The Equity Premium. Eugene F. Fama and Kenneth R. French * Abstract

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1 First draft: March 2000 This draft: July 2000 Not for quotation Comments solicited The Equity Premium Eugene F. Fama and Kenneth R. French * Abstract We compare estimates of the equity premium for from realized returns and the Gordon constant dividend growth model. The two approaches produce similar estimates of the real equity premium for , about 4.0 percent per year. For , however, the Gordon estimate, 3.40 percent per year, is only about forty percent of the estimate from realized returns, 8.28 percent. We argue that the difference between the realized return for and the Gordon estimate of the expected return is largely due to unexpected capital gains, the result of a decline in discount rates. Our analysis thus suggests that expected future stock returns are low. Eugene F. Fama Graduate School of Business University of Chicago 1101 East 58 th Street 50 Memoriral Drive Chicago, IL Cambridge, MA (773) (617) Eugene.fama@gsb.uchicago.edu KFrench@mit.edu Kenneth R. French Sloan School of Management Massachusetts Institute of Technology * Graduate School of Business, University of Chicago (Fama) and Sloan School of Management, MIT (French). Discussions with John Cochrane, Kent Daniel, John Heaton, and Tuomo Vuolteenaho and the comments of seminar participants at Dartmouth College, Purdue University, and the University of Chicago have been helpful.

2 The equity premium (the difference between the expected return on the market portfolio of common stocks and the riskfree interest rate) is important in portfolio allocation decisions, estimates of the cost of capital, the debate about the advantages of investing Social Security funds in stocks, and many other applications. Financial economists typically use the average return on a broad portfolio of stocks to estimate the expected market return. The average real return for on the S&P index (a common proxy for the market portfolio) is 9.13 percent per year. The real return on six-month commercial paper (a proxy for the riskfree interest rate) is 3.24 percent. This large spread (5.89 percent) between the average equity return and the interest rate is the source of the so-called equity premium puzzle: stock returns seem too high given the observed volatility of consumption (Mehra and Prescott, 1985). We use a valuation model, Gordon s (1962) constant dividend growth model, to estimate the expected equity premium. At a minimum, the model produces alternative estimates of the premium that can be compared to realized returns. More aggressively, we argue that shocks to long-term expected dividend growth or to the discount rate used to price expected dividends have less effect on expected return estimates from the Gordon model. As a result, Gordon estimates of the expected market return are likely to be closer to the true expected return than estimates from realized returns. Our work is part of a growing literature that uses valuation models to estimate expected stock returns (Blanchard, 1993, Claus and Thomas, 1999, Gebhardt, Khorana, Moyer, and Patel, 1999, Lee, and Swaminathan, 1999, Vuolteenaho, 2000). We extend the earlier work in two ways. (i) Many of the existing papers use forecasts by security analysts to estimate expected cash flows. Analyst forecasts are available only for short sample periods (e.g., in Claus and Thomas). Our tests use realized dividends and earnings from 1872 to This 128-year period provides a long perspective on the behavior of competing expected return estimates. (ii) We also push harder on the valuation model. Specifically, we evaluate the expected return estimates from the Gordon model and realized returns relative to the predictions of valuation theory about the relations among the book-to-market ratio, the income return on investment, the earnings-price ratio, and the cost of equity capital.

3 The Gordon estimate of the expected real equity premium for is 3.64 percent per year. The estimate from realized real returns, 5.73 percent, is almost 60 percent higher. The difference between the two is largely due to the last fifty years. The expected real equity premium for from the Gordon model, 3.79 percent per year, is close to the estimate from realized returns, 4.10 percent. In contrast, the realized equity premium for , 8.28 percent per year, is 2.44 times the Gordon estimate, 3.40 percent. Three pieces of evidence suggest that the lower Gordon equity premium for is a better estimate of the expected premium. (i) The Gordon estimate has more than twice the precision of the average realized return. (ii) The lower Gordon equity premium for corresponds to a decline in return volatility; the Gordon equity premium has the same Sharpe ratio for and In contrast, the Sharpe ratio for the equity premium from realized returns more than doubles from to (iii) Most important, other fundamentals line up with the Gordon equity premium in the manner predicted by valuation theory, but book-to-market ratios, income returns on investment, and earnings-price ratios suggest that the equity premium from realized returns is too high. Many papers find that valuation models produce lower estimates of expected returns than realized returns. And many papers speculate that the high realized returns are due to unexpected capital gains, the result of some combination of a decline in discount rates and high expected future dividend and earnings growth. What we add is evidence on the likely source of the unexpected gains. Specifically, the behavior of dividends and earnings provides no basis for projecting high long-term future growth rates. Expected returns, however, do seem to fall during the last half century, from unusually high values in 1950 to unusually low values in The result is a cumulative percent capital gain during the period more than six times the cumulative percent growth in dividends. Some of this extra capital gain is probably expected, the result of reversion of the high beginning-of-period expected returns toward the unconditional mean. But most of the excess capital gain seems to be due to the unexpected decline of the expected return to ending values far below the long-term mean. 2

4 Our story starts with a discussion of the Gordon model (section I). The equity premium estimates are dissected in sections II to V. Section VI concludes. I. The Gordon Model The simple version of the Gordon model assumes a world of perfect certainty. Each year the firm has infinitely-lived investments that are a constant fraction k of earnings. The firm s capital structure is all equity, and it finances all investment with retained earnings. Thus, its earnings retention rate is k, and its dividend payout ratio is 1 k. The income return on investment (earnings per dollar of assets) is R *, in perpetuity, and earnings and dividends grow at the constant rate G = kr *. These assumptions imply that the value of the firm at time t (the market value of the stream of future dividends) is, (1) P t = D t+1 /(R G), where D t+1 is the dividend at t+1, and R is the discount rate for dividends. Thus, R (the cost of capital and the required return on stock) is, (2) R = D t+1 /P t + G. The model makes predictions about the relations among the earnings-price ratio (Y t+1 /P t ), the income return on investment (R * = Y t+1 /B t ), the discount rate (R), and the book-to-market ratio (B t /P t ). The firm s current capital stock, B t, produces a constant stream of earnings, Y t+1 each year. The firm s value, P t, exceeds the value of this stream, Y t+1 /R, if and only if its future investments have a positive net present value, that is, if the income return on investment, R *, exceeds the discount rate, R, (3) P t > Y t+1 /R iff R * > R. Equivalently, the earnings-price ratio is less than R if R * is greater than R, (4) Y t+1 /P t < R iff R * > R. Similarly, the firm is worth more than the cost of its current capital stock, B t, if the return on investment exceeds the required return, (5) P t > B t iff R * > R. 3

5 Finally, the earnings-price ratio, Y t+1 /P t, is less than the income return on investment, Y t+1 /B t, if P t is greater than B t, which is true when R * is greater than R. Thus, using (4), (6) Y t+1 /P t < R < Y t+1 /B t iff R * > R. The simple relations summarized in (4) to (6) provide our framework for judging the credibility of estimates of expected stock returns from the Gordon model versus those from realized returns. We use the Gordon model to estimate the long-term or unconditional expected return on the market portfolio of U.S. equities. Specifically, applying (2) each year, we calculate the Gordon return as the dividend yield plus the rate of growth of dividends, (7) RG t+1 = D t+1 /P t + G t+1. The Gordon estimate of the expected equity return is then the average annual dividend yield during the estimation period, plus the average annual growth rate of dividends, (8) A(RG t+1 ) = A(D t+1 /P t ) + A(G t+1 ). A major advantage of (8) is that its expected return estimates are likely to be less sensitive than average realized returns to unexpected changes in the long-term dividend growth rate and the discount rate that prices expected dividends. For example, our data suggest that R G in (1) is about 4.0 percent. Suppose there is an unexpected 1.0 percent permanent increase in G, so R G drops to 3.0 percent. The shock produces an unexpected 33.3 percent capital gain. If the average realized equity return for a 33-year period is used to estimate the expected return, the 1.0 percent shock to G adds a positive 1.0 percent measurement error to the expected return estimate. In contrast, if the Gordon model is used to estimate the expected return, and if the market is quick to recognize the shock to G, the shock produces a 1.0 percent estimation error in the year it occurs. But it increases the 33-year estimate of the expected return from the Gordon model by only percent. The Gordon model provides a simple intuition for equation (8), but (8) produces approximately unbiased estimates of the long-term expected equity return under more general assumptions. The key condition is that all relevant stochastic processes (dividend yields, the dividend growth rate, and the stock return) are stationary. To see the point, express the realized return for year t+1, R t+1, as 4

6 Dt+ 1 Pt+ 1 Dt+ 1 D t/pt Dt+ 1 (9) 1+ Rt+ 1 = + = +. P P P D /P D t t t Since the dividend yield D t /P t is highly autocorrelated, the ratio of dividend yields in (9) is close to 1.0, and we can approximate the expression after the second equality as, (10) 1 + R t+1 D t+1 /P t + D t /P t D t+1 /P t+1 + D t+1 /D t. If the realized return, R t+1, the dividend growth rate, G t+1 = D t+1 /D t 1, and the two versions of the dividend yield (D t+1 /P t and D t /P t ) are stationary stochastic processes, equation (10) implies that the unconditional expected value of the realized return from (9) is approximately equal to the unconditional expected value of the Gordon return from (7). Thus, given the relevant stationarity conditions and a long estimation period, the Gordon estimate (8) provides an approximately unbiased estimate of the long-term (unconditional) expected stock return. Without showing the details (on which there is much existing evidence), we can report that for the time periods examined below, the autocorrelations of returns, dividend yields, and dividend growth rates are consistent with stationarity. The first three autocorrelations of annual returns on our proxy for the market portfolio (the S&P 500 and its antecedents) are random in sign and close to zero (less than 0.26 in magnitude), especially after the first lag. The same is true for annual dividend growth rates. The firstorder autocorrelations of annual dividend yields (D t+1 /P t and D t /P t ) are high (around 0.8), but the autocorrelations decay across longer lags. This is in line with previous evidence (e.g., Fama and French, 1988) that dividend yields are highly autocorrelated but slowly mean-reverting stationary processes. Given the relevant stationarity conditions, how close is E(RG), the expected value of the Gordon estimate (8), to the true long-term expected return, E(R)? We address this question in section V, after presenting the empirical results. For the moment, suffice it to say that bias is not an important factor in our inferences. Finally, it is appropriate to dwell a bit on what the Gordon expected return estimate (8) can and cannot do. There is much evidence that variation in the expected stock return generates autocorrelation in the dividend yield (e.g., Fama and French, 1988, or Lewellen, 2000). Variation in the expected dividend t+ 1 t+ 1 t 5

7 growth rate can also produce variation in the dividend yield. But if the stock return and the dividend growth rate in (7) are stationary (mean-reverting), the dividend yield is also stationary. Then, over long estimation periods, the averages of the sample dividend yields and the sample dividend growth rates provide good estimates of the long-term expected values, and the Gordon estimate (8) provides an approximately unbiased estimate of the long-term (unconditional) expected stock return. But if the expected stock return is time varying, (8) does not provide good estimates of near-term expected returns. To estimate near-term (conditional) expected returns, some other approach that explicitly models the return process is necessary. These comments return to haunt us toward the end of the paper, where, after focusing on the long-term, we dabble with estimates of near-term expected returns. II. Main Results Table 1 shows estimates of the S&P real equity premium and summary statistics for related variables for The deflator is the Producer Price Index until 1926 [from Shiller (1989)] and the Consumer Price Index thereafter (from Ibbotson Associates). The riskfree interest rate is the annual real return on six-month commercial paper, rolled over at mid-year. The riskfree rate and S&P earnings data are from Shiller (1989), updated by Vuolteenaho (2000) and us. We construct S&P book equity data (not available until 1925) from the book equity data in Davis, Fama, and French (2000), expanded to include NYSE utilities and transportation firms. The data on dividends, prices, and returns for are from Shiller (1989). Shiller s annual data on the level of the S&P (used to compute returns and other variables involving price) are averages of daily January values. The S&P dividend, price, and return data for are from Ibbotson Associates, and the returns for are true annual returns. One can derive the Gordon model in real or nominal terms. Since portfolio theory says the goal of investment is consumption, the real version of the model seems more relevant. There are, however, problems with the deflator used to estimate real variables. Prior to 1926 the deflator is a producer price index (largely raw materials), rather than a consumer price index. The price data for the early years of the 6

8 sample period also have limited coverage. Thus, to provide perspective and a basis for comparison, Table 2 shows results for nominal values of the variables. A. The Equity Premium For a large part of the period up to about 1950 the Gordon model and realized returns produce similar estimates of the expected stock return. Thereafter the estimates diverge. To illustrate, Tables 1 and 2 show summary statistics for the equity premium and other key variables for (78 years) and (50 years). The 1949 break year is arbitrary; moving it backward or forward several years has little effect. For more detail, Tables 3 and 4 show results for each decade, from the 1870 s to the 1990 s. For the earlier period, there is not much reason to favor the estimates of the expected stock return from the Gordon model over realized average returns. Precision is not a big issue; the standard errors of the estimates from the two approaches are similar, the result of similar standard deviations of annual dividend growth rates and returns. More important, the Gordon model and realized returns provide similar estimates of the expected stock return. The Gordon estimate of the expected real return for is 7.79 percent per year; the average realized real return is only slightly higher, 8.10 percent (Table 1). Given similar estimates of the expected return, the two approaches produce similar real equity premiums, 3.79 percent (Gordon) and 4.10 percent (realized returns). The estimates of expected nominal returns for tell a similar story (Table 2). The competition between the Gordon model and realized returns becomes more interesting after The Gordon estimate of the expected real stock return for , 5.45 percent, is about half the realized average return, percent. And the Gordon estimate of the real equity premium, 3.40 percent, is only about 40 percent of the estimate from realized returns, 8.28 percent (Table 1). Not surprisingly, the two approaches produce similarly disparate estimates of the expected nominal return and equity premium for (Table 2). 7

9 Which estimates of the expected stock return are closer to the true expected values? We think the Gordon model is better, for three reasons. (i) The Gordon estimates of real and nominal expected stock returns are about 2.5 times as precise. For example, the standard error of the mean of the annual Gordon real returns from (7) is 0.87 percent, versus 2.38 percent for the annual average realized real return. (ii) Table 1 shows Sharpe ratios for the real equity premiums from the Gordon model and realized returns. (Only the average premium in the numerator of the Sharpe ratio differs between the two approaches. The denominator for both is the standard deviation of the annual realized real return.) The Sharpe ratio for the Gordon real equity premium, 0.20, is similar to that produced by realized real returns, More striking, the Sharpe ratio for the Gordon equity premium, 0.20, is the same (up to two decimals) as that for Thus, the slight decline in the Gordon estimate of the expected equity premium, from 3.79 percent in to 3.40 percent in , matches a decline in return volatility (Table 1). The nearly identical Sharpe ratios for the and Gordon equity premiums suggest that aggregate risk aversion is similar in the two periods. In contrast, realized returns produce a large increase in the Sharpe ratio for Although return volatility falls a bit from to , the equity premium estimate from realized real returns jumps from 4.10 percent to 8.28 percent. As a result, the Sharpe ratio for the equity premium from realized returns more than doubles, from 0.22 for to 0.49 for It seems implausible that risk aversion increases so much from the earlier to the later period. The Sharpe ratios for nominal returns (Table 2) repeat the story. (iii) Most important, the behavior of fundamentals also favors the Gordon model. We have book equity data for , which allows us to estimate book-to-market ratios and income returns on investment. The average book-to-market ratio for is In the Gordon model, book-tomarket ratios less than 1.0 say that the income return on investment (R * = Y t+1 /B t ) is greater than the cost of equity capital (R), and the earnings-price ratio (Y t+1 /P t ) is less than the cost of equity. 8

10 These predictions are confirmed when the Gordon model is used to estimate the cost of equity (the expected stock return). The Gordon estimate of the expected real stock return for (5.45 percent, Table 1), is less than the average real income return on investment, Y t+1 /B t (7.60 percent), and greater than the average real earnings-price ratio, Y t+1 /P t (3.82 percent). Similarly, the Gordon estimate of the expected nominal stock return for (9.69 percent, Table 2), is less than the nominal income return on investment, y t+1 /b t (11.88 percent), and greater than the nominal earnings-price ratio, y t+1 /p t (7.98 percent). In sharp contrast, realized returns produce an estimate of the expected real stock return (10.33 percent per year, Table 1), that is much higher than the real income return on investment (7.60 percent) and the real earnings-price ratio (3.82 percent). The realized average nominal return for (14.56 percent per year, Table 2) is also larger than the nominal income return on investment (11.88 percent) and the nominal earnings-price ratio (7.98 percent). Estimates of the expected stock return that exceed the income return on investment are difficult to reconcile with an average book-to-market ratio much less than 1.0 and an average earnings-price ratio less than the income return on investment. All this evidence suggests that the realized average stock return for is high relative to the ex ante expected value. Additional support for this conclusion follows. B. Unexpected Capital Gains The Gordon estimate of the unconditional expected stock return is the average dividend yield plus the average annual dividend growth rate. The estimate from realized returns is the average dividend yield plus the average annual rate of capital gain. The two estimates thus differ by the spread between the average rate of capital gain and the average dividend growth rate. If, as we argue, the gap between the realized return and Gordon estimates arises because the realized average return is higher than the expected return, then the stock price grew by more than expected during the period. Our goal is to explain this apparently unexpected capital gain. 9

11 The Gordon price equation (1) suggests three possible sources of unexpected capital gain. (i) The level of the end-of-sample real dividend, D 1999, is higher than was expected in 1950; equivalently, the growth rate of dividends for is unexpectedly high. (ii) The discount rate, R, is unexpectedly low at the end of the sample period. (iii) The expected future (post-1999) growth rate of dividends, G, is unexpectedly high. We argue that dividend growth for does exceed its 1950 expected value. But this unexpected dividend growth produces similar positive measurement error in the realized average return and the Gordon estimate of the expected return. Thus, the unexpected part of the spread between the average return and the Gordon expected return must be explained by a high end-of-sample G and/or a low R. We argue that most of the action comes from R. If the prosperity of the U.S. over the last fifty years was not fully anticipated in 1950, dividend growth for exceeds its 1950 expectation. To isolate the effect of such unexpected dividend growth from the effects of changes in R and G, suppose R and G (and thus R G) are the same at the beginning and end of the period. The Gordon equation (2) implies that the dividend yield is also unchanged, so the total percent growth in dividends during the period is the same as the total percent growth in the stock price. But the realized average return and the Gordon estimate of the expected return only differ by the spread between the average rate of capital gain and the average dividend growth rate. In itself, then, unexpected dividend growth for creates similar positive measurement error in both estimates of the expected return. Unexpected dividend growth during the period thus cannot explain why the average return is so much higher than the Gordon expected return. It is worth dwelling on this point. There is probably survivor bias in the U.S. average stock return for , as well as for During the period, it was not a foregone conclusion that the U.S. equity market would survive several financial panics, the Great Depression, two world wars, and the cold war. The average return for a market that emerges as the winner of many potentially cataclysmic challenges is likely to be higher than the ex ante expected return (Brown, Goetzmann, and Ross, 1995). But if the positive bias shows up only as higher than expected dividend growth during the sample period (it does not affect future discount rates and expected dividend growth), then similar survivor 10

12 bias shows up in the Gordon estimate of the expected return. Thus, like average returns, Gordon expected return estimates are subject to market survivor bias a problem we do not resolve. Our more limited goal is to explain why the realized average return for seems to have positive measurement error (is unexpectedly high) relative to the Gordon estimate. This task falls to stories about the end-of-sample values of future discount rates (R) and/or expected dividend growth (G). Based on the historical behavior of dividends and earnings, there is little evidence that long-term expected dividend growth is high at the end of the sample period. The decadal average growth rates of annual nominal dividends are quite similar during the period (Table 4). If anything, the growth rate of real dividends declines during the period (Table 3). The average growth rates for the first two decades are higher than the average growth rates for the last three. The regressions in Table 5 are more formal evidence on the best forecast of the future (post-1999) dividend growth rate. Regressions are shown for forecasts one year ahead (the explanatory variables for year t+1 dividend growth are variables known at the end of year t) and two years ahead (the explanatory variables for year t+1 dividend growth are variables known at the end of year t-1). The regressions for suggest strong forecast power one year ahead. The slopes on the lagged payout ratio (D t /Y t ), the dividend yield (D t /P t ), and the stock return (R t ) are 1.95 to 3.45 standard errors from zero, so these variables have reliable marginal information about the following year s dividend growth (G t+1 ). The regressions to forecast dividend growth one year ahead capture an impressive 61 percent of the variance of nominal dividend growth, and 40 percent of the variance of real dividend growth. The signs of the regression slopes are also correct. If the payout ratio is mean reverting, G t+1 should be negatively related to the deviation of D t /Y t from its mean. If the stock price P t has information about future dividend growth, G t+1 should be positively related to R t, but the relation between G t+1 and D t /P t should be negative. The signs of the slopes confirm these predictions. The G of the Gordon model is, however, the long-term dividend growth rate. The regressions say that even in the period, power to forecast dividend growth does not extend much beyond a year. When the explanatory variables for year t+1 dividend growth are limited to variables known at the end of 11

13 year t-1, the regression R 2 falls from 0.40 to 0.07 for real dividend growth, and from 0.61 to 0.07 for nominal dividend growth. And without showing the details, we can report that extending the forecast horizon from two to three years causes all hint of forecast power to disappear. Thus, for , the best forecast of dividend growth more than a year or two ahead is the historical average growth rate. Even the short-term forecast power of the dividend regressions evaporates in the period. The payout ratio D t /Y t has some forecast power (t = -1.99) for nominal dividend growth one year ahead, and the lagged real return R t has some information (t = 2.07) about real dividend growth. But the regressions pick up only two percent of the variance of real and nominal dividend growth one year ahead. And forecast power does not improve for longer forecast horizons. Our evidence that dividend growth rates are essentially unpredictable during the last fifty years is consistent with the results in Campbell (1991) and Cochrane (1991a, 1991b). If near-term dividend growth and long-term dividend growth are unpredictable, historical average growth rates are optimal forecasts of future growth. Real earnings growth rates also suggest that long-term expected dividend growth is not unusually high in Consistent with the assumption of the Gordon model, the dividend payout ratio is roughly constant during the period. The average payout ratio for , 0.55 (Table 3), is not a lot higher than the average for , Figure 1 confirms that the payout ratio wanders about during the period, but shows no clear tendency to decline. If the long-term payout ratio is constant, the average growth rate of real earnings provides an alternative estimate of the long-term growth rate of real dividends. There is no apparent trend in real earnings growth rates during the period. The most recent decade, , indeed produces the highest real growth rate of the period, 5.91 percent per year. But earnings growth is volatile. The standard errors of the ten-year average growth rates of the period vary around 5.0 percent. It is thus not surprising that the decade immediately preceding produces the lowest average real earnings growth rate of the period, 0.46 percent per year. And the average growth rate of real earnings for the last 20 years of the period, 3.19 percent per year, is similar to that for the first 30, 2.95 percent. 12

14 The regressions in Table 6 are formal evidence on the predictability of real and nominal earnings growth. Unlike dividends, earnings growth is somewhat predictable one and two years ahead in both the and periods. In the regressions to forecast earnings growth one year ahead, the slopes on the lagged income return, Y t /B t-1, are negative, and the slope in the regression that uses real versions of the variables is standard errors from zero. In the one-year forecasts, the slopes on the last year s dividend payout ratio, D t /Y t, are positive and more than 2.8 standard errors from zero. The stock return, R t, also predicts next year s earnings growth rate, with t-statistics for that exceed 3.9. The negative Y t /B t-1 slopes suggest some predictability of earnings growth due to the mean reversion of profitability. [Regressions (not shown) of the change in Y t /B t-1 on the lagged level indeed suggest that Y t /B t-1 is mean-reverting.] The positive D t /Y t slopes say that next year s earnings growth is likely to be high when current earnings are low relative to current dividends. Mean reversion of profitability is again a possible explanation. The regressions show some power to forecast earnings growth two years ahead, but the regressions are difficult to interpret. In particular, all forecast power comes from the two-year lagged income return on investment. But the slopes on this variable have opposite signs, positive in the regression for real earnings growth and negative for nominal growth. Without showing the details, we can also report that forecast power in the period does not extend beyond two years. Since we are concerned with post-1999 expected earnings growth, the regressions for are more pertinent. Near-term profitability is again moderately predictable in the period. But predictability seems largely due to transitory variation in earnings that is irrelevant for forecasting longterm earnings. In the regressions to forecast earnings growth one year ahead, the slopes on the first lag of the stock return are positive, but the slopes on the second lag are negative and about the same magnitude. Thus, the prediction of next year s earnings growth from this year s stock return is just about reversed the following year. In the one-year forecast regressions, other than lagged returns, the only variable with reliable forecast power and then only in the regression for real earnings growth is 13

15 the third lag of the real earnings growth rate. But the slope is negative, so it predicts that the strong real earnings growth of recent years is soon to be reversed. The regressions to forecast earnings growth two years ahead produce a similar message. The only variables with reliable forecast power are the second lag of the stock return and (for real earnings growth) the third lag of the growth rate of earnings. But the slopes on these variables are negative, so again the 1999 prediction is that the strong earnings growth of recent years is soon to be reversed. And again, regressions (not shown) confirm that forecast power for the period does not extend beyond two years. Thus, beyond two years, the best forecast of earnings growth is the historical average growth rate. To summarize, the behavior of dividends during the period suggests that future growth rates are largely unpredictable in both the near-term and the long-term, so the historical mean of past growth rates is a nearly optimal forecast of expected future growth rates. Earnings growth for is somewhat predictable one and two years ahead, but the end-of-sample message is that the recent high growth is likely to revert quickly to the historical mean. And it is worth noting that the market survivor bias argument of Brown, Goetzmann, and Ross (1995) suggests that past average growth rates of dividends and earnings are, if anything, upward biased estimates of future growth rates. In short, we cannot find any evidence to support a forecast of strong long-term dividend growth at the end of our sample period. Now comes the punch line. If the end-of-sample expected future growth rate of dividends is not unexpectedly high, the task of explaining why the capital gain for is unexpectedly high relative to dividend growth falls to the end-of-sample value of the discount rate R. The evidence indeed points to a decline in the real discount rate. The Gordon estimate of real R falls from 7.79 percent in and 8.52 percent in to 5.45 percent in (Table 1). Table 3 shows that the estimate of the expected real return also declines within the period, from 8.07 percent in the first decade ( ) to 4.26 percent in the last ( ). We can give perspective on the likely magnitude of the unexpected capital gain due to the decline in the discount rate. The Gordon model says that the dividend yield, D t+1 /P t, equals R G. If the

16 expected future dividend growth rate, G, is not high (or low) relative to the value expected in 1950, any change in the dividend yield during the period can be attributed to R. The yield, D t+1 /P t, falls from 8.29 in 1950 to 1.32 in Thus, the growth in the stock price over the fifty-year period, P 1998 /P 1949, is 8.29/1.32 = 6.28 times the growth in dividends, D 1999 /D Equivalently, given D 1999, the stock price at the end of 1998 is 6.28 times what it would be if the dividend yield, D t+1 /P t = R G, had remained at its 1950 value. Part of the large difference between the capital gain and the growth in dividends was, however, probably anticipated in Figure 2 shows that the dividend yield D t+1 /P t for 1950, 8.29 percent, is unusually high. The average yield for is 5.30 percent. If the dividend yield is stationary (mean-reverting), then the expectation in 1950 of the yield in 1999 is close to the unconditional expected value, say 5.30 percent. The actual dividend yield for 1999 is 1.32 percent. Thus, given the 1999 dividend, the 1998 stock price is 5.30/1.32 = 4.02 times what it would be if the dividend yield for 1999 were the 5.30 percent projected in Roughly speaking, this unexpected capital gain adds about 2.8 percent per year to the compound return for the fifty-year period. In a nutshell, the low end-of-sample dividend yield is due to the large difference between the cumulative capital gain return and the cumulative dividend growth rate for Some of the decline in the yield was probably expected in 1950, but most of it is unexpected. Given the evidence that dividend growth is unpredictable, rational forecasts of long-term dividend growth are not unusually high in We conclude that the unexpected capital gains for are largely due to a decline in the discount rate. In other words, the low end-of-sample dividend yield implies low expected future stock returns. This conclusion does not hinge on the validity of the Gordon model. All valuation models say the dividend yield is driven by expected future returns (discount rates) and expectations about future dividends. If one accepts our inference that expected future dividend growth is not unusually high, all valuation models agree that the low D 1999 /P 1998 implies low expected future returns. 15

17 III. Looking Ahead What, then, does the Gordon model currently say about the equity premium? The real dividend yield for 1999 is 1.32 percent. If we use the average growth rate of real dividends for , 1.61 percent per year, to estimate the expected future growth rate, the expected real stock return is 2.93 percent. The riskfree real interest rate for 1999 is 2.24 percent, so the estimate of the expected real equity premium is 0.69 percent. If we replace the dividend growth rate with the higher average growth rate for , 2.15 percent per year, the expected real stock return rises to 3.56 percent, and the expected equity premium is 1.32 percent. These forward-looking estimates of the real equity premium are lower than the Gordon estimate for , 3.40 percent per year, but they are in the neighborhood of the estimates for and , 1.27 and 1.71 percent (Table 3). What is required to push the end-of-sample expected real equity premium up to the 3.40 percent average value for ? With a 2.24 percent real interest rate and a 1.32 percent real dividend yield, the expected growth rate of real dividends, for the indefinite future, has to be 4.32 percent per year. This is 2.7 times and about 3.35 standard errors above the average growth rate for , 1.61 percent per year (Table 1). In short, extreme optimism about long-term dividend growth is necessary to project an expected future equity premium near historical averages. Such optimism is difficult to justify, given the evidence that historical average growth rates of dividends and earnings seem to be the best forecasts of long-term growth rates. Where might the low end-of-sample Gordon estimate of the expected real equity premium go wrong? We consider three possibilities. (i) The growth rate of real dividends for 1999 is slightly negative, percent. This suggests that the actual dividend for 1999 undershoots the value expected at the end of 1998, so the dividend yield for 1999, D 1999 /P 1998, is lower than expected. But reasonable estimates of the expected 1999 dividend have little effect on the dividend yield, and thus on the Gordon estimate of the expected stock return. For example, suppose the expected real dividend for 1999 built into P 1998 assumes that the dividend growth rate 16

18 is equal to the average for , 1.61 percent per year. The expected dividend yield, 1.34, is then trivially different from the observed, (ii) Share repurchases surge after 1983 (Bagwell and Shoven, 1989, Dunsby, 1995). Repurchases are, however, irrelevant for our purposes. Simply put, the average stock return and the Gordon estimate of the expected return apply to shares that remain outstanding. A problem for the Gordon estimate of the expected return does arise if a repurchasing firm will never pay a cash dividend (or will never increase its cash dividend) but intends instead to eventually liquidate or be acquired by another firm. But this is not a problem in applying the Gordon model to the market portfolio. Confirming Fama and French (2000), Table 3 shows that the average cash dividend payout ratio for , which includes the period of strong repurchases, is a bit higher (0.49) than the payout ratio for the immediately preceding period (0.46), and a bit lower than payout ratio for (0.55). (iii) The final explanation for the low end-of-sample Gordon estimate of the expected stock return is that the average growth rate of real dividends for (used in the estimate) understates the expected future growth rate. The average growth rate of earnings for , 3.04 percent per year, is higher than the average dividend growth rate, 1.61 percent. If we use the average earnings growth rate as a proxy for the expected growth rate of dividends, the end-of-sample Gordon estimate of the expected real stock return is 4.36 percent and the expected equity premium rises to a still modest 2.14 percent. There is, however, a problem in using the average (simple) earnings growth rate to estimate the expected (simple) dividend growth rate. The dividend payout ratio does not decline much during the period. Yet the average simple growth rate of real earnings during the period, 3.04 percent per year, is almost double the growth rate of real dividends, 1.61 percent (Table 1). This is a variance effect. Earnings growth for is 2.4 times more volatile than dividend growth. Thus, although earnings and dividends have similar compound growth rates for , the average simple growth rate of earnings is much higher than the average simple growth rate of dividends. As a result, the average simple growth rate of earnings is an upward biased estimate of the expected simple dividend growth rate. 17

19 It is, of course, possible that historical average dividend growth rates are below expected future growth simply because of statistical estimation error. The standard error of the average growth rate is 0.81 percent. Adding two standard errors to the end-of-sample estimate of the Gordon equity premium that uses the average dividend growth rate raises the forward-looking equity premium from 0.69 percent to 2.31 percent. But subtracting two standard errors of the average dividend growth rate from the end-of-sample Gordon estimate of the equity premium pushes it below zero. Finally, at least for near-term forecasts, there is also a potential source of upward bias in our low Gordon estimates of the end-of-sample equity premium. We estimate the end-of-sample expected stock return as the 1999 dividend yield plus various estimates of the long-term expected dividend growth rate. Models like those in Campbell (1991) and Blanchard (1993) can be used to show that our 1999 expected returns are weighted averages of all future expected returns (with exponentially declining weights). Thus, the 1999 expected return estimates are only estimates of near-term expected returns if the expected stock return has fallen to a permanently lower level, expected to remain constant for the indefinite future. Many papers suggest that some of the decline in the equity premium is permanent, the result of (i) wider equity market participation by individuals and institutions and (ii) lower costs of obtaining diversified equity portfolios from mutual funds (e.g., Diamond, 1999, Heaton and Lucas, 1999). But there is also evidence that the dividend yield is slowly mean reverting, the result of variation in business conditions that generates a slowly mean-reverting expected stock return (e.g., Fama and French, 1989). If the expected stock return is mean reverting, it varies more than one-for-one with the dividend yield, and our low 1999 expected return estimates actually overstate near-term expected returns. The overestimate is larger the faster the rate of mean reversion. (Saying more than this requires a model for the conditional expected return, and specific parameter estimates a task fraught with measurement error.) Put a bit differently, the decline in the dividend yield in the later years of our sample period produces capital gains that exceed dividend growth. But if the dividend yield is slowly mean reverting, we face a protracted period where expected dividend growth exceeds expected capital gains. 18

20 IV. Robustness The main advantage of the S&P data used in Tables 1 to 4 is a long 128-year perspective on the behavior of the equity premium. The downside is limited coverage. The S&P Index includes only 90 stocks before Even after 1957, the 500 firms in the index are predominantly large and successful. To confirm that our results are not specific to this set of firms, we repeat our tests using the CRSP valueweight index of NYSE, AMEX, and NASDAQ stocks. The differences between the S&P and CRSP results are trivial. CRSP data are available from 1926 to The average nominal return on the CRSP index is 25 basis points higher than the average S&P return for , and 18 basis points higher for The Gordon estimates of the expected return are also quite similar for the two indices. The Gordon-CRSP estimates of the expected nominal market return, 9.29 percent for and 9.46 percent for , are only slightly lower than the Gordon-S&P estimates, 9.55 percent and 9.61 percent. In short, the CRSP value-weight index simply confirms the evidence about expected returns from the S&P data. What one takes to be the riskfree rate has a more substantial effect on estimates of the equity premium. Treasury bill rates are a common choice, but Treasury bills are not available for the entire period. To maintain comparability across periods, Tables 1 to 4 use the six-month commercial paper rate (reinvested at mid-year) as the riskfree rate. Substituting the one-month Treasury bill rate (reinvested monthly) for the commercial paper rate causes the estimate of the equity premium for to rise by 1.00 percent. To estimate the equity premium in annual returns, we would prefer a riskfree rate with a maturity of one year. In this respect, the six-month commercial paper rate is better than the one-month bill rate. (Long time series on one-year interest rates are not available.) Moreover, since defaults on high-grade commercial paper are rare, some of the spread between commercial paper rates and Treasury bill rates is probably due to the role of Treasury bills as secondary reserve assets for banks. If this special demand distorts Treasury bill rates relative to other open market interest rates, commercial paper rates are again preferable for estimating the equity premium. 19

21 V. Estimating Expected Returns: Some Issues We have left until last two questions about the inferences from our estimates of expected returns. (i) For a long sample period, the Gordon estimate of the expected annual return from (8) is near unbiased. How good is the approximation? (ii) The return concept in discrete time asset pricing models is a one-period simple return, and our empirical work focuses on the one-year return. But asset pricing theory says nothing about the relevant return horizon, and it is likely that most investors are concerned with long-term returns. What are the advantages and disadvantages of the Gordon model for estimating long-term returns? This section addresses these questions. There is downward bias in the Gordon estimate of the (unconditional) expected annual simple return the result of a variance effect. The expected value of the Gordon estimate is the expected value of the dividend yield plus the expected value of the annual simple dividend growth rate. The expected annual simple return is the expected value of the dividend yield plus the expected annual simple rate of capital gain. If the dividend yield is stationary, the compound annual rate of capital gain converges to the compound annual dividend growth rate as the sample period increases. But because the annual dividend growth rate is less volatile than the annual rate of capital gain, the expected simple dividend growth rate must be less than the expected simple capital gain rate. The standard deviation of the annual simple real rate of capital gain for is 2.92 times the standard deviation of the annual dividend growth rate (Table 1). The resulting downward bias of the average dividend growth rate as an estimate of the expected annual simple rate of capital gain is roughly 1.21 percent per year (half the difference between the variances of the two growth rates). The bias is smaller for (0.56 percent per year) since the volatility of the dividend growth rate for is closer to the volatility of the rate of capital gain. Despite its bias, the Gordon estimate, A(RG), can be a better measure of the expected annual simple return, E(R), for than the average realized return, A(R). First, because annual dividend growth rates are so much less volatile than annual rates of capital, the estimate of E(R) provided by A(RG) 20

22 may have lower mean squared error than A(R). Second, both survival bias and the large decline in the dividend yield from 1950 to 1999 suggest that the average annual return for the period over-estimates the expected return. Survival bias also affects the Gordon estimate of the expected annual return, but this bias offsets the downward volatility bias of A(RG) as an estimate of E(R). The Gordon estimate of the expected annual simple return is clearly superior to the average annual simple return if we are concerned with the long-term expected wealth generated by the market portfolio. Like Campbell (1991) and Cochrane (1991a, 1991b), we find that the dividend growth rate for is essentially unpredictable. If the annual dividend growth rate is serially uncorrelated, the expected value of the compounded dividend growth rate is the compounded expected simple growth rate, E[Π t T =1(1+G t )] = [1+E(G)] T. And if the dividend yield is stationary, for long horizons the expected compounded dividend growth rate equals the expected compounded rate of capital gain, E[Π t T =1(1+G t )] = E[Π t T =1(1+C t )]. Thus, when the horizon T is long, compounding the true expected annual simple return from the Gordon model produces an unbiased estimate of the unconditional expected long-term return, [1+E(RG)] T = E[Π t T =1(1+R t )]. In contrast, if the dividend growth rate is unpredictable and the dividend yield is stationary, part of the higher volatility of annual rates of capital gain is transitory, the result of a mean-reverting expected annual return. Thus, compounding even the true unconditional expected annual simple return, E(R), yields an upward biased measure of the expected compound return, [1+E(R)] T > E[Π t T =1(1+R t )]. Moreover, we only have estimates, A(R) and A(RG), of E(R) and E(RG). Compounding average values, rather than true expected values, adds a systematic upward bias to estimates of expected long-term returns (Blume, 1974). And the bias increases with the imprecision of the averages. This is another reason to favor the more precise Gordon A(RG) when estimating long-term expected returns. 21