A return-generating model tries to estimate the expected return of a security based on certain parameters. 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 forecast value and has an expected value of zero. The factors, such as GDP, interest rates, and 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.

Here is a two-factor macroeconomic model.

R_i = a_i + b_i1 F_GDP + b_i2 F_INT + ε_i

where

• R_i = the return for asset i.
• a_i = expected return for asset i in the absence of any surprises.
• b_i1 = GDP surprise sensitivity of asset i. This is a slope coefficient which is interpreted as the GDP factor sensitivity of asset i.
• F_GDP = surprise in GDP growth. This is the GDP factor surprise, the difference between the expected value and the actual value of the GDP.
• b_i2 = interest rate surprise sensitivity of asset i. This is the interest rate factor sensitivity of asset i.
• F_INT = surprise in interest rates. This is the interest rate factor surprise.
• ε_i = 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 these factors.