Theoretical Background: Valuation Theory

In an efficient market, asset prices reflect the present value of future profits, discounted at some discount rate. The discount rate used to value a company is exceptionally important to investors: it is the return that the investor expects to earn.

A higher discount rate is the same thing as a higher expected return. A higher discount rate indicates that the market is pricing more risk into the security. A risk that is reflected in asset prices is referred to as a priced risk.

The relationship between profits, asset prices, and expected return is easily illustrated by the dividend discount model. This is a model that relates expected future dividends and the discount rate to the price. The thinking follows that rational investors are, at some level, using this thinking when they interact with the market.

The dividend discount model appears as follows

Formual 1 Summary.png
Fama and French Summary.png

Empirical Evidence: The Fama French Five-Factor Model

With the framework of Equation (2), the theoretical relationships between relative price, profitability, and investment are obvious – even tautology. Theory is interesting but not necessarily directly applicable to managing real portfolios.

Fama and French (2015) introduced a five-factor model that incorporates the profitability and investment factors alongside the original market, size, and relative price factors used in the three-factor model. They chose these additional two factors – profitability and investment – specifically because they are so clearly identified in the valuation equation.

Notably, the size factor does not make an appearance in the valuation equation. Fama and French (2015) explain “If variables not explicitly linked to this decomposition, such as Size and momentum, help forecast returns, they must do so by implicitly improving forecasts of profitability and investment or by capturing horizon effects in the term structure of expected returns.” This statement is fascinating and it is in line with much of the research that has been critical of the size premium; there may not be a standalone size premium, but the other premiums tend to be magnified in smaller stocks.

The development of the five-factor model created a tool for testing the theory behind Equation (2). If the relationships in the valuation equation are, in fact, reliable, then we would expect a model that accounts for each of the factors to be able to explain the majority of differences in returns between diversified portfolios.

Testing the five-factor model

Factors are technically defined as a portfolio that is long one thing and short another. The market factor is the portfolio that is long the stock market and short one-month US treasury bills. The size factor is the portfolio that is long small stocks and short large stocks. Mathematically that’s the return of small stocks minus the return of big stocks. That’s how the size factor gets the name SmB (small minus big). Likewise, the value factor is defined as high book-to-market stocks minus low book-to-market, or HmL (high minus low); the profitability factor is firms with robust profitability minus firms with weak profitability, or RmW (robust minus weak); the investment factor is firms that invest conservatively minus firms that invest aggressively, or CmA (conservative minus aggressive).

Fama and French (2015) tested the five-factor model against a number of portfolios formed to produce large spreads in Size, B/M, profitability, and investment. They estimated that the model explains between 71% and 94% of the cross-section variance of expected returns.

In a separate paper from the same year, Fama and French set out to test the model against anomalies that their three-factor model had been unable to explain. They showed that the high average returns associated with low β stocks, share repurchases, and low volatility stocks are well explained by the addition of the profitability and investment factors.

These empirical results are compelling in their ability to support the theoretical underpinning of the five-factor model. The more confident that we can be in the theoretical underpinning, the more confident we can be that positive expected returns will be persistent. A risk-based premium should persist over the long-term.



Campbell, J., Shiller, R.J., 1988. The dividend-price ratio and expectations for future dividends and discount factors. Review of Financial Studies 1, 195–228.

Fama, E. F., French, K. R., 2006. Profitability, investment, and average returns. Journal of Financial Economics 82, 491–518.

Fama, E. F., French, K. R., 2015. A five-factor asset pricing model. Journal of Financial Economics 116: 1-22.

Fama, E. F., French, K. R., Dissecting Anomalies with a Five-Factor Model (June 2015). Fama-Miller Working Paper.

Miller, M., Modigliani, F., 1961. Dividend policy, growth, and the valuation of shares. Journal of Business 34, 411-433.