Verifying the Existence of Arbitrage

To learn whether the arbitrage pricing theory holds, do not look at graphs (which is

obviously impossible if there are more than two factors) or form tracking portfolios.

Instead, determine whether a single set of

s can generate the expected returns of all

the securities. For example, one can use the returns and factor sensitivities of K 1

securities to solve equation 6.7 for the K

s. If these

s also generate the expected

returns of all the other securities, the APTholds; if they do not, the APTis violated

and there is an arbitrage opportunity (assuming the factor model assumption holds).

Example 6.9 illustrates the procedure for testing whether a single set of

s have this

property.

Example 6.9:Determining WhetherArbitrage Exists

Let the following 2-factor model describe the returns to four securities:three risky securities

indexed 1, 2, and 3 and a risk-free security.

r˜ .05

f

r˜ .060˜.02˜

FF

1 12

r˜ .08.02F˜.01F˜

2 12

r˜ .15.04F˜.04F˜

3 12

Is there an arbitrage opportunity?

Answer:The APT risk-expected return equation says

r˜ r

ifi11i22

For securities 1 and 2, this reads

.06 .05 0

.02

12

and

.08 .05 .02

.01

1 2

The first of these two equations implies

.5.Substituting this value for

into the sec-

2 2

ond equation gives

1.25.Using these values for the APT equation of security 3, check

1

to see whether

.15 equals [.05 .04(1.25) .04(.5)]

Since the right-hand side term in brackets equals .12, which is less than the value on the

left-hand side, .15, there is arbitrage.A long position in security 3 (the high expected return

security) and an equal short position in its tracking portfolio, formed from securities 1, 2, and

the risk-free security, generates arbitrage.

In Example 6.9, if the expected return of security 3 had been equal to .12, there

would be no arbitrage. However, because its expected return of .15 exceeded the .12

expected return of its tracking portfolio, the no arbitrage risk-return relation of equa-

tion (6.7) is violated. While this provides a prescription for generating arbitrage, it does

not determine whether security 3 is underpriced or whether its tracking portfolio is over-

priced. Based on this example, all one knows is that security 3 is underpriced relative

to its tracking portfolio.

An alternative method for identifying the existence of arbitrage is to test directly

whether a unique set of

s generate the expected returns of the securities. In this case,

solve for the set of

s using one group of securities (the number of securities in the

Grinblatt423Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw423Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

Chapter 6

Factor Models and the Arbitrage Pricing Theory

203

set is one plus the number of factors). Then, solve again, using a different group of

securities. If the different sets of

s are the same, there is no arbitrage; if they differ,

there is arbitrage. Example 6.10 illustrates this technique.

Example 6.10:Determining WhetherFactorRisk Premiums Are Unique

Use the same data provided in Example 6.9 to determine whether there is an arbitrage

opportunity by comparing the pair of

s found in Example 6.9 with the pair of

s found by

using securities 2, 3, and the risk-free asset.

Answer:The APT risk-expected return equation says

r˜ r

ii11i22

f

Example 6.9 found that using the risk-free asset and securities 1 and 2 to solve for

and

1

yields

2

1.25	and	.5
1		2

Using securities 2 and 3 and the risk-free asset to solve for the

’s requires solving the fol-

lowing pair of equations

.08 .05 .02

.01

1 2

.15 .05 .04

.04

1 2

The first equation of the pair immediately above says

32

.Substituting this into the

21

second equation and solving for

yields

.5, which, when substituted back into the first

11

equation, yields

2.Since this pair of

s differs from the first pair, the APT equation does

2

not hold and there is arbitrage.

If security 3 in the last example had an expected return of .12, the second pair of

s would have been identical to the first pair. This would be indicative of no arbitrage.

Exhibits 6.7 and 6.8 illustrate this technique in a slightly more general fashion.

They graph the factor risk-premiums that are consistent with the APTrisk-expected

return equation (6.7) for each of the three risky securities in the last example. The

horizontal axis corresponds to

and the vertical axis corresponds to

. As you can

12

see, solving systems of equations is identical to finding out where lines cross. The

intersection of lines for securities 1 and 2 (2 and 3) represent the first (second) pair

of factor risk premiums. In Exhibit 6.7, the intersection point for the lines that cor-

respond to securities 1 and 2, at point A, is not on the line for security 3. To pre-

clude arbitrage, which would be the case if the expected return of security 3 was

.12, all three of these lines would have to intersect at the same point, point Ain

Exhibit 6.8. Any other securities that exist would also require corresponding lines

that go through this point.

The Risk-Expected Return Relation for Securities with Firm-Specific Risk

Up to this point, Chapter 6 has examined the risk-expected return relation of portfolios

and securities that have no firm-specific risk. With a sufficiently large number of secu-

rities, however, the APTrisk-expected return equation (6.7) also must hold, at least

approximately, for individual securities that contain firm-specific risk.