Chapter 12: Using Multifactor Models

USING MULTIFACTOR MODELS CHAPTER12 1 APT - The Carhart four-factor model The Carhart model posits the existence of three systematic risk factors beyond the market risk factor: • Small minus big (SMB) • High minus low (HML) • Winners minus losers (WML) The model, in which the excess return on the portfolio is explained as a function of the portfolio’s sensitivity to a; • Market index (RMRF), • Market capitalization factor (SMB), • Book-to-market factor (HML), which is essentially the reciprocal of the aforementioned price-to-book ratio, • Momentum factor (WML). Size, value, and momentum are common themes in equity portfolio construction, and all three factors continue to have robust uses in active management risk decomposition and return attribution. 17 APT - The Carhart four-factor model Equation 3a is the Carhart model: Where: Rp and RF = the return on the portfolio and the risk-free rate of return, respectively ap = “alpha” or return in excess of that expected given the portfolio’s level of systematic risk (assuming the four factors capture all systematic risk) bp = the sensitivity of the portfolio to the given factor RMRF = the return on a value-weighted equity index in excess of the one-month T-bill rate SMB = small minus big, a size (market capitalization) factor; SMB is the average return on smallcap portfolios minus the average return on large-cap portfolios HML = high minus low, the average return on high book-to-market portfolios minus the average return on low book-to-market portfolios WML = winners minus losers, a momentum factor; WML is the return on a portfolio of the past year’s winners minus the return on a portfolio of the past year’s losers. (Note that WML is an equally weighted average of the stocks with the highest 30% month returns lagged 1 month minus the equally weighted average of the stocks with the lowest 30% 11-month returns lagged 1 month.) εp = an error term that represents the portion of the return to the portfolio, p, not explained by the 18 model MULTIFACTORMODELS: TYPES,Factors and Types of Multifactor Models Macroeconomic factor model: A multifactor model in which the factors are surprises in macroeconomic variables that significantly explain equity returns. Fundamental factor models: A multifactor model in which the factors are attributes of stocks or companies that are important in explaining cross-sectional differences in stock prices. Statistical factor model: A multifactor model in which statistical methods are applied to a set of historical returns to determine portfolios that best explain either historical return covariances or variances. 19 The Structure of Macroeconomic Factor Models Suppose that Kmacro factors explain asset returns. Then in a macroeconomic factor model, Equation 4 expresses the return of asset i: where Ri = the return to asset i ai = the expected return to asset i bik = the sensitivity of the return on asset i to a surprise in factor k, k = 1, 2, . . ., K Fk = the surprise in the factor k, k . 1, 2, . . ., K εi = an error term with a zero mean that represents the portion of the return to asseti not explained by the factor model • When we have the parameters for the individual assets in a portfolio, we can calculate the portfolio’s parameters as a weighted average of the parameters of individual assets. • An individual asset’s weight in that calculation is the proportion of the total market value of the portfolio that the individual asset represents. • In macroeconomic factor models, the time series of factor surprises are constructed first. Regression analysis is then used to estimate assets’ sensitivities to the factors. • In practice, estimated sensitivities and intercepts are often acquired from one of the many consulting companies that specialize in factor models. 20 The Structure of Fundamental Factor Models In fundamental factor models, the factors are stated as returns rather than return surprises in relation to predicted values, so they do not generally have expected values of zero. This approach changes the meaning of the intercept, which is no longer interpreted as the expected return. Note that if the coefficients were not standardized, as described in the following paragraph, the intercept could be interpreted as the risk-free rate because it would be the return to an asset with no factor risk (zero factor betas) and no asset-specific risk (with standardized coefficients, the intercept is not interpreted beyond being an intercept in a regression included so that the expected asset-specific risk equals zero). Factor sensitivities are also interpreted differently in most fundamental factor models. In fundamental factor models, the factor sensitivities are attributes of the security. An asset’s sensitivity to a factor is expressed using a standardized beta: the value of the attribute for the asset minus the average value of the attribute across all stocks divided by the standard deviation of the attribute’s values across all stocks. 21 The Structure of Fundamental Factor Models We can represent the structure of fundamental factor models with the equation of a macroeconomic factor model , but we need to interpret the terms differently: Standardized beta With reference to fundamental factor models, the value of the attribute for an asset minus the average value of the attribute across all stocks, divided by the standard deviation of the attribute across all stocks. The factors of most fundamental factor models for equities: 1. Company fundamental factors: These are factors related to the company’s internal performance. 2. Company share-related factors: These factors include valuation measures and other factors related to share price or the trading characteristics of the shares. 3. Macroeconomic factors: Sector or industry membership factors fall under this heading. 22 Fixed-Income Multifactor Models, Macroeconomic Multifactor Models The return to bond i, Ri, can be modeled as: where Ri = the return to bond i ai = the expected return to bond i bi1 = the sensitivity of the return on bond i to inflation rate surprises FINFL = the surprise in inflation rates bi2 = the sensitivity of the return on bond i to GDPgrowth surprises FGDP = the surprise in GDPgrowth (assumed to be uncorrelated with FINFL) εi = an error term with a zero mean that represents the portion of the return to bond i not explained by the factor model 23 Risk and Style Multifactor Models Another category of multifactor approach incorporates risk, or style, factors, several of which can thematically apply across asset classes. Examples of such factors include momentum, value, carry, and volatility. Many of these are similar in construction to those commonly used in equity portfolios. Examples include defining value as real (inflation-adjusted) yield, momentum as the previous 12month excess return, and carry as the term spread. An illustrative example of risk factor approaches, in this case across asset classes, can be found in Exhibit 8. 25 Review of Examples Review Examples 3 to 6 in the textbook for the application of the aforementioned formulae and equations