¬le=capmrecipe-joint.tex: RP
Section 17·5. Value Creation and Destruction.

17·5.B. Avoiding Cost-of-Capital Mixup Blunders That Destroy Value

When Acquiring Another Company
Section 7·1.B dropped a cryptic hint that practitioners sometimes forget that NPVs are additive. Common misuse of
CAPM: a uniform cost of
You may have wondered what was meant. You are now ready to see why this is such a common
mistake. The most common abuse of the CAPM arises from the use of the ¬rm™s overall cost of
capital for individual projects, and here is an example why.
Assume the risk-free rate of return is 3% and the equity premium is 4%. Your old ¬rm, cleverly What happens if the ¬rm
uses its overall cost of
named O, is worth $100 and has a market beta of 0.5. An acquisition target (or just a new
capital for projects,
project), cleverly named N, costs $10 and is expected to pay o¬ $11 next year. (Its rate of rather than a project
return is therefore 10%.) The beta of this new project is 3. speci¬c cost of capital?
For IRR, see Chapter 8.
The simplest method to compute the value of project N relies on the fact that NPVs are additive.
The Solution: Compute
You value the new project using its own expected cash ¬‚ows and own cost of capital. Who owns
the NPV of the project.
it should matter little: the project is worth what it is worth. So, N should o¬er an expected
rate of return of

E (˜N ) = 3% + · = 15%
Correct Cost of Capital: 4% 3
E (˜N ) = rF + [E (˜M ) ’ rF ] · βN,M
r r ,

and the true NPV of the new project is

NPVN = ’$10 + ≈ ’$0.44 .
1 + 15%
Therefore, if ¬rm O adopts project N, N™s owners would be “$0.44 poorer than they would be
if their managers did not adopt it (i.e., $100 vs. $99.56).
Unfortunately, this is not obvious to some practitioners. In many ¬rms, it is standard policy to Bad Company Policy.
evaluate all projects by the ¬rm™s overall cost of capital. Would such a ¬rm take the N project?
Evaluated with a market beta of 0.5, the hurdle rate for the project would be

E (˜) = 3% + · = 5% .
Incorrect Cost of Capital: 4% 0.5
= rF + E (˜M ) ’ rF · βO,M
r .

With its internal rate of return of $11/$10 ’ 1 = 10%, a (poor) manager would indeed take this
If the O ¬rm did take project N, how would its value change? With a beta of 0.5, the old ¬rm The loss if the ¬rm
makes the mistake.
had an expected rate of return of 3% + 4%·0.5 = 5%. Its expected value next year would be $105.
Using PV, we see that the present value of the combined ¬rm would be

$105 $11
PVcombined = + ≈ $109.56 .
1 + 5% 1 + 15% (17.25)
= + .

This is $0.44 less than the original value of $100 plus the $10 acquisition cost of the new project.
Taking the project has made the N owners 44 cents poorer.
However, contrary to the perfect CAPM world, it is not always true in the real world that mergers Real World Exceptions.
never add value on the cost-of-capital side. If capital markets are not as e¬cient for small ¬rms
as they are for large ¬rms, it would be possible for a large acquirer to create value. For example,
if a target previously had no access to capital markets, as explained in Section 6·1 (Page 112),
then the cost of capital to the target can change when it is acquired. The correct cost of capital
for valuing the acquisition (the target), however, is neither the cost of capital of the acquirer,
nor the blended post-acquisition cost of capital of the ¬rm. Instead, the correct cost of capital
is that appropriate for the target™s projects, given the “now ordinary” access to capital markets.
For example, if an entrepreneur inventor of holographic displays previously had faced a cost
¬le=capmrecipe-g.tex: LP
442 Chapter 17. The CAPM: A Cookbook Recipe Approach.

of capital of, say, 303%, primarily due to access only to personal credit card and credit shark
¬nancing, and if this inventor™s business is purchased by IBM with its cost of capital of 6.5%
(market-beta of 1.5), the proper cost of capital is neither IBM™s (market-beta based) cost, nor a
blended average between 303% and 6.5%. Instead, if part of IBM, the holographic project division
should be evaluated at a cost of capital that is appropriate for projects of the market-beta risk
class “holographic display projects.” This can add value relative to the 303% earlier cost of
capital. (Of course, large corporations are often also very adept at destroying all innovation
and thereby value in the small companies that they are taking over.)

When Acquiring Another Project
It is important to realize that not only ¬rms-to-be-acquired, but also smaller projects themselves
Projects must be
discounted by their own consist of components with di¬erent market-betas, which therefore have di¬erent costs of
capital. For example, when ¬rms keep cash on hand in Treasury bonds, such investments
have a zero market-beta, which is lower than the beta for the ¬rms™ other projects. These
bonds should need not earn the same expected rate of return as investments in the ¬rm™s risky
projects. (The presence of this cash in the ¬rm lowers the average beta of the ¬rm and thus
the average cost of capital for the ¬rm by the just-appropriate amounts.)
Here is another application example: Assume that you consider purchasing a rocket to launch
Another Example
Problem. a Telecomm satellite next year. It would take you 1 year to obtain the rocket, at which point you
would have to pay $100 million. Then you launch it. If the rocket fails (25% chance), then your
investment will be lost. If the rocket succeeds, the satellite will produce a revenue stream with
an appropriate beta of 2. (Telecomm revenues tend to have a high covariance with the market.)
Telecomm™s expected cash ¬‚ows will be $20 million forever. Assume that the risk-free rate is
3% per year and the market equity-premium is 4%.
The correct solution is to think of the rocket as one project and of the Telecomm revenues as
The Example Solution.
another project. The rocket project has only idiosyncratic risk; therefore, its beta is close to
zero, and its discount factor is the same as the risk-free rate of return, 3%. The rocket value (in
millions of dollars today is)
PVrocket ≈ ≈ ’$97 .
1 + 3%
You can think of this as the cost of storing the $100 million in T-bills until we are ready to
try our second project. The Telecomm revenues, however, would be a risky perpetuity. With
a beta of 2, their cash ¬‚ows would be discounted at about 11%. However, the cash ¬‚ows will
only occur with a probability of 75%. Therefore,

E (Telecomm Pro¬ts) 75% · $20 $15
PVTelecomm ≈ = = = $136 . (17.27)
E (˜Telecomm Revenues ) 3% + 4% · 2 = 11%
r 11%

Consequently, this project has a net present value of about $39 million dollars.

17·5.C. Di¬erential Costs of Capital ” Theory and Practice!

There is no doubt that projects must be discounted by their project-speci¬c cost of capital. Yet,
In practice, a good
number of ¬rms do not Graham and Harvey found in their 2001 survey (the same survey you saw in Chapter 1) that
use differential costs of
just about half of surveyed CFOs always”and incorrectly”use the ¬rm™s overall cost of capital,
rather than the project-speci¬c cost of capital. And even fewer CFOs correctly discount cash
¬‚ows of di¬erent riskiness within projects. The easy conclusion is that CFOs are ignorant”and
though some CFOs may indeed use a uniform cost of capital because they are ignorant, some
intelligent CFOs are doing so quite deliberately.
You already know that it is very di¬cult to correctly estimate the cost of capital. In theory, you
Getting project costs of
capital is dif¬cult. just know the market-beta of every project and the other CAPM inputs. In practice, you do not.

1. Even the historical betas of publicly traded corporations are not entirely reliable and in-
dicative of the future. Di¬erent estimation methods can come up with di¬erent numbers.
This is why you may want to use the market betas of similar, publicly traded comparables
¬le=capmrecipe-joint.tex: RP
Section 17·5. Value Creation and Destruction.

or the market beta of an entire industry. But many of your projects may be so idiosyn-
cratic, so unusual, or in such far-away locales that no comparable may seem particularly

2. You could try to estimate your own market beta. To do so, you would need a time-series
of historical project values, not just historical project cash ¬‚ows. This is because you
cannot rely on historical cash ¬‚ow variation as a substitute for historical value variation.
You already know that the market values themselves are the present discount value of
all future cash ¬‚ows, not just of one period™s. Here is an example how this can go awry.
Consider a ¬rm whose cash ¬‚ows are perfectly known. Therefore, its appropriate true
discount rate would be close to the risk-free rate. However, if its cash ¬‚ows occur only
every other month ($200, $0, $200, etc.), this ¬rm would have in¬nite monthly cash ¬‚ow
volatility (’100% followed by +∞%). Its percent changes in cash ¬‚ows would not be in-
dicative of its value-based rates of returns. Plus, almost surely, it would have an extreme
market-beta estimate, indicating a wrong cost of capital. So in order to estimate your
market-beta, you would need to somehow obtain a time series of estimated market values
from the known time series of cash ¬‚ows. Of course, you already know that it is di¬cult
to estimate one market value for our ¬rm”but estimating a time-series of how this mar-
ket value changes every month is entirely beyond anyone™s capability. (When only cash
¬‚ows but not market-values are known, your estimates must necessarily be less accurate.
The best way to estimate an appropriate cost of capital relies on the certainty equivalence
formula in Section A.)

3. Many ¬rms may not have any historical experience that you can use, not just for market
values, but even for cash ¬‚ows. There would be nothing you could veri¬ably and credibly
use to estimate in the ¬rst place.

So, beta estimates are often di¬cult to estimate, equity premium estimates are very uncertain,
and the CAPM is not a perfect model. These uncertainties may not only distort the overall
corporate cost of capital, but also the relative costs of capital across di¬erent projects. Quite
simply, you must be cognizant of the painful reality that your methods for estimating the cost
of capital are often just not as robust as you would like them to be.
Consequently, the problem with assigning di¬erent costs of capital to di¬erent projects may Flexible costs of capital
can cause arguments
now become one of disagreement. Division managers can argue endlessly why their projects
and agency con¬‚icts.
have a lower cost of capital than the company™s. Is this how you want your division managers
to spend their time? Managers could even shift revenues from weeks in which the stock market
performed well into weeks in which the stock market performed poorly in order to produce a
lower market-beta. The cost of capital estimate itself becomes a piece in the game of agency
con¬‚ict and response”every manager would like to convince himself and others that a low
cost of capital for her own division is best. What the overall corporation would like to have
in order to suppress such “gaming of the system” would be one immutable good estimate of
the cost of capital for each division that cannot be argued with. In the reality of corporate
politics, however, it may be easier to commit to one and the same immutable cost of capital for
all divisions than it would be to have immutable but di¬erent costs of capital for each division.
This is not to argue that this one cost of capital is necessarily a good system, but just that there
are cases in which having this one cost of capital may be a necessary evil.
And ¬nally there is the forest. You know that each component must be discounted at its You will never get this
perfectly right. Get it
own discount rate if you want to get the value and incentives right. However, if you want to
right where it matters!
value each paperclip by its own cost of capital, you will never come up with a reasonable ¬rm
value”you will lose the forest among the trees. You need to keep your perspective as to what
reasonable errors are and what unreasonable errors are. The question is one of magnitude: if
you are acquiring a totally di¬erent company or project, with a vastly di¬erent cost of capital,
and this project will be a signi¬cant fraction of the ¬rm, then the choice of cost of capital
matters and you should di¬erentiate. However, if you are valuing a project that is uncertain,
and the project is relatively small, and its cost of capital is reasonably similar to your overall
cost of capital, you can probably live with some error. It all depends”your mileage may vary!
¬le=capmrecipe-g.tex: LP
444 Chapter 17. The CAPM: A Cookbook Recipe Approach.


• Theoretically, all projects must be discounted by their own cost of capital,
and not by the ¬rm™s overall cost of capital.

• Practically, sanity considerations prevent discounting every paper clip by its
own cost of capital.

Therefore, you must judge when it is important to work with di¬erent costs of
capital and when it is better to use just one cost of capital.

Solve Now!
Q 17.22 A $300 million ¬rm has a beta of 2. The risk-free rate is 4%, the equity premium is 3%.
A supplier has approached the ¬rm for a 1-year loan of $100 million that has a beta of 0. The
supplier is willing to pay 6% interest, and there is no default risk. The ¬rm has a policy of only
accepting projects with a hurdle rate of 10%.

(a) If the ¬rm changes its policy and extends the loan, how would its value change?

(b) If the ¬rm changes its policy and extends the loan, approximately how would its beta

(c) If the ¬rm changes its policy and extends the loan, approximately how would its cost of
capital change?

(d) If the ¬rm changes its policy and extends the loan, approximately what would its cash ¬‚ows
be expected to be?

(e) If the ¬rm changes its policy and extends the loan, can you compute the combined ¬rm™s
NPV by dividing its expected cash ¬‚ow by its combined cost of capital?

(f) Should the ¬rm change its policy?

Q 17.23 Some companies believe they can use the blended cost of capital post-acquisition as the
appropriate cost of capital. However, this also leads to incorrect decisions. We explore this now.

(a) What is the value of the new project, discounted at its true cost of capital, 15%? (Assume
that the combined ¬rm value is around $109.48.)

(b) What is the weight of the new project in the ¬rm?

(c) What is the beta of the new overall (combined) ¬rm?

(d) Use this beta to compute the combined cost of capital.

(e) Will the ¬rm take this project?

(f) If the ¬rm takes the project, what will the ¬rm™s value be?
¬le=capmrecipe-joint.tex: RP
Section 17·6. Empirical Reality.

17·6. Empirical Reality

Now you know what the world looks like and how securities should be priced in a perfect CAPM
world. What evidence would lead you to conclude that the CAPM is not an accurate description
of reality? And what is reality really like?

17·6.A. Non-CAPM Worlds and Non-Linear SMLs

What would happen from the CAPM™s perspective if a stock o¬ered more than its due expected What happens if a stock
offers too much or too
rate of return? Investors in the economy would want to buy more of the stock than would be
little expected rate of
available: its price would be too low. It would be too good a deal. Investors would immediately return?
¬‚ock to it, and because there would not be enough of this stock, investors would bid up its
price and thereby lower its expected rate of return. Eventually, the price of the stock would
equilibrate at the correct CAPM expected rate of return. Conversely, what would happen if a
stock o¬ered less than its due expected rate of return? Investors would not be willing to hold
enough of the stock: the stock™s price would be too high, and its price would fall.
Neither situation should happen in the real world”investors are just too smart. However, No CAPM would not
mean arbitrage, but it
you must realize that if a stock were not to follow the CAPM formula, buying it would still be
could imply good deals.
risky. Yes, such a stock would o¬er too high or too low an expected rate of return and thus
be a good or a bad deal, attracting too many or too few investors chasing a limited amount
of project”but it would still remain a risky investment, and no investor could earn risk-free
pro¬t by exploiting the pricing ine¬ciency.
Under what circumstances would you lose faith in the CAPM? Figure 17.4 plots what security The “Security Market
Line” if the CAPM is the
market relations could look like if the CAPM did not work. In Graph (A), the rate of return
wrong model (with
does not seem to increase linearly with beta if beta is greater than about 0.5. Because beta respect to its own
is a measure of risk contribution to your market portfolio, as an investor, you would not be functional form).
inclined to add stocks with betas greater than 1 or 2 to your (market) portfolio”these stocks™
risk contributions are too high, given their rewards. You would like to deemphasize these ¬rms,
tilting your portfolio towards stocks with lower betas. In Graph (B), the rate of return seems
unrelated to beta, but the average rate of return on the stock market seems quite a bit higher
than the risk-free rate of return. In this case, you again would prefer to tilt your portfolio
away from the overall market and towards stocks with lower beta risk. This would allow you
to construct a portfolio that has lower overall risk and higher expected rate of return than
the market portfolio. In Graph (C), higher beta securities o¬er lower expected rates of return.
Again, you should prefer moving away from your current portfolio (the market) by adding more
of stocks with lower market-betas.
Graphs (D) through (F) focus on a distinction between growth ¬rms and value ¬rms. In Graph (D), The “Security Market
Line” if the CAPM is the
even though each cluster has a positive relationship between beta and the expected rate of
wrong model (with
return, growth ¬rms have a di¬erent relationship than value ¬rms”but the CAPM says not respect to a speci¬c
only that market-beta should matter, but that market-beta is all that should matter. If you better alternative).
knew whether a ¬rm was a growth ¬rm or a value ¬rm, you could do better than if you relied
on market-beta. Rather than just holding the market portfolio, you would prefer tilting your
portfolio towards growth stocks and away from value stocks”for a given beta contribution
to your portfolio, you would earn a higher reward in growth ¬rms. Graphs (E) and (F) show
the same issue, but more starkly. If you could not identify whether a ¬rm was a growth ¬rm
or a value ¬rm, you would conclude that market-beta works”you would still draw a straight
positive line between the two clusters of ¬rms, and you would conclude that higher market-beta
stocks o¬er higher rewards. But, truly, it would not be beta that matters, but whether the ¬rm
is a growth ¬rm or a value ¬rm. After taking into account what type the ¬rm is, beta would not
matter in Graph (E), and even matter negatively in Graph (F). In either case, as an investor, you
could earn higher expected rates of return buying stocks based on ¬rm type rather than based
on beta.
¬le=capmrecipe-g.tex: LP
446 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Figure 17.4. The Security Market Line in non-CAPM Worlds

Historical Average Rate of Return

Historical Average Rate of Return

* **
* **
** * *

* * *
* ** * * * ** * *
** ***** * ** *M
** *
** * * **M * * * * ** *
* *
** *
* * * * * *** *
* * ** *
** * * * *
* * ** * * * * *
* * ** *
** * *
* ** *
* * **
** * ** *** * * *
* ** *
* **

** *
* *
**** *
* *** ** *
** *
* *
* *
* *

* *

’1 0 1 2 ’1 0 1 2

Historical Beta With The Stock Market Historical Beta With The Stock Market

(A) (B)
Historical Average Rate of Return

Historical Average Rate of Return

Growth+Firms+++ ++
+ + ++ + ++ +
+ ++ + ++
++++++ + +
+ ++ +
++ + +

+ + +++ +
** M M
* * ** * *
* *
* ** *
** * *
* * * *** ****** *

** * **
** * *
* *
* **
* * *** * * * * ** * *
* ** *** * *
F **
* ** ** *
* *
* ** * *
** * ** * * **
* *** * *
* *
* *

* * ** *** * *
* *
* * * ** Value Firms
** *



’1 0 1 2 ’1 0 1 2

Historical Beta With The Stock Market Historical Beta With The Stock Market

(C) (D)
Historical Average Rate of Return

Historical Average Rate of Return


Growth Firms Growth Firms
++++ +
++ +
+++++ +
++ +
+ +++ + ++
++ ++ + +
++++ +
+ ++
+ +++++ ++ + +
+ +++ +
++ +++ ++
+ ++++ ++++++
+ +


+ ++

* ** * ****
* * * **
* ******* ** *
* **** *
** *


* *F ** * * ** F
* ***
* ***
** *
** *


Value Firms Value Firms


’1 0 1 2 ’1 0 1 2

Historical Beta With The Stock Market Historical Beta With The Stock Market

(E) (F)

Each point is one stock (or project or fund)”its historical beta and its historical average rate of return. (The market
and risk-free rate are noted by a letter inside the circle.) Growth ¬rms are ¬rms with high market values and low
sales, earnings and/or book values. Value ¬rms are the opposite.
In these ¬gures, the security market line does not appear to be linear, as the CAPM suggests. Therefore, if these
patterns are not just statistical mirages, you should be able to invest better than just in the market: from the CAPM
perspective, there are “great deal” stocks that o¬er too much expected return given their risk contributions to your
(market) portfolio, which you would therefore want to overemphasize; and “poor deal” stocks that o¬er too little
expected return, given their risk contribution, which you would therefore want to underemphasize.
¬le=capmrecipe-joint.tex: RP
Section 17·6. Empirical Reality.

But be warned: these relationships could also appear if your procedures to estimate beta or Historical Patterns can
be deceptive.
expected rates of return are poor”after all, when you plot such ¬gures with real-world histori-
cal data, you do not have the true beta or true expected rates of return. Even if your statistical
procedures are sound, statistical noise makes this a hazardous venture. In particular, in real
life, although you can estimate market-betas pretty reliably, you can only roughly estimate
expected rates of return from historical rates of return.

17·6.B. How Well Does the CAPM Work?

A model is just a model”models are never perfect descriptions of reality. They can be useful A model is a model.
within a certain domain, even if on closer examination they are rejected. For example, we do not
live in a world of Newtonian gravity. Einstein™s model of relativity is a better model”though it,
too, is not capable of explaining everything. Yet no one would use Einstein™s model to calculate
how quickly objects fall. The Newtonion model is entirely appropriate and much easier to
use. Similarly, planetary scientists use Einstein™s model, even though we know it, too, fails to
account for everything”but it does well enough for the purposes at hand and there are as yet
no better alternatives (even though string theory is trying). This latter situation is pretty much
the situation in which corporations ¬nd themselves”the CAPM is not really correct, but there
are no clear better alternatives. However, ultimately, your concern has to be about the domain
within which the CAPM is useful, and it usually is very useful for corporate capital budgeting.

Figure 17.5. Average Historical Rates of Return Against Historical Market Beta, 1970“2000.
Average Monthly Rate of Return


’1 0 1 2 3

Average Beta Over All Years

Note: The returns are monthly and not annualized. Betas are with respect to the value-weighted stock market.
Extreme observations were cut: at “1 and +3 for beta, and at “3% and +4% for monthly returns. The solid black
line is “smoothed” to ¬t points locally, allowing it to show non-linearities. The dashed blue line indicates that this
smoothed line suggests that a “beta=0” security had an approximate rate of return of 64 basis points per month, or
about 8% per annum. The typical “beta=1” security had an approximate rate of return of 136 basis points per month,
or about 18% per annum.
¬le=capmrecipe-g.tex: LP
448 Chapter 17. The CAPM: A Cookbook Recipe Approach.

So, what does the security market line (SML) really look like? Figure 17.5 plots the relationships
The empirical relation
looks reasonably linear from 1970 to 2000. The typical stock with a beta of 0 earned a rate of return of about 8% per
and upward sloping.
annum, while the typical stock with a beta of 1 (i.e., like the market) earned a rate of return of
about 18% per annum. Not drawn in the ¬gure, the average stock with a beta of 2 earned about
217 basis points per month (30% per annum), and the average stock with a beta of 3 earned
about 354 basis points per month (50% per annum). You can see that these 30 years were a
very good period for ¬nancial investments! The ¬gure shows also how there was tremendous
variability in the investment performance of stocks. More importantly from the perspective
of the CAPM, the relationship between average rate of return and beta was not exactly linear,
as the CAPM suggests, but it was not far o¬. If we stopped now, you would conclude that the
CAPM was a pretty good model.
But look back at Figure 17.4. The empirical evidence is not against the CAPM in the sense of
But this is
deceptive”the CAPM the ¬rst three plots (linearity)”it is against the CAPM in the sense of the last three plots (better
fails against speci¬c
alternative classi¬cations). So, although you cannot see this in Figure 17.5, the CAPM fails
better alternatives.
when stocks are split into groups based on di¬erent characteristics. The empirical reality is
somewhat closer to the latter three ¬gures than it is to the idealized CAPM world. For example,
there is good empirical evidence that ¬rms that are classi¬ed as “growth ¬rms” (they have
low sales and book value but high market value) generally underperform “value ¬rms” (the
opposite)”but we do not really know why, nor do we know what we should recommend a
corporate manager should do about this fact. Maybe managers should pretend that their ¬rms
are growth ¬rms”because investors like this claim so much they are willing to throw money
at too cheap a cost of capital at growth ¬rms”but then act like value ¬rms and thereby earn
higher returns. In any case, the ¬rms that lie above the CAPM line are disproportionally value
¬rms, and those below the CAPM line are disproportionally growth ¬rms. Market beta seems
to matter only if we do not control for this growth-value and some other ¬rm characteristics.
The “only little problem” (irony warning) is that we ¬nance academics are not exactly sure what
all these characteristics are, why they matter, and how a CFO should work in such a world.
Di¬erent academics draw di¬erent conclusions from this evidence. Some recommend outright
My personal
opinion”and the world against using the CAPM, but most professors recommend “use with caution.” Here is my per-
out there.
sonal opinion:

For a Corporate Manager Although the CAPM is likely not to be really true, market-beta is still
a useful cost-of-capital measure for a corporate ¬nance manager. Why so? Look again at
the last three plots in Figure 17.4: If you have a beta of around 1.5, you are more than
likely a growth ¬rm with an expected rate of return of 10% to 15%; if you have a beta of
around 0, you are more than likely a value ¬rm with an expected rate of return of 3% to
7%. Thus, beta would still provide you with a decent cost of capital estimate, even though
it was not market-beta itself that mattered, but whether your ¬rm was a growth or a value
¬rm. (Market beta helped by indicating to you whether the ¬rm was a growth or a value
¬rm.) Admittedly, using an incorrect model is not an ideal situation, but the cost-of-capital
errors are often reasonable enough that corporate managers generally can live with them.
If you recall from the manager survey in Chapter 1, 73.5% of the CFOs reported that they
always or almost always use the CAPM. CAPM use was even more common among large
¬rms and CFOs with an MBA, and no alternative method was used very often. So, you
ultimately have no choice but to understand the model well”it is the benchmark model
that your future employer will expect you to understand and understand well. In any case,
if you cannot live with the fact that the CAPM is not perfectly correct, I really do not know
what to recommend to you as a clearly better alternative!

For An Investor In contrast, my advice to an investor would be not to use the CAPM for in-
vesting (portfolio choice). Although it is true that wide diversi¬cation needs to be an
important part of any good investment strategy, there are better investment strategies
than just investing in the market.
¬le=capmrecipe-joint.tex: RP
Section 17·7. Robustness: How Bad are Mistakes in CAPM Inputs?.

Solve Now!
Q 17.24 Draw some possible security markets relations which would not be consistent with the

17·7. Robustness: How Bad are Mistakes in CAPM Inputs?

You know that you do not really know the inputs for the CAPM perfectly. You can only make Where will we inevitably
go wrong?
educated guesses. And even after the fact, you will never be sure”you observe only actual
rates of returns, never expected rates of return. So, how robust is the CAPM with respect to
errors in its inputs?

The Risk-Free Rate Errors in the risk-free rate (rF ) are likely to be modest. The risk-free rate Errors in the risk-free
rate tend to be very
can be considered to be almost known for practical purposes. Just make sure to use a
risk-free rate of similar duration and maturity as your project.

This leaves you with having to judge the in¬‚uence of errors in estimating betas (βi,M ), errors
in estimating the expected market rate of return (E(˜M )), and model errors (i.e., that the CAPM
itself is false).

Market-Beta Reasonable beta estimates typically have some uncertainty, but good comparables Errors in beta estimates
tend to be modest.
can often be found in the public market. If due care is exercised, a typical range of
uncertainty about beta might be about plus or minus 0.4. For example, if the equity
premium is 3% and if you believe your beta is 2, but it is really 1.6 instead, then you
would overestimate the appropriate expected rate of return by 2 · 3% ’ 1.6 · 3% = 1.2%.
Although this level of uncertainty is not insigni¬cant, it is tolerable in corporate practice.

Equity Premium Estimates Reasonable equity premium estimates can range from about 2% per Disagreement on the
equity premium tends to
year to about 6% per year”a large range. To date, there is no universally accepted method
be large, and these
to estimate the expected rate of return on the market, so this disagreement cannot be easily differences in equity
settled with data and academic studies. Unfortunately, reasonable di¬erences of opinion premium estimates can
have a large in¬‚uence.
in estimating the expected rate of return on the market can have a large in¬‚uence on
expected rate of return estimates. For example, assume the risk-free rate is 3%, and take
a project with a beta of 2. The CAPM might advise this corporation that potential investors
demand either an expected rate of return of 5% per year (equity premium estimate of 1%)
or an expected rate of return of 19% per year (equity premium estimate of 8%), or anything
in between. This is”to put it diplomatically”a miserably large range of possible cost of
capital estimates. (And this range does not even consider the fact that actual future
project rates of return will necessarily di¬er from expected rates of return!) Of course,
in the real world, managers who want to take a project will argue that the expected rate
of return on the market is low. This means that their own project looks relatively more
attractive. Potential buyers of projects will argue that the expected rate of return on the
market is high. This means that they claim they have great opportunities elsewhere, so
that they can justify a lower price o¬er for this project.

Model Errors So, what about the CAPM as a model itself? First, you need to realize again Use the CAPM as
guidance, not as gospel!
that there are really no better alternatives. No matter how poor or imprecise the CAPM
estimates are, without a better alternative, you have little choice but to use it. Second,
as a CAPM user, you need to be aware of its limitations. The CAPM is a model that can
often provide a “reasonable expected rate of return,” but not an “accurate expected rate
of return.” Anyone who believes that CAPM expected rates of return should be calculated
with more than one digit after the decimal point is deluded. The CAPM can only o¬er
expected rates of returns that are of the “right order of magnitude.” The CAPM also often
tends to be better in ranking projects than in providing a good absolute cost of capital.
In this case, estimating the equity premium to be too low or too high tends to bias the
valuation of all projects”though not necessarily equally so.
¬le=capmrecipe-g.tex: LP
450 Chapter 17. The CAPM: A Cookbook Recipe Approach.

You will often use the CAPM expected rate of return as our cost of capital in an NPV calculation.
Put together NPV and
CAPM robustness Here, you combine errors and uncertainty about expected cash ¬‚ows with your errors and
uncertainty in CAPM estimates. What should you worry about? Recall that in Chapter 5, you
saw the relative importance of getting the inputs into the NPV formula correct. The basic
conclusion was that for short-term projects, getting the cash ¬‚ows right is more important
than getting the expected rate of return right; for long-term projects, getting both right is
important. We just discussed the relative importance of getting the equity premium and the
project beta right. Now recall that your basic conclusion was that the CAPM formula is ¬rst
and foremost exposed to errors in the market risk premium (equity premium), though it is also
somewhat exposed to beta estimates. Putting these two conclusions together suggests that for
short-term projects, worrying about exact beta estimates is less important than worrying about
estimating cash ¬‚ows ¬rst and the appropriate equity premium second. For long-term projects,
the order of importance remains the same, but the di¬erence in the relative importance of good
estimates of expected cash ¬‚ows and good estimates of the equity premium estimates shrinks.
In contrast, in most cases, honest mistakes in beta, given reasonable care, are relatively less
Solve Now!
Q 17.25 To value an ordinarily risky project, that is, a project with a beta in the vicinity of about
1, what is the relative contribution of your personal uncertainty (lack of knowledge) in: the risk-
free rate, the equity premium, the beta, and the expected cash ¬‚ows? Consider both long-term
and short-term investments.

Anecdote: “Cost of Capital” Expert Witnessing
When Congress tried to force the “Baby Bells” (the split up parts of the original AT&T) to open up their local
telephone lines to competition, it decreed that the Baby Bells were entitled to a fair return on their infrastructure
investment”with fair return to be measured by the CAPM. (The CAPM is either the de facto or legislated
standard for measuring the cost of capital in many other regulated industries, too.) The estimated value of the
telecommunication infrastructure in the United States is about $10 to $15 billion. A di¬erence in the estimated
equity premium of 1% may sound small, but even in as small an industry as local telecommunications, it meant
about $1,000 to $1,500 million a year”enough to hire hordes of lawyers and valuation consultants opining in
court on the appropriate equity premium. Some of my colleagues bought nice houses with the legal fees.
I did not get the call. I lack the ability to keep a straight face while stating that “the equity premium is exactly x
point y percent,” which was an important quali¬cation for being such an expert. In an unrelated case in which
I testi¬ed, the opposing expert witness even explicitly criticized my statement that my cost of capital estimate
was an imprecise range”unlike me, he could provide an exact estimate!
¬le=capmrecipe-g.tex: RP
Section 17·8. Summary.

17·8. Summary

The chapter covered the following major points:

• The CAPM provides an “opportunity cost of capital” for investors, which corporations can
use as the hurdle rate (or cost of capital) in the NPV formula. The CAPM formula is

E ( ri ) = rF + E ( rM ) ’ rF · βi,M .
˜ ˜

Thus, there are three inputs: the risk-free rate of return, the expected rate of return on
the stock market, and the project market-beta. Only the latter is project speci¬c.

• The line plotting expected rates of return against market-beta is called the security-
markets line (SML).

• Use a risk-free rate that is similar to the approximate duration or maturity of the project.

• There are a number of methods to estimate market-beta. For publicly traded ¬rms, it
can be computed from stock return data or obtained from commercial data vendors. For
private ¬rms or projects, a similar publicly traded ¬rm can often be found. Finally, man-
agerial scenarios can be used to estimate market-betas.

• The expected rate of return on the market is often a critical input, especially if market-beta
is high”but it is di¬cult to guess.

• The CAPM provides an expected rate of return, consisting of the time-premium and the
risk-premium. In the NPV formula, the default-risk and default-premium works through
the expected cash ¬‚ow numerator, not the cost of capital denominator.

• Corporations can reduce their risk by diversi¬cation”but if investors can do so as easily,
diversi¬cation per s© does not create value.

• To value a project, corporations should not use the cost of capital (market-beta) applicable
to the entire ¬rm, but the cost of capital (market-beta) applicable to the project. However,
because the e¬ort involved can be enormous, you should use individual, project-speci¬c
costs of capital primarily when it makes a di¬erence.

• Certainty equivalence is discussed in the appendix. You must use the certainty-equivalence
form of the CAPM when projects are purchased or sold for a price other than their fair
present market-value.
¬le=capmrecipe-g.tex: LP
452 Chapter 17. The CAPM: A Cookbook Recipe Approach.


A. Appendix: Valuing Goods Not Priced at Fair Value via Cer-
tainty Equivalence

The CAPM is usually called a pricing model”but then it is presented in terms of rates of return,
How to value a project if
the ef¬cient price today not prices. This turns out to have one perplexing consequence, which leaves us with one
is not known?
important and di¬cult conceptual issue best illustrated with a brainteaser: What is today™s
value of a gift expected to return $100 next year?

a. Finding The True Value of A Good That is Not Fairly Priced

How do you even compute the beta of the gift™s rate of return with the rate of return on the
At a price of zero, is the
appropriate cost of stock market? The price is $0 today, which means that your actual rate of return will be in¬nite!
capital in the CAPM
But we clearly should be able to put a value on this gift. Indeed, our intuition tells us that this
formula in¬nite?
cash ¬‚ow is most likely worth a little less than $100, the speci¬cs depending on how the cash
¬‚ow covaries with the stock market. But, how do we compute this value? The solution to this
puzzle is that the price of the gift may be $0 today, but its value today (PV0 ) is not”and it is
the latter, i.e., the fair value, that is used in the CAPM, not the former.


• The CAPM works only with expected rates of return (E(˜i ) = [E(Pi,t=1 ) ’
Pi,t=0 ]/Pi,t=0 ) that are computed from the true perfect market asset values
today (PVt=0 ) and tomorrow (E(Pi,t=1 )).

• If either the price today or next period is not fair, then you cannot work with
the standard CAPM formula, E(˜i ) = rF + E(˜M ) ’ rF · βi,M .
r r

Of course, in a perfect and e¬cient market, what you get is what you pay for (P0 = PV0 and
P1 = PV1 ), so this issue would never arise. But, if you buy an asset at a better or worse deal
(P0 < PV0 or P0 > PV0 ), e.g., from a benevolent or malevolent friend, then you can absolutely not
use such a P0 to compute the expected rate of return in the CAPM formula. The same applies
to E(P1 ): the expected value tomorrow must be the true expected value, not a sweetheart deal
value at which you may let go of the asset, or an excessive price at which you can ¬nd a desperate
Now, return to our question of how to value a gift. Our speci¬c computational problem is tricky:
We need to rearrange
the CAPM formula into we could compute a rate of return for the cash ¬‚ow if we knew PV0 , then from the rate of return
the Certainty
we could compute the project beta, which we could use to ¬nd the discount rate to translate
Equivalence Formula:
the expected cash ¬‚ow back into the price PV0 today. Alas, we do not know PV0 , so we cannot
We work out an expected
value that we can
compute a rate of return. To solve this dilemma, we must use an alternative form of the CAPM
discount with the
formula, called its certainty equivalence form. It is
risk-free rate.

˜ ˜˜
E (P1 ) ’ E (˜M ’ rF )/Var(˜M ) · Cov(P1 , rM )
r r
PV0 =
1 + rF
˜ ˜˜
E (P1 ) ’ » · Cov(P1 , rM )
= ,
1 + rF

where » is {[E(˜M ) ’ rF ]/Var(˜M )}, and all three quantities pertain to the period from time
r r
0 to time 1. So, if we believe that the expected annual equity premium is 5%, and that the
variance of the rate of return on the market is around 0.04 (a standard deviation of 20%), we
¬le=certaintyequivalence.tex: RP
Section A. Appendix: Valuing Goods Not Priced at Fair Value via Certainty Equivalence.

would choose a lambda of around 1.25. (It is the equity premium (8% ’ 3% = 5%) divided by the
variance of the rate of return on the stock market, [(28% ’ 8%)2 + (’12% ’ 8%)2 ]/2 = 0.04). If
the risk-free rate today is 3%, we would value projects as
˜ ˜˜ ˜ ˜˜
E (P1 ) ’ 1.25 · Cov(P1 , rM ) E (P1 ) 1.25 · Cov(P1 , rM )
PV0 = = ’ . (17.30)
1 + 3% 1 + 3% 1 + 3%
as-if-risk-free risk discount

The name “certainty-equivalence” is apt. The ¬rst form shows that, after we have reduced
the expected value of the future cash ¬‚ow (E(P1 )) by some number that relates to the cash
¬‚ow™s covariance with the market, we can then treat this reduced value as if it were a perfectly
certain future cash ¬‚ow and discount it with the risk-free rate. The second form shows that
we can decompose the price today into an “as-if-risk-neutral” value discounted only for the
time-premium (with the risk-free rate) and an additional discount for covariance risk with the
stock market.
The covariance between the future value P1 and the rate of return on the market is related, but Watch out: the
covariance is different in
not identical to the project™s market-beta. It is not the covariance of the project™s rate of return
this form!
with the market rate of return, either. It is the covariance of the project™s cash ¬‚ow with the
market rate of return, instead.
With the certainty equivalence formula, we can now value the $100 expected gift. Assuming Our problem solved with
zero covariance.
that the risk-free rate is 3% per annum, and that the lambda is the aforementioned 1.25,
$100 ’ 1.25 · Cov(P1 , rM )
PV0 =
1 + 3%
˜ ˜˜
E (P1 ) ’ » · Cov(P1 , rM )
PV0 = .
1 + rF

If we believe that the gift™s payout does not covary with the rate of return on the market, so
Cov(P1 , rM ) = 0, then
$100 ’ 1.25 · 0 $100
PV0 = = = $97.09
1 + 3% 1 + 3%
˜ ˜˜
E (P1 ) ’ » · Cov(P1 , rM )
= .
1 + rF

But what if we believe that our windfall does covary with the market? How can we guesstimate Our problem solved with
positive covariance.
the cash ¬‚ow™s covariance with the rate of return on the stock market? The answer is that we
need to write down some scenarios, and then use our covariance computation formula (from
Section 13·4.B). This is easiest to understand in an example. Let us assume that we believe that
if the market goes up by 28%, our gift will be $200; if the market goes down by 12%, our gift
will be $0. We also believe these two outcomes to be equally likely.

1/2 1/2
Prob :

Var Sdv
Bad Good Mean
Stock Market “12% +28% 8% 4% 20%
$2 10,000
Our Windfall $0 $200 $100 $100

Now use the covariance formula to compute the average product of deviations from the mean.
This is
($200 ’ $100) · (28% ’ 8%) + ($0 ’ $100) · (’12% ’ 8%)
Cov(P1 , rM ) = = $20
N ˜ ˜
j=1 [P1,outcome j ’ E (P1 )] · [˜
rM,outcome j ’ E (˜M )]
¬le=capmrecipe-g.tex: LP
454 Chapter 17. The CAPM: A Cookbook Recipe Approach.

Lambda is still 1.25, and we can now use the certainty equivalence formula to value our expected
windfall of $100 next year. It is worth
$100 ’ 1.25 · $20 $75
PV0 = = = $72.82
1 + 3% 1 + 3%
˜ ˜˜
E (P1 ) ’ » · Cov(P1 , rM )
= .
1 + rF

Finally, note that a di¬erent way to write the certainty equivalence formula is
An alternative method
to write the CEV formula.
˜ E (˜M ) ’ rF
E (P1 ) r
PV0 = ’ · bP1 ,˜M ,
1 + rF 1 + rF

where bP1 ,˜M is the beta of a regression in which the value (not the rate of return) is the depen-
dent variable.
Digging Deeper: Knowing the fair price of $72.82, we can now easily check that we have really just recom-
puted the CAPM formula. The project will either provide a rate of return of $200/$72.82 ’ 1 = 174%, or a rate
of return of ’100%, for an average rate of return of 37%. The beta computed with rates of return is

Cov(˜i , rM )
r˜ 0.274
= = = = 6.85 .
βi,M (17.36)
(+28%’8%)2 +(’12%’8%)2
Var(˜M )
r 0.04

The ordinary CAPM formula states that the expected rate of return, given this beta of 6.85, should be

E (˜i ) = rF + [E (˜M ) ’ rF ] · βi,M = 0.03 + (0.08 ’ 0.03) · 6.85 ≈ 0.37 ,
r r

which is indeed what we had computed above. Here is a proof of the certainty equivalence form. Start with the
CAPM formula:
E (˜i ) = rF + [E (˜M ) ’ rF ] · βi,m .
r r

Rewrite beta
Cov(˜i , rM )

E (˜i ) = rF + [E (˜M ) ’ rF ] · (17.39)
r r .
Var(˜M )
Rewrite the rate of return, ri = P1 /PV0 ’ 1,

Cov(P1 /PV0 ’ 1, rM )
˜ (17.40)
E (P1 /PV0 ’ 1) = rF + [E (˜M ) ’ rF ] ·
r .
Var(˜M )
We want to simplify Cov(P1 /PV0 ’ 1, rM ). Covariances are easy to manipulate: if a and b are known constants,
then Cov(a·x + b, y) = a·Cov(x, y). (The constant b does not comove with y, so it disappears.) 1/PV0 plays the
˜ ˜ ˜˜
˜ ˜˜
role of the constant a, ’1 plays the role of the constant b. So we can write Cov(P1 /PV0 ’1, rM ) = Cov(P1 , rM )/PV0 .
˜ ˜
We already know how to manipulate expectations, so E (P1 /PV0 ’ 1) = E (P1 )/PV0 ’ 1. Substitute back in these
unrolled expectation and covariances, and you get

Cov(P1 , rM )
E (P1 )/PV0 ’ 1 = rF + [E (˜M ) ’ rF ] · (17.41)
r .
PV0 · Var(˜M )
If you solve this equation for PV0 , you will arrive at Formula 17.29.

Solve Now!
Q 17.26 Although you are a millionaire, keeping all your money in the market, you have man-
aged to secure a great deal: if you give your even richer Uncle Vinny $10,000 today, he will
help you buy a Ferrari, expected to be worth $200,000, if his business can a¬ord it. He is an
undertaker by profession, so his business will have the money if the stock market drops, but not
if it increases. For simplicity, assume that the stock market drops 1 in 4 years and by “10% when
it does and increases by 18% per annum if it does not drop. (Write it out as four separate possible
state outcomes to make your life simpler.) The risk-free rate is 6%. What is your Uncle™s promise
worth at market value?
¬le=certaintyequivalence.tex: RP
Section A. Appendix: Valuing Goods Not Priced at Fair Value via Certainty Equivalence.

b. An Application of the Certainty Equivalence Method

You are asked to advise a ¬rm on its appropriate cost of capital. The owners of this ¬rm are The opportunity cost of
capital of a privately
very wealthy and widely diversi¬ed, so that their remaining portfolio is similar to the market
held corporation.
portfolio. (Otherwise, our investor™s opportunity cost of capital may not be well represented by
the CAPM”and therefore, the calculations here are not relevant for the typical cash-strapped
entrepreneur.) To make this a more realistic and di¬cult task, this ¬rm is either privately held
or only a division, so you cannot ¬nd historical public market values, and there are no obvious
publicly traded comparable ¬rms. Instead, the ¬rm hands you its historical annual cash ¬‚ows:

Year 1999 2000 2001 2002 2003 2004 Average
+21.4% ’5.7% ’12.8% ’21.9% +26, 4% +9.0% +2.7%
Cash Flows $8,794 $5,373 $8,397 $6,314 $9,430 $9,838 $8,024

In a perfect world, this is an easy problem: you could compute the value of this ¬rm every year,
then compute the beta of the ¬rm™s rate of return with respect to the market rate of return,
and plug this into the CAPM formula. Alas, assessing annual ¬rm value changes from annual
cash ¬‚ows is beyond my capability. You can also not presume that percent changes in the
¬rm™s cash ¬‚ows are percent changes in the ¬rm™s value”just consider what would happen
to your estimates if the ¬rm had earned zero in one year. So, what cost of capital are you
recommending? Having only a time series of historical cash ¬‚ows (and no rates of return) is a
very applied and not simply an obscure theoretical problem, and you might ¬rst want to re¬‚ect
on how di¬cult it is to solve this problem without the certainty equivalence formula.
First, we have to make our usual assumption that our historical cash ¬‚ows and market rates of Let us attempt to value
returns are representative of the future. To solve our problem, we begin by computing the beta
of the ¬rm™s cash ¬‚ows with respect to the S&P500. This is easier if we work with di¬erences
from the mean,

Year 1999 2000 2001 2002 2003 2004 Average
+0.187 ’0.084 ’0.155 ’0.246 +0.237 +0.063
S&P500 0
+$770 ’$2, 651 +$373 ’$1, 710 +$1, 406 +$1, 814
Cash Flows $0

To compute the covariance of the S&P500 returns with our cash ¬‚ows, we multiply these and
take the average (well, we divide by N ’ 1, because this is a sample, not the population, but it
won™t matter in the end),
(+0.187) · (+$770) + (’0.084) · (’$2, 651) + · · · + (+0.063) · (+$1, 814)
Cov CF,˜M =
5 (17.42)
≈ ,

and compute the variance

(+0.187)2 + (’0.084)2 + · · · + (0.063)2
Var(˜M ) = ≈ 0.0373 .
The cash ¬‚ow beta is the ratio of these,
Cov CF,˜M $235.4
βCF,M = = ≈ $6, 304 . (17.44)
Var(˜M )
r 0.03734

It is easiest now to proceed by considering the historical mean cash ¬‚ow of $8,024. We need
an assumption of a suitable equity premium and a suitable risk-free rate. Let us adopt 3% and
4%, respectively. In this case, the value of our ¬rm would be
$8, 024 4%
PV0 = ’ · $6, 304
1 + 3% 1 + 3%

≈ $7, 791 ’ = $7, 546

˜ E (˜M ) ’ rF
E (CF) r
= ’ · bCF,˜M
1 + rF 1 + rF
¬le=capmrecipe-g.tex: LP
456 Chapter 17. The CAPM: A Cookbook Recipe Approach.

The certainty equivalence formula tells us that because our ¬rm™s cash ¬‚ows are correlated
with the market, we shall impute a risk discount of $245. We can translate this into a cost of
capital estimate”at what discount rate would we arrive at a value of $7,546?
$8, 024
$7, 546 = ’ E (˜) = 6.3% .
1 + E (˜)
E (CF)
1 + E (˜)

We now have an estimate of the cost of capital for our cash ¬‚ow for next year. We can also
translate this into an equivalent returns-based market-beta, which is

3% + 4% · βi,M = 6.3% ’ β ≈ 0.8 .
rF + [E (˜M ) ’ rF ] · βi,M

Now I can reveal who the ¬rm in this example really was”it was IBM. Because it is publicly
Are we close?
traded, we can see how our own estimate of IBM™s cost of capital and market beta would have
come out if we had computed it from IBM™s annual market values. Its rates of return were

Year 1999 2000 2001 2002 2003 2004 Average
+17.5% ’20.8% +43.0% ’35.5% +20.5% +7.2% +5.3%
IBM™s Rate of Return

If you compute the market-beta of these annual returns, you will ¬nd an estimate of 0.7”very
close to the estimate we obtained from our cash ¬‚ow series. (For IBM, this is a very low market-
beta estimate. If we used monthly cash ¬‚ows or monthly stock returns, we would obtain a
considerably higher market-beta estimate.)
Solve Now!
Q 17.27 A ¬rm reported the following cash ¬‚ows:

Year 1999 2000 2001 2002 2003 2004 Average
+21.4% ’5.7% ’12.8% ’21.9% +26, 4% +9.0% +2.7%
+$2, 864 +$1, 666 ’$1, 040 +$52 +$1, 478 ’$962 +$997
Cash Flows

(Note that the cash ¬‚ows are close to nothing in 2002 and even negative in 2004, the latter
preventing you from computing percent changes in cash ¬‚ows.) What cost of capital would you
recommend for this ¬rm?
¬le=certaintyequivalence.tex: RP
Section A. Appendix: Valuing Goods Not Priced at Fair Value via Certainty Equivalence.

Solutions and Exercises

You can use the CAPM formula here: 4% + (16% ’ 4%)·1.5 = 22%.
E (˜) = 4% + (7% ’ 4%)·3 = 13%.
E (˜) = 4% + (12% ’ 4%)·3 = 28%.
Solve E (˜) = 4% + (7% ’ 4%)·βi,M = 5%. Therefore, βi,M = 1/3.
5. Do it!
E (˜M ) ’ rF is the premium that the stock market expects to o¬er, above and beyond the rate that risk-free
investments o¬er.

7. It does not matter what you choose as the per-unit payo¬ of the bond. $100 is expected to return $99. So,
the price of the bond is
PV = ≈ $95.19 (17.48)
1 + (3% + 5% · 0.2)
Therefore, the promised rate of return on the bond is $100/$95.19 ’ 1 ≈ 5.05%.
8. The risk-free rate is 3%, so this is the time premium. The expected risk premium is 1%. The remaining 1.05%
is the default premium.
9. The cost needs to be discounted with the current interest rate. Since payment is upfront, this cost is $30,000
now! The appropriate expected rate of return for cash ¬‚ows (of your earnings) is 3% + 5% · 1.5 = 10.5%. You


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