Better Factor Portfolios and Pricing Book-to-Market Characteristics with the Fama-French Factor Model

This paper suggests forming portfolios by optimizing an objective function instead of by sorting. This is more parsimonious and flexible, and makes better use of the data. Empirically, our paper confirms the Davis, Fama, and French (2000) conjecture that the Daniel and Titman (1997) result was unique to their 1973–1993 sample period. The latter’s evidence is obsolete: From 1973–2008, the Fama-French model can price their sort-based incongruence portfolio (spreading HML exposures vs. book-to-market characteristics) almost perfectly. However, we show that it could never price optimized incongruence portfolios. Moreover, one can also construct optimized benchmark factors in lieu of the original Fama-French benchmark factors. These alternatives have higher Sharpe ratios, and some models based on them can price their own incongruence portfolios. Tests of asset-pricing models often rely on multidimensional sorting techniques. For example, the common “dependent sorts” technique sorts stocks by market capitalization first into groups of k stocks each. Within each of these groups, stocks are then sorted by a variable of interest, say x. A zero-investment test portfolio is then formed by going long in the stock with the smallest x and short in the stock with the biggest x (i.e., within each group). The next step is to regress (in time-series) the rates of return of the resulting test portfolio on a benchmark factor model. The most prominent is the Fama and French (1993) model, which is based on a book-to-market, a marketcap and a stock-market factor, often augmented by a momentum factor (Jegadeesh and Titman (1993), Carhart (1997)). If the intercept is not zero, the benchmark model is said to be unable to price the stock-return related influence of x. Sorting techniques have drawbacks. They require an ex-ante choice about the number of groups. They usually do not take advantage of the fact that stocks have different values of x within the same portfolio leg. They do not use information that some stocks have higher residual variance than others. And they make it almost impossible to control for more than a few dimensions. Even a dependent 4-dimensional sort on 6,000 stocks has only 9 stocks per portfolio. Independent sorts struggle even more with finding stocks in the extreme corner portfolios. Thus, it is by necessity that researchers often control their test portfolios only for one variable, usually market capitalization. Our paper suggests replacing sorts as the method for forming portfolios with optimization of an objective function. For example, one can form a portfolio designed to produce reliable inference—it could minimize the expected standard error of alpha, subject to a number of constraints, such as balance with respect to market capitalization and large exposure with respect to the book-to-market ratio. Under a set of assumptions, the portfolio maximizing such an objective function can be the same as a portfolio implicit in the coefficients of the (cross-sectional) Fama-Macbeth tests (Fama (1976)). However, we use the rates of return on some of these portfolios as the dependent variable in a Fama and French (1993) time-series regression with respect to a benchmark factor pricing model. Our test is then whether the alpha of this portfolio is zero. Empirically, our paper uses optimized portfolios to revive a disagreement about whether the Fama-French model can price the empirical regularity that it was designed to explain— the superior performance of value stocks. Daniel and Titman (1997), henceforth DT, identify stocks that have incongruous characteristics and exposures: either growth firms (low bookto-market characteristics) with high HML exposures, or value firms (high book-to-market characteristics) with low HML exposures. This allows them to determine that stocks’ own book-to-market ratios are responsible for their higher returns, and not their HML factor