APPROXIMATE ANALYTICAL SOLUTION TO CONSUMPTION AND LONG-TERM SECURITY INVESTMENT OPTIMIZATION PROBLEM WITH EPSTEIN-ZIN UTILITY

Abstract

Campbell and Viceira examine the consumption and long-term security investment optimization problem of the consumer, who has Epstein-Zin utility, in which relative risk aversion and elasticity of intertemporal substitution are inseparable, under a securities market composed of the short-term bond and a long-term fixed-maturity inflation indexed bond. They derive an approximate analytical solution applying log-linear approximation to a non-homogeneous term in the Hamilton-Jacobi-Bellman equation. This paper explores the problem under a general securities market in which all maturity inflation-indexed bonds and main indices including stock index are traded, and under the assumption that these security prices are subject to latent state variables. Approximate analytical candidate solutions are derived from the log-linear approximation, and it is revealed that optimal approximate investment depends on state variables. Also, a sufficient condition is presented in order to identify the optimal solution among candidate solutions. Finally, precisions of the approximate optimal consumption-wealth ratio and of the approximate optimal investment are evaluated by a simplified method, and it is confirmed that the approximations are overall highly accurate.