Factor Model

What is a factor model?

A factor model is a framework that explains an asset’s or portfolio’s returns through its exposures to selected drivers, called factors. Return-based models use factor portfolio returns such as market, size and value; other models use economic variables, characteristics or statistical components. A historical factor regression splits returns into a fitted intercept, factor contributions and a period-specific residual.

In that regression the intercept is alpha. A residual is the remaining observation-specific difference after including that intercept and the factor contributions.

How factor contributions are calculated

Multiply a factor loading by that period’s factor return. Add the contributions and intercept to obtain the fitted excess return.

Averaged over the sample, the residuals net to zero, so mean excess return equals alpha plus each loading times its mean factor return. That is why the example’s 11% splits exactly into 13.55 points of factor contributions and a −2.55-point alpha.

Fama-French three-factor and five-factor models

The three-factor model includes market excess return, SMB and HML. SMB represents small-minus-big portfolio returns; HML represents high-minus-low book-to-market portfolio returns.

The five-factor model adds profitability, RMW, and investment, CMA. It also rebuilds SMB across three sets of size sorts. Use the exact SMB series belonging to the selected model.

Subtract the matched risk-free return from the asset and market returns. The long-short factor spreads already subtract one portfolio return from another, so do not subtract the risk-free return again.

Momentum is a separate factor. It ranks stocks on prior months 2 through 12 and skips the most recent month. Add it for a fund that chases trends, so the trend-following return is priced as exposure before alpha is judged.

Factor model example: CAPM alpha vs three-factor alpha

The matched annualized arithmetic means are 14% for the fund and 3% for the risk-free return, giving 11% mean excess return.

Market contribution is 1.05 × 7% = 7.35 percentage points. Size contributes 0.80 × 4% = 3.20 points. Value contributes 0.60 × 5% = 3.00 points.

The three contributions total 13.55 points, leaving a −2.55-point intercept. With the example’s market-only beta also fixed at 1.05, that regression’s intercept is 3.65 points. The 6.20-point difference equals size and value contributions here.

Market beta stays at 1.05 after adding factors only when the size and value terms have no net covariance with the market, as in this example. Otherwise beta shifts as well, so run both regressions on the same sample and compare the intercepts directly.

Factor attribution vs factor forecasts

Historical attribution assigns a measured return record to fitted components. A forward scenario combines selected exposures with forecast factor returns. Those forecasts can have either sign.

On the same sample, every added factor raises or holds R-squared, so a better fit is expected and alpha is the number to watch; it rises or falls with the fund’s tilts. A high R-squared still leaves any omitted common driver in the residual, so check what the residual tracks before calling it company-specific.

Factor exposure in a trading portfolio

Weight each position’s factor loadings by its share of the book and add them up. The total shows the bets your positions share: several tech longs that each load on the market and on growth sum to one large market-and-growth position. Options need scenarios as well, because their delta moves with price, so test the book across price, volatility and date changes before trusting a single loading.

Limitations of factor models

• Name the model before quoting alpha. The example shows a 3.65-point market-only alpha turning into a −2.55-point three-factor alpha on the same record.
• Loadings drift. They are estimated from past returns with sampling error, so re-estimate them whenever a portfolio’s holdings or style change.
• Fixed loadings miss option risk. An option’s sensitivity moves with price, so run price and volatility scenarios on any book that holds options.
• Omitted factors sit in the residual, even at a high R-squared. Before calling the leftover return stock-specific, check whether it tracks a factor the model left out, such as momentum.
• The fit supplies loadings; the forward view is yours. Multiply the loadings by the factor returns you forecast, and a negative value premium turns the 0.60 value loading into a drag.

Bottom line

A factor model breaks returns into the exposures that drove them and the part left over. Use it to measure what a portfolio is really betting on and to judge alpha against the right benchmark. Name the model, factor series and sample behind any loading or alpha you quote.