Abstract
The aim of this paper is to study a classic problem in actuarial mathematics, namely, an optimal reinsurance-investment problem, in the presence of stochastic interest and inflation rates. This is of relevance since insurers make investment and risk management decisions over a relatively long horizon where uncertainty about interest rate and inflation rate may have significant impacts on these decisions. We consider the situation where three investment opportunities, namely, a savings account, a share, and a bond, are available to an insurer in a security market. In the meantime, the insurer transfers part of its insurance risk through acquiring a proportional reinsurance. The investment and reinsurance decisions are made so as to maximize an expected power utility on terminal wealth. An explicit solution to the problem is derived for each of the two well-known stochastic interest rate models, namely, the Ho–Lee model and the Vasicek model, using standard techniques in stochastic optimal control theory. Numerical examples are presented to illustrate the impacts of the two different stochastic interest rate modeling assumptions on optimal decision making of the insurer.
Highlights
Ere has been a large amount of work on optimal investment of an insurance company
Zhang and Siu [32] discussed an optimal proportional reinsurance and investment problem using the criterion of maximizing utility function on terminal wealth
Liu et al [34] considered an optimal insurance-reinsurance problem in the presence of dynamic risk constraint and regime switching. Both the stochastic interest rates and inflation cannot be ignored for decision makers when dealing with long-term decision problems. erefore, the incorporation of both stochastic interest rate and inflation makes our problems more practicable for the long-term decision making and this is the first contribution of our paper
Summary
As usual, we describe uncertainties in the continuous-time economy, finance, and insurance markets as well as their information flows by a complete filtered probability space (Ω, F, F , p), where F Ftt∈[0,T] is right continuous, p is the complete filtration, and p is a realworld probability. For all t ∈ [0, T], Π(0) Π0, where W0(t)t∈[0,T] is a standard Brownian motion; σ0(t) is the volatility of the price index; and the stochastic drift I(t) represents the instantaneous expected inflation rate whose evolution over time is assumed to be governed by the following time-dependent Ornstein–Uhlenbeck (OU) process: dI(t) β(t)[α(t) − I(t)]dt + σ0(t)dW0(t). Without loss of generality, it is assumed that the surplus process of the company R(t)t∈[0,T] evolves over time according to a diffusion approximation model with the impact of the price index being incorporated as follows: dR(t) Π(t)c(t)dt + Π(t)σ3(t)dW3(t),. Suppose that the insurer is allowed to continuously purchase proportional reinsurance and invests all of his (or her) wealth in the financial market over the time [0, T] with T < T1. We denote Θ by the set of all admissible controls
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