6.5FactorBetas

The magnitudes of a security’s factor betas describe how sensitive the security’s return

is to changes in the common factors.

What Determines Factor Betas?

Consider how the stock prices of General Motors (GM), an automobile manufacturer,

and AMC Entertainment, Inc. (AMC), a chain of movie theaters, react differently to

factors. Auto sales are linked highly to overall economic activity. Therefore, the returns

to holding GM stock should be very sensitive to changes in industrial production. In

contrast, movie attendance is not as related to the business cycle as car purchases, so

AMC should prove less sensitive to this factor. Hence, GM should have a larger fac-

tor beta on the industrial production factor than AMC. If consumers go to more movies

during recessions than at other times, substituting cheap theater entertainment for

expensive vacations during tough times, AMC might even have a negative factor beta

on the industrial production factor.

Factor Models for Portfolios

Multifactor betas, like single-factor betas, have the property that portfolio betas are the

portfolio-weighted averages of the betas of the securities in the portfolio (see exercise

6.1). For example, if stock A’s beta on the inflation factor is 2 and stock B’s is 3, a

portfolio that has weights of .5 on stock Aand .5 on stock B has a factor beta of 2.5

on this factor.

Result 6.2

The factor beta of a portfolio on a given factor is the portfolio-weighted average of the indi-vidual securities’betas on that factor.

Given the K-factor model (or factor model with Kdistinct factors) of equation (6.4)

for each stock i, a portfolio of Nsecurities with weights xon stock i and return,

i

˜ xr˜xr˜. . .xr˜,has a factor equation of

R

p1122N N

˜˜˜. . .F˜˜

R FF

ppp11p22pKKp

where

. . .

x x x

p 11 22 NN

. . .

xxx

p1 111 221 NN1

. . .

xxx

p2 112 222 NN2

....

.

.

.

.

....

xx. . . x

pK 11K 22K NNK

˜ x˜x˜. . . x˜

p 11 22 NN

Example 6.1 shows that not only is the factor beta a portfolio-weighted average of the

factor betas of the stocks in the portfolio, but also the alphas ( ) and the epsilons (˜)

of the portfolios are the portfolio-weighted averages of the alphas and epsilons of the

stocks.

Grinblatt394Titman: Financial	II. Valuing Financial Assets	6. Factor Models and the	© The McGraw394Hill
Markets and Corporate		Arbitrage Pricing Theory	Companies, 2002
Strategy, Second Edition

188Part IIValuing Financial Assets

Example 6.1:Computing FactorBetas forPortfolios

Consider the following two-factor model for the returns of three securities:Apple Computer

(security A), Bell South (security B), and Citigroup (security C).

˜ .03F˜˜˜

r4F

A 12A

˜˜˜˜

r .053F2F

B 12B

˜˜˜˜

r .101.5F0F

C 12C

Using Result 6.2, write out the factor equation for a portfolio that (a) equally weights all three

securities and (b) has weights X .5, X 1.5, and X 0.

ABC

Answer:

111

(a) (.03)(.05)(.10) .06

^p333

111

(1)(3)(1.5) 1.833

^p1333

111

(4)(2)(0).667

^p2333

Thus

˜˜˜˜

R .061.833F .667F

p12p

where

˜is an average of the three ’s

p

(b) .5(.03)1.5(.05)0(.10) .06

p

.5(1)1.5(3)0(1.5) 4

p1

.5(4)1.5(2)0(0) 5

p2

Thus

˜˜˜˜

R .064F5F

p12p

where

˜ .5˜1.5˜

pAB