Optimal Investment for Insurers with the Extended CIR Interest Rate Model

Abstract

A fundamental challenge for insurance companies (insurers) is to strike the best balance between optimal investment and risk management of paying insurance liabilities, especially in a low interest rate environment. The stochastic interest rate becomes a critical factor in this asset-liability management (ALM) problem. This paper derives the closed-form solution to the optimal investment problem for an insurer subject to the insurance liability of compound Poisson process and the stochastic interest rate following the extended CIR model. Therefore, the insurer’s wealth follows a jump-diffusion model with stochastic interest rate when she invests in stocks and bonds. Our problem involves maximizing the expected constant relative risk averse (CRRA) utility function subject to stochastic interest rate and Poisson shocks. After solving the stochastic optimal control problem with the HJB framework, we offer a verification theorem by proving the uniform integrability of a tight upper bound for the objective function.

Optimal investment strategies for asset-liability management with affine diffusion factor processes and HARA preferences

“…F,P (0, T ; R) and verifies (13). In the same vein, solutions to linear BSDEs ( 14) and ( 15) are two triplets of stochastic processes (G…”

“…Notice that BSDEs (13)-( 15) constitute a coupled BSDE system, and this coupling is recursive. More specifically, the generator of linear BSDE (14) involves the solution of (Y t , Z 1,t , Z 2,t ) to BSRE (13), while the generator of linear BSDE (15) includes both the solutions of (Y t , Z 1,t , Z 2,t ) to BSRE (13) and of (G 1,t , Λ 1,t , Λ 2,t ) to BSDE (14). This observation implies that BSDEs ( 13)-( 15) shall be solved recursively forwards.…”

“…Besides the mean-variance criterion, ALM problems under the framework of expected utility maximization have attracted the attention of quite a few researchers over the last several years. For example, Chiu and Wong [13] studied an ALM problem with stochastic interest rates for an insurer under power and logarithmic utility, where the interest rate was modeled by an extended Cox-Ingersoll-Ross (CIR) process and the liability followed a risk model of compound Poisson process. Liang and Ma [33] incorporated mortality and salary risks into an ALM problem under power and exponential utility and derived the optimal approximation investment strategy by using the martingale approach and the dynamic programming approach.…”

Insurer Optimal Asset Allocation in a Small and Closed Economy: The Case of Iran’s Social Security Organization

“…Many scholars have recently worked on ALM-related issues. Chiu and Wong (2014) and Zhang et al (2017) studied an ALM problem with statedependent risk aversion under the mean-variance criterion. Wei et al (2013) and Wei and Wang (2017) investigated the time-consistent strategies of meanvariance ALM problems under the Markov regime-switching model and random coefficients setting.…”

“…In the financial market, there were also some financial instruments used to hedge against the inflation risk, such as inflation-indexed bonds (i.e., Treasury Inflation-Protected Securities (TIPS) and UK inflation-linked gilt-edged securities (ILGS)). Chiu and Wong (2014) considered the ALM problem with a stochastic interest rate under the CRRA utility framework, where the liability followed a risk model of compound Poisson process, and the interest rate was assessed based on Cox-Ingersoll-Ross (CIR) model. Since the ALM plan may take quite a long time for an investor, it would be reasonable to take into account the risks of interest rate and inflation in particular.…”