The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 533 Wednesday, February 4, 2004 1:10 PM

Multifactor Models and Common Trends for Common Stocks 533

Once the descriptors that are statistically significant in explaining

stock returns are identified, they are grouped into “risk indices” to cap-

ture related company attributes. For example, descriptors such as market

leverage, book leverage, debt-to-equity ratio, and company’s debt rating

are combined to obtain a risk index referred to as “leverage.” Thus, a risk

index is a combination of descriptors that captures a particular attribute

of a company. The Barra fundamental multifactor risk model, the “E3

model” being the latest version, has 13 risk indices and 55 industry

groups. Exhibit 18.1 lists the 13 risk indices in the Barra model.^3

Also shown in the exhibit are the descriptors used to construct each

risk index. The 55 industry classifications are further classified into sec-

tors. For example, the following three industries comprise the energy

sector: energy reserves and production, oil refining, and oil services. The

consumer noncyclicals sector consists of the following five industries:

food and beverages, alcohol, tobacco, home products, and grocery

stores. The 13 sectors in the Barra model are basic materials, energy,

consumer noncyclicals, consumer cyclicals, consumer services, industri-

als, utility, transport, health care, technology, telecommunications, com-

mercial services, and financial.

Given the risk factors, information about the exposure of every

stock to each risk factor (βi,Fj) is estimated using statistical analysis. For

a given time period, the rate of return for each risk factor (RFj) also can

be estimated using statistical analysis. The prediction for the expected

return can be obtained from the above equation for any stock. The non-

factor return (ei) is found by subtracting the actual return for the period

for a stock from the return as predicted by the risk factors.

Moving from individual stocks to portfolios, the predicted return for

a portfolio can be computed. The exposure to a given risk factor of a

portfolio is simply the weighted average of the exposure of each stock in

the portfolio to that risk factor. For example, suppose a portfolio has 42

stocks. Suppose further that stocks 1 through 40 are equally weighted in

the portfolio at 2.2%, stock 41 is 5% of the portfolio, and stock 42 is

7% of the portfolio. Then the exposure of the portfolio to risk factor j is

0.022 β1,Fj + 0.022 β2,Fj + ... + 0.022 β40,Fj + 0.050 β41,Fj + 0.007 β42,Fj

The nonfactor error term is measured in the same way as in the case of

an individual stock. However, in a well diversified portfolio, the nonfactor

error term will be considerably less for the portfolio than for the individ-

(^3) For a more detailed description of each descriptor, see Appendix A in Barra, Risk
Model Handbook United States Equity: Version 3 (Berkeley, CA: Barra, 1998). A
listing of the 55 industry groups is provided in Exhibit 13.9.