Fama-French five-factor model: formula, factors and how to run it

Berkshire Hathaway's class A shares returned 16.1% a year compounded from March 1985 to September 2026. Measured against the capital asset pricing model, which adjusts for market risk alone, the stock earned an alpha of 8.0% a year, with a t-statistic of 2.74. Measured against the Fama-French five-factor model, the alpha falls to 4.5% a year and the t-statistic to 1.62, below the level of about 2 that researchers treat as statistically significant. The model assigns part of Berkshire's return to its tilt toward cheap stocks and highly profitable companies.

The Fama-French five-factor model is an asset pricing model that explains the returns of a stock, fund or portfolio with five factors: the market's return above Treasury bills, size, value, profitability and investment. Eugene Fama and Kenneth French published it in the Journal of Financial Economics in 2015, adding profitability and investment to the three-factor model they had introduced in 1993.

• Mkt-RF, the market factor: the value-weighted return of US stocks minus the one-month Treasury bill rate.
• SMB, small minus big: small stocks minus big stocks.
• HML, high minus low: stocks with a high book-to-market ratio, or value stocks, minus stocks with a low one, or growth stocks.
• RMW, robust minus weak: firms with high operating profitability minus firms with low operating profitability.
• CMA, conservative minus aggressive: firms whose total assets grow slowly minus firms whose assets grow fast.

The 2015 paper and the valuation equation behind it

The paper covers US stocks from July 1963 to December 2013, 606 months. Fama and French started from the dividend discount model, which sets a share's price equal to the discounted value of its expected dividends. Following Merton Miller and Franco Modigliani, they restated it for the whole firm and divided both sides by book equity:

Hold everything else fixed and the equation makes three predictions. A lower price relative to book equity means a higher expected return. Higher expected earnings mean a higher expected return. Faster expected growth in book equity means a lower expected return. Those three predictions became the value, profitability and investment factors.

Fama and French pointed out that the predictions are the same whether prices are rational or irrational. A premium for cheap or profitable stocks can therefore reflect risk or mispricing, the question at the center of the debate over the efficient market hypothesis.

Size and momentum have no place in the equation. If they predict returns, the authors argued, they must do so by improving forecasts of profitability and investment, or by capturing differences between short-term and long-term expected returns. On the empirical side, Robert Novy-Marx had shown in 2013 that gross profitability predicted returns about as strongly as book-to-market, and Aharoni, Grundy and Zeng had found a weaker but statistically reliable relation between investment and average returns.

The five-factor regression formula

To test the model, or to analyze a fund with it, the monthly excess return of a portfolio is regressed on the five factor returns:

R_it − RF_t

The return of stock, fund or portfolio i in month t minus the one-month Treasury bill rate: its excess return.

RM_t − RF_t, SMB_t, HML_t, RMW_t, CMA_t

The five factor returns for month t, published by Kenneth French.

b, s, h, r, c

The factor loadings, or slopes, estimated by the regression. A loading of 0.5 on HML means the portfolio's return rose by about half a percentage point for each point of HML return, with the other factors held constant. Loadings can be negative, and a growth fund usually has a negative HML loading.

a

The intercept, or alpha: the average monthly return the five factors leave unexplained.

e

The residual, the part of each month's return specific to the portfolio.

If the five factors capture every difference in expected returns, the intercept is indistinguishable from zero for any portfolio. The Gibbons, Ross and Shanken test, or GRS test, checks that jointly across a set of portfolios.

How SMB, HML, RMW and CMA are built

At the end of each June, all NYSE, AMEX and NASDAQ stocks with the required data are sorted twice. The first sort splits them into small and big at the median market capitalization of NYSE stocks. The second sort, done independently, splits them at the 30th and 70th percentiles of NYSE stocks on book-to-market, on operating profitability or on investment. Each pair of sorts produces six value-weighted portfolios, held from July to the following June, and the factors are differences between their returns.

Written out, the profitability factor is RMW = ½(Small Robust + Big Robust) − ½(Small Weak + Big Weak), and HML and CMA follow the same pattern. Averaging the small-stock and big-stock spreads keeps each factor roughly neutral to size. SMB draws on all 18 portfolios: it is the average of the size spreads from the book-to-market, profitability and investment sorts.

The accounting data come from the fiscal year that ended in the previous calendar year, so the June 2026 sort uses 2025 accounts. The risk-free rate changed source in June 2024. The one-month Treasury bill rate comes from Ibbotson Associates through May 2024 and from the ICE BofA US 1-Month Treasury Bill Index after that.

Factor returns from 1963 to 2013 and since

The table compares the paper's sample with the 153 months from January 2014 to September 2026. Both columns use the current version of the Kenneth French Data Library. The library is rebuilt from the CRSP database at each update, so its figures for past years differ slightly from the published ones, and it keeps an archive of each July version for anyone reproducing an older study.

Annual figures compound the monthly returns. The t-statistics test whether the average monthly return differs from zero.

Since 2013 only the market and profitability factors have made money, and none of the four non-market factors has earned a premium distinguishable from zero. HML's annualized volatility was 12.4% over those 153 months, which puts the standard error of its average annual return at about 3.5 points: a true premium of 4% a year would have produced a t-statistic of little more than 1.

RMW, the one non-market factor with a positive return after 2013, lost 10.2% in 2025 and another 12.6% in the first nine months of 2026. At the end of September 2026 it stood 23.6% below its October 2023 peak. Its deepest fall on record was 41.8%, from August 1998 to February 2000, during the late-1990s technology rally. The investment factor's worst drawdown is recent: CMA lost 27.6% from December 2022 to October 2025.

Our guide to factor investing covers the longer record back to 1927, the value factor's 57.8% fall from 2006 to 2020 and the momentum, quality and low-volatility factors used by fund providers.

Why HML became redundant in the five-factor model

Fama and French regressed each of the five factors on the other four. A factor whose average return is fully explained by its exposure to the others has an intercept of zero, and adding it to the model does not improve the description of average returns. Four factors had intercepts more than three standard errors from zero. HML did not.

Intercepts are in percent a month. The published figures are for the 2x3 factors in Table 6 of the paper. The current-data column is our regression on the version of the library released with September 2026 data.

HML's average return was absorbed mostly by CMA, with a slope of 1.04 and a t-statistic of 23, and by RMW, with a slope of 0.23. The CMA slope fits the accounting: value firms tend to invest little, and HML and CMA returns have a correlation of about 0.70. The RMW slope is less intuitive, since value stocks are on average less profitable than growth stocks. With the other factors held fixed, though, value stocks behave like stocks with robust profitability.

The authors warned that the result might be specific to their sample. Twelve more years of data leave it intact: from July 1963 to September 2026, HML's intercept is minus 0.03% a month and its CMA slope is 1.00. The factor that weakened in the update is size, whose intercept is no longer significant.

Fama and French kept HML in the model anyway. Money managers often target the value premium, and the HML loading still measures a portfolio's value tilt. The paper also offers an orthogonal version, HMLO, equal to HML's intercept plus its residual from the regression on the other four factors. Using HMLO in place of HML leaves every intercept and the value loading unchanged, and the other four factors then carry the explanation of average returns.

Where the five-factor model fails

Every model in the paper failed the GRS test, for every set of test portfolios. For four of the six sets the p-values rounded to zero at three decimals, and even in the best case, 25 portfolios sorted on size and profitability, they stayed below 0.04. Fama and French judged the models on how much of the spread in average returns they left unexplained:

• In units of return, the five-factor model's average absolute intercept was 42% to 54% of the dispersion of average excess returns across portfolios. For the three-factor model it was 54% to 68%. The intercepts of the capital asset pricing model were 1.26 to 1.55 times as dispersed as the returns themselves.
• As a share of the cross-sectional variance of expected returns, the five-factor model left about 28% unexplained for 25 portfolios sorted on size and investment, under 25% for size and book-to-market, and 6% to 12% for size and profitability. For 32 portfolios sorted on size, profitability and investment it left about 20%, against 61% to 69% for the three-factor model.
• The five-factor model did no better than the four-factor version without HML on any measure.

The largest errors come from one kind of portfolio: small stocks whose returns behave like those of firms that invest heavily despite low profitability, with negative loadings on both RMW and CMA. They earn far less than the model predicts. In each sort that produced such a portfolio, its intercept alone was negative enough to reject the model for the whole set of 25 or 32 portfolios.

A 2016 follow-up tested the model on anomalies it was not designed to explain. Positive loadings on RMW and CMA, returns that behave like those of profitable firms investing conservatively, captured the high average returns of low-beta stocks, share repurchases and low-volatility stocks. Negative loadings on the same two factors helped explain the low returns of high-beta stocks, large share issues and highly volatile stocks. Accruals and momentum were the exceptions: for both, the five-factor model did no better than the three-factor model, and for accruals it did worse.

Momentum, the factor the model leaves out

The five-factor model has no momentum factor, the return of recent winners minus recent losers, a pattern that conflicts with the random walk picture of stock prices. Fama and French left it out because, for portfolios sorted on size, value, profitability and investment, momentum loadings were close to zero and adding the factor barely changed the results. For portfolios sorted on past returns, they wrote, a momentum factor is crucial.

Regressing the library's momentum factor on the five factors from July 1963 to September 2026 gives an R² of 0.09 and an intercept of 0.75% a month, with a t-statistic of 4.92. That intercept is larger than momentum's own average return of 0.60% a month, because momentum loads negatively on HML, at minus 0.47, and a value tilt and a momentum tilt partly offset each other.

Fama and French added momentum three years later. In "Choosing factors", published in 2018, they ranked models by the maximum squared Sharpe ratio their factors can reach in combination. The models they compared run from CAPM through the three-factor and five-factor models to a six-factor model that adds a momentum factor. The same paper compared profitability factors built on operating profitability with factors built on cash profitability. The library publishes its momentum factor, labeled Mom, in a separate file, so running the six-factor version on a fund takes one more column.

Criticisms of the five-factor model and its rivals

In 2018 four researchers at the asset manager Robeco, David Blitz, Matthias Hanauer, Milan Vidojevic and Pim van Vliet, set out five concerns with the model:

1. It keeps the CAPM relation in which a higher market beta brings a higher return, while the measured relation is flat or even negative.
2. It leaves out momentum.
3. The two new factors raise robustness questions, including whether they existed before 1963 and why investment is measured as asset growth.
4. The economic rationale for the profitability and investment factors is much less clear.
5. It is unlikely to settle the main debates in asset pricing or produce a consensus.

Two studies rebuilt US accounting data from before 1963 out of Moody's manuals, and they partly disagree. Juhani Linnainmaa and Michael Roberts found no statistically reliable profitability or investment premium in the decades before 1963, and concluded that most accounting anomalies, investment included, are probably the product of data snooping. Sunil Wahal, with hand-collected data for 1940 to 1963, found that the profitability premium emerges once value is controlled for. Neither found a reliable investment premium before 1963.

Outside the US, Fama and French found in 2017 that average returns in North America, Europe and Asia Pacific rise with book-to-market and profitability and fall with investment, and that the five-factor model largely absorbs those patterns. Japan was the exception, with a strong value effect and little relation between average returns and profitability or investment. The model's main problem abroad was the same as at home: small stocks whose returns behave like those of unprofitable firms investing aggressively.

The closest rival is the q-factor model of Kewei Hou, Chen Xue and Lu Zhang, built on an investment-based approach to asset pricing and published in 2015. Its four factors are market, size, investment and profitability, with profitability measured by return on equity, and it has no value factor. In 2019 the same authors, joined by Haitao Mo, reported that the q-factor model largely subsumes the Fama-French five-factor and six-factor models.

How to run the five-factor model on a fund

1. Download the factors. On the Kenneth French Data Library, the file "Fama/French 5 Factors (2x3)" holds monthly factor returns from July 1963, in percent, with the risk-free rate in the last column. The CSV opens with a few lines of notes and contains a monthly table followed by an annual one: keep the rows dated YYYYMM. A daily version exists, and separate files cover developed markets, developed markets outside the US, North America, Europe, Japan, Asia Pacific outside Japan and emerging markets, for funds that hold foreign stocks.
2. Get the fund's total returns. Use monthly returns that include reinvested dividends and capital gain distributions, from adjusted prices or the fund company's published total returns. Price returns understate the result by the distribution yield.
3. Align units and dates. Express the fund's returns in percent, match them to the same months, and subtract RF from each month's fund return. Standard errors shrink with the square root of the number of months, so four times as much data halves them.
4. Run the regression. Regress the fund's excess return on Mkt-RF, SMB, HML, RMW and CMA with an intercept. In Excel, LINEST or the Regression tool of the Analysis ToolPak accepts the five factor columns as X values. In Python, with pandas and statsmodels:

   # ff: monthly factor table, fund: monthly fund returns,
   # both in percent and indexed by the same months
   data = ff.join(fund.rename("fund"), how="inner")
   y = data["fund"] - data["RF"]
   X = sm.add_constant(data[["Mkt-RF", "SMB", "HML", "RMW", "CMA"]])
   result = sm.OLS(y, X).fit()
   print(result.summary())

5. Read the output. The constant is the monthly alpha: multiply it by 12 for an annual figure, and check its t-statistic before reading anything into it. The five slopes are the loadings, and the R² is the share of the fund's monthly variance the factors explain.

Ordinary least squares t-statistics assume the residuals are uncorrelated from month to month. Newey-West standard errors relax that assumption, and in statsmodels they take one argument: fit(cov_type="HAC", cov_kwds={"maxlags": 6}). Loadings also drift when a manager changes style, which a rolling regression over 36 months shows.

Five-factor results for four real portfolios

We ran the regression on four portfolios with long records and very different styles: the DFA US Small Cap Value Portfolio (DFSVX), the ARK Innovation ETF (ARKK), Berkshire Hathaway class A shares (BRK-A) and the Vanguard 500 Index Fund Admiral Shares (VFIAX). Returns are monthly total returns from adjusted prices, from each fund's first full month, or March 1985 for Berkshire, through September 2026. Annual alphas are monthly intercepts times 12.

The small-cap value fund loads as its name says: 0.85 on size and 0.44 on value, each with a t-statistic above 20, and the five factors explain 97% of its monthly variance. Its five-factor alpha of minus 1.23% a year is lower than its three-factor alpha of minus 0.35%, because the fund also holds profitable companies, with an RMW loading of 0.15, and the five-factor model credits that exposure with part of the return. Fama and French disclose in the 2015 paper that they are consultants to, board members of and shareholders in Dimensional Fund Advisors, which manages the fund.

ARK Innovation has loadings that resemble the portfolios the model prices worst: 0.69 on size, minus 0.77 on value, minus 1.02 on profitability and minus 0.35 on investment. Its returns behave like those of small, expensive, unprofitable companies that invest heavily. Market exposure contributed 18.5 points of its 17.7% average annual excess return, and the negative profitability loading took away 2.0 points. With a t-statistic of 0.37, its alpha cannot be told apart from zero over 142 months.

Berkshire Hathaway has a market beta of 0.81, a negative size loading typical of a large company, and positive loadings on value and profitability. Those two exposures account for 2.2 points of its 14.1% average annual excess return. The R² of 0.35 is low because a single stock carries far more company-specific risk than a diversified fund, and that noise is why the remaining 4.5% alpha falls short of significance.

The S&P 500 index fund tracks the market factor closely, with a beta of 1.00 and an R² of 0.997. Its size loading of minus 0.13 reflects 500 large companies set against a market factor built from all NYSE, AMEX and NASDAQ stocks. Its intercept of minus 0.32% a year is the gap between the fund and a mix of factor portfolios that exist only on paper and pay no fees, trading costs or taxes.

Factor contributions to a fund's return

Multiplying each loading by the factor's average return over the same months splits a fund's average excess return into factor contributions. For DFSVX from April 1993 to September 2026:

The figures are arithmetic averages of monthly returns times 12, so they add up. The market supplied 9.58 of the fund's 10.48 points a year. Size and value, the two premiums the fund is built to capture, added 1.44 points between them: over the fund's lifetime SMB averaged 0.8% a year and HML 1.7%, against 3.4% and 4.6% from 1963 to 2013.

This article is general information, not investment advice.

Questions about the five-factor model

What are the five factors in the Fama-French model?

The market factor (Mkt-RF) is the return of US stocks minus the one-month Treasury bill rate. Size (SMB, small minus big) is the return of small stocks minus big stocks. Value (HML, high minus low) is the return of stocks with a high book-to-market ratio minus those with a low one. Profitability (RMW, robust minus weak) is the return of firms with high operating profitability minus firms with low profitability. Investment (CMA, conservative minus aggressive) is the return of firms whose assets grow slowly minus firms whose assets grow fast. Eugene Fama and Kenneth French added the last two to their three-factor model in a paper published in 2015.

What is the formula of the Fama-French five-factor model?

It is a time-series regression: R minus RF = a + b(RM minus RF) + s SMB + h HML + r RMW + c CMA + e. R is the monthly return of the stock or fund, RF the one-month Treasury bill rate and RM the market return. The slopes b, s, h, r and c are the factor loadings estimated by the regression, and the intercept a is the alpha, the average return the five factors leave unexplained.

What is the difference between the Fama-French three-factor and five-factor models?

The three-factor model of 1993 uses market, size and value. The five-factor model of 2015 adds profitability (RMW) and investment (CMA). In US data from July 1963 to December 2013, the five-factor model left 42% to 54% of the dispersion of average portfolio returns unexplained, against 54% to 68% for the three-factor model. It also made the value factor redundant: HML's average return was fully explained by its exposure to the other four factors.

What does RMW mean in finance?

RMW stands for robust minus weak. It is the Fama-French profitability factor: the return of stocks with high operating profitability minus the return of stocks with low operating profitability, averaged across small and big stocks. Operating profitability is revenue minus cost of goods sold, selling, general and administrative expenses and interest expense, divided by book equity. From July 1963 to September 2026, RMW returned 2.7% a year compounded.

What is the Fama-French six-factor model?

It is the five-factor model plus a momentum factor: the return of stocks with high returns from 12 months to 2 months ago minus stocks with low returns over the same window. Fama and French included it in Choosing Factors, a 2018 paper that ranks models by the maximum squared Sharpe ratio their factors can produce. From July 1963 to September 2026, the five factors explain only 9% of the variance of the momentum factor's monthly returns and leave 0.75% a month of its return unexplained.

Is the five-factor model better than CAPM?

In Fama and French's 2015 tests, the intercepts left by the capital asset pricing model were 1.26 to 1.55 times as dispersed as the average returns they were meant to explain, against 0.42 to 0.54 times for the five-factor model. Formal statistical tests reject both models, and the five-factor model still predicts too high a return for small stocks whose returns behave like those of firms that invest heavily despite low profitability.