Fama–French three-factor model
In asset pricing and portfolio management the Fama–French three-factor model is a model designed by Eugene Fama and Kenneth French to describe stock returns. Fama and French were professors at the University of Chicago Booth School of Business, where Fama still resides. In 2013, Fama shared the Nobel Memorial Prize in Economic Sciences. The three factors are (1) market risk, (2) the outperformance of small versus big companies, and (3) the outperformance of high book/market versus small book/market companies. However, the size and book/market ratio themselves are not in the model. For this reason, there is academic debate about the meaning of the last two factors.
The traditional asset pricing model, known formally as the capital asset pricing model (CAPM) uses only one variable to describe the returns of a portfolio or stock with the returns of the market as a whole. In contrast, the Fama–French model uses three variables. Fama and French started with the observation that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a high book-to-market ratio (B/P, customarily called value stocks, contrasted with growth stocks).
Here r is the portfolio's expected rate of return, Rf is the risk-free return rate, and Rm is the return of the market portfolio. The "three factor" β is analogous to the classical β but not equal to it, since there are now two additional factors to do some of the work. SMB stands for "Small [market capitalization] Minus Big" and HML for "High [book-to-market ratio] Minus Low"; they measure the historic excess returns of small caps over big caps and of value stocks over growth stocks.
These factors are calculated with combinations of portfolios composed by ranked stocks (BtM ranking, Cap ranking) and available historical market data. Historical values may be accessed on Kenneth French's web page. Moreover, once SMB and HML are defined, the corresponding coefficients bs and bv are determined by linear regressions and can take negative values as well as positive values.
The Fama–French three-factor model explains over 90% of the diversified portfolios returns, compared with the average 70% given by the CAPM (within sample). They find positive returns from small size as well as value factors, high book-to-market ratio and related ratios. Examining β and size, they find that higher returns, small size, and higher β are all correlated. They then test returns for β, controlling for size, and find no relationship. Assuming stocks are first partitioned by size the predictive power of β then disappears. They discuss whether β can be saved and the Sharpe-Lintner-Black model resuscitated by mistakes in their analysis, and find it unlikely.
Griffin shows that the Fama and French factors are country specific (Canada, Japan, the U.K., and the U.S.) and concludes that the local factors provide a better explanation of time-series variation in stock returns than the global factors. Therefore, updated risk factors are available for other stock markets in the world, including the United Kingdom, Germany and Switzerland. Eugene Fama and Kenneth French also analysed models with local and global risk factors for four developed market regions (North America, Europe, Japan and Asia Pacific) and conclude that local factors work better than global developed factors for regional portfolios. The global and local risk factors may also be accessed on Kenneth French's web page. Finally, recent studies confirm the developed market results also for emerging markets.
A number of studies have reported that when the Fama–French model is applied to emerging markets the book-to-market factor retains its explanatory ability but the market value of equity factor performs poorly. In a recent paper, Foye, Mramor and Pahor (2013) propose an alternative three factor model that replaces the market value of equity component with a term that acts as a proxy for accounting manipulation. 
Fama–French five-factor model
In 2015, Fama and French extended the model, adding a further two factors -- profitability and investment. Defined analogously to the HML factor, the profitability factor (RMW) is the difference between the returns of firms with robust (high) and weak (low) operating profitability; and the investment factor (CMA) is the difference between the returns of firms that invest conservatively and firms that invest aggressively. In the US (1963-2013), adding these two factors makes the HML factors redundant since the time series of HML returns are completely explained by the other four factors (most notably CMA which has a -0.7 correlation with HML).
Whilst the model still fails the Gibbons, Ross & Shanken (1989) test, which tests whether the factors fully explain the expected returns of various portfolios, the test suggests that the five-factor model improves the explanatory power of the returns of stocks relative to the three-factor model. The failure to fully explain all portfolios tested is driven by the particularly poor performance (i.e. large negative five-factor alpha) of portfolios made up of small firms that invest a lot despite low profitability (i.e. portfolios whose returns covary positively with SMB and negatively with RMW and CMA). If the model fully explains stock returns, the estimated alpha should be statistically indistinguishable from zero.
Whilst a momentum factor wasn't included in the model since few portfolios had statistically significant loading on it, Cliff Asness, former PhD student of Eugene Fama and co-founder of AQR Capital has made the case for its inclusion.. Foye (2018) tested the five-factor model in the UK and raises some serious concerns. Firstly, he questions the way in which Fama and French measure profitability. Furthermore, he shows that the five-factor model is unable to offer a convincing asset pricing model for the UK. .
- Returns-based style analysis, a model that uses style indices rather than market factors
- Carhart four-factor model (1997) — extension of the Fama–French model, containing an additional momentum factor (MOM), which is long prior-month winners and short prior-month losers
- Size premium
- Petkova, Ralitsa (2006). "Do the Fama–French Factors Proxy for Innovations in Predictive Variables?". Journal of Finance. 61 (2): 581–612. doi:10.1111/j.1540-6261.2006.00849.x.
- Fama, E. F.; French, K. R. (1993). "Common risk factors in the returns on stocks and bonds". Journal of Financial Economics. 33: 3–56. CiteSeerX 10.1.1.139.5892. doi:10.1016/0304-405X(93)90023-5.
- Fama, E. F.; French, K. R. (1992). "The Cross-Section of Expected Stock Returns". The Journal of Finance. 47 (2): 427. doi:10.1111/j.1540-6261.1992.tb04398.x. JSTOR 2329112.
- Griffin, J. M. (2002). "Are the Fama and French Factors Global or Country Specific?" (PDF). Review of Financial Studies. 15 (3): 783–803. doi:10.1093/rfs/15.3.783. JSTOR 2696721.[permanent dead link]
- Fama, E. F.; French, K. R. (2012). "Size, value, and momentum in international stock returns". Journal of Financial Economics. 105 (3): 457. doi:10.1016/j.jfineco.2012.05.011.
- Cakici, N.; Fabozzi, F. J.; Tan, S. (2013). "Size, value, and momentum in emerging market stock returns". Emerging Markets Review. 16 (3): 46–65. doi:10.1016/j.ememar.2013.03.001.
- Hanauer, M.X.; Linhart, M. (2015). "Size, Value, and Momentum in Emerging Market Stock Returns: Integrated or Segmented Pricing?". Asia-Pacific Journal of Financial Studies. 44 (2): 175–214. doi:10.1111/ajfs.12086.
- Pahor, Marko; Mramor, Dusan; Foye, James (2016-03-04). "A Respecified Fama French Three Factor Model for the Eastern European Transition Nations". SSRN 2742170. Cite journal requires
- Fama, E. F.; French, K. R. (2015). "A Five-Factor Asset Pricing Model". Journal of Financial Economics. 116: 1–22. CiteSeerX 10.1.1.645.3745. doi:10.1016/j.jfineco.2014.10.010.
- Gibbons M; Ross S; Shanken J (September 1989). "A test of the efficiency of a given portfolio". Econometrica. 57 (5): 1121–1152. CiteSeerX 10.1.1.557.1995. doi:10.2307/1913625. JSTOR 1913625.
- "Our Model Goes to Six and Saves Value from Redundancy Along the Way".
- Foye, James (2018-05-02). "Testing Alternative Versions of the Fama-French Five-Factor Model in the UK". Risk Management. 20 (2): 167–183. doi:10.1057/s41283-018-0034-3.
- Carhart, M. M. (1997). "On Persistence in Mutual Fund Performance". The Journal of Finance. 52 (1): 57–82. doi:10.1111/j.1540-6261.1997.tb03808.x. JSTOR 2329556.
- The Dimensions of Stock Returns: Videos, paintings, charts and data explaining the Fama–French Five Factor Model, which includes the two factor model for bonds.