Return Predictability and Stock Market Crashes in a Simple Rational Expectations Model

Analyzing the annual S&P 500 real price and price-dividend data for the
time period 1871-2002 ¹⁴, one finds that the price-dividend ratio is reason-
ably approximated by AD = 7.27ADt'53 with an R² of almost 40 percent
where ADt = 12Dt is the annual and Dt the monthly dividend . This implies
for the numerical simulation based on monthly data St+h = 325.6D_t¹₊^.5_h³ .The
real interest rate is set to 2.5 percent p.a. which is consistent with the his-
torical average (see Brennan and Xia, 2002). Consistent with the historical
mean and volatility of real monthly dividend growth we choose σD = 0.037
and μD = 0.002. The initial dividend D₀ is set to 1 or 4. The asset price is
given by the present value of future dividends:

t+h

St = ^ exp(rf (t - s))E (DsΦt,sDt)               (7)

s=t

+ exp(-rf h)E (dp D+ hΦt,t+h∖Dt)

Using the stochastic discount factor polynomial, E(S_t+_hΦ_t,_t+_h∖D_t) is a weighted
average of means of power functions of St+h.SinceSt+h is a power func-
tion of D_t+h and D_t+h is lognormally distributed, E(S_t+hΦ_t,t+h∖Dt) can
be derived analytically. The same is true of E(DsΦt,s∖Dt) since Φt,s =
E(Φt,t+h∖Ds) by no-arbitrage. Hence the price St can be derived analyti-
cally as a function of the dividend Dt as shown in the appendix. Since the
stochastic discount factor Φt,t+h is assumed to be time-homogeneous, the
asset pricing function (7) is also time-homogeneous. The asset price is a
function of the dividend only.

We use simulation to derive the properties of the price process. In each
simulation run, we generate 240 observations of the dividend process. This
corresponds to 20 years of monthly data. Given a constant investment hori-
zon of h = 240 months, we obtain a sequence of 240 asset prices, derived from
equation (7). For every model specification we run 1000 simulations. The
parameters of every specification 1 to 3 are given in Table 1, approximating
aggregate RRA in Figures 1 to 3.

4.3 Simulation Results

First, consider Figure 4 illustrating the relationship between the asset price
and the concurrent dividend.

- insert Figure 4 here -

this paper support this view.

¹⁴Source: Shiller (http://www.econ.yale.edu/ shiller/data.htm)

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