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1. Introduction

There is much evidence that average stock returns are related to the book-to-market equity ratio, B/M. There is also evidence that profitability and investment add to the description of average returns provided by B/M. We can use the dividend discount model to explain why these variables are related to average returns. The model says the market value of a share of stock is the discounted value of expected dividends per share,

(approximately) the long-term average expected stock return or, more precisely, the internal rate of return on expected dividends.

Eq. (1) says that if at time t the stocks of two firms have the same expected dividends but different prices, the stock with a lower price has a higher (long-term average) expected return. If pricing is rational, the future dividends of the stock with the lower price must have higher risk. The predictions drawn from (1), here and below, are, however, the same whether the price is rational or irrational.

mt ∑1

τ ¼ 1

Eðdt þτÞ=ð1 þrÞτ: ð1Þ

With a bit of manipulation, we can extract the implica- tions of Eq. (1) for the relations between expected return

In this equation, mt is the share price at time t, E(dtþτ) is the expected dividend per share for period t þτ, and is

and expected profitability, expected investment, and B/M. Miller and Modigliani (1961) show that the time t total market value of the firm's stock implied by (1) is,

☆ Fama and French are consultants to, board members of, and share- holders in Dimensional Fund Advisors. Robert Novy-Marx, Tobias Mos- kowitz, and Ľuboš Pástor provided helpful comments. John Cochrane,

Savina Rizova, and the referee, Kent Daniel, get special thanks.

n Corresponding author.

E-mail address: kfrench@dartmouth.edu (K.R. French).

http://dx.doi.org/10.1016/j.jfineco.2014.10.010 0304-405X/& 2014 Elsevier B.V. All rights reserved.

In this equation, Ytþτ, is total equity earnings for period

t þτ and dBtþτ ¼ Btþτ — Btþτ— 1 is the change in total book

equity. Dividing by time t book equity gives,

We test the performance of the five-factor model in two steps. Here we apply the model to portfolios formed

1EðYt þτ — dBt þτÞ=ð1 þ rÞτ

Bt

on size, B/M, profitability, and investment. As in FF (1993), the portfolio returns to be explained are from finer versions of the sorts that produce the factors. We move

Eq. (3) makes three statements about expected stock

returns. First, fix everything in (3) except the current value of the stock, Mt, and the expected stock return, r. Then a lower value of Mt, or equivalently a higher book-to-market equity ratio, Bt/Mt, implies a higher expected return. Next, fix Mt and the values of everything in (3) except expected future earnings and the expected stock return. The equa- tion then tells us that higher expected earnings imply a higher expected return. Finally, for fixed values of Bt, Mt, and expected earnings, higher expected growth in book equity – investment – implies a lower expected return. Stated in perhaps more familiar terms, (3) says that Bt/Mt is a noisy proxy for expected return because the market cap Mt also responds to forecasts of earnings and investment. The research challenge posed by (3) has been to identify proxies for expected earnings and investments. Novy-Marx (2013) identifies a proxy for expected profit- ability that is strongly related to average return. Aharoni, Grundy, and Zeng (2013) document a weaker but statisti- cally reliable relation between investment and average return. (See also Haugen and Baker, 1996; Cohen, Gompers, and Vuolteenaho, 2002; Fairfield, Whisenant, and Yohn, 2003; Titman, Wei, and Xie, 2004; and Fama and French, 2006, 2008.) Available evidence also suggests that much of the variation in average returns related to profitability and investment is left unexplained by the three-factor model of Fama and French (FF, 1993). This leads us to examine a model that adds profitability and investment factors to the market, size, and B/M factors of the FF three-factor model.

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