Bond and Stock Returns in a Simple Exchange Model

Abstract

In this paper I analyze a simple "representative agent" exchange model of general equilibrium, and derive closed form solutions for returns on stocks and real and nominal bonds. The model restricts the representative agent's utility function to be time-separable with isoelastic period utility, and the endowment to be conditionally lognormal. These assumptions allow me to examine a general stationary stochastic process for the log of the endowment. Money and nominal prices are modelled by means of a Clower constraint. Risk premia on stocks and real and nominal discount bonds are simple functions of the coefficient of relative risk aversion, the variance of the innovation to the log endowment, and the weights in the moving average representation of the log endowment. One-period holding premia on real bonds may be positive or negative, but the limit as maturity increases is positive. When the money supply is deterministic, stocks and nominal bonds are perfect substitutes. Their expected returns to maturity are higher than those on real bonds of equal maturity, but need not be higher over other holding periods. Nominal interest rates vary positively with prices (the "Gibson paradox") if the coefficient of relative risk aversion is greater than one. In the last section of the paper I consider random shocks to the agent's utility function. These shocks may generate risk premia even when the agent is risk-neutral.