Reinsurance-investment Strategy Research Articles

Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk

In this paper, we consider the time-consistent reinsurance–investment strategy under the mean–variance criterion for an insurer whose surplus process is described by a Brownian motion with drift. The insurer can transfer part of the risk to a reinsurer via proportional reinsurance or acquire new business. Moreover, stochastic interest rate and inflation risks are taken into account. To reduce the two kinds of risks, not only a risk-free asset and a risky asset, but also a zero-coupon bond and Treasury Inflation Protected Securities (TIPS) are available to invest in for the insurer. Applying stochastic control theory, we provide and prove a verification theorem and establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) equation. By solving the extended HJB equation, we derive the time-consistent reinsurance–investment strategy as well as the corresponding value function for the mean–variance problem, explicitly. Furthermore, we formulate a precommitment mean–variance problem and obtain the corresponding time-inconsistent strategy to compare with the time-consistent strategy. Finally, numerical simulations are presented to illustrate the effects of model parameters on the time-consistent strategy.

Robust investment–reinsurance optimization with multiscale stochastic volatility

This paper investigates the investment and reinsurance problem in the presence of stochastic volatility for an ambiguity-averse insurer (AAI) with a general concave utility function. The AAI concerns about model uncertainty and seeks for an optimal robust decision. We consider a Brownian motion with drift for the surplus of the AAI who invests in a risky asset following a multiscale stochastic volatility (SV) model. We formulate the robust optimal investment and reinsurance problem for a general class of utility functions under a general SV model. Applying perturbation techniques to the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation associated with our problem, we derive an investment–reinsurance strategy that well approximates the optimal strategy of the robust optimization problem under a multiscale SV model. We also provide a practical strategy that requires no tracking of volatility factors. Numerical study is conducted to demonstrate the practical use of theoretical results and to draw economic interpretations from the robust decision rules.

Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process

This paper studies an optimal investment–reinsurance problem for an insurer with a surplus process represented by the Cramér–Lundberg model. The insurer is assumed to be a mean–variance optimizer. The financial market consists of one risk-free asset and one risky asset. The market price of risk depends on a Markovian, affine-form, square-root stochastic factor process, while the volatility and appreciation rate of the risky asset are given by non-Markovian, unbounded processes. The insurer faces the decision-making problem of choosing to purchase reinsurance, acquire new business and invest its surplus in the financial market such that the mean and variance of its terminal wealth is maximized and minimized simultaneously. We adopt a backward stochastic differential equation approach to solve the problem. Closed-form expressions for the efficient frontier and efficient strategy of the mean–variance problem are derived. Numerical examples are presented to illustrate our results in two special cases, the constant elasticity of variance model and Heston’s model.

Time-consistent reinsurance–investment strategy for an insurer and a reinsurer with mean–variance criterion under the CEV model

This paper is devoted to derive the time-consistent reinsurance–investment strategy for an insurer and a reinsurer under mean–variance criterion. We aim to maximize the weighted sum of the insurer’s and the reinsurer’s objectives with different risk averse coefficients. The claim process is assumed to follow a Brownian motion with drift and the insurer can purchase proportional reinsurance from the reinsurer. Moreover, both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. In order to reduce risks, the insurer and the reinsurer can invest in different risky assets whose price processes are described by the constant elasticity of variance (CEV) models. Consideration of the CEV model and the profits of both the insurer and the reinsurer, the proof of the verification theorem becomes difficult and the solution becomes complex. We first formulate a general problem and prove the verification theorem. By solving an extended Hamilton–Jacobi–Bellman (HJB) equation, we obtain the time-consistent reinsurance–investment strategy and the corresponding value function explicitly. Finally, sensitivity analysis and numerical simulation are presented to show the effects of parameters on the time-consistent strategy and illustrate the economic meaning.

Time-Consistent Investment-Reinsurance Strategies Towards Joint Interests of the Insurer and the Reinsurer Under CEV Models

The present paper studies the time-consistent solutions to an investment-reinsurance problem under a mean-variance framework. The paper is distinguished from other literature by taking into account the interests of both the insurer and reinsurer jointly. The claim process of the insurer is governed by a Brownian motion with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated on a weighted sum of the insurer’s and reinsurer’s surplus processes, where the weight can be viewed as a regularization parameter to balance the importance of each party in the decision. Upon a verification theorem, which is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are given to demonstrate those derived results as well as the economic meanings behind.

Optimal Investment and Reinsurance for Insurers with Uncertain Time-Horizon

This paper considers the investment-reinsurance problems for an insurer with uncertain time-horizon in a jump-diffusion model and a diffusion-approximation model. In both models, the insurer is allowed to purchase proportional reinsurance and invest in a risky asset, whose expected return rate and volatility rate are both dependent on time and a market state. Meanwhile, the market state described by a stochastic differential equation will trigger the uncertain time-horizon. Specifically, a barrier is predefined and reinsurance and investment business would be stopped if the market state hits the barrier. The objective of the insurer is to maximize the expected discounted exponential utility of her terminal wealth. By dynamic programming approach and Feynman-Kac representation theorem, we derive the expressions for optimal value functions and optimal investment-reinsurance strategies in two special cases. Furthermore, an example is considered under the diffusion-approximation model, which shows some interesting results.

Time-Consistent Investment-Reinsurance Strategy for Mean-Variance Insurers With a Defaultable Security

This paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean-variance criterion in a jump-diffusion risk model. The insurer is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market consisting of a risk-free bank account, a stock and a defaultable bond. From a game theoretic perspective, the extended Hamilton-Jacobi-Bellman systems are established for the post-default case and the pre-default case, respectively. Furthermore, for the two cases, closed-form expressions for the optimal time-consistent investment-reinsurance strategies and the corresponding optimal value functions are derived, and some properties of the strategies are analyzed. Finally, some special cases of our model are presented, and numerical analysis is provided to illustrate our results.

Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model

In this paper, we study the optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk model. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price process satisfies the Heston model. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. By applying stochastic optimal control approach, we obtain the optimal strategy and value function explicitly. In addition, a verification theorem is provided and the properties of the optimal strategy are discussed. Finally, we present a numerical example to illustrate the effects of model parameters on the optimal investment–reinsurance strategy and the optimal value function.

Optimal investment-reinsurance with dynamic risk constraint and regime switching

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.

Optimal reinsurance-investment strategies for insurers under mean-CaR criteria

This paper considers an optimal reinsurance-investment problem foran insurer, who aims to minimize the risk measured byCapital-at-Risk (CaR) with the constraint that the expected terminalwealth is not less than a predefined level. The surplus of theinsurer is described by a Brownian motion with drift. The insurercan control her/his risk by purchasing proportional reinsurance,acquiring new business, and investing her/his surplus in a financialmarket consisting of one risk-free asset and multiple risky assets.Three mean-CaR models are constructed. By transforming these modelsinto bilevel optimization problems, we derive the explicitexpressions of the optimal deterministic rebalancereinsurance-investment strategies and the mean-CaR efficientfrontiers. Sensitivity analysis of the results and a numericalexample are provided.

On proportional reinsurance with a linear transaction rate

We study a model of an insurance company whose surplus is represented by a pure diffusion. The company is allowed to buy proportional reinsurance and invest its surplus in a Black-Sholes financial market. Further, it is assumed that transaction cost rate of the reinsurance decreases linearly while the insurance company buys more reinsurance. We consider two optimization criteria - minimizing probability of ruin and maximizing expected exponential utility of terminal wealth for a fixed time. Corre- sponding Hamilton-Jacobi-Bellman (HJB) equations are analyzed; consequently we find explicit expressions of the minimal ruin probability, maximal expected terminal utility, and their associated optimal reinsurance-investment strategies via various param- eter conditions. We observe from the explicit results that for some special parameter cases, the optimal investment-reinsurance strategies coincide under the two optimization criteria; i.e., Ferguson's longstanding conjecture on the relation between the two problems holds.

Optimal proportional reinsurance and investment with minimum probability of ruin

The paper concerns a problem of optimal reinsurance and investment in order to minimizing the probability of ruin. In the whole paper, the cedent’s surplus is allowed to invest in a risk-free asset and a risky asset and the company’s risk is reduced through proportional reinsurance, while in addition the claim process is assumed to follow a Brownian motion with drift. By solving the corresponding Hamilton–Jacobi–Bellman equations, the optimal reinsurance–investment strategy is derived. The presented results generalize those by Taksar [1].

Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model

We consider an optimal reinsurance-investment problem of an insurer whose surplus process follows a jump-diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a “simplified” financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The objective of the insurer is to choose an optimal reinsurance-investment strategy so as to maximize the expected exponential utility of terminal wealth. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Explicit forms for the optimal reinsuranceinvestment strategy and the corresponding value function are obtained. Numerical examples are provided to illustrate how the optimal investment-reinsurance policy changes when the model parameters vary.