The aim of this paper is twofold: to provide a theoretical framework and to
give further empirical support to Shiller's test of the appropriateness of
prices in the stock market based on the Cyclically Adjusted Price Earnings
(CAPE) ratio. We devote the first part of the paper to the empirical analysis
and we show that the CAPE is a powerful predictor of future long run
performances of the market not only for the U.S. but also for countries such as
Belgium, France, Germany, Japan, the Netherlands, Norway, Sweden and
Switzerland. We show four relevant empirical facts: i) the striking ability of
the logarithmic averaged earning over price ratio to predict returns of the
index, ii) how this evidence increases switching from returns to gross returns,
iii) moving over different time horizons, the regression coefficients are
constant in a statistically robust way, and iv) the poorness of the prediction
when the precursor is adjusted with long term interest rate. In the second part
we provide a theoretical justification of the empirical observations. Indeed we
propose a simple model of the price dynamics in which the return growth depends
on three components: a) a momentum component, naturally justified in terms of
agents' belief that expected returns are higher in bullish markets than in
bearish ones; b) a fundamental component proportional to the log earnings over
price ratio at time zero, from which the actual stock price may deviate as an
effect of random external disturbances, and c) a driving component ensuring the
diffusive behaviour of stock prices. Under these assumptions, we are able to
prove that, if we consider a sufficiently large number of periods, the expected
rate of return and the expected gross return are linear in the initial time
value of the log earnings over price ratio, and their variance goes to zero
with rate of convergence equal to minus one.