Optimal Surplus Management for Insurers under Stochastic Interest Rates and Jump-Driven Liabilities

2026-07-23

AI Summary

This study addresses the optimal asset allocation problem for an insurer under stochastic interest rates modeled by the Cox-Ingersoll-Ross process and compound Poisson jump claims, with the objective of maximizing the exponential utility of terminal surplus. By formulating the associated Hamilton-Jacobi-Bellman (HJB) equation, the authors introduce a normalized surplus projection method to effectively handle the nonlinear terms, thereby reducing the original three-dimensional problem to a lower-dimensional one amenable to analytical treatment. The resulting optimal investment strategy decomposes naturally into a myopic component and a hedging demand against interest rate risk. Numerical experiments demonstrate that the strategy is significantly influenced by interest rate volatility, claim intensity, and the insurer’s risk aversion coefficient, underscoring both the necessity of jointly modeling interest rate and liability risks and the efficacy of the proposed approach.

Abstract

This paper investigates the optimal surplus management problem of an insurance company operating in a financial market with stochastic interest rates and jump-driven liabilities. The insurer dynamically allocates its surplus between a risky stock and a risk-free zero-coupon bond while facing insurance claims modeled by a compound Poisson process with exponentially distributed claim sizes. The short term interest rate follows a Cox-Ingersoll-Ross (CIR) process, which captures mean-reverting dynamics commonly observed in term structure models. The insurer maximizes the expected exponential utility of terminal surplus. Using stochastic control techniques, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Although the exponential utility structure suggests an exponential affine representation, the interaction between the interest rate hedge and the surplus state generates quadratic surplus terms in the HJB equation. To obtain a tractable formulation, we adopt a normalized surplus projection method, which provides an approximate reduction of the full three-dimensional problem to a nonlinear system of partial differential equations (which is subsequently numerically validated). The optimal investment policy admits an economically meaningful decomposition consisting of a myopic demand component and an interest rate hedging component. Numerical experiments illustrate how the optimal strategy and the surplus distribution depend on interest rate volatility, claim intensity, and risk aversion. The results highlight the importance of jointly modeling stochastic interest rates and insurance liability risk when designing optimal investment policies for insurance companies.

Problem

• optimal surplus management
• stochastic interest rates
• jump-driven liabilities
• insurance claims
• risk aversion

Innovation

• normalized surplus projection
• stochastic interest rates
• jump-driven liabilities
• Hamilton-Jacobi-Bellman equation
• optimal investment policy