In the dividend discount valuation model, the market value of a share of stock is the present value of expected dividends per share,
where Mt is the price at time t, E(Dt+τ) is the expected dividend for t+τ, and r is (approximately) the long-term average expected stock return or, more precisely, the internal rate of return on expected dividends. With clean surplus accounting, the time t dividend, Dt, is equity earnings per share, Yt, minus the change in book equity per share (retained earnings), dBt-1,t=Bt−Bt−1. (Note that here and only here dBt−1,t is the change in levels, rather than logs.) The dividend discount model is then
or, dividing by time t book equity,
Equation (4) says that, controlling for expected equity cashflows (earnings minus changes in book equity, measured relative to current book equity), a higher B/M equity ratio, Bt/Mt, implies a higher average expected stock return, r. This is the motivation for using the B/M ratio as a proxy for expected returns. It is clear from (4), however, that dispersion across stocks in expected cashflows can act like noise that obscures the information in Bt/Mt about the cross section of expected stock returns. Thus, using variables in addition to Bt/Mt to predict returns will improve estimates of expected returns if the additional variables help disentangle expected cashflows and expected returns.
Fama and French (2006) search extensively for proxies for expected cashflows that improve estimates of expected returns from Bt/Mt, with limited success. We take a simpler approach here. Specifically, we argue that the evolution of the B/M ratio itself provides interesting candidates.
Recalling equation (1), the log of the B/M ratio at time t, BMt, is the log ratio at t−k, BMt−k, plus the difference between the change in the log of book equity, dBt−k,t, and the change in the log of price, dMt−k,t. The lagged growth in book equity, dBt−k,t, is related to past and expected future cashflows. Past growth in book equity tends to be high for growth (low BMt) stocks and low to negative for value (high BMt) stocks, the result of high earnings and reinvestment for growth stocks and low earnings and reinvestment for value stocks (Lakonishok et al. (1994), Fama and French (1995)). This behavior of fundamentals tends to persist in the years after t, making dBt−k,t an interesting proxy for expected cashflows. Similarly, the logic of the dividend discount equation (2) is that the lagged change in price, dMt−k,t, summarizes changes from t−k to t in expected future returns and expected cashflows.
The interplay between dBt−k,t and dMt−k,t is also important in explaining how stocks migrate between value and growth. Stocks typically move to high expected return value portfolios as a result of poor past profitability and low (often negative) growth in book equity accompanied by even sharper declines in stock prices (Fama and French (1995)). Conversely, stocks that move to lower expected return growth portfolios typically have high past profitability and growth in book equity along with even sharper past increases in stock prices.
In short, using BMt to forecast returns may bury independent information in its components about expected cashflows and expected returns. There is thus reason to expect that using the components of BMt in (1) to predict returns provides better estimates of expected returns than BMt alone.
An extension of this logic suggests that more distant changes in book equity and price have less information about expected cashflows and returns than more recent changes; that is, old news is less relevant than new news. A simple way to test this prediction is to examine the estimates of expected returns provided by the components of BMt in (1) for different lags k. If old news is less relevant, the slopes in regressions of returns on the components of BMt should decay as the lag k for changes in price and book equity is increased. Moreover, the slope for the lagged B/M ratio BMt−k (which summarizes the history of growth in book equity and price preceding dBt−k,t and dMt−k,t) should be weaker than the slopes for one or both of dBt−k,t and dMt−k,t.
The B/M ratio is the same whether we use price and book equity per share or total market cap and total book equity. The choice of total or per share growth rates of book equity and market value, however, does affect the decomposition of BMt in (1) because total changes include net share issues (issues minus repurchases). But net share issues is itself an interesting candidate to capture variation in the cross section of BMt due to expected cashflows, to better estimate expected returns. Thus, prior work says that firms that issue stock tend to have large (past and future) investments relative to earnings, while the opposite is true for firms that repurchase (Fama and French (2005)). To disentangle the effects of net share issues from the effects of per share changes in market and book values, we use per share growth rates and include net share issues as a separate explanatory variable for returns.
The success of our predictions is not guaranteed. Thus, it is plausible that newer information about expected cashflows is more relevant for enhancing estimates of expected returns, but it is also possible that information about expected cashflows cumulates in a martingale fashion, so old news is as relevant as new news. Similarly, it is plausible that lagged changes in price and book equity have different mixes of information about expected cashflows and returns, but it is also possible that information in the two variables combines in such a way that 1% changes in dBt−k,t and dMt−k,t have similar marginal effects on estimates of expected returns. And there is no reason that all this works out in the same way for different kinds of firms. We shall see that the results on old versus new news are similar for Micro and ABM stocks, but (for whatever reason) the marginal information in dBt−k,t and dMt−k,t about expected returns is different for the two size groups.
Finally, it is worth emphasizing that in the framework of the valuation equation (4), the cross section of BMt is determined entirely by differences across stocks in expected cashflows and expected returns. Thus, when we later infer that a variable helps forecast returns because it helps disentangle expected cashflows and expected returns, we are not talking loosely, even though we do not present direct evidence about how the marginal information splits between expected cashflows and returns.
B. Regression Setup
Fama and French (1992) use cross-section regressions of individual stock returns on market cap and the B/M ratio to identify differences in average returns related to the size and value-growth characteristics of firms. Here, we test whether the origins of the B/M ratio, in terms of share issues and changes in price and book equity per share, can be used to improve estimates of expected returns. In the tradition of Fama and MacBeth (1973), our tests center on the average slopes from monthly cross-section regressions of stock returns on five variables,
In this regression, Rt+n is a stock's simple return for month t + n in excess of the 1-month Treasury bill rate; MCt is the stock's market cap at time t; BMt−k is its B/M ratio for t−k (with k= 12, 36, or 60); and dMt−k,t, dBt−k,t, and NSt−k,t are the change in price per split-adjusted share, the change in book equity per split-adjusted share, and the change in split-adjusted shares outstanding for the preceding 1, 3, and 5 years. The explanatory variables in the regression are updated at the end of June each year, and they are used in the monthly regressions for July through the following June. Thus, the subscript n runs from 1 to 12 and the subscript t jumps in increments of 12, from one June to the next. The data start in June 1926, so when k= 12, t starts in June 1927 and the first regression explains returns for July 1927. For lags k= 36 and k= 60, t starts in June 1929 and June 1931. To simplify the notation, the subscript j (for stock j) that should appear on the dependent return and all the explanatory variables in (5) is omitted. The variables MCt and BMt−k are natural logs of market cap and the B/M ratio, and the three change variables are changes in logs.
To ensure that book equity per share, Bt, is known in June (time t), we use the fiscal year value reported during the previous calendar year. As in Fama and French (1992), the stock price in MCt is for the end of June (time t), but the price in BMt is for the end of the preceding December. The three change variables line up with BMt. Thus, dBt−k,t is the change in book equity per share for the preceding 1, 3, or 5 fiscal years, and dMt−k,t and NSt−k,t are the changes in price per share and split-adjusted shares outstanding for the k months that end in December. To be precise, dMt−k,t is the continuously compounded without-dividend return, from CRSP, over the preceding 1, 3, or 5 calendar years; NSt−k,t is the difference between the continuously compounded growth in total market equity, computed using the price and shares outstanding reported by CRSP, and the continuously compounded capital gain (dMt−k,t) over the same period; and dBt−k,t is the difference between the continuously compounded growth in total book equity over the preceding 1, 3, or 5 fiscal years and the matching NSt−k,t computed from fiscal year-end to fiscal year-end. Thus, with these definitions, total changes in market cap and book equity are per share changes plus our measure of share issues. Book equity data are from Compustat, with missing data for NYSE stocks filled in by us as in Davis, Fama, and French (2000).
Our null hypothesis is that breaking the B/M ratio into its components does not enhance the estimates of expected returns provided by BMt. In terms of regression (5), the null is that the true slopes for BMt−k, dBt−k,t, and dMt−k,t have the same magnitude, with positive slopes for BMt−k and dBt−k,t and a negative slope for dMt−k,t. The alternative hypothesis is that the components of BMt help isolate information about expected cashflows, thus improving estimates of expected returns. Under the alternative, the true slopes for the components of BMt differ because the three components capture different mixes of information about expected cashflows and expected returns. For example, if old information is less relevant than new information, the true slopes on the components of BMt decline as the lag k for changes in price and book equity increases. And the true slope for BMt−k, which summarizes older forecasts of cashflows and returns, is closer to zero than the slope for dBt−k,t and/or the slope for dMt−k,t.
In general, however, beyond predicting that the average slopes for the three components of BMt do not have the same magnitude, the alternative hypothesis that breaking BMt into its components improves expected return estimates does not make strong predictions about average slopes. The reason is that the three components of BMt may each be a different mix of forecasts of cashflows and returns, so there are no clear predictions about marginal effects. On the other hand, if the average slopes for the components of BMt are equal in magnitude, positive for BMt−k and dBt−k,t and negative for dMt−k,t, the unavoidable conclusion is that forecasts of returns from the components collapse to forecasts from BMt, so there is no additional information in the origins of BMt beyond that in BMt alone.
There is a simple way to test the hypothesis that the true slopes for dMt−k,t and dBt−k,t in regression (5) are equal in magnitude to the slope for BMt−k. Consider an alternative regression that substitutes the most recent B/M ratio, BMt, for the lagged BMt−k in (5),
Since BMt=BMt−k+dBt−k,t−dMt−k,t, the slopes in (6) link directly to those in (5). The slopes for MCt and NSt−k,t do not change in going from (5) to (6). Similarly, the slope for BMt−k in (5) is the slope for BMt in (6). Intuitively, the slope for BMt in (6) is the marginal effect of BMt, given the lagged changes in price and book equity, dMt−k,t and dBt−k,t, but this is also the marginal effect of BMt−k in (5). The slopes for dMt−k,t and dBt−k,t are, however, different in (5) and (6). The slope for dBt−k,t in (6) is the slope for dBt−k,t in (5) minus the slope for BMt−k. And the slope for dMt−k,t in (6) is the slope for dMt−k,t in (5) plus the slope for BMt−k.1 The average slopes for dMt−k,t and dBt−k,t in (6) thus provide a formal test of whether the true slopes for the two changes in (5) have the same magnitude as the slope for BMt−k. More directly, if the true slopes on dMt−k,t and dBt−k,t in (6) are zero, the origins of BMt in terms of lagged changes in price and book equity add nothing to forecasts of returns from BMt.
We estimate the monthly regressions (5) and (6) separately for ABM stocks (market cap above the 20th percentile of market cap for NYSE stocks) and for Microcaps (market cap below the 20th NYSE percentile). Before 1963, the tests cover only NYSE stocks and there are on average just 141 Microcap stocks. Amex stocks are added in 1963 and Nasdaq stocks in 1973. Amex and Nasdaq stocks are mostly Microcaps. In the estimates of (5) and (6) that use 1-year (k= 12) versions of dMt−k,t and dBt−k,t, the ABM sample on average has 1,274 stocks during July 1963 to December 2006. There are on average 1,709 Microcap stocks, but they account for only about 3% of the market cap of NYSE–Amex–Nasdaq stocks. Because they are so numerous and their returns and explanatory variables in (5) and (6) have more dispersion, Microcaps are influential in regressions that use all stocks. Estimating the regressions on ABM stocks thus provides a more reliable view of the cross section of average returns for the stocks that account for the lion's share of stock market wealth. And we shall see that in one important respect, Microcaps do produce a different story.
We have also examined regressions that break the ABM sample between big stocks (market cap above the 50th NYSE percentile) and small stocks (market cap between the 20th and 50th NYSE percentiles). Skipping the details, we can report that the average slopes in (5) and (6) are similar for big and small stocks. Several readers have suggested that we split ABM stocks into even finer groups, such as the top four NYSE market cap quintiles. The problem here is small sample sizes that would destroy any power to identify differences in true slopes across size quintiles. Moreover, the fact that average regression slopes are similar for big and small stocks, but different for Micro stocks, suggests that the split of stocks into the ABM and Micro samples captures the main differences in results for size groups.
There are, of course, potentially interesting splits of the regression results on variables other than market cap. And there are many variables beyond those we break out of the B/M ratio that may have information about expected cashflows that can enhance estimates of expected returns. (Examining a broad range of candidates is the approach in Fama and French (2006).) The simple approach taken here, however, produces quite a full plate for this paper.
The initial tables show results only for the full July 1927 to December 2006 period (henceforth, 1927 to 2006). We have examined results for two periods that split in July 1963, the start date in many previous studies. There are differences in nuance between the results for the two subperiods, but except for net share issues, NSt−k,t, the average slopes for July 1927 to June 1963 are mostly within a standard error of the average slopes for July 1963 to December 2006. Thus, there seems little point in spending table space on the subperiods, at least in the discussion of results for variables other than NSt−k,t. We do, however, occasionally interject comments about the subperiod results to reinforce or shade inferences from the results for the full sample period. Finally, when we consider net share issues in Section V, the subperiod average slopes for NSt−k,t are the thrust of the story and they are examined in detail.