The Carhart Four-Factor Model

Run a Fama-French three-factor regression on a fund that buys recent winners and sells recent losers, and something odd happens: it still shows a large, statistically significant alpha. The three factors, market, size, value, cannot explain why momentum strategies made money. Either momentum managers are geniuses the model cannot see, or the model is missing a factor. Carhart's answer, in 1997, was to add one.

The analogy

Imagine you are grading a runner's marathon time using only two variables: how much they trained, and their natural build. Most of the variation across runners is explained. But one group of runners keeps beating the model's prediction, every single one of them happens to have been on a hot streak of races lately. You could call it unexplained talent, or you could admit "recent form" is a real, systematic variable you left out. Carhart looked at Momentum strategies the same way: not skill, a missing risk factor.

Building the model

Take the The Fama-French Factor Models three-factor regression, market minus risk-free ($R_m - R_f$), size (SMB, small minus big), and value (HML, high minus low book-to-market), and add a fourth term:

$R_i - R_f = \alpha_i + \beta_1(R_m - R_f) + \beta_2\,\text{SMB} + \beta_3\,\text{HML} + \beta_4\,\text{UMD} + \epsilon_i.$

UMD stands for "up minus down": a portfolio that goes long the stocks with the highest returns over the past 12 months (skipping the most recent month) and short the stocks with the worst. Every symbol here is a slope, $\beta_1$ through $\beta_4$, telling you how much of stock $i$'s return moves with each of the four factor portfolios' returns. In plain words: a stock's or fund's excess return is explained by four separate bets, on the market, on smallness, on cheapness, and now on recent winning streaks, and $\alpha_i$ is whatever is left over once all four are accounted for.

Picture the horizontal axis as UMD's return in a given month and the vertical axis as a momentum fund's return that same month. Before Carhart's factor existed, that co-movement showed up as pure alpha, an unexplained edge. Adding UMD lets the regression absorb it as a factor loading instead.

Worked example 1: three-factor alpha versus four-factor alpha

A fund's monthly excess returns are regressed and produce these three-factor loadings: $\beta_1 = 1.0$, $\beta_2 = 0.2$, $\beta_3 = -0.1$, with an average unexplained monthly residual of $0.6\%$ (this is the alpha under the three-factor model, roughly $7.2\%$ annualized). The fund is known to tilt heavily toward recent winners. Adding UMD to the regression, its loading comes out at $\beta_4 = 0.4$, and UMD itself averaged $0.5\%$ per month over the sample. That loading alone explains $0.4 \times 0.5\% = 0.2\%$ of the fund's monthly return. The new four-factor alpha is $0.6\% - 0.2\% = 0.4\%$ per month, smaller, though still positive. Roughly a third of what looked like manager skill was actually a momentum tilt.

Worked example 2: a fund with no momentum tilt

A second fund has $\beta_4 = 0.02$, essentially no exposure to UMD. Its three-factor alpha was $0.3\%$ per month. Multiplying $0.02 \times 0.5\% = 0.01\%$, adding the fourth factor barely moves its alpha at all, it stays at roughly $0.29\%$. This is the useful contrast: Carhart's factor only reclassifies alpha for funds that were actually riding momentum. A genuinely momentum-neutral value fund's performance evaluation is essentially unchanged by adding UMD.

What this means in practice

Mutual fund and hedge fund performance evaluation almost always benchmarks against the four-factor model (or Fama-French's later five-factor extension) rather than the original three, precisely because momentum is common enough among active managers that ignoring it inflates apparent skill. A fund's four-factor alpha is a stricter, more honest hurdle than its three-factor alpha.