6.10No Arbitrage and Pricing: The Arbitrage Pricing Theory

Because firm-specific risk is fairly unimportant to investors who hold well-diversified port-

folios, it is reasonable at this point to pretend that firm-specific risk does not exist and to

analyze the risk of securities by focusing only on their factor betas. If most investors do

not have to bear firm-specific risk because they hold well-diversified portfolios, our analy-

sis of the relation between risk and return will be unaffected by this omission.

If two investments perfectly track each other and have different expected returns,

then, in the absence of transaction costs and related market frictions, an investor can

achieve riskless profits by purchasing the investment with the higher expected return

and selling short the investment with the lower expected return. It is possible to demon-

strate that such arbitrage opportunities will exist only if securities returns do not sat-

isfy an equation that relates the expected returns of securities to their factor betas. As

noted previously, this risk-expected return relation is known as the arbitrage pricing

theory (APT).

The Assumptions of the Arbitrage Pricing Theory

The APTrequires only four assumptions:

1.Returns can be described by a factor model.

2.There are no arbitrage opportunities.

3.There are a large number of securities, so that it is possible to form portfolios

that diversify the firm-specific risk of individual stocks. This assumption

allows us to pretend that firm-specific risk does not exist.

4.The financial markets are frictionless.

This section derives the APT. To keep the analysis relatively simple, consider invest-

ments with no firm-specific risk.

Arbitrage Pricing Theory with No Firm-Specific Risk

Consider investment iwith returns generated by the K-factor model represented by

˜˜˜. . .˜
r FFF	(6.6)
iii11i22iKK

Note that equation (6.6) has no ˜term; thus, there is no firm-specific risk. As Result

i

6.5 noted, one way to track the return of this investment is to form a portfolio with

K

weights of 1 on the risk-free security, o n factor portfolio 1, on factor

iji1i2

j 1

portfolio 2,..., and finally on factor portfolio K.Recall that these factor portfo-

iK

lios can be generated either from a relatively small number of securities with no firm-

specific risk or from a very large number of securities where the firm-specific risk is

diversified away.

²¹Moreover,

even a risk-free security could have been formed from securities c, g, and s in Example

6.6, so it is possible to break this down to an even more basic level.

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200Part IIValuing Financial Assets

The expected return of the portfolio that tracks investment i is

r

fi11i22iKK

where

,...

are the risk premiums of the factor portfolios.

1K

It should be immediately apparent that an arbitrage opportunity exists—unless the

original investment and its tracking portfolio have the same expected return—because

a long position in investment i and an offsetting short position in the tracking portfo-

lio has no risk and no cost. For example, if the common stock of Dell Computer is

investment i, buying $1 million of Dell and selling short $1 million of the tracking

portfolio would require no up-front cash. Moreover, since the factor betas of the long

and short positions match exactly, any movements in the value of Dell’s stock due to

factor realizations would be completely offset by exactly opposite movements in the

value of the short position in the tracking portfolio. Hence, if the expected return of

Dell’s stock exceeds the expected return of Dell’s tracking portfolio, an investor obtains

a riskless positive cash inflow at the end of the period. For example, if Dell’s expected

return exceeds the tracking portfolio’s by 2 percent, the investor would receive

$1,000,000 .02 $20,000

Since this cash does not require any up-front money and is obtained without risk, buy-

ing Dell and shorting its tracking portfolio represents an arbitrage opportunity. Simi-

larly, if the expected return of Dell stock was smaller than the expected return of the

tracking portfolio, a short position in Dell stock and an equal long position in its track-

ing portfolio would provide an arbitrage opportunity. To prevent arbitrage, the expected

return of Dell and its tracking portfolio must be equal.

Result 6.6 states this formally:

Result 6.6An arbitrage opportunity exists for all investments with no firm-specific risk unless

. . .
r r	(6.7)
ifi11i22iKK

where

,...

applies to all investments with no firm-specific risk.

1K

The equation of the arbitrage pricing theory, equation (6.7), is a relation between

risk and expected return that must hold in the absence of arbitrage opportunities. On

the left-hand side of the equation is the expected return of an investment. On the right-

hand side is the expected return of a tracking portfolio with the same factor betas as

the investment. Equation (6.7) thus depicts a relationship where there is no arbitrage:

The equal sign merely states that the expected return of the investment should be the

same as that of its tracking portfolio.

Graphing the APT Risk Return Equation

In the one-factor case, the graph of equation (6.7) observed in Exhibit 6.5 is very

similar to the graph of the securities market line (depicted in panel B of Exhibit 5.5

on page 146). On one axis is the beta or factor beta of a security; on another axis

is its mean return. In this case, the risk-return relation graphs as a straight line.

According to the results in this section, if there is no arbitrage, all investments must

lie on this line.

In the two-factor case, equation (6.7) graphs as a plane in three dimensions (see

Exhibit 6.6). The location and slope of the plane are determined by the risk-free return,

which is the height of the plane above the origin, and the two risk premiums, or

s of

the pure factor portfolios. All investments must lie on this plane if there is no arbitrage.

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

201

EXHIBIT6.5

The APTRelation between the Mean Return of a Stock and Its FactorBetas

Mean

return

r f +
	Factor
	portfolio

r f

			Factor Beta
0	.5	1

EXHIBIT6.6

Relation between the Mean Return of a Stock and Its FactorBetas in a MultifactorModel

Mean

return

	Factor 2
r f	Beta

Factor 1

Beta

Grinblatt421Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw421Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

202Part IIValuing Financial Assets