Equity risk premium and time horizon: What do the U.S. secular data say?

Abstract

An ex-ante equity risk premium is the difference between the expected return of a risky asset at time t for a given future time horizon and an equivalent maturity risk-free interest rate. Using annual US secular data from 1871 to 2008, this study aims to model simultaneously the measures and the explanations of ex-ante equity risk premia for two polar horizons: the one period ahead horizon (i.e. the "short term" premium) and the infinite time horizon (i.e. the "long term" premium). Expectations being represented by traditional adaptive processes, large disparities in the dynamics of the two premia are evidenced. According to the conditional CAPM, each premium is at time t explained by the product of the price of risk by the expected variance of returns, these two magnitudes being horizon dependant. The expected variances depend on the past values of the centered squared returns (we found 5 and 8 years for the one year and the infinite horizon, respectively). For each horizon, the price of risk is determined by a spread of interest rates capturing economic factors of uncertainty and by an unobservable variable determined according to the kalman filter methodology (i.e. a state variable). The state variables are supposed to capture the influence of hidden variables and of non directly measurable psychological effects. The model gives a valuable representation of the "short term" and "long term" premia. Résumé-Une prime de risque ex-ante est définie par la différence entre la rentabilité anticipée à la date t d'un actif risqué pour un horizon de placement donné, et le taux d'intérêt d'un titre de créance sans risque ayant une maturité égale à cet horizon. En utilisant les données séculaires annuelles américaines sur la période 1871-2008, cet article a pour objectif de modéliser simultanément les mesures et les explications des primes de risque ex-ante des actions pour deux horizons pôlaires : l'horizon d'une période (« prime « de court terme ») et horizon infini (prime « de long terme »). Les anticipations étant représentées par des processus adaptatifs traditionnels, on constate des différences importantes entre ces deux primes au cours de la période. Suivant le MEDAF conditionnel, à la date t, chaque prime est expliquée par le prix du risque multiplié par la variance anticipée des rentabilités, ces grandeurs étant paramétrées par l'horizon. Les variances anticipées sont supposées dépendre des valeurs passées du carré des rentabilités centrées (on trouve une durée de 5 ans et 8 ans pour la prime de court terme et la prime de long terme, respectivement). Pour chaque horizon, le prix du risque est déterminé par un « spread » de taux d'intérêt capturant les facteurs économiques de l'incertitude et par une variable inobservable déterminée suivant la méthode du filtre de Kalman (i.e. une variable d'état). Les variables d'état sont supposées capturer à la fois les influences de variables cachées et d'effets psychologiques non directement mesurables. Le modèle donne une représentation valable des primes de court et de lon...

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