6.7FactorModels and Tracking Portfolios

Having learned about several applications of factor models, such as estimating covari-

ances and decomposing variances, we now turn to what is perhaps the most important

application of these models: Designing a portfolio that targets a specific factor beta

configuration in order to track the risk of an asset, a liability, or a portfolio.¹⁷The track-

ing application is not only useful for hedging and for allocating capital, but it is the

foundation of the no-arbitrage risk-return relation derived in Section 6.10.

Tracking Portfolios and Corporate Hedging

Assume that Disney, which has extensive operations in Japan, knows that for every 10

percent appreciation in the Japanese yen, its stock declines by 1 percent, and vice versa.

Similarly, a weakening of the Japanese economy, which would reduce turnout at the

Disney theme park in Tokyo and dampen sales of Disney’s videos, might result in Dis-

ney’s stock price dropping by 5 percent for every 10 percent decline in the growth of

Japanese GDP. Hence, Disney has two sources of risk in Japan to worry about: cur-

rency risk and a slowing of the Japanese economy.

Disney can hedge these sources of risk by selling short a portfolio that tracks the

sensitivity of Disney’s equity to these two sources of risk. Ashort position in such a

tracking portfolio, which might be composed of U.S. and Japanese stocks, as well as

currency instruments, would (1) appreciate in value by 1 percent for every 10 percent

appreciation of the yen and (2) increase in value by 5 percent when Japan experiences

a 10 percent decline in the growth of its GDP. Afactor model allows Disney to measure

the sensitivity of all securities to these two sources of risk and identify the portfolio

weights needed to form this type of tracking portfolio.

¹⁷Chapter

5 introduced tracking portfolios.

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

193

Generally, in a context where factor models are used, tracking portfolios are well

diversified. That is, they have little or no firm-specific risk.

Capital Allocation Decisions of Corporations and Tracking Portfolios

The tracking portfolio strategy also has value for advising corporations about how to

allocate investment capital. Acentral theme of this text is that corporations create value

whenever they allocate capital for real investment projects with returns that exceed

those of the project’s tracking portfolio in the financial markets. Moreover, the corpo-

ration does not have to actually sell short the tracking portfolio from the financial mar-

kets to create wealth. That can be achieved by the investors in the corporation’s equity

securities if they find that such arbitrage is consistent with their plans for selecting opti-

mal portfolios. What is important is that the tracking portfolio be used as an appropri-

ate benchmark for determining whether the real investment is undervalued.

Designing Tracking Portfolios

Atracking portfolio is constructed by first measuring the factor betas of the investment

one wishes to track. Having identified the target configuration of factor betas, how do

we construct a portfolio of financial securities with the target configuration?

Knowledge of how to compute the factor betas of portfolios from the factor betas

of the individual investments enables an analyst to design portfolios with any targeted

factor beta configuration from a limited number of securities. The only mathematical

tool required is the ability to solve systems of linear equations.

AStep-by-Step Recipe.To design a tracking portfolio, one must follow a sequence

of steps.

1.Determine the number of relevant factors.¹⁸

2.Identify the factors with one of the three methods discussed in Section 6.4

and compute factor betas.

3.Next, set up one equation for each factor beta. On the left-hand side of the

equation is the tracking portfolio’s factor beta as a function of the portfolio

weights. On the right-hand side of the equation is the target factor beta.

4.Then, solve the equations for the tracking portfolio’s weights, making sure

that the weights sum to 1.

For example, to target the beta with respect to the first factor in a K-factor model,

the equation would be

x x ... x target beta on factor 1

111 221 NN1

The betas on the left-hand side and target beta on the right-hand side would appear as

numbers, and the x’s (the portfolio weights), would remain as unknown variables that

have to be solved for. The equation targeting the beta with respect to the second fac-

tor would be

x x ... x target beta on factor 2

112 222 NN2

Proceed in this manner until each factor has one target beta equation. Then, add an

additional equation that forces the portfolio weights to sum to 1.

¹⁸The

number of factors, which can often be found in the finance research literature, is based on

statistical tests.

Grinblatt406Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw406Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
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194Part IIValuing Financial Assets

x x ... x 1

1 2 N

Solving all of these equations for the portfolio weights, x, x, . . . x,designs a track-

12N

ing portfolio with the proper factor betas. Example 6.5 illustrates how to do this.

Example 6.5:Designing a Portfolio with Specific FactorBetas

Consider the three stocks in Examples 6.1, 6.3, and 6.4.You are informed that the Wilshire

5000 Index, a broad-based stock index, has a factor beta of 2 on the first factor and a fac-

tor beta of 1 on the second factor.Design a portfolio of stocks A, B, and C that has a fac-

tor beta of 2 on the first factor and 1 on the second factor and thus tracks the Wilshire 5000

Index.

Answer:To design a portfolio with these characteristics, it is necessary to find portfolio

weights, x, x, x, that make the portfolio-weighted averages of the betas equal to the tar-

ABC

get betas.To make the weights sum to one, x, x, and xmust satisfy

ABC

x x x 1

A BC

To have a factor beta of 2 on the first factor implies

1x 3x 1.5x 2

A B C

To have a factor beta of 1 on the second factor implies

4x 2x 0x 1

A B C

Substituting the value of xfrom the first equation into the other two equations implies

C

(i) 1x3x1.5(1xx) 2

ABAB

(ii)4x2x 1

AB

Equation (i), immediately above, is now solved for x.This value, when substituted into equa-

B

tion (ii), eliminates Xfrom equation (ii), so that it now reads

B

2

4x (x1) 1

^A3^A

This equation has X .1 as its solution.Since equation (i) reduces to

A

x1

x ^A

^B 3

x .3

B

implying that

x .8

C

The Numberof Securities Needed to Design Portfolios with Specific Target Beta

Configurations.Example 6.5 could have made use of any configuration of target

betas on the two factors and derived a solution. Hence, it is possible to design a port-

folio with almost any factor beta configuration from a limited number of securities. In

a two-factor model, only three securities were needed to create investments with any

factor beta pattern. In a five-factor model, six securities would be needed to tailor the

factor risk. In a K-factor model, K1securities would be needed.

An Interesting Target Beta Configuration.An important application of the design

of portfolios with specific factor configurations is the design of pure factor portfolios.

These portfolios, discussed in the next section, can be thought of as portfolios that track

Grinblatt408Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw408Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

Chapter 6

Factor Models and the Arbitrage Pricing Theory

195

the factors. They make it easier to see that factor models imply a useful risk-expected

return relation.