In brief

The Carhart four-factor model extends the Fama–French three-factor model by adding momentum: the historical tendency for recent relative winners to continue outperforming recent relative losers for a time.

The model explains a portfolio’s return using four measured exposures:

  1. the broad stock market;
  2. small companies versus large companies;
  3. value stocks versus growth stocks; and
  4. recent winners versus recent losers.

Its standard time-series regression is:

Rₚ − Rf = α + βm(MKT − RF) + s(SMB) + h(HML) + m(MOM) + ε

The model is particularly useful for evaluating active funds. A fund that recently owned winning stocks may look skillful under CAPM or the three-factor model. After momentum is included, some of that apparent alpha may be reclassified as exposure to a systematic return pattern.

That does not make the four-factor model a forecasting machine or prove why momentum exists. It is a historical attribution framework. Its results depend on the data, factor definitions, period, return frequency, benchmark universe, and regression assumptions.

Where the fourth factor came from

The model is named for Mark Carhart’s 1997 paper, “On Persistence in Mutual Fund Performance”.

Carhart was investigating a practical puzzle. Funds with strong recent performance sometimes continued to rank well over short horizons. Was that evidence of managers with persistent stock-picking skill—a “hot hand”—or could common portfolio characteristics explain it?

His survivorship-bias-free database covered diversified U.S. equity funds from January 1962 through December 1993. Carhart found that common return factors, investment expenses, and transaction costs explained nearly all persistence in mutual-fund returns. The one-year persistence previously attributed to hot hands was largely associated with the momentum effect documented by Narasimhan Jegadeesh and Sheridan Titman.

In their 1993 paper, “Returns to Buying Winners and Selling Losers”, Jegadeesh and Titman studied strategies that ranked stocks by past returns and then bought recent winners while selling recent losers. They documented intermediate-horizon continuation in their U.S. sample. Carhart added a momentum return factor to the existing three-factor framework so fund performance could be evaluated after accounting for that tendency.

The historical sequence is useful:

Year Research development Contribution to the four-factor model
1964 Sharpe publishes a foundational CAPM paper Market beta as the central systematic exposure
1993 Fama and French publish “Common Risk Factors in the Returns on Stocks and Bonds” Market, size, and value factors
1993 Jegadeesh and Titman publish evidence on winner-minus-loser strategies Empirical foundation for intermediate-horizon momentum
1997 Carhart studies mutual-fund persistence Adds momentum to a three-factor performance model

The four-factor model is therefore not simply “CAPM plus three more variables.” It combines two research paths: multifactor asset pricing and performance evaluation.

The equation, one term at a time

The dependent variable is the portfolio’s return above the risk-free rate:

Rₚ − Rf

The right side contains an intercept, four factor returns multiplied by estimated sensitivities, and a residual.

Alpha: α

Alpha is the regression intercept. It is the average return not explained by the four included factors over the fitted sample.

A positive estimated alpha does not automatically prove skill. The estimate may reflect:

  • sampling noise;
  • an omitted factor;
  • a mismatched factor universe;
  • stale or inaccurate returns;
  • changing exposures;
  • data mining;
  • leverage or nonlinear positions the regression does not capture; or
  • genuine security-selection ability.

Alpha should be accompanied by statistical uncertainty, economic magnitude, costs, and a clearly chosen model. “Alpha” is always alpha relative to something.

Market excess return: MKT − RF

This is the broad equity market return minus the risk-free rate. Its coefficient, βm, is market beta.

  • βm near 1 suggests market-like sensitivity.
  • βm above 1 suggests amplified market sensitivity in the fitted sample.
  • βm below 1 suggests less market sensitivity.

The factor is broader than the S&P 500 in common academic datasets. Kenneth French’s U.S. research factors use a broad value-weighted stock market, so substituting an S&P 500 return without documenting the change produces a different model.

Size: SMB

SMB means Small Minus Big. It measures the return spread between portfolios of smaller and larger companies under the dataset’s construction rules.

  • Positive s indicates a small-company tilt.
  • Negative s indicates a large-company tilt.

SMB is a long-short research factor, not the return of a small-cap fund. Its construction diversifies across value and growth groups to isolate size more cleanly.

Value: HML

HML means High Minus Low, referring to high versus low book-to-market equity.

  • Positive h indicates value-like exposure.
  • Negative h indicates growth-like exposure.

Book-to-market is an accounting-based classification, not a judgment that every high-book-to-market company is undervalued. A portfolio can load negatively on HML because it owns expensive growth companies, even when those companies are profitable and financially strong.

Momentum: MOM or PR1YR

The fourth factor is commonly labeled MOM, UMD (Up Minus Down), WML (Winners Minus Losers), or PR1YR in discussions of Carhart’s model. Names and construction details vary by dataset.

Kenneth French’s current U.S. momentum data form six portfolios from two size groups and three prior-return groups. The prior return used for the monthly portfolios is cumulative return from month t−12 through t−2, deliberately skipping the most recent month. The momentum factor averages the two high-prior-return portfolios and subtracts the average of the two low-prior-return portfolios. See the Kenneth French Data Library and its factor descriptions.

  • Positive m indicates recent-winner exposure.
  • Negative m indicates recent-loser or contrarian exposure.

The factor is a hypothetical long-short portfolio. An ordinary long-only fund will not literally hold the complete academic factor, but its return can still covary with it.

Residual: ε

The residual is the portion of one observation’s return not captured by the intercept and four factor contributions. Alpha summarizes the average unexplained component; residuals show the unexplained variation from period to period.

CAPM, three-factor, and four-factor models compared

Each model asks a different attribution question.

Model Regression factors What it can reveal What it can misclassify
CAPM Market excess return Whether performance differs from what market beta alone explains Size, value, momentum, and other systematic tilts may appear as alpha
Fama–French three-factor Market, size, value Whether size and value exposures explain returns beyond market beta Momentum exposure may remain in alpha
Carhart four-factor Market, size, value, momentum Whether recent-winner exposure explains apparent persistence or alpha Profitability, investment, quality, low volatility, nonlinear strategies, and other omitted effects may remain

CAPM is the simplest and easiest to communicate. The CAPM guide explains its equilibrium logic, beta, and Security Market Line.

The Fama–French three-factor guide explains how size and value expanded the market-only framework.

Carhart’s model asks one more question: Did the manager outperform because of stock-selection skill, or because the portfolio held recent winners during a period when momentum paid?

Adding a factor does not guarantee a better explanation out of sample. A more complex model can fit historical data more closely while adding estimation error and interpretation risk.

One worked example across all three models

The following numbers are hypothetical monthly observations created to show the mechanics. They are not estimates for a real fund and are not expected returns.

Assume one month produced:

Input Hypothetical value
Portfolio return above the risk-free rate 0.82%
Market excess return 0.50%
SMB return −0.10%
HML return 0.20%
MOM return 0.30%
Estimated market loading 1.05
Estimated SMB loading −0.20
Estimated HML loading 0.35
Estimated MOM loading 0.45

For teaching purposes, suppose these loadings came from a historical regression and remain fixed for this attribution.

CAPM attribution

CAPM uses only market exposure:

Explained excess return = 1.05 × 0.50% = 0.525%

Unexplained return = 0.82% − 0.525% = 0.295%

Viewed only through CAPM, the portfolio appears to have 0.295% of positive unexplained return for the month.

Three-factor attribution

Add size and value:

Market contribution = 1.05 × 0.50% = 0.525%

Size contribution = −0.20 × −0.10% = 0.020%

Value contribution = 0.35 × 0.20% = 0.070%

Total explained = 0.525% + 0.020% + 0.070% = 0.615%

Unexplained return = 0.82% − 0.615% = 0.205%

The apparent unexplained return falls because size and value account for another 0.09 percentage point.

Four-factor attribution

Now add momentum:

Momentum contribution = 0.45 × 0.30% = 0.135%

Total explained = 0.615% + 0.135% = 0.750%

Unexplained return = 0.82% − 0.750% = 0.070%

The same observed portfolio return now produces three different unexplained amounts:

Model Explained excess return Unexplained portion
CAPM 0.525% 0.295%
Fama–French three-factor 0.615% 0.205%
Carhart four-factor 0.750% 0.070%

This single-month decomposition is not a regression alpha estimate. In a real analysis, alpha and loadings are estimated jointly across many observations. The example demonstrates the intuition: omitted systematic exposure can look like skill under a simpler model.

How the regression is actually estimated

A standard analysis follows these steps:

  1. Choose a portfolio and observation period.
  2. Obtain total returns at a consistent frequency, commonly monthly.
  3. Match each portfolio return with the same month’s risk-free rate and four factor returns.
  4. Calculate portfolio excess return.
  5. Regress excess return on MKT−RF, SMB, HML, and MOM with an intercept.
  6. Review coefficients, standard errors, t-statistics, residuals, and model fit.
  7. Test stability across subperiods and alternative factor definitions.

The fitted equation is:

Excess returnₜ = α + βm·MKTₜ + s·SMBₜ + h·HMLₜ + m·MOMₜ + εₜ

Ordinary least squares chooses coefficients that minimize the sum of squared residuals in the sample. This mathematical fit does not establish causation.

What a coefficient means

Suppose the fitted momentum loading is 0.45. The model estimates that when MOM was one percentage point higher in a month, the portfolio’s excess return tended to be about 0.45 percentage point higher, holding the other included factor returns constant.

It does not mean:

  • 45% of the portfolio is invested in a momentum fund;
  • momentum caused every observed return;
  • the loading will remain 0.45 next year; or
  • the portfolio is expected to earn an extra 0.45% annually.

Statistical versus economic significance

A positive alpha can be economically interesting but statistically uncertain. A t-statistic compares the estimate with its estimated standard error. Longer samples can improve precision, but they also increase the chance that the strategy, manager, holdings, or market regime changed during the period.

Statistical significance is not a binary certificate of skill. Testing many funds, periods, and model variations increases the chance of finding an apparently impressive result by accident.

R-squared

R-squared measures how much of the variation in the dependent variable the fitted model explains in sample.

A high R-squared can indicate that factor exposures closely describe the portfolio. It does not prove that the model is economically correct. A low R-squared may be expected for a concentrated, option-driven, market-neutral, or rapidly changing strategy.

How to read a hypothetical regression output

Consider this invented educational output from 120 monthly observations:

Term Estimate Standard error Illustrative interpretation
Alpha 0.08% per month 0.11% Positive estimate, but small relative to its uncertainty
Market 1.03 0.05 Slightly more market-sensitive than the factor portfolio
SMB −0.18 0.07 Large-company tilt
HML 0.31 0.09 Value tilt
MOM 0.42 0.08 Meaningful recent-winner exposure
R-squared 0.91 — Factors explain most monthly variation in this sample

The alpha t-statistic is approximately:

0.08 ÷ 0.11 = 0.73

That would not provide strong statistical evidence that true alpha differs from zero under conventional inference. Meanwhile, the factor estimates describe a portfolio with market-like beta, a large-cap tilt, value exposure, and positive momentum exposure.

This is exactly the kind of result for which “the fund beat its benchmark” is less informative than a factor analysis. Much of the return pattern may be replicable through systematic exposures, although obtaining those exposures in a real portfolio involves products, fees, turnover, taxes, and tracking differences.

What Carhart found about mutual funds

Carhart’s study included 1,892 diversified equity funds and 16,109 fund-years from 1962–1993. It excluded sector, international, and balanced funds.

The paper’s central result was not that momentum funds are guaranteed to win. It was that common factor exposure and costs explained most persistence in fund performance. In particular:

  • the earlier hot-hands result was largely explained by one-year stock momentum;
  • expense ratios and transaction costs helped explain longer-run differences;
  • past winner funds did not produce reliably positive future four-factor alpha; and
  • persistent poor performance among the worst funds remained more evident than persistent superior performance among the best.

The sample predates modern ETFs, zero-commission brokerage, decimal pricing, and today’s fund market. Its fee and transaction-cost estimates should not be applied directly to current products. Its research design remains important because it demonstrates how a richer benchmark can change the interpretation of performance.

This connects directly to the broader evidence in Why Most Investors Cannot Beat the S&P 500: past relative winners are visible after the fact, but persistent outperformance is difficult to identify in advance and must clear costs.

Why might momentum exist?

The four-factor model measures momentum exposure; it does not settle its cause. Explanations generally fall into three overlapping groups.

Behavioral explanations

Investors may underreact to new information, update beliefs slowly, or initially anchor on older expectations. Prices then continue moving as information is incorporated. At longer horizons, extrapolation and crowding may produce overreaction and reversal.

Risk-based explanations

Momentum strategies may earn compensation for risks that are not obvious in ordinary volatility. Their losses can cluster during sharp market rebounds and changing regimes, when short positions in prior losers rise rapidly.

Kent Daniel and Tobias Moskowitz’s 2016 paper, “Momentum Crashes”, studies infrequent but severe momentum losses, particularly after market declines when volatility is high and the market rebounds. This downside is important because an attractive long-run average can hide a difficult return path.

Market-friction explanations

Information diffusion, trading constraints, institutional mandates, transaction costs, and limits to arbitrage may allow continuation to persist without offering a simple, costless profit.

These explanations are not mutually exclusive. A factor can be associated with behavioral errors, economic risk, and implementation frictions at the same time.

Cross-sectional momentum is not trend following

The terms are often mixed together, but they describe different comparisons.

Cross-sectional momentum ranks assets relative to one another. It buys relative winners and sells or underweights relative losers.

Time-series momentum, often called trend following, compares each asset with its own past return or trend. It can be long many assets, short many assets, or in cash depending on their individual signals.

Carhart’s fourth factor is cross-sectional stock momentum. A managed-futures trend strategy should not be treated as though its returns are fully captured by Carhart MOM.

Momentum factor versus a momentum ETF

An academic momentum factor and an investable fund are not the same thing.

The research factor is generally:

  • long recent winners;
  • short recent losers;
  • reconstituted frequently;
  • constructed with academic breakpoints; and
  • reported before the full costs and constraints faced by a particular investor.

A long-only momentum ETF may:

  • own winners without shorting losers;
  • use a proprietary score;
  • apply volatility or quality adjustments;
  • limit turnover or constituent weights;
  • rebalance on a different schedule;
  • charge a management fee; and
  • incur taxes, spreads, and tracking differences.

Its regression loading on MOM can confirm exposure, but it will not usually equal one and may change substantially.

Momentum and value together

Momentum and value often select different securities. A stock can be inexpensive relative to fundamentals but have poor recent performance; another can be expensive and rising.

Clifford Asness, Tobias Moskowitz, and Lasse Pedersen documented value and momentum premia across multiple markets and asset classes in “Value and Momentum Everywhere”. They also found that value and momentum returns were negatively correlated in their samples.

That historical relationship can make the two exposures complementary in a model or portfolio. It does not guarantee diversification in every period. Both factors can lose, correlations can change, and implementation costs differ.

The HML and MOM coefficients should therefore be read independently. A fund can have:

  • positive HML and positive MOM exposure;
  • positive HML and negative MOM exposure;
  • negative HML and positive MOM exposure; or
  • little exposure to either.

Labels such as “value fund” or “growth fund” do not determine the regression result.

Strengths of the four-factor model

It improves performance attribution

Momentum can explain return persistence that looks like alpha under CAPM or the three-factor model.

It is transparent

The equation and factor datasets are public. An analyst can reproduce the regression when the return series and exact data choices are documented.

It separates outcomes from exposures

A positive return is not automatically stock-picking skill. The model estimates how much historical performance moved with common factors.

It connects fund holdings with behavior

The signs and magnitudes of loadings provide a compact summary of market, size, value, and momentum tendencies.

It creates a fairer peer comparison

Two funds with the same raw return may have very different factor exposures and residual risk.

Limitations and failure modes

The model is descriptive, not a law

Factor premiums vary over time and can remain negative for long periods. Historical exposure does not forecast future reward.

Factor definitions differ

MOM, UMD, WML, and PR1YR are not automatically interchangeable. Global, developed, regional, and U.S. datasets use different universes and breakpoints.

The coefficients can change

A manager can alter holdings, a fund can drift, and market relationships can shift. One full-sample loading may conceal major subperiod differences.

Momentum is costly to implement

Momentum signals change as rankings change. Turnover, spreads, market impact, taxes, shorting costs, and capacity can reduce an investable result.

Momentum can crash

Long-run average performance does not show the timing or severity of losses. A long-short momentum strategy can suffer when prior losers rebound abruptly.

The model is linear

Options, leverage, dynamic hedging, stop-loss rules, and tactical allocation can produce nonlinear returns that constant linear loadings describe poorly.

Alpha inherits model error

An omitted factor or inappropriate benchmark can leave systematic return in the intercept. Adding momentum reduces one omission; it does not eliminate all of them.

Multiple testing creates false discoveries

Trying many start dates, funds, factor variants, and specifications makes an impressive backtest easier to find by chance.

Statistical fit is not investability

A factor return may involve short positions and frequent rebalancing. A regression does not show that an investor could have captured the modeled return cheaply, at scale, or after tax.

Four common interpretation mistakes

1. Calling every positive alpha skill

Alpha is an estimate with uncertainty. The model, benchmark, sample, and costs determine what it means.

2. Treating MOM as “the market went up”

The market factor measures the broad market. Momentum measures the spread between relative winners and losers.

3. Comparing coefficients from incompatible datasets

A U.S. fund analyzed with developed-market factors can produce different loadings from the same fund analyzed with U.S. factors.

4. Assuming a factor premium is guaranteed

Factor evidence is historical. Momentum has experienced sharp losses, and no coefficient promises a future return.

A reliable analysis checklist

Before accepting a four-factor result, I would document:

  1. Return definition: total return, net asset value return, or investor return?
  2. Frequency: daily, weekly, or monthly?
  3. Currency: do portfolio and factors use the same currency?
  4. Factor universe: U.S., developed, emerging, or global?
  5. Risk-free series: is it the one bundled with the factor dataset?
  6. Sample period: does it include enough observations and more than one regime?
  7. Costs: are returns gross or net of expenses, trading, and loads?
  8. Missing data: were months dropped consistently across all series?
  9. Inference: are standard errors appropriate for the residual behavior?
  10. Stability: do rolling or split-sample coefficients tell the same story?
  11. Benchmark alternatives: does alpha survive CAPM, three-factor, four-factor, and other reasonable models?
  12. Economic meaning: could the factor exposure have been implemented at the assumed cost?

How I would use the model

I would use Carhart’s model for three tasks.

First, I would diagnose a fund’s historical style. The coefficients can reveal exposures that the name or category does not make obvious.

Second, I would test a claim of persistent skill. If apparent alpha disappears after adding momentum, the result is better described as factor exposure than unexplained performance.

Third, I would compare models rather than crown one. CAPM provides a clean market-risk baseline. The three-factor model adds size and value. Carhart adds momentum. If the conclusion changes materially each time a factor is added, that instability is part of the answer.

I would not use the model to predict next year’s fund return, set a guaranteed expected return, or choose an investment from one regression table. It is one lens on historical returns, not a complete investment decision.

Bottom line

The Carhart model extends CAPM and the Fama–French three-factor model by asking whether momentum exposure explains part of a portfolio’s historical return. It is most useful as a diagnostic tool: factor coefficients describe past co-movement, while alpha depends on the model, sample, and assumptions and should not be treated as proof of repeatable skill.

Evidence map and further reading

The cited studies describe historical samples and research portfolios. They do not guarantee that momentum, value, size, or the market factor will earn a positive premium in any future period.