How to use the factor exposure calculator
The Factor regression tool estimates exposures from return history:
- Paste a table with a header row. One column holds the portfolio’s returns, the others the factor returns: Mkt-RF, SMB, HML and, for the four-factor model, MOM (UMD or WML also work). Returns are in percent, as Kenneth French’s data library publishes them. Columns can be in any order; a Date column is ignored. The example contains 60 months of simulated data for a fictional fund, Fund B.
- Choose the model: the Fama-French three-factor model or Carhart’s four-factor model with momentum.
- Say whether the portfolio column is already an excess return. If it is a total return, include an RF column and the calculator subtracts it row by row.
- Choose the frequency so alpha and volatilities are annualized correctly.
The Risk from known betas tool skips the regression. Enter each factor’s beta and annual volatility, a correlation matrix and the residual volatility, and it splits total risk by source. Use it for exposures from a risk system or a fund report, or to test what a change in tilt would do.
How the regression and risk split are calculated
Contribution of factor f = βf × (Σβ)fΣ is the sample covariance matrix of the factors. Each factor's contribution is its beta times its covariance with the whole factor exposure, so the contributions add up exactly to β′Σβ. Divided by the portfolio's variance, the factor shares add up to R² and the residual takes 1 − R².
A worked example
Fund B’s 60 simulated monthly returns give a market beta of 1.00, a size beta of 0.36, a value beta of -0.20 and a momentum beta of 0.08. In plain terms: market-like risk, a clear tilt toward small companies, a mild tilt toward growth and a slight momentum tilt. All four have t-statistics above 2.
The factors explain 94.7% of the variance. The fund earned an average excess return of +10.04% a year; the market exposure accounts for +6.98% of it, size for +1.31%, value for -1.97% and momentum for +1.25%. What remains is an alpha of +2.46% a year (0.21% a month), with a t-statistic of only 1.08: not distinguishable from zero.
On the risk side, the market supplies 84.6% of the variance, size 6.6% and value 4.4%. Momentum’s share is -0.8%: its small positive beta slightly offsets risk from the other exposures, because in this sample momentum tends to move against the fund’s combined market, size and value exposure. The residual, the fund’s own stock selection, adds 5.3%, an idiosyncratic volatility of 4.27% a year against total volatility of 18.56%.
Three factors or four: why the model changes the alpha
Fama and French (1993) showed that two factors beyond the market, size and value, explain much of the variation in returns across diversified stock portfolios that the CAPM’s single beta leaves unexplained. Carhart (1997) added momentum, the return of recent winners over recent losers, while studying mutual fund performance, and found that a large part of what looked like persistent manager skill was simply exposure to momentum.
The example shows the same effect on a small scale:
| Three-factor | Four-factor | |
|---|---|---|
| Alpha (annual) | +4.35% | +2.46% |
| Alpha t-statistic | 2.00 | 1.08 |
| Market beta | 0.97 | 1.00 |
| Value (HML) beta | -0.25 | -0.20 |
| R² | 94.3% | 94.7% |
Without momentum, the fund’s alpha sits right at the conventional significance line. Add the momentum factor and part of that return is reassigned to a known, cheaply replicable exposure: the alpha shrinks from 4.35% to 2.46% and its t-statistic nearly halves. An alpha is always relative to a model, so a fund’s “skill” is only as convincing as the factors it has been tested against.
Fama and French later extended their model to five factors (2015) by adding profitability and investment. The regression tool reads any three or four of the columns described above; for other models the same arithmetic applies, but this page only labels the standard factors.
Reading factor risk contributions
Contributions measure variance, and variance is the quantity that adds up. Volatilities do not: a factor with a standalone volatility of 3% does not add 3% to a portfolio’s volatility.
Correlations matter as much as betas. In the second tool’s example, the market supplies 84.5% of the variance and the portfolio’s total volatility is 18.65%, about $1,865,176 a year on $10 million. Remove the correlation matrix and treat the factors as independent, and total volatility becomes 17.98% while the momentum share changes sign, from -0.9% to 0.7%. A negative contribution is not an error: it means that exposure hedges part of the others.
For a long-only equity fund the market usually dominates, as here. For a market-neutral or long-short fund, the market share should be small, and a large size, value or momentum share reveals a style bet that may not be intended. That makes the risk split a practical check on whether a portfolio’s risk comes from the decisions its manager claims to make.
What this calculator does not include
- Live factor data. You paste the returns. The example data are simulated from a known model, not Kenneth French’s published series.
- Time-varying betas. Exposures are constant over the sample. Rolling regressions show drift in style.
- Robust standard errors. t-statistics assume independent, constant-variance errors; Newey-West errors are often wider for overlapping or autocorrelated returns.
- Holdings-based analysis. Exposures are inferred from returns, not from what the portfolio owns.
- Positive definiteness checks beyond rejecting a negative factor variance in the second tool. A correlation matrix copied from different sources can be inconsistent.