6.9Tracking and Arbitrage

The previous sections illustrated how to construct portfolios with any pattern of factor

betas from a limited number of securities. With more securities, the additional degrees

of freedom in the selection of portfolio weights make the task of targeting a specific

factor beta configuration even easier. Because a sufficiently large number of securities

in the portfolio makes it likely that these portfolios will have negligible firm-specific

risk, it is usually possible to construct portfolios that perfectly track investments that

have no firm-specific risk. This is done by forming portfolios of the pure factor port-

folios that have the same factor betas as the investment one wishes to track. The fac-

tor equations of the tracking portfolio and the tracked investment will be identical

except for the s. By assumption, there are no ˜terms in these factor equations. Hence,

the return of the tracking portfolio and the tracked investment can, at most, differ by

a constant: the difference in their s, which is the difference in their expected returns.

If the factor betas of the tracking portfolio and the tracked investment are the same

but their expected returns differ, then there will be an opportunity for arbitrage. For

example, if the tracking portfolio has a higher expected return, then investors can buy

that portfolio and sell short the tracked investment and receive risk-free cash in the

future without spending any money today.

Using Pure Factor Portfolios to Track the Returns of a Security

Example 6.8 illustrates how to use factor portfolios and the risk-free asset to track the

returns of another security.

Example 6.8:Tracking and Arbitrage with Pure FactorPortfolios

Given a two-factor model, find the combination of a risk-free security with a 4 percent return

and the two pure factor portfolios from Example 6.6 that tracks stock i, which is assumed

to have a factor equation of

r˜˜˜

.0862F.6F

12

Then, find the expected return of the tracking portfolio and determine if arbitrage exists.

Recall that the factor equations for the two pure factor portfolios (see Example 6.7) were

respectively given by

˜˜˜

R .06F0F

p112

˜˜˜

R .040FF

p212

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198Part IIValuing Financial Assets

Answer:To track stock i’s two factor betas, place a weight of 2 on factor portfolio 1 and

a weight of .6 on factor portfolio 2.Since the weights now sum to 1.4, a weight of .4

must be placed on the risk-free asset.The expected return of this portfolio is the portfolio-

weighted average of the expected returns of the risk-free asset, factor portfolio 1, and fac-

tor portfolio 2, which is

.4(.04) 2(.06) .6(.04) .08

An arbitrage opportunity exists because this expected return of 8 percent differs from the

8.6 percent return of the tracked security, as indicated by the .086 intercept (and the com-

mon convention, discussed earlier, that the means of the factors are zero).

The Expected Return of the Tracking Portfolio

In Example 6.8, the tracking portfolio for stock i is a weighted average of the two fac-

tor portfolios and the risk-free asset. Factor portfolio 1 is used solely to generate the

target factor 1 beta. Factor portfolio 2 is used solely to generate the target sensitivity

to factor 2. The risk-free asset is used only to make the weights of the tracking port-

folio sum to one. Note that the expected return of this tracking portfolio is

Expected return (1 )r (r

) (r

)

i1i2f i1f 1i2f2

where is the factor beta of the tracked investment on factor j (and thus also the

ij

weight on pure factor portfolio j), and

is the risk premium of factor portfolio j (mak-

j

ing r

its expected return). The expression above for the expected return is also

f j

written in an equivalent form

Expected return r

f i11 i22

Result 6.5 generalizes Example 6.8 as follows:

Result	6.5	An investment with no firm-specific risk and a factor beta of on the jthfactor in a
		ij
		K-factormodel is tracked by a portfolio with weights of on factor portfolio 1, on
		i1i2
		K
		factor portfolio 2,..., on factor portfolio K,and 1 on the risk-free asset.
		iK ij
		j 1
		The expected return on this tracking portfolio is therefore

. . .

r

fi11i22iKK

where

,

, .. .

denote the risk premiums of the factor portfolios and ris the risk-

12Kf

free return.

Decomposing Pure Factor Portfolios into Weights on More Primitive Securities

Factor portfolios are themselves combinations of individual securities, like stocks and

bonds. In Example 6.8, the portfolio with weights of .4 on the risk-free security, 2

on factor portfolio 1, and .6 on factor portfolio 2, can be further broken down. Recall

from Example 6.6 that factor portfolio 1 has respective weights of (2, 1/3, 4/3) on

the three individual securities while factor portfolio 2 has corresponding weights of (3,

2/3, 4/3). Hence, a weight of 2 on factor portfolio 1 is really a weight of 4 on secu-

rity c, 2/3 on security g, and 8/3 on security s. Aweight of 0.6 on factor port-folio

2 is really a weight of 1.8 on security c, 0.4 on security g, and 0.8 on security s.

Summing the weights for each security found in Example 6.6 implies that at a more

basic level, our tracking portfolio in Example 6.8 consisted of weights of 2.2, 16/15,

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Strategy, Second Edition

Chapter 6

Factor Models and the Arbitrage Pricing Theory

199

and 28/15 on securities c, g, and s, respectively, plus a weight of 0.4 on the risk-

free asset.²¹

Thus, it makes no difference in the last example whether one views the

tracking portfolio as being generated by stocks c, g, and s or by pure factor portfolios.