Factor models and Fama-French

RiskAtoms Value Income beat CAPM by 3.0% a year. Is its manager brilliant, or did the fund just hold the kind of shares that tend to do well? A factor regression answers that in one line, and the same idea lets a risk team describe 2,000 shares with a handful of numbers each.

L3.1.06: Many assets and the covariance matrix showed that $n$ assets need $n(n-1)/2$ covariances. For 100 shares that is 4,950 numbers, each estimated with error. A factor modelA model that explains each asset's return by its exposures to a few common factors plus its own noise. cuts this down by assuming that shares move together only because they respond to a few common factors.

The single-index model. William Sharpe proposed it in 1963: one factor, the market. Each asset's returnA one-factor model with the market as the factor: Rᵢ = αᵢ + βᵢRm + εᵢ, with unrelated εᵢ across assets. is $R_i = \alpha_i + \beta_i R_m + \varepsilon_i$, as in L3.1.09: CAPM, beta and the security market line, with one extra assumption: the specific parts $\varepsilon_i$ of different assets are unrelated. Then any two assets covary only through the market, and the whole matrix follows from three numbers per asset.

The price of the shortcut is the assumption. Two banks share more than the market: they respond to the same interest rate and credit news. The single-index model leaves that out. In the data below, the two banks' correlation is 0.54, but the model gives only 0.45.

Multi-factor models. Add more factors and the model catches more of the co-movement: $R_i = \alpha_i + b_{i1}F_1 + \dots + b_{iK}F_K + \varepsilon_i$. Each loadingAn asset's sensitivity to a factor: the slope on that factor in a regression. Also called a factor exposure or beta. $b_{ik}$ is an exposure. Factors can be macroeconomic (growth, inflation), statistical (principal components), industry and country groups, or long-shortA factor built as the return on one group of shares minus another, such as small minus big: it costs nothing to hold in principle. portfolios built from firm characteristics. The last kind is the Fama-French family.

Fama and French. After finding that size and book-to-market explained average returns better than beta (L3.1.10: Using and testing CAPM), Eugene Fama and Kenneth French built factors from them in 1993. Carhart added momentum in 1997, and Fama and French added profitability and investment in 2015:

Two uses, one regression. Regress a fund's excess return on the factors (M-D03: Linear regression). For performance, the intercept is the return the factors do not explain: a stricter alpha than CAPM's, because tilts that anyone can buy cheaply (value, size, momentum) are no longer counted as skill. For risk, the loadings say which factors the fund is betting on, and $\Sigma = B\,\Omega\,B' + D$ gives its covariance with every other position.

Factor correlations in these data. As in published data, value and investment move together, and momentum leans against value:

Multi-factor covariance. Stack the loadings in an $n \times K$ matrix $B$, write $\Omega$ for the $K \times K$ covariance matrix of the factors and $D$ for the diagonal matrix of specific variances. With $\mathbf R = \boldsymbol\alpha + B\mathbf F + \boldsymbol\varepsilon$, factors unrelated to the $\varepsilon$s, and $\varepsilon$s unrelated to each other:

With one factor, $B$ is the column $\boldsymbol\beta$ and $\Omega = \sigma_m^2$, which gives back the single-index formula. Counting inputs: the full model needs $n$ means, $n$ variances and $n(n-1)/2$ covariances; a $K$-factor model needs $n$ alphas, $nK$ loadings, $n$ specific variances, $K$ factor means and $K(K+1)/2$ factor covariances:

The regressions. With excess returns on the left and the long-short factors on the right:

Least squares in matrix form gives all the coefficients at once, $\hat{\mathbf b} = (X'X)^{-1}X'\mathbf y$, with standard errors from the diagonal of $s^2(X'X)^{-1}$ and $s^2 = \sum_t \hat\varepsilon_t^{\,2}/(n - K - 1)$ (M-D03: Linear regression). Since the factors are excess returns on portfolios, the model also gives an expected return: $\mathbb{E}[R_i] - r_f = \sum_k b_{ik}\,\mathbb{E}[F_k]$, the APT form of L3.1.12: Arbitrage pricing theory.

Start with Value Income and the CAPM model: an alpha of 3.0% a year. Switch to Fama-French 3 and watch HML take most of it. Then try the Momentum fund: FF3 and FF5 leave a large alpha and a negative HML loading; Carhart 4 finds the momentum loading and the alpha shrinks. Reveal the true loadings to check.

1. A 3 × 3 single-index covariance matrix. RiskAtoms Bank, the water utility and the software company, with market volatility 16% a year. Illustrative inputs.

2. Reading a factor regression. RiskAtoms Value Income, 60 months, CAPM against Fama-French 3.

All five models for all three funds are in the Try it tool: 3 funds × 5 models.

Commercial equity risk models used by banks and asset managers are multi-factor models of exactly this form, with industry, country and style factors. Regulators meet the same structure in the market risk standard, where positions are mapped to prescribed risk factors with prescribed correlations (M4.3: Market risk capital (FRTB) and CVA), and in internal models, where factor choice is a model risk question (M6.2: Market risk modelling, M6.8: Model risk, validation and machine learning). Mapping positions to factors is the first step in M3.2: Market risk.

• The single-index model assumes assets covary only through the market: Σ = ββ′σm² + D, three numbers per asset.
• Multi-factor models generalise this to Σ = BΩB′ + D and cut the inputs from about n²/2 to a few per asset.
• Fama-French 3 adds size (SMB) and value (HML) to the market; Carhart adds momentum; Fama-French 5 adds profitability (RMW) and investment (CMA).
• Factor loadings come from a multiple regression of excess returns on the factors; the intercept is the alpha the factors cannot explain.
• Leaving out a factor the fund loads on pushes that factor's return into alpha: a tilt can look like skill.

Factor model

A model that explains each asset's return by its exposures to a few common factors plus its own noise.

Single-index model

A one-factor model with the market as the factor: Rᵢ = αᵢ + βᵢRm + εᵢ, with unrelated εᵢ across assets.

Factor loading

An asset's sensitivity to a factor: the slope on that factor in a regression. Also called a factor exposure or beta.

Long-short factor

A factor built as the return on one group of shares minus another, such as small minus big: it costs nothing to hold in principle.

SMB

Small minus big: small firms' average return minus large firms'.

HML

High minus low: the return on high book-to-market (value) shares minus low book-to-market (growth) shares.

MOM

Momentum: the return on the past year's winners minus the past year's losers.

RMW

Robust minus weak: high operating profitability minus low.

CMA

Conservative minus aggressive: firms that grow their assets slowly minus those that grow them fast.