A Gentle Introduction to Factor Models

Modern investment theory is based largely on two ideas. The first, due to the work of Harry Markowitz, is that in an efficient financial market, higher return expectations require higher risk exposures. The second, established by William F. Sharpe, is that since the risks associated with individual securities tend to cancel each other out in diversified portfolios, exposure to higher "specific" risks is not associated with higher expected returns. However, there is a payoff for exposure to greater "systematic" risk (risk that cannot be diversified away). Sharpe's theory is known as the Capital Asset Pricing Model (CAPM). Both Markowitz and Sharpe received Nobel Prizes for their seminal work.

Sharpe's CAPM defined systematic risk as the risk associated with exposure to the volatile, unpredictable swings of the stock market, and quantified systematic risk as the slope (beta) obtained from a regression of a securities' returns against the market's returns. The CAPM was the first factor model. While the CAPM attributed risk to a single systematic factor, arbitrage pricing theory (APT), first presented by Stephen Ross, established a firm theoretical foundation for the existence of multiple systematic sources of risk and return, and paved the way for the multi-factor models of today.

Contemporary factor models are generally classified in three groups: fundamental, statistical, and econometric. Each of type of model has its strengths and weaknesses, but with the proper statistical approach, an econometric factor model can combines all three types of factors. Due to it's intuitive appeal and its generality, we are going to focus on (and build) an econometric factor model.

Econometric factor models are based on the idea that the returns of individual assets are influenced by the market itself and other broadly influential factors like unemployment rates, interest rates, consumer sentiment, business activity, and factors which have been given names like "size" or "value/growth" that are thought to be related to, for example, the risk associated with companies that are small, not commonly know, or poorly researched companies (size), and companies that are recently troubled, out of favor, not currently valued at a premium (value/growth).

In an econometric factor model, while the values of the economic factors are known, the "exposure" of individual stocks to the economic factors are not know. The exposures for each stock must be estimated with a multi-factor regression of the stock's returns over some time period to the values of the economic factors during that time period. The regression model is a conventional one, resulting in an alpha, the exposures (also called the factor betas), and an error term for each stock. The error term is a normally distributed variable with a zero mean and a different size for each stock in the regression model.

The table below shows the factor exposure estimates for the stocks discussed in QEPM's chapter on econometric models (Table 7.4, page 224). These estimates are based on regressions for the period from January 2000 to December 2003.
The factor exposures should make economic sense. For example, Wal-Mart goes up when unemployment increases. Exxon Mobil is least affected by consumer sentiment. Microsoft is the most sensitive to the market. The returns of all of these large company stocks are hurt during periods when large stocks do poorly.

It's easy to write the equation for the expected return of a portfolio of stocks one you have estimated the alpha and factor exposures of the stocks in the portfolio, because portfolios have the alpha and factor exposures of a weighted average of the alphas and exposures of the stocks it holds. The expected return of a portfolio equals its alpha plus the sum of its factor exposures times the values either observed or forecast for the economic time series variables that make up the model.

The risk equation is more complicated (we'll discuss it later), but once you have the risk and return equations you can maximize a portfolio's return given a specific level of risk, minimize its tracking error to a benchmark, or minimize its exposure to one risk while maintaining its exposure to others.

For a less gentle introduction to factor models, see: Estimating A Combined Linear Factor Model. (There is a large literature on factor models. I'll add a few more links when I find good ones.)
Having finished this post, I'll get back to building one.