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Rate of return

statistics

A $10 25.0% 29.58% 1.00 0.15 0.29 0.68

B 10 20.0 33.91 0.15 1.00 0.87 0.38

C 10 32.5 48.15 0.29 0.87 1.00 0.22

D 10 22.25 8.58 0.68 0.38 0.22 1.00

High Real Low Real

TA B L E 7.12 Interest Rates Interest Rates

Rate of return

Rate of Inflation Rate of Inflation

projections

High Low High Low

Equally weighted portfolio:

A, B, and C 23.33 23.33 20.00 36.67

Dreck (D) 15.00 23.00 15.00 36.00

Suppose we sell short 300,000 shares of Dreck and use the $3 million proceeds to buy

100,000 shares each of Apex, Bull, and Crush. The dollar profits in each of the four scenarios

will be as follows.

High Real Interest Rates Low Real Interest Rates

Inflation Rate Inflation Rate

Dollar

Stock Investment High Low High Low

Apex $ 1,000,000 $ 200,000 $ 200,000 $ 400,000 $ 600,000

Bull 1,000,000 0 700,000 300,000 200,000

Crush 1,000,000 900,000 200,000 100,000 700,000

Dreck 3,000,000 450,000 690,000 450,000 1,080,000

Portfolio $ 0 $ 250,000 $ 10,000 $ 150,000 $ 20,000

The first column verifies that the net investment in our portfolio is zero. Yet this portfolio

yields a positive profit in all scenarios. It is therefore a money machine. Investors will want to

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

244 Part TWO Portfolio Theory

take an infinite position in such a portfolio, for larger positions entail no risk of losses yet yield

ever-growing profits.6 In principle, even a single investor would take such large positions that

the market would react to the buying and selling pressure: The price of Dreck would come

down, and/or the prices of Apex, Bull, and Crush would go up. The pressure would persist un-

til the arbitrage opportunity was eliminated.

>

5. Suppose Dreckâ€™s price starts falling without any change in its per-share dollar pay-

Concept

offs. How far must the price fall before arbitrage between Dreck and the equally

CHECK weighted portfolio is no longer possible? (Hint: Account for the amount of the

equally weighted portfolio that can be purchased with the proceeds of the short

sale as Dreckâ€™s price falls.)

The critical property of an arbitrage portfolio is that any investor, regardless of risk aver-

sion or wealth, will want to take an infinite position in it so that profits will be driven to an in-

finite level. Because those large positions will force some prices up and/or some down until

the opportunity vanishes, we can derive restrictions on security prices that satisfy the condi-

tion that no arbitrage opportunities are left in the marketplace.

The idea that equilibrium market prices ought to be rational in the sense that they rule out

arbitrage opportunities is perhaps the most fundamental concept in capital market theory. Vio-

lation of this principle would indicate the grossest form of market irrationality.

There is an important distinction between arbitrage and CAPM risk-versus-return domi-

nance arguments in support of equilibrium price relationships. A dominance argument, as in

the CAPM, holds that when an equilibrium price relationship is violated, many investors will

make portfolio changes. Each individual investor will make a limited change, though, de-

pending on wealth and degree of risk aversion. Aggregation of these limited portfolio changes

over many investors is required to create a large volume of buying and selling, which restores

equilibrium prices.

When arbitrage opportunities exist, by contrast, each investor wants to take as large a po-

sition as possible; in this case, it will not take many investors to bring about the price pressures

necessary to restore equilibrium. Implications derived from the no-arbitrage argument, there-

fore, are stronger than implications derived from a risk-versus-return dominance argument,

because they do not depend on a large, well-educated population of investors.

The CAPM argues that all investors hold mean-variance efficient portfolios. When a se-

curity (or a bundle of securities) is mispriced, investors will tilt their portfolios toward the

underpriced and away from the overpriced securities. The resulting pressure on prices comes

from many investors shifting their portfolios, each by a relatively small dollar amount. The

assumption that a large number of investors are mean-variance optimizers, is critical; in con-

trast, even few arbitrageurs will mobilize large dollar amounts to take advantage of an arbi-

trage opportunity.

Well-Diversified Portfolios and the Arbitrage Pricing Theory

arbitrage pricing

theory (APT)

The arbitrage opportunity described in the previous section is further obscured by the fact that

A theory of risk-return it is almost always impossible to construct a precise scenario analysis for individual stocks

relationships derived

that would uncover an event of such straightforward mispricing.

from no-arbitrage

Using the concept of well-diversified portfolios, the arbitrage pricing theory, or APT,

considerations in

resorts to statistical modeling to attack the problem more systematically. By showing that

large capital markets

6

We have described pure arbitrage: the search for a costless sure profit. Practitioners often use the terms arbitrage and

arbitrageurs more loosely. An arbitrageur may be a professional searching for mispriced securities in specific areas

such as merger-target stocks, rather than one looking for strict (risk-free) arbitrage opportunities in the sense that no

loss is possible. The search for mispriced securities is called risk arbitrage to distinguish it from pure arbitrage.

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Fifth Edition Theory

245

7 Capital Asset Pricing and Arbitrage Pricing Theory

mispriced portfolios would give rise to arbitrage opportunities, the APT arrives at an expected

returnâ€“beta relationship for portfolios identical to that of the CAPM. In the next section, we

will compare and contrast the two theories.

In its simple form, just like the CAPM, the APT posits a single-factor security market.

Thus, the excess rate of return on each security, Ri ri rf, can be represented by

Ri iRM e (7.5)

i

where alpha, i, and beta, i, are known, and where we treat RM as the single factor.

Suppose now that we construct a highly diversified portfolio with a given beta. If we use

enough securities to form the portfolio, the resulting diversification will strip the portfolio of

nonsystematic risk. Because such a well-diversified portfolio has for all practical purposes well-diversified

zero firm-specific risk, we can write its returns as portfolio

A portfolio sufficiently

RP PRM (7.6)

P

diversified that

(This portfolio is risky, however, because the excess return on the index, RM, is random.) nonsystematic risk

is negligible.

Figure 7.11 illustrates the difference between a single security with a beta of 1.0 and a well-

diversified portfolio with the same beta. For the portfolio (Panel A), all the returns plot exactly

on the security characteristic line. There is no dispersion around the line, as in Panel B,

because the effects of firm-specific events are eliminated by diversification. Therefore, in

Equation 7.6, there is no residual term, e.

Notice that Equation 7.6 implies that if the portfolio beta is zero, then RP P. This im-

plies a riskless rate of return: There is no firm-specific risk because of diversification and no

factor risk because beta is zero. Remember, however, that R denotes excess returns. So the

equation implies that a portfolio with a beta of zero has a riskless excess return of P, that is,

a return higher than the risk-free rate by the amount P. But this implies that P must equal

zero, or else an immediate arbitrage opportunity opens up. For example, if P is greater than

zero, you can borrow at the risk-free rate and use the proceeds to buy the well-diversified

zero-beta portfolio. You borrow risklessly at rate rf and invest risklessly at rate rf P, clear-

ing the riskless differential of P.

Return (%) Return (%)

10 10

RM RM

0 0

A: Well-diversified portfolio B: Single stock

F I G U R E 7.11

Security characteristic lines

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

246 Part TWO Portfolio Theory

Suppose that the risk-free rate is 6%, and a well-diversified zero-beta portfolio earns (a sure)

rate of return of 7%, that is, an excess return of 1%. Then borrow at 6% and invest in the zero-

7.8 EXAMPLE beta portfolio to earn 7%. You will earn a sure profit of 1% of the invested funds without put-

ting up any of your own money. If the zero-beta portfolio earns 5%, then you can sell it short

Arbitrage with

and lend at 6% with the same result.

a Zero-Beta

Portfolio

In fact, we can go further and show that the alpha of any well-diversified portfolio in Equa-

tion 7.6 must be zero, even if the beta is not zero. The proof is similar to the easy zero-beta

case. If the alphas were not zero, then we could combine two of these portfolios into a zero-

beta riskless portfolio with a rate of return not equal to the risk-free rate. But this, as we have

just seen, would be an arbitrage opportunity.

To see how the arbitrage strategy would work, suppose that portfolio V has a beta of v and

an alpha of v. Similarly, suppose portfolio U has a beta of u and an alpha of u.

Taking advantage of any arbitrage opportunity involves buying and selling assets in

proportions that create a risk-free profit on a costless position. To eliminate risk, we buy

portfolio V and sell portfolio U in proportions chosen so that the combination portfolio

(V U) will have a beta of zero. The portfolio weights that satisfy this condition are

u v

wv wu

v u v u

Note that wv plus wu add up to 1.0 and that the beta of the combination is in fact zero:

u v

Beta(V U) 0

v u

v u v u

Therefore, the portfolio is riskless: It has no sensitivity to the factor. But the excess return of

the portfolio is not zero unless v and u equal zero.

u v

R(V U) 0

v u

v u v u

Therefore, unless v and u equal zero, the zero-beta portfolio has a certain rate of return that

differs from the risk-free rate (its excess return is different from zero). We have seen that this

gives rise to an arbitrage opportunity.

Suppose that the risk-free rate is 7% and a well-diversified portfolio, V, with beta of 1.3 has an

alpha of 2% and another well-diversified portfolio, U, with beta of 0.8 has an alpha of 1%. We

7.9 EXAMPLE go long on V and short on U with proportions

Arbitrage with 1.3

0.8

wv wu

1.6 2.6

Mispriced 1.3 0.8

1.3 0.8

Portfolios These proportions add up to 1.0 and result in a portfolio with beta 1.6 1.3 2.6

0.8 0. The alpha of the portfolio is: 1.6 2% 2.6 1% 0.6%. This means that the

riskless portfolio will earn a rate of return that is less than the risk-free rate by .6%. We now

complete the arbitrage by selling (or going short on) the combination portfolio and investing

the proceeds at 7%, risklessly profiting by the 60 basis point differential in returns.

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Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

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7 Capital Asset Pricing and Arbitrage Pricing Theory

We conclude that the only value for alpha that rules out arbitrage opportunities is zero.

Therefore, rewrite Equation 7.6 setting alpha equal to zero

RP PRM

rP rf P(rM rf)

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