Optimal portfolio for a highly risk-averse investor: A differential game interpretation

Abstract

Risk-sensitive portfolio optimization is treated with a linear-Gaussian-factor model. The main interest is how a highly risk-averse investor controls his/her interest rate risk. A simple risk-averse limit with increasing risk-averse parameter γ↑∞ is not appropriate: the associated maximized risk-sensitized expected growth rate goes to −∞ as γ↑∞, and a “breakdown” occurs in the limit. Instead, a small-noise and large-risk-aversion limit is considered, assuming the factor-noise has a small parameter ε<<1, taking ε-dependent risk-averse parameter γ(ε)=O(ε⁻²)>>1, and letting ε↓0. The limit value is characterized as the value of a linear-quadratic differential game. A sequence (π^˜(ε))_ε>0 of ε-dependent dynamic investment strategies is constructed from a saddle point of the game, and its asymptotic optimality is shown as ε↓0.