6.6Using FactorModels to Compute Covariances and Variances

This section demonstrates that the correlation or covariance between the returns of any

pair of securities is determined by the factor betas of the securities. It then discusses

how to use factor betas to compute more accurate covariance estimates. When using

mean-variance analysis to identify the tangency and minimum variance investment port-

folios, the more accurate the covariance estimate, the better the estimate of the weights

of these critical portfolios.

Computing Covariances in a One-Factor Model

Since the ˜s in the factor equations described in the last section are assumed to be

uncorrelated with each other and the factors, the only source of correlation between

securities has to come from the factors. The next example illustrates the calculation of

a covariance in a one-factor model.

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189

Example 6.2:Computing Covariances from FactorBetas

The following equations describe the annual returns for two stocks, Acorn Electronics and

Banana Software, where F˜is the change in the GDP growth rate and A and B represent

Acorn and Banana, respectively

r˜ .102F˜˜

AA

r˜ .153F˜˜

BB

The ˜’s are assumed to be uncorrelated with each other as well as with the GDP factor, and

the factor variance is assumed to be .0001.Compute the covariance between the two stock

returns.

Answer:

cov (.102˜˜, .153˜˜)

FF

AB AB

cov (2F˜,˜,3F˜˜)

A B

since constants do not affect covariances.Expanding this covariance, using the principles

developed in Chapter 4, yields

cov (2F˜, 3F˜)cov (2F˜, ˜)cov (˜, 3F˜)cov (˜, ˜)

ABBAAB

cov (2F˜, 3F˜)000

Thus, the covariance between the returns is the covariance between 2˜and 3˜,which is

FF

6var(˜), or .0006.

F

The pair of equations for ˜and r˜in Example 6.2 represents a one-factor model

r

AB

for stocks Aand B. Notice the subscripts in this pair of equations. The ˜s have the

same subscripts as the returns, implying that they represent risks specific to either stock

Aor B. The value each takes on provides no information about the value the other

acquires. For example, , taking on the value .2, provides no information about the

A

value of . The GDPfactor, represented by ˜,has no Aor B subscript, implying that

F

B

this macroeconomic factor is a common factor affecting both stocks. Since the firm-

specific components of these returns are determined independently, they have no effect

on the covariance of the returns of these stocks. The common factor provides the sole

source of covariation. As a result, the covariance between the stock returns is deter-

mined by the variance of the factor and the sensitivity of each stock’s return to the

factor. The more sensitive the stocks are to the common factor, the greater is the covari-

ance between their returns.

Computing Covariances from Factor Betas in a Multifactor Model

Example 6.3illustrates how return covariances are calculated within a two-factor

model.

Example 6.3:Computing Covariances from FactorBetas in a Two-Factor

Model

Consider the returns of three securities (Apple, Bell South, and Citigroup), given in Exam-

ple 6.1.Compute the covariances between the returns of each pair of securities, assuming

that the two factors are uncorrelated with each other and both factors have variances of

.0001.

Answer:Since the two factors, denoted F˜and F˜, are uncorrelated with each other, and

12

since the ˜s are uncorrelated with each of the two factors and with each other

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cov (r˜, r˜) 3var (F˜)8var (F˜) .0005

AB12

cov (r˜, r˜) 1.5var (F˜) .00015

AC1

cov (r˜, r˜) 4.5var (F˜) .00045

BC1

In Example 6.3, the covariances between the returns of any two securities are deter-

mined by the sensitivities of their returns to factor realizations and the variances of the

factors. If some of the factors have high variances, or equivalently, if a number of stock

returns are particularly sensitive to the factors, then those factors will account for a

large portion of the covariance between the stocks’returns. More generally, covariances

can be calculated as follows:

Result 6.3

Assume that there are Kfactors uncorrelated with each other and that the returns of secu-rities i and j are respectively described by the factor models:

˜ F˜F˜. . .˜˜

rF

iii11i22iKKi

˜ F˜F˜. . .˜˜

rF

jjj11j22jKKJ

Then the covariance between r˜and r˜is:

ij

var(˜)var(F˜. . .˜)	(6.5)
F)var(F
iji1j11i2j22iKjKK

Result 6.3 states that covariances between securities returns are determined entirely

by the variances of the factors and the factor betas. The firm-specific components, ˜

i

and ˜,play no role in this calculation. If the factors are correlated, the ˜’s are still

j

irrelevant for covariance calculations. In this case, however, additional terms must be

appended to equation (6.5) to account for the covariances between common factors.

Specifically, the formula becomes:

KK

cov(F˜, F˜)

ijim jn mn

m 1n 1

Factor Models and Correlations between Stock Returns

In a multifactor model, the returns of stocks that have similar configurations of factor

betas are likely to be highly correlated with each other while those that have differing

factor beta patterns are likely to be less correlated with each other.

In an examination of Deere, General Motors (GM), and Wal-Mart, one is likely to

find that the returns of GM (primarily a manufacturer of automobiles) and Deere (a

manufacturer of farm equipment, especially tractors) have the largest correlation while

Wal-Mart has less correlation with the other two. Indeed, monthly returns from 1982

through 2000 bear this out. The correlation between GM and Deere is .37, while Wal-

Mart’s correlations with these two firms are .27 and .33, respectively.

The greater correlation between Deere and GM occurs not because they both man-

ufacture transportation equipment—consumer demand for automobiles is not quite the

same as the farmers’demand for tractors—but because both companies are highly sen-

sitive to the interest rate factor and the industrial production factor. Wal-Mart, on the

other hand, while highly sensitive to the industrial production factor, is not particularly

sensitive to the interest rate factor.

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191

Applications of Factor Models to Mean-Variance Analysis

Result 6.3 is used by portfolio managers who estimate covariances to determine opti-

mal portfolio weights. For example, computing the tangency portfolio or the minimum

variance portfolio in mean-variance analysis requires the estimation of covariances for

each possible pairing of securities. The universe of securities available to most investors

is large. The more than 8,000 common stocks listed on the NYSE, AMEX, and Nasdaq

markets toward the end of 1999 had more than 30 million covariances between differ-

ent securities in addition to over 8,000 variances. Calculating more than 30 million

numbers is a herculean task. If a five-factor model is accurate enough as a description

of the covariance process, only five factor betas per security, or about 40,000 calcula-

tions, would be needed in addition to variance calculations for each of 8,000 securities

(and five factors). While 48,000 calculations is a daunting task, it is far less daunting

than 30 million calculations.

One of the original reasons for the development of the one-factor market model

was to reduce the computational effort needed to determine covariances. Researchers,

however, discovered that the market model added more than computational simplicity.

The correlations, and consequently the covariances, estimated from the one-factor mar-

ket model were, on average, better predictors of future correlations than the correla-

tions calculated directly from past data (see Elton, Gruber, and Urich (1978)). The cor-

relations and covariances based on multiple factor models might do even better.¹⁶

Using Factor Models to Compute Variances

Like the market model, factor models provide a method for breaking down the vari-

ance of a security return into the sum of a diversifiable and a nondiversifiable compo-

nent. For a one-factor model, where

˜ F˜˜

r

iii

return variance can be decomposed as follows:

var (˜2˜)var (˜)

r) var (F

iii

The first term in the variance equation algebraically defines factor risk; the second term

is firm-specific risk. The fraction of risk that is factor related is the R-squared statistic

from a regression of the returns of security i on the factor. Result 6.4 summarizes this

more generally in a multifactor setting.

Result 6.4When Kfactors are uncorrelated with each other and security i is described by the factor model

˜ F˜F˜. . .˜˜

rF

iii11i22iKKi

the variance of rcan be decomposed into the sum of K1 terms

˜

i

var(˜2˜2˜. . .2˜)var(˜)

r) var(F)var(F)var (F

ii1 1i22iK Ki

¹⁶Recall

that mean-variance analysis ideally requires the true covariances that generate securities

returns. However, just as a fair coin does not turn out to be heads 50 percent of the time in a series of

tosses, historical covariances based on a few years of data also will deviate from the true covariances. In

the experience of modern science, parsimonious models that capture the underlying structure of a

phenomenon are more accurate at prediction than mere extrapolations of data. Here, factor models—to

so dominate the inferences drawn from chance correlations based on past data—must be capturing some

of the underlying structure of the true covariances.

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192Part IIValuing Financial Assets

In this variance decomposition, the sum of the first Kterms is the factor risk of

the security while the last term is the firm-specific risk. Example 6.4 applies the decom-

position given in Result 6.4.

Example 6.4:Decomposing Variance Risk

Assume that the two factors in Example 6.1 each have a variance of .0001 and that the s of

˜

the three securities have variances of .0003, .0004, and .0005, respectively.Compute the factor

risk and the firm-specific risk of each of the three securities in Example 6.1.Then compute the

return variance.The factor equations for the three securities in the example are repeated here.

˜ .03F˜4F˜˜

r

A12A

˜ .053˜2˜˜

rFF

B12B

˜ .101.5F˜0F˜˜

r

C12C

Answer:The variance equation in Result 6.4 implies

	(1)	(2)	(3) (1) (2)
Security	Factor Risk	Firm-Specific Risk	Return Variance

A	1(.0001)16(.0001) .0017	.0003	.002
B	9(.0001)4(.0001) .0013	.0004	.0017
C	2.25(.0001)0 .000225	.0005	.000725