Risk Aversion and the Martingale Property of Stock Prices

Abstract

gale property has received theoretical support from recent work by Samuelson [10].2 However, Samuelson's result depends on the assumption that investors require an exogenously given expected rate of return. It is natural to inquire whether the martingale property can be derived when the assumption of a given expected rate of return is relaxed. That question will be discussed in this paper. It is no longer assumed that the expected rate of return may be taken as given. thien it becomes necessary to consider how the expected rate of return is determined, and this involves analyzing the relation between the riskiness of stock and the risk-aversion of investors. We are led to consider models of portfolio selection of the type developed by Tobin [13], [14] and Markowitz [6], and the associated models of capital market equilibrium of Sharpe [12] and Lintner [5], since these deal explicitly with this question. However, it is apparent that models of the Sharpe-Lintner type, though they do relate the expected rate of return to the optimizing behavior of risk-averse investors, can cast no light on the martingale question. This is so because these models assume a one-period framework, with the expected value and variance of the next-period price being taken as given, and therefore cannot generate an intertemporal probability distribution. It is necessary somehow to dynamize the Sharpe-Lintner model; i.e., to modify the model so that it generates an intertemporal distribution of asset prices. In order to do this it is assumed that investors bid for a financial asset (stock) which constitutes a claim to a random intertemporal stream of earnings with known probability distribution, rather than for a security which matures in the next period as in the Sharpe-Lintner one-period portfolio models. It is assumed that investors have a choice between risky stock and a riskless asset earning a constant exogenous rate of return, as in the one-period portfolio models. In Section 2 such a model is developed, though only for a restricted class of