Multifactor Models

The CAPM is a single factor model. 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.

r = R_F + Risk premium 1 + Risk premium 2 + ... + Risk premium k

where Risk premium i = factor sensitivity i x factor risk premium i.

1. The Fama French Model

Fama and French started with the observation that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a high book-value-to-price ratio (customarily called "value" stocks; their opposites are called "growth" stocks). They then added two factors to CAPM to reflect a portfolio's exposure to these two classes:

r = R_F + β_mkt (R_M - R_F) + β_S SMB + β_V HML

Here r is the portfolio's return rate, R_F is the risk-free return rate, and R_M is the return of the whole stock market. The "three factor" beta is analogous to the classical beta but not equal to it, since there are now two additional factors to do some of the work. SMB and HML stand for "small [cap] minus big" and "high [book/price] minus low"; they measure the historical excess returns of small caps and "value" stocks over the market as a whole.

The corresponding coefficients β_S and β_V take values on a scale of roughly 0 to 1: β_S = 1 would be a small cap portfolio, β_S = 0 would be large cap, β_V = 1 would be a portfolio with a high book/price ratio, etc.

2. The Pastor-Stambaugh Model

The Fama-French model is augmented with a proxy for the Pastor-Stambaugh liquidity factor.

r = R_F + β_mkt (R_M - R_F) + β_S x SMB + β_V x HML + β_L x LIQ

3. Macroeconomic Factor 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.

• BIRR 5-variable Model: confidence, time horizon, inflation, business cycle and market timing.

4. Statistical Factor Models

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.

Build-Up Method Estimates

The build-up method is widely used by the income approaches to evaluate the market values of closely held businesses. The method begins with the rate of return required by investors for risk-free assets and adds one or more premiums for different risk factors to account for the fact that the investment is not risk-free.

r = R_F + R_E + IRP + SRP + CSRP

where:

• R_E: equity risk premium. Investments in common stocks are riskier than investments in government bonds.
• IRP: industry risk premium. Certain industries (e.g., software and biotech industries) are viewed to have more risk than others.
• SRP: size risk premium. Investments in small companies are riskier than investments in large companies so the size risk premium is added to account for this incremental risk.
• CSRP: company specific risk premium. This takes into account risk factors specific to the subject company: lack of management depth, or lack of customer diversification.

The risk-free rate, equity risk premium and the size risk premium are known collectively as the "systematic risks".

Bond Yield Plus Risk Premium

As a company's business risk rises it is immediately reflected in bond yields. This method uses normal spreads of equity returns over corporate bond yields, say around 3-4%.

r = R_D + Risk Premium

For example, suppose that ABC, Inc.'s interest rate on long-term debt is 10%. Assume the risk premium is 5%. ABC's cost of retained earnings is 10% + 5% = 15%.

International Issues

Emerging markets pose special challenges to required return estimation.

• The country spread model estimates the equity risk premium as the equity risk premium for a developed market plus a country premium. The country premium is the difference between the country's government bond yield and U.S. treasury bond yield.
• The country risk rating model approach uses risk ratings for developed markets to infer risk ratings and equity risk premiums for emerging markets.