Factor Model

A factor model is a mathematical description of asset returns in which the return of each asset is split into two parts. One part comes from a handful of influences shared across many assets, the factors. The other is specific to the asset. An asset's sensitivity to each factor is called its exposure, loading or beta. Two stocks move together, in this view, because they are exposed to the same factors.

Factor models are a basic tool of quantitative finance. They are used to estimate risk, to explain where a portfolio's performance came from, to build portfolios with chosen characteristics, and to judge whether a manager has added value. They also provide the theoretical basis of factor investing.

A linear factor model with $K$ factors is written:

• $R_i$ is the return of asset $i$, usually in excess of the risk-free rate.
• $F_k$ is the return of factor $k$ over the same period.
• $\beta_{ik}$ is the exposure of asset $i$ to factor $k$.
• $\alpha_i$ is the average return not explained by the factors.
• $\varepsilon_i$ is the asset-specific return, assumed unrelated to the factors and to the specific returns of other assets.

Take a three-factor model with market, size and value factors. A stock has exposures of 1.1, 0.4 and −0.2. In a given month the factors return 2 percent, 1 percent and −0.5 percent. The return explained by the factors is (1.1 × 2) + (0.4 × 1) + (−0.2 × −0.5) = 2.2 + 0.4 + 0.1 = 2.7 percent. If the stock's actual excess return was 3.2 percent, the remaining 0.5 percent is the sum of its alpha and specific return for the month.

• Single-factor models. The market model uses one factor, the market return. The capital asset pricing model adds the claim that alpha should be zero.
• Macroeconomic models. The factors are observable economic series such as inflation surprises, industrial production, interest rates and credit spreads. Exposures are estimated by regression.
• Fundamental models. The exposures are observable characteristics of each company, such as industry, size, valuation ratios and past returns, and the factor returns are estimated each period from a regression across stocks. Commercial equity risk models are of this kind. The Fama-French models are close relatives, with factors formed as portfolios sorted on characteristics.
• Statistical models. Factors are extracted from the returns themselves by techniques such as principal component analysis. They explain the data well but have no built-in economic meaning.

The arbitrage pricing theory supplies the link to expected returns: if returns follow a factor structure, then an asset's expected return should be a linear function of its exposures.

For a model with known factor returns, the exposures of an asset are found by regressing its returns on the factors, using linear regression for one factor or multivariate linear regression for several. The intercept is the estimate of alpha, the slopes are the betas, and the coefficient of determination shows the fraction of the asset's variance the factors explain. For a diversified equity fund that fraction is often above 90 percent. For a single stock it is much lower.

The first use is risk estimation. With $N$ assets there are far too many pairwise relationships to estimate directly. A factor model replaces them with a compact structure:

Here $\Sigma$ is the covariance matrix of asset returns, $B$ is the table of exposures, $\Sigma_F$ is the covariance matrix of the factors, and $D$ holds the specific variances on its diagonal. For 1,000 stocks and 50 factors this cuts the number of parameters from about half a million to about fifty thousand, and the result is stable enough for portfolio optimization and value at risk.

The model also divides risk into two kinds. Factor risk is shared and cannot be removed by holding more assets; it corresponds to systematic risk. Specific risk, or unsystematic risk, shrinks as holdings are added.

The second use is performance attribution. A fund's return is broken into the part due to each factor exposure and the remainder. The remainder is the fund's alpha. A manager who beat the index simply by holding small, cheap stocks in a period when those did well has shown factor exposure, not skill, and that exposure can be bought cheaply elsewhere. The third use is portfolio construction. Managers hedge unwanted exposures, and in statistical arbitrage the specific returns left after removing factor effects are themselves the object of trading.

A factor model is an approximation. The choice of factors is a judgment, and a model that omits an important one will attribute its effect to alpha or to specific risk. Exposures are estimated with error and change over time as companies change. The relationship is assumed to be linear, which may not hold in extreme markets.

The assumption that specific returns are unrelated across assets is often violated, especially in a crisis, when assets that looked independent fall together. Risk estimates built on calm periods then prove too low. Statistical models can fit past data closely and still describe the future poorly. And alpha is always relative to a model: a return that looks like skill against one set of factors may vanish when another factor is added.