A Stylized Model for Long-Run Index Return Dynamics

We introduce a discrete-time model of stock index return dynamics grounded on the ability of Shiller’s Cyclically Adjusted Price-to-Earning ratio to predict long-horizon market performances. Specifically, we discuss a model in which returns are driven by a fundamental term and an autoregressive component perturbed by external random disturbances. The autoregressive component arises from the agents’ belief that expected returns are higher in bullish markets than in bearish markets. The fundamental term, driven by the value towards which fundamentalists expect the current price should revert, varies in time and depends on the initial averaged price-to-earnings ratio. The actual stock price may deviate from the perceived reference level as a combined effect of an idiosyncratic noise component and local trends due to trading strategies. We demonstrate both analytically and by means of numerical experiments that the long-run behavior of our stylized dynamics agrees with empirical evidences reported in literature.