RATIONAL EXPECTATIONS VERSUS PERFECT FORESIGHT ASSET PRICING.
Larry Fogelberg, TroyUniversity
ABSTRACT
This study demonstrates that, when the length of the supranormal growth period is not known with certainty, all rational expectations pricing models result in some degree of overpricing, when compared ex post facto to perfect foresight models. Further, this study demonstrates that such overpricing may become extreme near the end of the supranormal growth period and may ultimately result in a catastrophic price adjustment when the supranormal growth period finally comes to an end. This catastrophic drop in stock price is not necessarily the result any irrational “bubble.”
THE PERFECT FORESIGHT MODEL OF ASSET PRICING
To begin, we consider the perfect foresight model of stock pricing, where the length of the supranormal growth period T is known with absolute certainty. Consider the standard continuous time model of stock price:
(1)
Where k is the (instantaneous) required rate of return, r is the projected (instantaeous) rate of return on equity for the next T periods, Etis the earnings rate at time t, Dt is the dividend rate, and Pt is the price of the stock at time t, is the retention rate and (1) is the dividend payout ratio. Assuming that the return on equity returns to normal levels
(rk) at the end of the supranormal growth period, we can calculate the earnings, the dividend rate, and the future value of the stock as follows:
, (2)
, (3)
. (4)
Substituting these values into formula (1) and integrating yields the following closed form expression for stock price under conditions of perfect foresight,
(5)
Although this expression may appear somewhat unwieldy, it has the advantage of total generality, as it can accommodate all retention rate values between 0 and 1. Further, it simplifies nicely in the limit when we let the retention rate 0, when we let 1, or when we let return on equity rk. Note that
, (6)
, (7)
and , (7b)
so that all extreme values of and r produce intuitive and simple results. Of particular interest is the case were 1, since it is optimal for the firm to retain all earnings during a period of supranormal growth.
Assuming that the firm retains all earnings as it moves through the supranormal growth period, from 0 to T, the perfect foresight model predicts a time t price of
. (8)
A RATIONAL EXPECTATIONS MODEL OF ASSET PRICING
Next, we consider a model of rational expectations pricing, where T is not known with absolute certainty. For simplicity, we assume that the supranormal growth periodfollows a random stopping process with a constant stopping rate , such that the expected duration of the supranormal growth period is T. Under this formulation, the length of the supranormal growth period follows the cumulative distribution , where ; and this rational expectations pricing model produces a stock price Pt*, where
. (9)
Note that this integral achieves convergence only where
. (10)
TIME PATH OF STOCK PRICE
Next, we can compare the perfect foresight price Pt with the rational expectations price Pt* at various points in time during the supranormal growth period. For purposes of illustration, we present a simple numerical example where T = 10 (or = .10), r = .15 and k = .10. Under perfect foresight pricing, the initial P/E ratio is 16.4. From that point forward, the stock generates returns of exactly 10 percent per period during the entire supranormal growth period.
=.10 (11)
Earnings growth of 15 percent is offset by an annual decline of 5 percent in the P/E ratio, resulting in a net return of exactly 10 percent per period. Of course, this assumes that the investor is able to continually update his estimate of the time remaining in the supranormal growth period, and that he is able to project that only (10 – t) periods of supranormal growth remain at each time t.
Next, we consider the rational expectations model, where = .10. Under this set of assumptions, the intitial pricing of the stock is 20 times earnings, as opposed to 16.4 times earnings.
As we move through the supranormal growth period, the pricing disparity between Ptand Pt* gets larger. Under our rational expectations model with a constant stopping rate, the investor cannot continuously decrement his estimate of the number of remaining periods of supranormal growth, and so he continues to project a “rolling window” of 10 periods of supranormal remaining, at all times. During that period, the stock price continues to grow at 15 percent until the end of the supranormal growth period.
(12)
Then, at the end of the supranormal growth period, the P/E ratio instantaneously drops from 20 to 10 and the stock loses 50 percent of its value.
By the end of the supranormal growth period, both pricing models produce the same terminal value, but they get there by drastically different time paths. The perfect foresight time path involves a steady 10 percent return each year, while the rational expectations model involves a 15 percent return from year to year, followed by a catastrophic 50 percent drop in price at the end of the supranormal growth period.
GENERALIZATION OF RESULTS
In the numerical example presented above, we compared the perfect foresight model of asset pricing with a rational expectations model involving a stopping process with a constant stopping rate . While that particular rational expectations pricing model may be the simplest and most intuitive model, it is not the only possible rational expectations pricing model. Nevertheless, it is possible to generalize our results. Consider the general case where the period of time remaining w is not known with absolute certainty, but which follows some distribution F(w) which has a mean of (T – t), then the rational expectations price Pt* could be stated as
. (13)
And since the integrand is convex in w, Jensen’s inequality states that
. (14)
In other words, for all points t in [0,T), so long as any uncertainty exists regarding the length of time remaining in the supranormal growth period, the rational expectations price will be greater than the perfect foresight price, even if we were able to obtain continuously updated and unbiased estimates of the length of time remaining.
INSIGHTS GAINED FROM THIS STUDY
This study demonstrates that, when the length of the supranormal growth period is not known with certainty, all rational expectations pricing models result in some degree of overpricing, when compared ex post facto to perfect foresight models. We provided a numerical example of how this could ultimately result in a catastrophic drop in stock price, when it is finally discovered that the supranormal growth period had come to an end. Our numerical example generated a catastrophic price decline of 50 percent, without positing any irrational “bubble.”
Another insight of this study is that the trading activity of “insiders,” with a better estimate of the length of time remaining in the supranormal growth period, would tend to exacerbate, rather than mitigate, any departures from perfect foresight pricing.
In the early stages of the supranormal growth period, they would be tend to be net buyers, even with full knowledge of the extent of the overpricing, since they would still be able to properly anticipate above normal holding period returns. These same “insiders” would then become net sellers near the end of the supranormal growth period, since they would properly anticipate below normal, or even negative, holding period returns. Thus, the trading activity of insiders may tend to amplify any departures from fundamental values.
A third insight of this study is that stock prices which are growing at a rate equal to the return on equity r, and which maintain a constant P/E ratio during the runup in price are most probably utilizing a constant stopping rate model to estimate the length of time remaining in the supranormal growth period. Under our analysis, these stocks present a significant risk of a catastrophic price adjustment, when the supranormal growth period finally comes to an end. Our model provides a method of quantifying the downside exposure on such stocks. In the numerical example provided, the downside risk was 50 percent.
A fourth insight of this study is the light it sheds on studies such as Fama and French (1988). In that study, Fama and French concluded that stock returns exhibit high negative autocorrelation, implying there is a slow mean-reverting movement in stock returns. Of course, under the weak-form Efficient Markets Hypothesis, asset returns should be serially uncorrelated. However, our study presents a rational expectations pricing model in which there can be significant negative autocorrelation. One interpretation of mean-reversion offered by Fama and French is that expectations are still rational, but that stock returns are subject to random shocks. These shocks do not affect future dividends, hence stock returns eventually revert to their "normal" long-run value. Another interpretation, however, is that investors become inexplicably bullish or bearish. If these fads persist for a while before eventually dying out, then stock market prices will exhibit "bubbles" (divergence from fundamental value) in the short run and mean reversion in the long run; see Summers (1986).
Shiller (1981,1989) challenges the EMH, based upon the "excess volatility" of financial markets. He found that stock prices diverged systematically from fundamental values, and he interpreted this finding as a violation of the EMH. This was the start of the "bubble literature," ultimately resulting in the literature review of Shiller (1989).
Since the studies enumerated above were all done on an ex post facto basis, what they may be measuring are departures from perfect foresight pricing (or perfect hindsight pricing, if you will) rather than departures from rational expectations pricing.
Perhaps it all boils down to a simple existential question. If the supranormal growth period really does follow a random stopping process, with a constant stopping rate, , then maybe is all that can be known during the duration of the supranormal growth period. Maybe there exists no better way of predicting the time remainining in the supranormal growth period. And if that is true, then most certainly there will be catastrophic price adjustments when we finally discover that the supranormal growth period has come to an end. Rational expectations presumes that we know all that is knowable. But what if is all that can be known?
REFERENCES
Fama, Eugene and K. French (1988),"Permanent and Temporary Components of Stock Prices," Journal of Political Economy 96, 246-273.
Fama, Eugene (1991), Efficient Capital Markets II, Journal of Finance.
Shiller, Robert (1990) "Speculative Prices and Popular Models," Journal of Economic Perspectives, 4(2): 55–65.
Shiller, Robert (1981) "Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?" American Economic Review 71, 421-436.
Robert Shiller (1989), Market Volatility, MIT Press, Cambridge, Mass.
Lawrence Summers (1986), "Does the stock Market Rationally Reflect Fundamental Values?" Journal of Finance 41, 591-601.