In 1992 Eugene Fama and Kenneth French published a paper called "The Cross-Section of Expected Stock Returns". They started the paper with the analysis of Sharpe-Lintner-Black model which proposed that the market portfolio was mean-variance efficient in Markowitz terms. This prediction implied that the expected returns on stocks were linearly dependent from their betas (the slope of the stock return to the market return) and that betas were sufficient to explain the expected returns. Fama and French claimed that there were a few contradictions to the Sharpe-Lintner-Black model. Banz had already discovered the size effect (Banz, 1981). He found out that the Market Equity (ME) was another significant factor along with betas in the explanation of the expected returns. The returns on stocks with high ME were on average too low for their betas and the returns on stocks with low ME were on average too high for their beta. Bhandari later found out that leverage was not included into beta as SLB model claimed but it should had been included as the variable itself along with the size factor and beta (Bhandari, 1988). Stattman, Rosenberg, Reid, and Lanstein analyzed the U. S. stock market and discovered the influence of book-to-market ratio, BE/ME, on expected returns (Rosenberg, Reid, Lanstein, 1985). Later Chan, Hamao, and Lakonishok also found this positive relation on Japanese stock market (Chan, Hamao, Lakonishok, 1991). It was widely argued whether price-earnings ratio, E/P, should be included as a factor. Basu and Ball claimed that E/P absorbs the other factors as a catch-all proxy. No matter what these factors are, high E/P represents higher risk and greater expected return (Basu, 1983). So, the goal of Fama-French 1992 paper was to analyze which of 5 factors - beta, size, book-to-market, leverage and E/P were redundant in predicting expected returns. They found out that size and book-to-market factors were significant in explaining expected returns. They concluded that beta alone did not explain the expected returns on stock and size and BE/ME factors absorbed the effects of leverage and E/P. Fama and French suggested that the stock risks were multidimensional. While ME represented one dimension of risk, BE/ME represented the other dimension of risk. Book-to-market ratio in their opinion could be the factor of relative distress. Firms with bad prospects estimated by the market tend to have higher expected returns than the good prospects. The economic interpretation of the results could be discussed but the statistical result was strong - size and book-to-market were significant factors in explaining expected returns on stocks.

Let's move to the methodology which Fama and French used. They did not include financial firms into data because these firms have too high leverage which is not typical in terms of risk. They also make a two-period gap between accounting data and the returns to ensure that the accounting information is known in the latter period. They claim that the choice of the year-end month does not alter the results. The authors estimate betas the following way: first they divide the stocks into portfolios then estimate betas and assign each beta to the individual stocks from the corresponding portfolios. This approach was introduced by Fama and Macbeth in 1973. Fama and French find that fixing the size betas does not explain the average stock returns. When they included book-to-market ratio and E/P factor, they found that across the portfolios E/P factor has got a U-shape and book-to-market factor shows strong positive relation. Influence of these factors did not steal the influence of beta, because beta alone showed no difference between portfolios. The authors explain that leverage and BE/ME factor are closely linked and suggest two ways to interpret book-to-market effect in average returns. A high BE/ME means that the firms has poor prospects compared to the firms with lower BE/ME. Hence, book-to-market factor accounts for financial distress. Also high BE/ME signals as an involuntary leverage effect. Ball claimed that E/P ratio is a factor which catches all the other risk factors in expected returns (Ball, 1978). However, the size factor eliminates the significance of price earnings factor. The conclusions of Fama and French 1992 paper are quite intermediate: they conclude that beta alone does not explain stock returns and also say that size and book-to-market factors absorb the other factors influencing price. The authors do not suggest the best model explaining stock returns.

However, in 1993 Fama and French published a paper called `Common risk factors in returns on stocks and bonds'. In this paper they identified three factors on stock market: overall market factor, firm size factor and book-to-market equity factor. The authors extended their previous paper in three ways:

They expanded the set of assets returns adding bonds to common stock (U. S.government and corporate bonds)

They expanded the set of explanatory variables. They added term-structure factors to explain bond returns. They hoped that bond and stock returns overlaped and there would be some chance to explain each other.

They changed the approach of testing. Fama and Macbeth approach was useless when applied to bonds because the cross-section returns with size and book-to-market factors were irrelevant for bonds.

Instead they used the approach of Black, Jensen, and Scholes (1972). Returns on stocks and bonds were regressed on returns of mimicking portfolios for size, book-to-market and term-structure factors. The slopes of the regression have much clearer interpretation for both stocks and bonds than just pure size and BE/ME factors.

The authors block the stocks the same as in 1992. Blocking by size divides the stocks to small and big (S and B) by medium size on NYSE. They also break the stock into three groups by book-to-market equity. Lowest 30% are called Low, medium 40% are called Medium, and the highest 30% on NYSE are called High. The intersection of the blocks create six portfolios (S/L, S/M, S/H, B/L, B/M, B/H). Then they create a size factor - SMB (small minus big), which is equal to monthly difference between the average returns on the small stocks and average returns on the big stocks. This factor is free of book-to-market influence because each part of the difference has approximately the same BE/ME. The next factor is book-to-market factor which is called HML (High minus Low). It is equal to the difference of monthly average returns on high BE/ME stocks minus average returns of the low BE/ME stocks. This factor is free from size influence.

In their data Fama and French use 25 portfolios of stocks which are formed by dividing stocks into 5 quintiles by size and 5 quintiles by book-to-market ratio, then intersecting these groups. The smallest size quintile has the most stocks but has the lowest value. The biggest size quintile has the least stocks and is the biggest in value.

Fama and French find that SMB and HML factors alone do not have a high explaining factor, however if we add market risk premium to the regression, R squared increases and all factors become significant. Adding market risk premium almost eliminates the intercept which is high in a two-factor case. So when the intercepts become close to zero it means that the three factors - market risk premium, SML, and HML include almost all common variation in returns and explain the cross-section of average stock returns really well.

So, Fama and French find out that book-to-market ratio is connected to the firms profitability. On average, firms, that have low BE/ME ratio, have higher earnings than the firms with high BE/ME ratio.

Fama and French summarize their paper by saying that the choice of factors is empirical and quite arbitrary: there is no economic theory behind. So there remain open question on why these three factors are chosen to explain average stock returns and what are any other possible combinations of factors which improve the model.

Further, the three-factor model was waiting for the extension. Several economists proposed to add momentum factor into the model, however they suggested different forms of this coefficient. In 1995, Mark Grinblatt, Sheridan Titman, and Russ Wermers published a paper called `Momentum investment strategies, portfolio performance, and herding: a study of mutual fund behaviour'. The paper mainly focused on the behaviour of the mutual funds and how the copied each other. However in the beginning of the paper there is an introduction of the momentum factor, which was later added to Fama and French model. The authors measure the momentum as the average increase in the stock's weight multiplied by its lagged return. The theory behind this measure implies that if a price of a stock has been increasing, it will continue growing. So for the positive return the investors increase the weight of a stock in their portfolio, increasing the momentum factor and for the negative return investors decrease the weight of the stock, causing momentum to increase, too. Grinblatt, Titman and Wermers did not add the momentum factor into the multi-factor model, however they inspired Mark Carhart to do it. Carhart in his paper `On persistence in mutual funds performance' (1997) defines the momentum factor as the equally weighted average of firms with the highest 30 percent eleven-month returns lagged one month minus equally weighted average of firms with lowest 30 percent eleven-month returns lagged one month. Carhart names this factor PR1YR and adds it to Fama and French three-factor model. Carhart finds that high variance in the factors and low cross-correlation help to account for sizeable variation in stocks returns with no multicollinearity effect. Carhart also finds that the four-factor model outperforms the CAPM and the three-factor model in terms of pricing errors. He claims that the three-factor model pricing errors are strongly positive for the winner stock portfolios in the previous year and that the errors are strongly negative for the loser stock portfolios. The four-factor model fixes this problem and reduces mean absolute error from 0.31 percent in the three-factor model to 0.14 percent in the four-factor model.

In 2012 Fama and French published a paper called `Size, value, and momentum in international stock returns', where in terms of momentum they divided the stocks into three groups - winners, neutral and losers (top 30 percent momentum, middle 40 percent, and lowest 30 percent accordingly). The momentum factor is called WML (winners minus losers) and is equal to weighted average of winner stocks minus the weighted average of loser stocks in the previous eleven months. Fama and French constructed the portfolios of all possible combinations of factor types for the deeper analysis. Some of their results were quite disappointing for those who adopted the four-factor model. Examining the returns in the four region which sum up the vast majority of trade (North America, Europe, Japan and Asia Pacific) there was a strong presence of the momentum factor in all regions except Japan. They analyzed the influence of the size factor on momentum returns and found that in Japan there was no momentum for any stock size. The three global models - CAPM, three-factor model, and the four-factor model performed badly when asked to explain average returns on size/BM or size/momentum regional portfolios. Speaking of local models, when the local model is acceptable, the four-factor model outperforms the CAPM and the three-factor models. The local models experience some issues with capturing momentum-size portfolios returns due to extreme values. However, the authors claim that these extreme tilts are in reality rare and would not cause a serious problem in application of the model.

In 2015 Fama and French published a new paper called `A five-factor asset pricing model', where they introduced profitability and investment factors instead of momentum. They state that these factors are present in the dividend discount model which says that the market value of a stock is equal to the discounted value of expected dividends per share. The evidence by Titman, Wei and Xei showed that the three-factor model lacked the variation due to profitability and investment (Titman, Wei, Xei, 2003), so Fama and French introduced these factors. Factor RMW is the difference between the returns on diversified portfolios with robust and weak profitability. Factor CMA is the difference between the returns on diversified portfolios with conservative (low) and aggressive (high) investment firms. They create value-weighted replicating portfolios and combine all factors in three-,four-, and five-factor models. To ease the process they increase the quantiles in most cases to break the factors in less groups. One of the interesting results is that in five-factor model HML factor never improves the model compared to four-factor model without HML. The authors leave the question if this is an anomaly in their data or it would happen internationally. They conclude that the parsimony may be an issue with the five factors and they tend to divide the factors by 2 or 3 in sorting because it both eases the process and shows better results. Fama and French suggest omitting HML variable and checking if it is redundant. However, if it is important for the analysis one can leave value in the model.

The most problematic types of stocks in the five-factor model were the small stock with negative exposures to risk and profitability. It is difficult to explain the average returns of the small firms that invest a lot despite low profitability. Another serious issue is that the intercept of the big stocks which invest a lot despite low profitability is positive. Economic logic struggles to explain some of the results by Fama and French and the authors conclude that again the small stock are the biggest problem in asset pricing model because they show the controversial results.

A three-factor model