Carhart Four-Factor Model
The Carhart four-factor model extends the Fama-French three-factor model by adding a momentum factor (WML — Winners Minus Losers), providing a more comprehensive framework for explaining the cross-sectional variation in mutual fund and portfolio returns.
Key takeaways
- The four factors are: market excess return (MKT), size (SMB — Small Minus Big), value (HML — High Minus Low book-to-market), and momentum (WML — Winners Minus Losers).
- Mark Carhart (1997) demonstrated that mutual fund performance persistence was largely explained by momentum exposure, not genuine stock-picking skill.
- The model provides a four-dimensional benchmark for performance attribution, allowing investors to assess alpha after controlling for systematic factor exposures.
- WML is constructed as the return of the top 30% momentum stocks minus the bottom 30%, rebalanced monthly based on prior 12-month returns (excluding the most recent month).
- The model has been criticized for the momentum factor's theoretical fragility — it lacks a clear risk-based explanation and has shown instability in periods like 2009 and 2020.
Explanation
The Carhart model emerged from empirical research on mutual fund performance persistence. Carhart (1997) found that prior-year winners continued to outperform in the subsequent year, but this apparent persistence almost entirely reflected mechanical momentum exposure — managers who happened to hold recent winners were predicted by the momentum factor, not by superior stock selection. After controlling for the WML factor, genuine alpha was elusive.
The model equation is: R_i − R_f = α + β_1 × MKT + β_2 × SMB + β_3 × HML + β_4 × WML + ε, where R_i − R_f is the portfolio's excess return, MKT is the market excess return, SMB (Small Minus Big) captures the size premium, HML (High Minus Low) captures the value premium, and WML (Winners Minus Losers) captures momentum. All four factor portfolios are long-short zero-investment portfolios, and their returns are published monthly by Ken French on his data library website.
The momentum factor (WML or UMD — Up Minus Down) is constructed by ranking stocks on their prior 12-month return (skipping the most recent month to avoid short-term reversal effects). The top decile (past winners) is bought and the bottom decile (past losers) is sold. The academic momentum premium has been documented across global markets, asset classes (commodities, bonds, currencies), and time periods. However, momentum exhibits severe crash risk: in sharp market reversals (March 2009, April-May 2020), past winners become liquidity sources while past losers rebound, causing the WML factor to suffer catastrophic drawdowns.
The Carhart model is widely used in performance attribution and factor analysis for institutional investors. When an allocator receives a fund manager's track record, they regress the returns against the four Carhart factors to extract the true alpha — the portion unexplained by passive factor exposure. A fund reporting 15% gross returns might yield an alpha of only +0.5% per annum once its SMB, HML, and WML loadings are accounted for, meaning the fund is primarily delivering beta to known factors at active management fees.
Subsequent research has added factors beyond Carhart's four — the Fama-French five-factor model (2015) adds profitability (RMW) and investment (CMA), and the q-factor model further refines the investment and profitability proxies. Nevertheless, the Carhart four-factor model remains a standard workhorse in empirical asset pricing and serves as the benchmark for most academic mutual fund performance studies.
Formula
R_i − R_f = α + β_1 × MKT + β_2 × SMB + β_3 × HML + β_4 × WML + ε
Example
An institutional investor evaluates two equity hedge funds over a three-year period. Fund A reported 16.2% gross annualized returns; Fund B reported 13.8%. Running Carhart four-factor regressions reveals that Fund A has high WML loading (β_4 = 0.65) and minimal alpha (α = 0.3% per annum), while Fund B has low factor loadings across all four dimensions and an alpha of 2.8% per annum (t-statistic of 2.4). Despite lower raw returns, Fund B is the more compelling manager: its outperformance reflects genuine skill, while Fund A is essentially delivering systematic momentum exposure that could be replicated by a factor ETF at a fraction of the cost.
Related terms
Alpha Beta Correlation Matrix Diversification Equity Esg Environmental Social Governance Factor Model Fama French Three Factor Model Five Factor Model Liquidity Portfolio Optimization Premium