Miuch smaller than the stock price, so a given proportional change in the dividend has a much smaller effect on the return than the same proportional hangc in the price.

Apjrroximation Accuracy

the approximation (7.1.19) holds exactly when the log dividend-price ratio is constant, for then d,+ i and p/+i move together one-for-one and equation 7.1.19) is equivalent to equation (7.1.17). Like any other Taylor expansion, the approximation (7.1.19) will be accurate provided that the variation in tjhe log dividend-price ratio is not too great. One can get a sense for the accuracy of the approximation by comparing the exact return (7.1.17) with the approximate return (7.1.19) in actual data. Using monthly nominal dividends and prices on the CRSP value-weighted stock index over the period 1926:1 to 1994:12, for example, the exact and approximate returns have means of 0.78% and 0.72% per month, standard deviations of 5.55% and 5.56% per month, and a correlation of 0.99991. The approximation с rror-the difference between the approximate and the exact return-has ;< mean of-0.06%, a standard deviation of 0.08%, and a correlation of 0.08 With the exact return. Using annual nominal dividends and prices on the CRSP value-weighted stock index over the period 1926 to 1994, the exact and approximate returns have means of 9.20% and 9.03% per year, standard deviations of 19.29% and 19.42% per year, and a correlation of0.99993. The approximation error has a mean of -0.17%, a standard deviation of 0.20%, and a correlation of 0.51 with the exact return. Thus the approximation misstates the average stock return but captures the dynamics of stock returns well, especially when it is applied to monthly data.7

Implications for Prices

Equation (7.1.19) is a linear difference equation for the log slock price, analogous to the linear difference equation for the level of the stock price lhat wc obtained in (7.1.4) under the assumption of constant expected returns. Solving forward and imposing the condition that

lim ppl+i = 0, (7.1.20)

wc obtain

k 00

p, = -- + У(> [{\ - M*u-j- r,+\+j]. (7-1-21)

P ;=()

7On<- can also compare- cxacl anil approximate real rriiirns. The roircciinn tin iullalioti lias no important effects on the comparison. See Campbell and Shiller (IWHa) lor a more detailed evaluation of approximation accuracy at short and long hori/.ons.

Equation (7.1.21) is a dynamic accounting identity; it has been obtained merely by approximating an identity ami solving forward subject to a terminal condition. The terminal condition (7.1.20) rules out rational bubbles that would cause (he log stock price to grow exponentially forever at rate I jp or faster. Equation (7.1.21) shows lhat if the slock price is high today, (hen there must be some combination of high dividends and low slock returns in the future.

Equation (7.1.21) holds ex post, but it also holds ex ante. Taking expectations of (7.1.21), and noting that = \:.t\p,\ because is known al time (, we obtain

l>i = i-+ К,

J=°

(7.1.22)

I his should be thought of as a consistency condition for expectations, analogous to the statement that the expectations of random variables X and Y should add up to the expectation of the sum X+Y. If the slock price is high today, then investors must be expecting some combination of high future dividends and low future returns. Equation (7.1.22) is a dynamic generalization of the Gordon formula for a stock price with constant required returns and dividend growth. Campbell and Shiller (1988a,b) call (7.1.22)-and (7.1.24) below-the dynamic Gordon growth model or the dividend-ratio model.

Like the original Gordon growth model, the dynamic Gordon growth model says that slock prices arc high when dividends are expected to grow rapidly or when dividends are discounted at a low rate; but the clfect on the stock price of a high dividend growth rale (or a low discount rate) now depends on how long the dividend growth rate is expected to be high (or how long the discount rate is expected to be low), whereas in the original model these rates are assumed to be constant at their initial levels forever. One can use the definitions of p and k lo show that the dynamic Gordon growth model reduces lo the original Gordon growth model when dividend growth rates and discount rates arc constant.

For future convenience, we can simplify the notation in (7.1.22), rewriting il as

Ii = уГ + I1 ~ (7.1.23)

where /;, is the expected discounted value of (I - p) times Inline log dividends in (7.1.22) and p is the expected discounted value of future log stock returns. This parallels the notation we used for the constant-cxpcclcd-return case in Section 7.1.1.

Campbell and Shiller (I.IHHa) evaluate ihe accuracy olilie appiosiiiiaiiou in (7.1.21).

Equation (7.1.22) ran be rewritten in terms of the log dividend-price ralio rather than die log sloe к price:

+ F,

/1-1 + /1

(7.1.2-1)

The- log dividend-price ratio is high when dividends are expected to grow only slowly, or when stock returns are expected lo he high. This equation is useful when the dividend follows a loglincar unit-root process, so that log dividends and log prices are uonstalionary. In this case changes in log dividends are stationary, so from (7.1.24) the log dividend-price ralio is stationary provided thai the expected stock return is stationary. Thus log stock prices and dividends are cointegratcd, and the stationary linear combination of these variables involves no unknown parameters since it is just the log ratio. This simple structure makes the loglincar model easier to use in empirical work than the linear cointegrated model (7.1.13).

So far we have written asset prices as linear combinations of expected future dividends and returns. We can use the same approach to write asset returns as linear combinations of revisions in expected future dividends and returns (Campbell MIDI 1). Substituting (7.1.22) into (7.1.19), we obtain

)+1 - E, [ rH

£/>M .

L;=

к,

L/=i

1 + 1

.;=

(7.1.2Г))

This equation shows that unexpected slock returns must be associated with changes in .expectations of future dividends or real returns. An increase in expected future dividends is associated with a capilal gain loday, while an increase in expected future returns is associated with a capital loss today. The reason is that with a given dividend stream, higher future returns can only he generated by future price appreciation from a lower current price. For convenience, we can simplify the notation of (7.1.2Г>) to

= /mi

4,1.1 ( I ~ >h.H I.

(7.1.2b)

when- i is the unexpected slock relurn, /,; + i is the change in expectations of future dividends in (7.1.25), and / M. is the change in expectations of future returns.

7. /.-/ Prices and Returns in n Simple Example

The formulas developed in die previous see lion maybe- easier to understand in the- eeintexl of a simple- example. Later we- will argue that the example

is not only simple, but also empirically relevant. Suppose that the expected log stock return is a constant r plus an observable zero-mean variable x,:

E,[r,+ 1] = r + x,. (7.1.27)

We further assume that x, follows the first-order autoregressive (AR(1))

process

x,+, = 0*, + tJ,+i. -1 < Ф < 1. (7.1-28)

When the AR coefficient ф is close to one, we will say that the x, process is highly persistent. Equation (7.1.28) implies that the variance of x, and its innovation which we write as a* and cr respectively, are related by

а* = (1 -ф>*.

Under these assumptions, it is straightforward to show that

pn = Ei

X>;r,+i+Ji

+ -L . (7.1.29)

1 - p l-рф

Equation (7.1.29) gives the effect on the stock price of variation through time in the expected stock return. The equation shows that a change in the expected return has a greater effect on the stock price when the expected return is persistent: Since, p is close to one, a 1% increase in the expected return today reduces the stock price by about 2% if ф - 0.5, by about 4% if j ф = 0.75, and by about 10% if Ф = 0.9.

This example illustrates an important point. The variability of expected stock returns is measured by the standard deviation of x,. If this standard deviation is small, it is tempting to conclude that changing expected returns have little influence on stock prices, in other words, that variability in pr, is small. Equation (7.1.29) shows that this conclusion is too hasty: The standard deviation of pr, is the standard deviation of x, divided by (1 - рф), so if expected returns vary in a persistent fashion, p can be very variable even when x, itself is not. This point was stated by Summers (1986), and particularly forcefully by Shtller (1984):

Returns on speculative assets are nearly unforecastable; this fact is the basis of the most important argument in the oral tradition against a role for mass psychology in speculative markets. One form of this argument claims that because real returns are nearly unforecastable, the real price of stocks is close to the intrinsic value, that is, the present value with constant discount rate of optimally forecasted future real dividends. This argument ... is one of the most remarkable errors in the history of economic thought.

ln enir example the stock price can be written as the sum of two terms. The first term is the expected discounted value of future dividends, рц; this

is not quite a random walk for the reasons given in Section 7.1.1 above, but it is close to a random walk when the dividend stream is not too large or variable. The second term is a stationary AR(1) process, -j) . This two-component description of slock prices is often found in the literature (see Summers [1986], Fama and French [ 1988b], Potcrba and Summers [1988], and Jegadecsh [1991]).

The AR(1) example also yields a particularly simple formula for the one-period stock return r,+t. The general stock-return equation (7.1.25) simplifies because the innovation in expected future stock returns, t/r + i, is given by p£/+i/(l - рф). Thus wc have

r,+i = r+ x, + iu.t+i ~ (7.1.30)

1 - рф

To understand the implications of this expression, assume for simplicity that news about dividends and about future returns, tj.z+i and are uncorrelated.4 Then using the notation Var[ ,; + i ] = ald (so а1л represents the variance of news about all future dividends, not the variance of the current dividend), and using the fact thatcr?2 = (1 -ф1), we can calculate the variance of r,+i as

Var[r,+il =

1 -tV-2p0

+ (7.1.31)

where the approximate equality holds when ф <K p and p is close lo one. Persistence in the expected return process increases the variability of realized returns, for small but persistent changes in expected returns have large effects on prices and thus on realized returns.

Equations (7.1.28) and (7.1.30) can also be used to show that realized stock returns follow an ARMA(1,1) process and to calculate the autocorrelations of this process. There arc offsetting effects: The positive autocorrelations of expected returns in (7.1.28) appear in realized returns as well, but a positive innovation to future expected returns causes a contemporaneous capital loss, and this introduces negative autocorrelation into realized returns. In the ARMA(1,1) representation the AR coefficient is ihe positive persistence parameter ф, but the MA coefficient is negative Problem 7.3 explores these effects in detail, showing that the latter effect dominates provided that ф < p. Thus there is some presumption that changing expected returns create negative autocorrelations in realized returns.

Problem 7.3 also generalizes the example to allow for a nonzero covariance between dividend news and expected-rcturn news. Slock returns can

This might 1ж ilie case, for example, il expected returns arc determined by the volatility of the dividend growth process, and dividend volatility is driven by a ( .ARCH model of the type discussed in Chapter 12 so lhat shocks to volatility are uncorrelated witli shocks to the level of tlividends.

be positively aulocorrclatcd if dividend news and cxpcclcd-rclui n news have a sufficiently large positive covariance. The covariance between dividend news and expected-relurn news can also be chosen lo make stock returns serially uncorrelated. This case, in which .stock returns follow a serially uncorrelated white noise process while expected stock returns follow a persistent AR(1) process, illustrates the possibility that an asset market may be weak-form efficient (returns are unlorecaslable from the history of returns themselves) but notsemistrong-foi in efficient (returns are lorecastable from the information variable x,).

This possibility seems to be empirically relevant for the US stock market. The statistically insignificant long-horizon autocorrelations reported at the end of Chapter 2 imply thai there is only weak evidence for predictability of long-horizon stock returns given past stock returns; but in the next section we show that there is stronger evidence for predictability of long-horizon returns given other information variables.

7.2 Present-Value Relations and US Stock Price Behavior

We now use the identities discussed in the previous section to interpret recent empirical findings on the time-series behavior of US stock prices. Section 7.2.1 discusses empirical work that predicts slock returns over long horizons, using forecasting variables other than past returns themselves. We present illustrative empirical results when dividend-price ratios and interest rate variables are used to forecast stock returns. Section 7.2.2 relates long-horizon return behavior to price behavior, in particular stock price volatility. Section 7.2.3 shows how time-series models can be used to calculate the long-horizon implications of short-horizon asset market behavior.

7.2.1 Long-Horizon Regressions

Recently there has been much interest in regressions of returns, measured over various horizons, onto forecasting variables. Popular forecasting variables include ratios of price to dividends or earnings (see Campbell and Shiller [1988a,b], Fama and French [1988a], Hodrick [1992), and Shiller [1984]), and various interest rale measures such as the yield spread between long- and short-term rales, die quality yield spread between low- and high-grade corporate bonds or commercial paper, and measures of recent changes in the level of short rates (see Campbell [1987), Fama and French [1989], Hodrick [1992], and Keim and Siambaugh [1986)).

Here we concentrate on the dividend-price ratio, which in US data is the niosi successful forecasting variable for long-horizon returns, and on a short-term nominal inlerest-rate variable. We start with prices and dividends on