40

$5, 000 1

· 1’ = $47, 619 · 98.2% ≈ $46, 741.46 . (17.49)

1 + 10.5%

10.5%

With 90% probability, you will do so, which means that the appropriate risk-adjusted and discounted cash

¬‚ow is about $42,067.32. The NPV of your education is therefore about $12,067.

10. An estimate between 2% and 8% per year is reasonable. Anything below 0% and above 10% would be unrea-

sonable. For reasoning, please see the di¬erent methods in the chapter.

11. Use the Treasury rate for the 1-year project, e.g., from the Wall Street Journal. Because the 10-year project

could have a duration of ¬‚ows anywhere from 5 to 10 years, depending on use, you might choose a risk-free

Treasury rate that is between 5 and 10 years.

12. A 5-year interest rate is a reasonably good guess. You should not be using a 30-day Treasury bill, or a 30-year

Treasury bond.

13. You could run a time-series regression of project rates of return on the stock market.

(17.50)

rp = ± + β · rM .

˜ ˜

(PS: Although not discussed, a slightly better regression would be rp ’ rF = ± + β · rM ’ rF .) The

˜ ˜

statistical package (or Excel) would spit out the project beta. The digging-deeper note explains that if you

want to improve the forward-looking estimate, you might want to shrink this beta towards 1.

14. Beta can be found in Yahoo!Finance™s “Pro¬le.” In June 2003, IBM™s beta was 1.48. Most upstart growth

companies have higher betas.

E (˜i ) ’ rF

r 20% ’ 4%

15. Rearrange the CAPM formula to βi,M = ≈ 2.7.

. So, the ¬rst beta estimate would be

E (˜M ) ’ rF 10% ’ 4%

r

’20% ’ 4%

The second beta estimate would be βi,M = = 2.7. Therefore, a reasonable estimate of beta would

’5% ’ 4%

be the average, here 2.7.

16. You can compute an unlevered beta.

2 1

βP ,M = ·0+ · 2.5

3 3 (17.51)

βP ,M = wDT · βDT,M + wEQ · βEQ,M .

Therefore, βP ,M = 0.83. We assume our project has the same beta, but a smaller debt ratio:

1 2

0.833 = ·0+ · βEQ,M ’ βEQ,M = 1.25.

3 3 (17.52)

βP ,M = wDT · βDT,M + wEQ · βEQ,M .

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

The comparable project™s equity expected rate of return would be 3% + 2% · 2.5 = 8%. Our own equity™s

17.

expected rate of return would be 3% + 2% · 1.25 = 5.5%

18. A debt/equity ratio of 2 is the same as the debt asset ratio of 2/3: two parts debt, one part equity. A

debt/equity ratio of 1/2 is the same as the debt asset ratio of 1/3: one part debt, two parts equity. To convert

a debt-equity ratio into a debt-asset ratio, recognize that

D+E E

1 1

= = = 1+

D/ D/ D D

(D + E)

A

D 1 1

’ = = (17.53)

1 1

E ’1 ’1

D/A 2/3

1 1

= = = .

2

3 1

’1

2 2

19. This is the same as above, too.

20. βcombined,M = (3/4) · 2.4 + (1/4) · 0.4 = 1.9.

21.

(17.54)

βi,M = wp · 0.2 + (1 ’ wp ) · 1 = 0 ’ wp = 1.25 .

So, if your wealth is $100, you want to short $25 in the fund to increase your own portfolio holdings to $125.

The net portfolio has zero market beta.

22. The CAPM cost of capital is 10%. Its current projects are expected to provide $30, the new project would

provide $6, $2 above the risk-free rate. Therefore, the value of the ¬rm would go up by $2 next year, which

has to be discounted to today. The new project “loan” would be about 1/4 of the new ¬rm. Therefore, the

new beta of the ¬rm will be

(17.55)

βFM,M = 3/4 · 2 + 1/4 · 0 = 1.5 .

The ¬rm™s cost of capital would therefore decline from 10% to 8.5%. At this cost of capital, the extra $2 would

add about $1.84 to the ¬rm value. The ¬rm™s cash ¬‚ows would change from $430 to $330 + $106 = $436.

Discounted at the 8.5% interest rate, this comes to about $401.84. Subtracting o¬ the $100 cost of the loan

con¬rms the NPV. The ¬rm should change its policy.

23.

(a) The value is $9.57.

(b)

$11

PVN 1+15% (17.56)

wN = = ≈ 8.73% .

PVcombined $109.48

(c)

(17.57)

βcombined = wO ·βO + wN ·βN ≈ 91.26%·0.5 + 8.73%·3 = 0.718 .

(d)

(17.58)

E (˜combined ) = 3% + 4%·0.718 = 5.872% ,

r

(e) Yes! The IRR of N is 10%. 10% is above the blended cost of capital of 5.872%.

(f) Firm value would be

$105 + $11

PV = = $109.56

1 + 5.872%

(17.59)

E (CFN ) + E (CFO )

= .

1 + E (˜combined )

r

So, again, the ¬rm has destroyed $0.44.

24. See Figure 17.4.

25. For short investments, the expected cash ¬‚ows are most critical to estimate well (see Section 5·4). For long-

term projects, cost of capital becomes more important to get right. Betas and risk-free rates are usually

relatively trouble free, having only modest degrees of uncertainty. The equity premium will be the most

important problem factor.

¬le=certaintyequivalence.tex: RP

459

Section A. Appendix: Valuing Goods Not Priced at Fair Value via Certainty Equivalence.

26. This is a certainty equivalence question. Although it is not a gift per se, you cannot assume that $10,000 is a

fair market value, so that you can compute a rate of return of 1,900%”after all, it is your Uncle trying to do

something nice for you. There are four outcomes:

1/4 1/4 1/4 1/4

Prob :

Crash No-Crash No-Crash No-Crash Mean

Stock Market “10% +18% +18% +18% 11%

Ferrari $200 $0 $0 $50

Plug this into the formula

˜˜ 1/4 · $150, 000 · (’21%) + (’$50, 000) · (+7%) +

Cov(P1 , rM ) =

(17.60)

(’$50, 000) · (+7%) + (’$50, 000) · (+7%) = ’$21, 000 .

We also need to determine the variance of the market. It is

(’21%)2 + (+7%)2 + (+7%)2 + (+7%)4

Cov(˜M , rM ) = = 147%%

r˜

4

(17.61)

(which incidentally comes to a standard deviation of 12% per annum, a bit low.) With the risk-free rate of 6%,

lambda (») is (11% ’ 6%)/147%% ≈ 3.4.

So, we can now use the certainty equivalence formula: the expected value of the Ferrari is $50,000. If it were

a safe payo¬, it would be worth $47,170. Because you get more if the rest of your portfolio goes down, it is

actually great insurance for you. So, you value it 3.4 · (’$21, 000)/(1 + 6%) ≈ $67, 358 higher than $47,170:

the Ferrari is worth $114,528. You have to pay $10,000 today, of course, so you have managed to secure a

deal for $104,528.

27. The beta of these cash ¬‚ows is $5,104. Therefore, the risk discount on $997 is about 4%/1.03·$997 ≈ $38.718,

which corresponds to a cost of capital of about 4% (a beta of about 0.25). This ¬rm is Sony. It had returns

of ’42% (in 2001, computed with the 2000 price), ’39%, ’10%, +2%, and ’8%. (Even these returns depend

sensitively on how dividends are reinvested.) The beta computed from market values comes out to just below

0.6. In the real world, the di¬erence of 0.35 would result in about 1% di¬erence in the cost of capital”a

reasonable amount of error, especially given that we had to estimate a cost of capital without knowing Sony™s

historical market values! Yahoo!Finance lists a Sony beta of around 1, but also computes this from monthly

stock returns.

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

¬le=capmrecipe-g.tex: LP

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

CHAPTER 18

The CAPM: The Theory and its Limits

Where does the CAPM equation come from?

last ¬le change: Feb 18, 2006 (15:55h)

last major edit: Sep 2004

This chapter appears in the Survey text only.

This chapter explains the CAPM theory in detail and its shortcomings. Actually, because we

have already discussed all the necessary ingredients”mostly in Chapter 16 which is absolutely

necessary background for this chapter”this chapter is surprisingly simple. Really.

461

¬le=capmtheory-g.tex: LP

462 Chapter 18. The CAPM: The Theory and its Limits.

18·1. The Theory

18·1.A. The Logic and Formula

We have already covered every necessary aspect of the CAPM theory in earlier chapters:

Here are the underlying

mathematical facts,

which were already

• In Section 12·3.D, we learned that the portfolio held by investors in the aggregate is the

explained.

(market-capitalization) value-weighted portfolio.

• In Section 16·3, we learned that the combination of Mean-Variance E¬cient portfolios is

itself MVE.

• In Section 16·5, we learned that each stock i in a MVE-portfolio must o¬er a fair expected

rate of return for its risk contribution to E. Formula 16.31 states that for each and every

security i in the MVE portfolio E, this means that

(18.1)

E (˜E ) = rF + E (˜E ) ’ rF · βi,E .

r r

The CAPM is a theory that has only one hypothesis and only one implication.

Here is the CAPM Theory.

CAPM Hypothesis The CAPM hypothesis is that each and every investor chooses an MVE port-

folio.

Unlike the method for choosing weights for an MVE portfolio”which are just mathemat-

Investors may not hold

MVE portfolios. ical optimization techniques and which have to be correct”the CAPM is a real theory

that may not hold in practice: investors might have di¬erent investment objectives and

therefore not hold MVE portfolios.

CAPM Implication If the CAPM hypothesis is correct, then the mathematical implication is that

the aggregate value-weighted market portfolio is also MVE.

If the CAPM hypothesis is not correct, there is no particular reason to expect the value-

The CAPM could be

wrong in practice. weighted portfolio to lie on the MVE Frontier. Instead, the value-weighted market portfolio

could have risk-reward characteristics that place it far inside the MVE Frontier.

This is it: the Capital-Asset Pricing Model, which won the 1990 Nobel Prize in Economics.

Important: The CAPM conjectures that all investors purchase MVE portfolios.

As a necessary mathematical consequence, the value-weighted market portfolio is

also MVE.

Put altogether again, the logic of the CAPM is:

OK, let™s do with algebra

and logic.

Mathematical Fact To enter an MVE Frontier portfolio E, each stock in E has to o¬er an appro-

priate reward for its risk. This formula, which relates its expected rate of return to its

covariation with E, is

(18.2)

E ( ri ) = rF + E ( rE ) ’ rF · βi,E .

˜ ˜

Stocks that o¬er expected rates of return higher than suggested by this formula would

generate too much aggregate investor demand; stocks that o¬er lower expected rates of

return would generate too little aggregate investor demand.

CAPM Theory Assumption Every investor purchases an MVE portfolio.

¬le=capmtheory-g.tex: RP

463

Section 18·1. The Theory.

’ Implication The value-weighted market portfolio is MVE.

’ Implication that

(18.3)

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

˜ ˜

In this case, the market portfolio consisting only of risky securities must be the tangency

portfolio T .

Digging Deeper: This is the “Sharpe-Lintner” CAPM. There is also a “Fisher Black” CAPM, in which there is

no risk-free rate of return. Everything works the same, except that the risk-free rate is replaced by the constant

a. All stocks in the economy follow the CAPM formula

(18.4)

E ( ri ) = a + [E ( rM ) ’ a] · βi,M ,

˜ ˜

with respect to the value-weighted market portfolio M.

18·1.B. Some Odds and Ends

• The theoretically correct market portfolio in the CAPM is the market-capitalization value- The market portfolio is

theoretically clear, but

weighted portfolio, consisting of all possible investment assets in the economy, not just

practically dif¬cult to

stocks in the U.S. stock market. identify.

However, in traditional investments use, the assumed goal is to pick the best portfolio

among publicly traded stocks. It is as if the CAPM theory was restated to conjecture

that investors seek to optimize the risk/reward relationship only within their portfolios

of publicly traded domestic stocks. In this case, the proper market portfolio would be

the market-capitalization value-weighted U.S. stock market portfolio. However, even this

portfolio is di¬cult to obtain every day. Fortunately, the value-weighted U.S. stock market

portfolio has high correlation with other broad stock market indexes, such as the S&P500.

Thus, the S&P500 is often used as a reasonably good substitute.

Although this is reasonable use for CAPM testing and corporate CAPM application, it is

not good investment advice: you should de¬nitely diversify across more assets than just

domestic stocks. International stocks, commodities, real-estate, and even education repre-

sent other investment classes that are readily available to join smart investors™ portfolios.

In the real world, di¬erent investors have di¬erent investment opportunities. For example,

you can invest in your house or you can invest your education and reap the rewards”

but I cannot invest in your house and your education. The CAPM ignores this issue

altogether. But if you have too much invested in some untraded assets (education or your

speci¬c house), you might not want the same kind of stocks to diversify your educational

investment risk away. That is, you might not want stocks to minimize your portfolio

variance only. Instead, you might want to purchase relatively more stocks that go up

when your other untraded assets go down, and vice-versa.

• Transaction costs would be high enough to prevent ordinary small investors from them- Mutual Funds facilitate

holding the market

selves purchasing the widely diversi¬ed portfolio prescribed by the CAPM. Mutual Funds

portfolio.

(Section 12·3.B), however, have made this relatively easy.

• What if the market portfolio does not sit on the MVE Frontier? Then there will be stocks If the market portfolio

is not MVE, then the

for which the CAPM Formula 18.4 does not hold. And then it should not be assumed that

CAPM does not hold and

the CAPM formula is the appropriate relation predicting stocks™ expected rates of return. the CAPM formula

cannot be used.

Indeed, then anything else could conceivably explain higher or lower required rates of

return. For example, instead of ¬rms with higher market betas, it may be ¬rms with

higher P/E ratios or ¬rms with older CEOs or ¬rms containing the letter “Z” that may end

up having to o¬er higher expected rates of return in order to induce investors to willingly

purchase and hold them in equilibrium. Naturally, such a situation would make it very

di¬cult for corporations to determine the appropriate cost of capital that their projects

should o¬er.

¬le=capmtheory-g.tex: LP

464 Chapter 18. The CAPM: The Theory and its Limits.

• Because E( rM ) > rF , the intuition of the CAPM formula is that stocks with higher stock-

˜

Higher beta implies

higher expected rate of market betas have to o¬er higher expected rates of return. Stocks with high betas are less

return.

helpful in reducing the risk of an investor who already holds the market portfolio.

• A risky project that has a beta of zero need only o¬er an expected rate of return no higher

From the perspective of

a widely diversi¬ed than the risk-free rate itself. This is because each investor would only hold a tiny amount

investor, a tiny bit of

of this security. Having one cent in the risk-free rate or in a security whose return is

zero-beta stock is just as

expected to be the same and with no correlation to the rest of the portfolio is practically

good as investment in

the risk-free security.

the same. In the real and naturally not perfectly CAPM world, this is not exactly correct”

but it is often still a reasonably good approximation.

Solve Now!

Q 18.1 What is the main CAPM hypothesis, and what is the main CAPM implication?

Q 18.2 Write down the CAPM formula without looking at the text. You must memorize it!

Q 18.3 Under what circumstances is the market-beta a good measure of risk?

Q 18.4 What can plotting both the stock-market portfolio and the risk-free rate into the MVE

Frontier graph tell you about the model and about the security markets line?

Q 18.5 What can checking whether every single security follows the CAPM formula (i.e., that

every single possible investment is on the security markets line in a graph of expected rate of

return against beta) tell you about the MVE Frontier (a graph of expected rate of return against

standard deviation)?

Q 18.6 If all but one security are right on the security-markets line, is the market portfolio MVE?

Q 18.7 Is the CAPM market portfolio the value-weighted or the equal-weighted portfolio?

Q 18.8 Can the S&P500 be used as a proxy for the market?

Q 18.9 In the formula, why do higher beta stocks o¬er higher expected rates of return?

Q 18.10 If there is a risk-free security, does the stock market portfolio have to be the tangency

portfolio for the CAPM to hold, or can it be any portfolio on the mean-variance frontier of risky

assets?

Q 18.11 Is it possible for a risky stock to o¬er an expected rate of return that is less than the

risk-free rate?

¬le=capmtheory-g.tex: RP

465

Section 18·2. Does the CAPM Hold?.

18·2. Does the CAPM Hold?

18·2.A. Listing All the CAPM Assumptions

We have sneaked in the CAPM assumptions one at a time, perhaps to make them appear more Let™s judge the CAPM

Model Assumptions.

palatable. It is excusable if they have slipped your mind by now. So, to help you judge how

likely it is that the CAPM assumptions are reasonably satis¬ed in the real world, it is worthwhile

to repeat them all at once. These are all the assumptions that we have used to conjecture that

each and every investor buys an MVE portfolio:

Market Assumptions • The market is perfectly competitive.

• There are no transaction costs.

• There are no taxes.

• The investment opportunity choices are identical for each investor. Therefore, the

value-weighted market portfolio is identical for all investors. It consists of all assets

that can be invested in. It includes such assets as real estate, bonds, international

markets, etc.

• The previous assumption means that are no untraded assets that only some, but not

other investors can hold. That is, you cannot have your own house, or your own

children, or your own education, or your own labor income, or your own executive

stock options.

Informational Assumptions • Financial markets are e¬cient: as we mentioned in Section 6·1.C,

and as we will discuss further in Chapter 19, this means that the market does not

ignore information in the setting of ¬nancial prices, which an investor could use to

outperform the market securities line. You cannot use public information to pick

stocks better than the market can.

• There is no actual inside information that allows some investors to pick stocks better

than the market can.

• There is no perceived inside information. That is, investors do not believe that they

are able to pick stocks better than the market can.

• All investors share the same “opinion” about security expected rates of return and

security variance/covariances.

• All model parameters are perfectly known:

“ The expected rate of return on each and every stock (including the market port-

folio) is known.

“ All covariances (and thus betas) are known.

“ If you as a researcher want to test the CAPM, i.e., whether other variables mat-

ter, you must also know possible alternative return factors (e.g., ¬rm size) that

investors can use.

• There are no managerial agency problems (i.e., managers enriching themselves) that

can be changed if a large investor holds more of the particular stock with agency

problems.

Preference Assumptions • Investors care only about their portfolio performance. They

do not care about other characteristics of their holdings (e.g., socially responsible

corporate behavior).

• Investors like mean and dislike variance. This fully describes their portfolio pref-

erences. (It does assume that investors do not care, e.g., about skewness of their

portfolio™s future investment rate of return, as we discussed on Page 358.)

• Agents maximize their portfolio performance independently each period. (Although

this may appear innocuous, it implies that investors do not choose their portfolios to

insure themselves against changes in future investment opportunities. You cannot

buy stocks that you believe to pay o¬ more in a recession, because you believe then

will be a good time to double up investment.)

¬le=capmtheory-g.tex: LP

466 Chapter 18. The CAPM: The Theory and its Limits.

• Investors do not care about how their portfolios perform relative to other variables,

including variables related to their own personal characteristics (except for risk-

tolerance). For example, investors must not care about earning a di¬erent rate of

return in states in which they ¬nd themselves ill or states in which in¬‚ation is high.

If these assumptions hold, the CAPM will hold. Of course, it could also be that the value-

The CAPM is just a

helpful model, not weighted market portfolio just happens to lie on the MVE frontier, in which case the CAPM

perfect reality.

security markets line would just happen to be a line by accident. Fortunately, the market

need not lie perfectly on the MVE frontier, nor must the assumptions hold perfectly in order

for the CAPM to be a useful model. After all, the CAPM is only a model, not reality. Only

mathematics works the same in theory and in practice”economic models do not. The real

question is whether the CAPM assumptions are su¬ciently badly violated to render the CAPM

a useless model”and in what context.

18·2.B. Is the CAPM a good representation of reality?

In real life, we know that some investors purchase portfolios other than MVE portfolios. This

You do not hold the

stock market portfolio, should not come as a big surprise: chances are that you yourself are one of these investors,

therefore the CAPM is

even if you are only a small investor. Therefore, in the strictest sense, we already know that

incorrect. True, but...

the CAPM does not hold. But, again, the CAPM is only a model. The question is whether the

CAPM simpli¬cation of reality is helpful in understanding the world, i.e., whether the model is

su¬ciently close to reality to be useful. Perhaps most of the real big investors “who matter”

do invest in portfolios su¬ciently similar to MVE portfolios, so that the value-weighted market

portfolio is on the MVE Frontier, meaning that the CAPM “roughly” works.

Unfortunately, the empirical evidence suggests that the market portfolio does not lie too closely

The empirical

relationship is graphed. on the MVE Frontier, that the security markets line is not perfectly linear, and that expected

security returns relate to other characteristics in addition to beta. Figure 18.1 plots average

monthly returns against stock betas. Care must be exercised when viewing these ¬gures, be-

cause there are few stocks with betas below 0 and above 2. Therefore, the smoothed lines are

not too reliable beyond this range. In the 1970s, the relationship between beta and average

(monthly) return was generally upward-sloping, though not perfectly linear. In the eighties, al-

though stocks with betas below 0.5 and 1.5 had similar average rates of return, stocks outside

the 0.5 to 1.5 beta range (not too many!) had a clear positive relationship. In the 1990s, ¬rms

with positive betas showed a nice positive relationship, though the (few) stocks below β = 0

did not.

The most interesting plot is the overall graph, plotting the relationships from 1970 to 2000. The

typical stock with a beta of 0 earned a rate of return of about 8% per 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). These 30 years were an amazingly 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, we 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 we cannot see this in Figure 18.1, the CAPM fails when

better alternatives.

we split these stocks into groups based on di¬erent characteristics”and academics seem to

come up with about ¬ve new di¬erent characteristics per year. So, the empirical reality is

somewhat closer to the latter three ¬gures than it is to the idealized CAPM world. This implies

that market beta seems to matter if we do not control for certain other ¬rm characteristics.

For example, some evidence suggests that ¬rms that are classi¬ed as “small growth ¬rms”

by some metrics generally underperform “large value ¬rms””but neither do we really know

why, nor do we know what we should recommend a corporate manager should do about this

fact. Maybe managers should pretend that they are growth ¬rms”because investors like this

¬le=capmtheory-g.tex: RP

467

Section 18·2. Does the CAPM Hold?.

Figure 18.1. Average Historical Rates of Return Against Historical Market Beta.

1970“1980 1980“1990

4

4

Average Monthly Rate of Return

Average Monthly Rate of Return

3

3

2

2

1

1

0

0

’1

’1

’2

’2

’3

’3

’1 0 1 2 3 ’1 0 1 2 3

Average Beta Over All Years Average Beta Over All Years

1990“2000 1970“2000

4

4

Average Monthly Rate of Return

Average Monthly Rate of Return

3

3

2

2

1

1

0

0

’1

’1

’2

’2

’3

’3

’1 0 1 2 3 ’1 0 1 2 3

Average Beta Over All Years Average Beta Over All Years

The web chapter on empirical asset pricing examines, among other issues, noise and

non-stationarity problems when testing investments models.

so much they are willing to throw money at growth ¬rms”but then act like value ¬rms and

thereby earn higher returns?! But we still have another “little problem” (irony warning)”we

¬nance academics are not exactly sure what these characteristics are, and why they matter.

18·2.C. Professorial Opinions on the CAPM

Di¬erent academics draw di¬erent conclusions from this evidence”and it should leave you You need to know pros

and cons.

with a more subtle perspective than we academics would like. See, we professors would be

happy if we could tell you that the CAPM is exactly how the world works, and then close the

chapter. We would probably even be happy if we told you that the CAPM is de¬nitely not the

right model to use, and then close the chapter. But it is not so simple. Yes, some professors

recommend outright against using the CAPM, but most professors still recommend “use with

caution.” Now, before I give you my own view, be aware that this is just my own assessment,

not a universal scienti¬c truth. In any case, you positively need to be aware of the advantages

and disadvantages of the CAPM, and know when to use it and when not to use it.

¬le=capmtheory-g.tex: LP

468 Chapter 18. The CAPM: The Theory and its Limits.

My own personal opinion is that although the CAPM is likely not to be really true, market beta

My personal opinion.

is still a useful cost-of-capital measure for a corporate ¬nance manager. Why so? Look at the

last three plots in Figure 17.4 again: 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 beta itself that

mattered, but whether the ¬rm was a growth or a value ¬rm. (Market beta helped in indicating

to you whether the ¬rm was a growth or a value ¬rm in the ¬rst place.) 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. And the fact is, if corporations

cannot live with these errors, we really do not know what to recommend as a better alternative

to the CAPM!

In sum, although the CAPM formula is mistaken, it continues to be the dominant model for the

Why the CAPM continues

to be and should be used. following reasons:

1. Although the market portfolio does not seem to lie directly on the MVE Frontier, it does

not appear to be too far away.

2. The CAPM provides good intuition for the characteristics that should determine the ex-

pected rates of return that have to be o¬ered by stocks: stocks that help diversi¬ed

investors achieve lower risk should be sellable even if they o¬er lower expected rates of

return.

3. For many purposes and many stocks, the CAPM provides reasonable expected rates of

return, not too far o¬ from those gleaned from more complex, di¬cult, and/or arbitrary

models.

4. There is no good alternative model that is either simpler to use or convincingly better in

providing appropriate expected rates of return for investments.

5. The CAPM is the gold standard for cost-of-capital estimations in the real world. It is in

wide use. Understanding it (and its shortcomings) is therefore crucial for any student

with an interest in ¬nance.

In contrast, my advice to an investor would be not to use the CAPM for investment portfolio

choices. There are better investment strategies than just investing in the market, although wide

diversi¬cation needs to be an important part of any good investment strategy.

Another interesting question is why the CAPM fails. In my own opinion, most investors do not

invest scienti¬cally. They buy stocks that they believe to be undervalued and sell stocks that

they believe to be overvalued. Why do even small and relatively unsophisticated investors”

often relying on investment advice that quali¬es as the modern equivalent of hocus-pocus”

believe they are smarter than the ¬nancial markets, whilst even professional investors have

di¬culties beating the market? This is perhaps the biggest puzzle in ¬nance.

Although people™s tendencies to categorize choices might have given us some hope that people

at least maximize MVE within the domain of their own pure market investment portfolios, per-

haps subject to their idiosyncratic estimates of expected rates of return, the evidence suggest

that investors are not even internally consistent. Even among the stocks for which they hold

opinions about high or low expected rates of return, investors tend not to choose securities to

reach their own best MVE Frontiers.

Unfortunately, if every investor behaves di¬erently, it is also not clear what the expected rate

of return on a particular investment really should be or has to be”other than that it need

not necessarily be that suggested by the CAPM. It could be high, it could be low, it could be

anything.

This phenomenon is the domain of behavioral ¬nance, eventually to be further expounded in

a web chapter.

¬le=capmtheory-g.tex: RP

469

Section 18·3. Portfolio Benchmarking with the CAPM?.

18·2.D. Why not Optimization instead of the CAPM?

So, why could we not just rely on optimal portfolio theory (Chapter 16) instead of on the What are our

alternatives to the

CAPM? From the perspective of an investor, this is a feasible and probably better choice. The

CAPM? Investors have

only drawback is that instead of believing that the market portfolio is MVE, i.e., that a part good, although tedious

of the investor™s portfolio should be allocated to it, the investor has to solve the more general alternatives...

problem of where the MVE Frontier is. This is tedious, but possible. Even so, in many situations,

i.e., when not too much money is at stake, even though the stock market portfolio is not exactly

MVE, it is probably still close enough for most investors to just choose it anyway in order to

avoid the pain of computation. Nevertheless, the CAPM should not be used for investment

purposes where more than $100 million is at stake. There may be better investment strategies.

Again, for smaller investment portfolios, the advice of purchasing a combination of the value-

weighted market portfolio (including international stocks and other assets) and a risk-free rate

is probably reasonably close to an MVE portfolio.

From the perspective of a ¬rm, however, relying on optimal portfolio theory is not a feasible ...while corporations

really have few or no

choice. Corporate managers need to be able to compute what expected rate of return investors

alternatives.

would demand for a particular project. If they do not know what investors want in equilibrium,

managers are in a guessing game. So, take the CAPM for what it is useful for: it is a model

that provides decent guesstimates of appropriate cost of capital for non-¬nancial projects, in

which the exact discount rate is not too critical. Do not use it if a lot of money can be lost if

the discount rate is misestimated by a percent or two.

Solve Now!

Q 18.12 List as many assumptions about the CAPM that you can recall. Which ones are most

problematic?

18·3. Portfolio Benchmarking with the CAPM?

The performance evaluation of investment managers is an example of how problematic the Can the CAPM Formula

be beaten in a CAPM

application of the CAPM formula can be. The typical CAPM use in portfolio benchmarking

world?

has investors compute the beta of portfolios held by their investment managers, and check

if the manager™s portfolio return over the year beat the securities market line. Unfortunately,

if the CAPM holds, the security markets line cannot be beaten. After all, the de¬nition of the

MVE Frontier is that it is the set of portfolios that o¬ers the best possible combination of

risk and return. If the CAPM does not hold, then the security markets line has no reason for

living. It would be totally ad-hoc. So, strictly speaking, the CAPM cannot be used for portfolio

benchmarking.

Nevertheless, the CAPM formula gives good intuition on what constitutes good performance There is a good intuitive

reason to use the CAPM,

contribution to a widely diversi¬ed investor. It also works if the fund manager with the presum-

anyway.

ably better information and stock picking ability is too small to move prices. This explains why

the use of the CAPM formula for performance evaluation continues”despite all its problems.

¬le=capmtheory-g.tex: LP

470 Chapter 18. The CAPM: The Theory and its Limits.

18·4. Summary

The chapter covered the following major points:

• The logic of the CAPM is that if every investor holds a mean-variance e¬cient portfolio,

then the (value-weighted) market portfolio is mean-variance e¬cient. This in turn means

that stocks follow the linear relationship between the expected rate of reward and market-

beta (the security markets line).

• The CAPM relies on many assumptions. It is only a model.

• The CAPM is a useful simpli¬cation in certain contexts (such as capital budgeting), because

it is intuitive, easy to use, and often gives a reasonable enough cost-of-capital estimate.

• The CAPM is not a reasonable description of reality in many other contexts. It should not

be used for stock investing purposes.

• There are no good alternatives to the CAPM. This is why the CAPM, with all its faults, is

still the predominant model in most corporate contexts.

¬le=capmtheory-g.tex: RP

471

Section A. Advanced Appendix: The Arbitrage Pricing Theory (APT) Alternative.

Appendix

A. Advanced Appendix: The Arbitrage Pricing Theory (APT) Al-

ternative

The Arbitrage Pricing Theory (APT) is an alternative to the CAPM. This Appendix provides a

brief treatment.

Assume that the returns of stocks”here S1, S2 and S3”are driven by some (say two) time-series The price-generation

process: a mean plus

factors:

random factors

˜ ˜

rS1 = + (+4) · fA + (+1) · fB + «S1

˜ ˜

24% multiplied.

˜ ˜

rS2 = + (’1) · fA + (+3) · fB + «S2

˜ ˜

5%

(18.5)

˜ ˜

rS3 = + (+2) · fA + (’1) · fB + «S2

˜ ˜

8%

˜ ˜

rSi = E ( rSi ) + aSi · fA + bSi · fB + «Si .

˜ ˜ ˜

What exactly is a factor? A factor is just an expected economy-wide outcome (de-facto some

˜

time-series) after it has been de-meaned. So, factor fA could be the actual rate of return on the

˜

S&P500 net of the typical (or expected) rate of return on the S&P500. Factor fB could be the

economy™s growth rate, or the percent change in the price of a barrel of oil, or the rate of return

on IBM. Graham and Harvey surveyed CFOs and found that they are concerned with interest

rate risk (especially small ¬rms), exchange rate risk (especially large Fortune-500 type ¬rms),

business cycle risk, and in¬‚ation risk. To use the APT, we ¬rst subtract out the expected value

to obtain a “surprise”:

˜ ˜ (18.6)

E ( fA ) = 0 E ( fB ) = 0 .

This is just a normalization, and can always be done. To give a feeling for returns that are

consistent with our speci¬c APT model, Table 18.1 shows some historical rates of return that

would have led to this APT model.

Table 18.1. A Set of Rates of Return Well Described By Our APT Model

Factors Rates of Return

˜ ˜ ˜ ˜

raw fA raw fB fA fB rS1 rS2 rS3

˜ ˜ ˜

Year

’0.26

1980 0.30 0.31 0.58 0.09 2.65 1.15

’0.37 ’0.82 ’0.09 ’1.04 ’1.14 ’2.98

1981 0.95

1982 1.20 0.85 1.48 0.63 6.79 0.47 2.41

’0.54 ’0.26 ’0.38 ’0.83

1983 0.62 0.40 1.51

’1.81 ’1.53 ’0.10 ’5.97 ’2.88

1984 0.12 1.29

’0.93 ’0.65 ’2.01 ’1.55

1985 0.56 0.34 1.71

’0.07 ’0.29 ’2.50

1986 1.39 1.67 6.63 3.72

’1.50 ’1.21 ’0.03 ’4.64 ’2.32

1987 0.19 1.17

’0.28 +0.22 +0.24 +0.05 +0.08

Mean 0.00 0.00

¬le=capmtheory-g.tex: LP

472 Chapter 18. The CAPM: The Theory and its Limits.

Common stock variation when all/many stocks are formed into a portfolio should be only in

Epsilons should be

uncorrelated. Common «

the factors, not the pricing error, so the pricing error (˜Si ) for each stock should not only be zero

variation should be in

on average, but should outright become zero if the portfolio has a very large number of stocks.

factors.

If the errors are still not idiosyncratic to each ¬rm, but continue to have common variation with

other errors, you need to introduce a new factor that captures this common variation.

˜

With the assumptions of de-meaned factors (E( f ) = 0) and mean-zero errors (E(˜Si ) = 0), the

«

Our return generation

process intercept is intercept is indeed the expected rate of return on the stock. To see this, take the expectation

indeed the expected rate

of both sides of Formula 18.5:

of return.

˜ ˜

E ( rSi ) = E Intercept + aSi · fA + bSi · fB + «Si

˜ ˜

˜ ˜ (18.7)

= Intercept + aSi · E ( fA ) + bSi · E ( fB ) + E ( «Si )

˜

= E ( rSi ) .

˜

The aSi and bSi coe¬cients are called factor loadings or factor exposures. A stock™s factor

loadings describe how the stock responds to factor surprises. In our example, stock S1 responds

˜

to surprises in the ¬rst factor more and in an opposite way as stock S2: if fA were to turn out as

5%, it would change the actual rate of return on stock S1 by +4 · 5% = 20% and change the actual

rate of return on stock S2 by ’1 · 5% = ’5%. Each stock can be described by how it responds to

the two factors”aside from the epsilon, which is just some error or noise that is not supposed

to correlate with anything else. Again, this is important: the factors are supposed to capture

everything that makes two stocks covary with one another.

The question that the APT is trying to answer is the same question that the CAPM is trying to

The APT question is

almost like the CAPM answer: given how a stock moves together with something, what should its expected rate of

question.

return be? In the CAPM context, this translates into: given the beta of a stock with respect to

the stock market, what should its expected rate of return be? In the APT context, this translates

into: given the factor loadings of a stock (which describe how a stock comoves with the factors),

what should its expected rate of return be?

The APT trick is to form a portfolio P that has zero exposure with the factors. For example, if we

The APT trick is to

˜

diversify away all risk. were to invest $100 into S1 and $100 into S2, a 10% factor realization by fA would increase our

S1 investment by $100 · (+4) · 10% = $40, and our S2 investment by $100 · (’1) · 10% = ’$10.

Net, we would gain $30. The rate of return on such a portfolio de¬nitely still depends on the

˜

factor fA . But, if we were to invest $100 into S1 and $400 into S2, a 10% factor realization

˜

by fA would increase our S1 investment by $100 · (+4) · 10% = $40, and our S2 investment

˜

by $400 · (’1) · 10% = ’$40. For this particular portfolio, no matter what the fA factor will

turn out to be in the future, it will not a¬ect its value. The portfolio has zero net exposure to

˜

factor fA .

Now assume that there is also a fourth stock available for investment. This fourth stock has

A portfolio if one stock

fails to follow the APT factor exposures of (+6) and (’2), respectively, but we do not yet know its expected rate of

pricing equation.

return.

˜ ˜

rS4 = E ( rS4 ) + (+6) · fA + (’2) · fB + «S3

˜ ˜ ˜

(18.8)

˜ ˜

rSi = E ( rSi ) + aSi · fA + bSi · fB + «Si .

˜ ˜ ˜

Our ultimate goal is to determine what expected rate of return it should o¬er. Our ¬rst step is

to determine a portfolio of four stocks (de¬ned by wS1 , wS2 , wS3 , and wS4 ) which has zero net

exposure to the factors. What is the actual rate of return of such a portfolio?

rP = wS1 ·˜S1 + wS2 ·˜S2 + wS3 ·˜S3 + wS4 ·˜S4

r r r r

˜

˜ ˜

= wS1 · E ( rS1 ) + (+4) · fA + (+1) · fB + «S1

˜ ˜

˜ ˜ (18.9)

+ wS2 · E ( rS2 ) + (’1) · fA + (+3) · fB + «S2

˜ ˜

˜ ˜

+ wS3 · E ( rS3 ) + (+2) · fA + (’1) · fB + «S3 .

˜ ˜

˜ ˜

+ wS4 · E ( rS4 ) + (+6) · fA + (’2) · fB + «S4 .

˜ ˜

¬le=capmtheory-g.tex: RP

473

Section A. Advanced Appendix: The Arbitrage Pricing Theory (APT) Alternative.

Collect the factor-related terms:

rP = wS1 · E ( rS1 ) + wS2 · E ( rS2 ) + wS3 · E ( rS3 ) + wS4 · E ( rS4 )

˜ ˜ ˜ ˜ ˜

˜

+ wS1 · (+4) + wS2 · (’1) + wS3 · (+2) + wS4 · (+6) ·fA

(18.10)

˜

+ wS1 · (+1) + wS2 · (+3) + wS3 · (’1) + wS4 · (’2) ·fB

+ wS1 · «S1 + wS2 · «S2 + wS3 · «S3 + wS4 · «S4 .

˜ ˜ ˜ ˜

To ¬nd a portfolio whose return does not depend on the factors, the net exposure on each

factor must be zero:

wS1 · (+4) + wS2 · (’1) + wS3 · (+2) + wS4 · (+6) =0,

(18.11)

wS1 · (+1) + wS2 · (+3) + wS3 · (’1) + wS4 · (’2) =0.

We also do not want to spend any money on this portfolio today, so quote our investment in

dollars (rather than in percents), and choose a portfolio so that it costs no money:

(18.12)

wS1 + wS2 + wS3 + wS4 = $0 .

There is not just one such portfolio, but many. For example, the portfolio

(18.13)

wS1 = ’$92.86 , wS2 = +$71.43 , wS3 = ’$78.57 , wS4 = +$100

happens to be one such a portfolio. It has zero net exposure to each factor and it costs nothing

today. Check this:

(’$92.86) · (+4) + (+$71.43) · (’1) + (’$78.57) · (+2) + (+$100) · (+6) = $0 ,

(18.14)

(’$92.86) · (+1) + (+$71.43) · (+3) + (’$78.57) · (’1) + (+$100) · (’2) = $0 ,

+ + + = $0 .

(’$92.86) (+$71.43) (’$78.57) (+$100)

Other portfolios that have zero net exposure and zero costs are multiplies of this portfolio. For

example, the following portfolio works equally well:

(18.15)

wS1 = ’$92.86/2 , wS2 = +$71.43/2 , wS3 = ’-$78.57/2 , wS4 = +$100/2 .

To recap, we have determined a portfolio that costs nothing today and has no factor exposure.

Regardless of what the factors will turn out to be, the portfolio will be una¬ected.

Now, what is the actual rate of return on this portfolio? If so, our portfolio

allows us to earn an

arbitrage!

rP = wS1 · E ( rS1 ) + wS2 · E ( rS2 )

˜ ˜ ˜

+ wS3 · E ( rS3 ) + wS4 · E ( rS4 )

˜ ˜

+ wS1 · «S1 + wS2 · «S2

˜ ˜

(18.16)

+ wS3 · «S3 + wS4 · «S4 .

˜ ˜

= (’$92.86) · E ( rS1 ) + (+$71.43) · E ( rS2 )

˜ ˜

+ (’$78.57) · E ( rS3 ) + (+$100) · E ( rS4 )

˜ ˜

+ wS1 · «S1 + wS2 · «S2 + wS3 · «S3 + wS4 · «S4 .

˜ ˜ ˜ ˜

We now need to make the big leap of faith that the APT requires: Assume that the epsilons

are not just expected to be zero, but that they are actually always zero. In a large portfolio of

thousands of stocks, the epsilons should be noise that should diversify away. (PS: In reality,

¬le=capmtheory-g.tex: LP

474 Chapter 18. The CAPM: The Theory and its Limits.

even the 5,000 stocks traded in the United States are not enough to make this come true. This

APT leap of faith is de¬nitely an approximation.)

rP ≈ (’$92.86) · E ( rS1 ) + (+$71.43) · E ( rS2 ) + (’$78.57) · E ( rS3 ) + (+$100) · E ( rS4 )

˜ ˜ ˜ ˜ ˜

= (’$92.86) · 24% + (+$71.43) · 5% + (’$78.57) · 8% + (+$100) · E ( rS4 )

˜

(18.17)

= + + + (+$100) · E ( rS4 )

(’$22.29) (+$3.57) (’$6.29) ˜

= ’$25.00 + (+$100) · E ( rS4 ) ,

˜

≈ wS1 · E ( rS1 ) + wS2 · E ( rS2 ) + wS3 · E ( rS3 ) + wS4 · E ( rS4 ) .

˜ ˜ ˜ ˜

Note that rP is an actual rate of return, while E( rS4 ) is just a number, known in advance.

˜ ˜

For this special portfolio, however, we have managed to eliminate all risk! The factor risk was

eliminated with clever portfolio choice, and the epsilon risk was assumed to be zero (diversi¬ed

away). Therefore, we can eliminate the tilde over the rate of return, and write

(18.18)

rP = ’$25.00 + $100 · E ( rS4 ) .

˜

Now, what would you do if the expected rate of return on the fourth stock were 30% instead of

25%? If you purchased this portfolio, you would be guaranteed a rate of return of

(18.19)

rP = ’$25.00 + $100 · 30% = +$5 .

Remember that this portfolio costs nothing and has no risk. It is an arbitrage portfolio. With a

positive payo¬, such a portfolio would constitute an arbitrage! Indeed, the only expected rate

of return on the fourth stock that does not permit arbitrage is 25%. It can be shown that the

expected rates of return on each and every stock that prevents the presence of arbitrage must

follow the formula

E ( rSi ) = 1% + 5% · aSi + 3% · bSi

˜

(18.20)

E ( rSi ) = rF + »a · aSi + »b · bSi .

˜

If even one stock does not follow this formula, you can construct a pro¬table arbitrage portfolio.

The intercept in this formula must be the risk-free rate, because the risk-free rate does not

covary with either factor, and therefore has zero aSi and zero bSi . (It has no exposure to either

factor.)

Side Note: The Arbitrage Pricing Theory is a misnomer. It should be called the Non-Arbitrage Pricing Theory,

because it determines the expected rates of returns if there are no arbitrage opportunities.

Note the similarity of Formula 18.20 to the CAPM. If »b is zero (meaning that a factor exposure

The APT equation looks

like a generalized form to the second factor does not in¬‚uence expected rates of return), and if »a is the equity premium

of the CAPM.

E( rM ’rF ), then the APT formula is the CAPM formula. So, in some sense, the CAPM can almost

˜

be considered to be a special case of the APT, in which there is only factor, which is the excess

rate of return on the stock market (“almost” because the two models di¬er by some technical

conditions). But APT ¬‚exibility and generality come at a price: you need to specify what the

important pricing factors are”and the APT can o¬er no guidance here.

In the real world, application of the arbitrage pricing theory requires determining what the right

The APT™s main problem:

what are the factors? factors are, so that the errors are small and diversify away. Some users do so by specifying

macroeconomic factors, or industry portfolios; other users rely on a statistical technique called

principal component analysis (or factor extraction).

¬le=capmtheory-g.tex: RP

475

Section A. Advanced Appendix: The Arbitrage Pricing Theory (APT) Alternative.

Many academic papers use four factors as predictors of expected rates of returns. (There is a

Side Note:

debate about whether these four factors are risk factors, ine¬cient stock market behavior, or just too many

academic researchers ¬nding random past coincidences.) At a minimum, controlling for such factors helps to

control for expected rate of return di¬erences that are already known.

The three factors additional to the stock market are book-to-market ratio, market value, and momentum

(recent stock returns). It appears as if value ¬rms (with high book-to-market ratios) command a premium

over growth ¬rms, as if small ¬rms command a premium over large ¬rms, and as if ¬rms that have recently

appreciated command an expected rate of return premium over stocks that have recently depreciated.

There are also some commercial vendors that rely on dozens of factors, and will sell corporations APT estimates

of their cost of capital.

Solve Now!

Q 18.13 Add the rates of return from security 4 (factor exposures of +6 and ’2) with an expected

rate of return of 25% to Table 18.1. Then write down the annual rates of return of the arbitrage

portfolio.

Q 18.14 Assume that you have determined that three stocks are always behaving as follows:

˜ ˜

rS1 = + (+1) · fA + (’1) · fB

˜ 9%

˜ ˜

rS2 = ’4% + (+2) · fA + (+3) · fB

˜

(18.21)

˜ ˜

= 15% + (+3) · fA + (’1) · fB

rS3

˜

A new stock has just arrived on the scene, and it has a factor exposure to each of the two factors

of +4. What is its expected rate of return?

Q 18.15 If the new stock has an expected rate of return of +2%, how could you make money?

¬le=capmtheory-g.tex: LP

476 Chapter 18. The CAPM: The Theory and its Limits.

Solutions and Exercises

1. Because each and every investor purchases an MVE portfolio, the value-weighted market portfolio is MVE.

2. See Formula 18.4

3. It is a measure of risk contribution to an investor™s portfolio, if this portfolio is the market portfolio.

4. If the market portfolio lies on the MVE Frontier, then the CAPM holds and all securities must lie on the security

markets line.

5. Whether the stock market portfolio is on the MVE Frontier.

6. No.

7. The value-weighted portfolio.

8. Maybe yes, maybe no”but it certainly is commonly used as a proxy for the value-weighted domestic stock

market portfolio.

9. Because E ( rM ) > rF : the stock market is assumed to have a higher expected rate of return than risk-free

˜

Treasury bonds.

10. It has to be the tangency portfolio.

11. Yes, if this stock helps exceptionally well to diversify the market portfolio (beta is negative).

12. See Subsection 18·2.A.

13. The revised table is

Factors Rates of Return

˜ ˜ ˜ ˜

raw fA raw fB fA fB rS1 rS2 rS3 rS4

˜ ˜ ˜ ˜

Year P¬o

+0.580 +0.089 ’0.26

1980 0.30 0.31 2.65 1.15 3.60 0.05

’0.37 ’0.82 ’0.085 ’1.038 ’1.14 ’2.98

1981 0.95 1.87 0.05

+1.480 +0.632

1982 1.20 0.85 6.79 0.47 2.41 7.92 0.05

’0.54 ’0.255 +0.401 ’0.38 ’0.83 ’2.03

1983 0.62 1.51 0.05

’1.81 ’1.529 ’0.096 ’5.97 ’2.88 ’8.68

1984 0.12 1.29 0.05

’0.93 ’0.648 +0.337 ’2.01 ’1.55 ’4.26

1985 0.56 1.71 0.05

’0.07 +1.671 ’0.294 ’2.50

1986 1.39 6.63 3.72 10.91 0.05

’1.50 ’1.213 ’0.030 ’4.64 ’2.32 ’6.92

1987 0.19 1.17 0.05

’0.28 +0.22 +0.24 +0.05 +0.08 +0.25

Mean 0.000 0.000 0.05

14. The formula for the expected rates of return is

E ( ri ) = 2% + 3% · 4% ’ 4% · 4% = ’2% ,

˜

(18.22)

E ( ri ) = 2% + 3% · ai ’ 4% · bi .

˜

15. A sample arbitrage portfolio is

(18.23)

wS1 = $112.50 , wS2 = ’$125 , wS3 = ’$87.50 , wS4 = +$100 .

Any portfolio that consists of multiples of these weights also works. The known rate of return on this portfolio

is

(18.24)

rP = $2 + $100 · E ( rS4 ) .

˜

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

CHAPTER 19

Ef¬cient Markets, Classical Finance, and

Behavioral Finance

Or, Do You Get What You Pay For?

last ¬le change: Feb 24, 2006 (16:56h)

last major edit: Jul 2004

This chapter revisits the concept of competitive, perfect, e¬cient markets, ¬rst mentioned in

Section 6·1.C. It develops three basic concepts of ¬nance in more depth: arbitrage, good bets,

and e¬cient markets (E-M). No study of ¬nance is complete without an understanding of these

concepts.

This chapter also discusses the consequences of the E-M concept: what e¬cient markets mean

for predicting stock performance; how to interpret the success of famous investors; and how

to use the e¬cient markets concept to run an event study to help assess the valuation impact

of some corporate events.

477

¬le=e¬mkts.tex: LP

478 Chapter 19. E¬cient Markets, Classical Finance, and Behavioral Finance.

19·1. Arbitrage and Great Bets

Although you may have an intuitive notion of what arbitrage is, it is important that you know

precisely what it is:

Important: An arbitrage is a business transaction

• that o¬ers positive net cash in¬‚ows in some states of the world,

• and under no circumstance”either today or in the future”a negative net

cash out¬‚ow. Therefore it is risk-free.

Let™s ¬rst be clear about what arbitrage is not. It is not the same as “earning money without

Arbitrage is the

“Perpetuum Mobile” of risk”: after all, we know that investments in Treasury securities earn a positive risk-free rate

economics. It is de¬ned

of return. But buying safe bets like T-bills requires you to lay out cash today. Arbitrage is also

in terms of cash outlays

not the same as “money in today”: if you are willing to accept risk, you can often receive cash

and risk.

today. For example, insurance companies take money from you in exchange for the possibility

that they may have to pay you in the future.

In theory, what would a hypothetical arbitrage opportunity look like? For example, if you can

In a sense, positive NPV

projects under certainty purchase an item for $1, borrow at an interest rate of 9% (all costs, including your time included),

are arbitrage.

and sell the item tomorrow for $1.10 for sure, you earn 1 cent for certain today without any

possible negative out¬‚ows in the future. If you ever stumble upon such an opportunity, please

execute it”it is a positive NPV project! More than this, it is an arbitrage because you cannot

lose money under any scenario (it is without risk!), but it is obviously not a very important

arbitrage. Now, in ¬nancial markets, many transactions can be scaled up. If you could repeat

this transaction one billion times, then you could earn $10 million. Of course, it is even more

unlikely that you can ¬nd such an arbitrage opportunity that works for one billion items than it

is to ¬nd an arbitrage that works for one item. After all, you are not the only one searching! True

arbitrage opportunities are di¬cult or outright impossible to ¬nd in the real world, especially

in very competitive ¬nancial markets.

Another hypothetical example of arbitrage involves a violation of the law of one price, that is,

Arbitrage could

conceivably occur that the same good should cost the same amount. Assume that PEP shares are quoted for $51

between different

on the Frankfurt Stock Exchange, and for $50 on the New York Stock Exchange. The arbitrage

¬nancial markets.

could be executed by selling short one share at a price of $51 in Frankfurt, taking the $51, and

then investing $50 in one share of PEP on the N.Y.S.E.. You pocket $1 today. What you had to

promise to the Frankfurt buyer, which is all PEP payouts (such as dividends), will be covered

by your ownership of the N.Y.S.E. PEP share. If you can do this with 20,000 PEP shares worth

$1 million, you earn $20,000 without e¬ort or risk.

But before you conclude that this is an arbitrage, you still have to make sure that you have not

Consider the hindrances.

forgotten costs or risks. The arbitrage may be a lot more limited than it seems”or not even

present. Consider the following issues:

1. There are the direct and indirect transaction costs. How much commission do you have

to pay? Do you have to pay extra fees to short a stock? Is $51 the Frankfurt bid price at

which you can sell, and $50 the NYSE ask price at which you can buy? Have you accounted

for the value of your own time watching the screen for opportunities?

2. Share prices can move when you want to transact a signi¬cant amount of shares. Only the

¬rst 100 shares may be available for $50 for a net pro¬t of $100. The next 900 shares may