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### Subject 7. Factors and types of multifactor models

Both the market model and CAPM are single factor models. The common, single factor is the return on the market portfolio. Multifactor models describe the return on an asset in terms of the risk of the asset with respect to a set of factors. Such models generally include systematic factors, which explain the average returns of a large number of risky assets. Such factors represent priced risk, risk which investors require an additional return for bearing.

According to the type of factors used there are three categories of multifactor models:

• In macroeconomic factor models, the factors are surprises in macroeconomic variables that significantly explain equity returns. Surprise is defined as actual minus forecasted value and has an expected value of zero. The factors, such as GDP, interest rates, inflation, can be understood as affecting either the expected future cash flows of companies or the interest rate used to discount these cash flows back to the present.

• In fundamental factor models, the factors are attributes of stocks or companies that are important in explaining cross-sectional differences in stock prices. Among the fundamental factors are book-value-to-price ratio, market cap, P/E ratio, financial leverage, and earnings growth rate.

• In statistical factor models, statistical methods are applied to a set of historical returns to determine portfolios that explain historical returns in one of two senses. In factor analysis models, the factors are the portfolios that best explain (reproduce) historical return covariances. In principal-components models, the factors are portfolios that best explain (reproduce) the historical return variances.

Macroeconomic Factor Models

The key to understanding macroeconomic factor models is that the variables that explain returns are the surprises (the unexpected part) because the predicted values have been reflected in asset prices.

Here is a two-factor macroeconomic model.

Ri = ai + bi1 FGDP + bi2 FINT + εi

where

• Ri = the return for asset i.
• ai = expected return for asset i in absence of any surprises.
• bi1 = GDP surprise sensitivity of asset i. This is a slope coefficient which is interpreted as the GDP factor sensitivity of the asset i.
• FGDP = surprise in the GDP growth. This is the GDP factor surprise, the difference between the expected value and the actual value of the GDP.
• bi2 = Interest rates surprise sensitivity of asset i. This is the interest rates factor sensitivity of the asset i.
• FINT = surprise in interest rates. This is the interest rates factor surprise.
• εi = firm-specific surprises (the portion of the return to asset i not explained by the factor model).

The model says stock returns are explained by surprises in GDP growth and interest rates. The regression analysis is usually used to estimate assets' sensitivities to the factors.

Using the macroeconomic factor model, the expected return for a stock equals the intercept. The expected return on a portfolio of two stocks is thus: E(RP) = wi E(Ri) + wj E(Rj)

The calculation is straightforward. See the example 10 of the reading for details.

Macroeconomic Factor Models versus Fundamental Factor Models

• In fundamental factor models the factors are stated as returns rather than surprises.
• The factor sensitivities (attributes) are usually specified first in fundamental factor models, and then the factor returns are estimated through regressions. In contrast, in macroeconomic factor models the factor (surprise) series are first developed and then the factor sensitivities are estimated through regressions.
• Macroeconomic factors are usually small in number. Fundamental factors are often large in number, providing a more detailed picture of risk in terms that are easily related to company and security characteristics.
• The intercept in a macroeconomic model is the stock's expected return. The intercept in a fundamental model is the expected return for stocks with factor sensitivities equal to the market-wide averages.

The factors of most fundamental factor models may be classified as company fundamental factors, company share-related factors, or macroeconomic factors.

#### Practice Question 1

The macroeconomic models assume that the returns to each asset are correlated with:

A. only the surprises in some factors.
B. only the actual values of some factors.
C. only the expected values of some factors.

A surprise is the difference between actual value and expected value.

#### Practice Question 2

In the macroeconomic factor models,

I. The "b's" are very similar to betas in the market model.
II. A one percentage point surprise in factor 1 will contribute bi1 percentage points to the return to stock i.
III. The error term ε i represents unsystematic risk related to firm-specific events.
IV. The mean of the error term ε i is zero.
V. Different assets have different factor sensitivities.

Correct Answer: I, II, III, IV and V