Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

In this paper, based on equilibrium control law proposed by Björk and Murgoci (2010), we study an optimal investment and reinsurance problem under partial information for insurer with mean–variance utility, where insurer’s risk aversion varies over time. Instead of treating this time-inconsistent problem as pre-committed, we aim to find time-consistent equilibrium strategy within a game theoretic framework. In particular, proportional reinsurance, acquiring new business, investing in financial market are available in the market. The surplus process of insurer is depicted by classical Lundberg model, and the financial market consists of one risk free asset and one risky asset with unobservable Markov-modulated regime switching drift process. By using reduction technique and solving a generalized extended HJB equation, we derive closed-form time-consistent investment–reinsurance strategy and corresponding value function. Moreover, we compare results under partial information with optimal investment–reinsurance strategy when Markov chain is observable. Finally, some numerical illustrations and sensitivity analysis are provided.     

Optimal mean–variance investment and reinsurance problem for an insurer with stochastic volatility

This paper considers an optimal investment and reinsurance problem for an insurer under the mean–variance criterion. The stochastic volatility of the stock price is modeled by a Cox-Ingersoll-Ross (CIR) process. By applying a backward stochastic differential equation (BSDE) approach, we obtain a BSDE related to the underlying investment and reinsurance problem. Then solving the BSDE leads to closed-form expressions for both the efficient frontier and the efficient strategy. In the end, numerical examples are presented to analyze the economic behavior of the efficient frontier.     

Robust optimal investment–reinsurance strategies for an insurer with multiple dependent risks

This paper considers a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who is uncertain about the model parameters. We assume that the surplus of the insurance company can be allocated to the financial market consisting of one risk-free asset and one risky asset whose price process satisfies square root factor process. Under the objective of maximizing the expected utility of the terminal surplus, by adopting the technique of stochastic control, closed-form expressions of the robust optimal strategy and the corresponding value function are derived. The verification theorem is also provided. Finally, by presenting some numerical examples, the impact of some parameters on the optimal strategy is illustrated and some economic explanations are also given. We find that the robust optimal reinsurance strategies under the generalized mean–variance premium are very different from that under the variance premium principle. In addition, ignoring model uncertainty risk will lead to significant utility loss for the AAI.     

Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps

This paper analyzes the equilibrium strategy of a robust optimal reinsurance-investment problem under the mean–variance criterion in a model with jumps for an ambiguity-averse insurer (AAI) who worries about model uncertainty. The AAI’s surplus process is assumed to follow the classical Cramér–Lundberg model, and the AAI is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. By applying stochastic control theory, we establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples.     

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.     

Time-consistent mean–variance proportional reinsurance and investment problem in a defaultable market

In this paper, we consider an optimal time-consistent reinsurance-investment problem incorporating a defaultable security for a mean–variance insurer under a constant elasticity of variance (CEV) model. In our model, the insurer’s surplus process is described by a jump-diffusion risk model, the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset, a defaultable bond and a risky asset whose price process is assumed to follow a CEV model. Using a game theoretic approach, we establish the extended Hamilton–Jacobi–Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we obtain the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function in both cases. Finally, we provide numerical examples to illustrate the impacts of model parameters on the optimal time-consistent strategy.     

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 mean–variance investment and reinsurance problems for the risk model with common shock dependence

In this paper, we study the optimal investment–reinsurance problems in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Under the criterion of mean–variance, two cases are considered: One is the optimal mean–variance problem with bankruptcy prohibition, i.e., the wealth process of the insurer is not allowed to be below zero at any time, which is solved by standard martingale approach, and the closed form solutions are derived; The other is the optimal mean–variance problem without bankruptcy prohibition, which is discussed by a very different method—stochastic linear–quadratic control theory, and the explicit expressions of the optimal results are obtained either. In the end, a numerical example is given to illustrate the results and compare the values in the two cases.     

Optimal investment–reinsurance with delay for mean–variance insurers: A maximum principle approach

This paper is concerned with an optimal investment and reinsurance problem with delay for an insurer under the mean–variance criterion. A three-stage procedure is employed to solve the insurer’s mean–variance problem. We first use the maximum principle approach to solve a benchmark problem. Then applying the Lagrangian duality method, we derive the optimal solutions for a variance-minimization problem. Based on these solutions, we finally obtain the efficient strategy and the efficient frontier of the insurer’s mean–variance problem. Some numerical examples are also provided to illustrate our results.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

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.     

Time-consistent investment-proportional reinsurance strategy with random coefficients for mean–variance insurers

In this study, we consider an insurer who manages her underlying risk by purchasing proportional reinsurance and investing in a financial market consisting of a risk-free bond and a risky asset. The objective of the insurer is to identify an investment–reinsurance strategy that minimizes the mean–variance cost function. We obtain a time-consistent open-loop equilibrium strategy and the corresponding efficient frontier in explicit form using two systems of backward stochastic differential equations. Furthermore, we apply our results to Vasiček’s stochastic interest rate model and Heston’s stochastic volatility model. In both cases, we obtain a closed-form solution.     

Robust optimal reinsurance–investment strategy with price jumps and correlated claims

This paper considers the robust optimal reinsurance–investment strategy selection problem with price jumps and correlated claims for an ambiguity-averse insurer (AAI). The correlated claims mean that future claims are correlated with historical claims, which is measured by an extrapolative bias. In our model, the AAI transfers part of the risk due to insurance claims via reinsurance and invests the surplus in a financial market consisting of a risk-free asset and a risky asset whose price is described by a jump–diffusion model. Under the criterion of maximizing the expected utility of terminal wealth, we obtain closed-form solutions for the robust optimal reinsurance–investment strategy and the corresponding value function by using the stochastic dynamic programming approach. In order to examine the influence of investment risk on the insurer’s investment behavior, we further study the time-consistent reinsurance–investment strategy under the mean–variance framework and also obtain the explicit solution. Furthermore, we examine the relationship among the optimal reinsurance–investment strategies of the AAI under three typical cases. A series of numerical experiments are carried out to illustrate how the robust optimal reinsurance–investment strategy varies with model parameters, and result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.     

A characterization of equilibrium strategies in continuous-time mean–variance problems for insurers

In this work, we study the equilibrium reinsurance/new business and investment strategy for mean–variance insurers with constant risk aversion. The insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where the surplus of the insurers is assumed to follow a jump–diffusion model and the financial market consists of one riskless asset and a multiple risky assets whose price processes are driven by Poisson random measures and independent Brownian motions. By using a version of the stochastic maximum principle approach, we characterize the open loop equilibrium strategies via a stochastic system which consists of a flow of forward–backward stochastic differential equations (FBSDEs in short) and an equilibrium condition. Then by decoupling the flow of FSBDEs, an explicit representation of an equilibrium solution is derived as well as its corresponding objective function value.     

Optimality of excess-loss reinsurance under a mean–variance criterion

In this paper, we study an insurer’s reinsurance–investment problem under a mean–variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance–investment strategy by solving the extended Hamilton–Jacobi–Bellman equation.     

Constrained investment–reinsurance optimization with regime switching under variance premium principle

This paper studies optimal investment and reinsurance problems for an insurer under regime-switching models. Two types of risk models are considered, the first being a Markov-modulated diffusion approximation risk model and the second being a Markov-modulated classical risk model. The insurer can invest in a risk-free bond and a risky asset, where the underlying models for investment assets are modulated by a continuous-time, finite-state, observable Markov chain. The insurer can also purchase proportional reinsurance to reduce the exposure to insurance risk. The variance principle is adopted to calculate the reinsurance premium, and Markov-modulated constraints on both investment and reinsurance strategies are considered. Explicit expressions for the optimal strategies and value functions are derived by solving the corresponding regime-switching Hamilton–Jacobi–Bellman equations. Numerical examples for optimal solutions in the Markov-modulated diffusion approximation model are provided to illustrate our results.     

Investment strategies and compensation of a mean–variance optimizing fund manager

This paper introduces a general continuous-time mathematical framework for solution of dynamic mean–variance control problems. We obtain theoretical results for two classes of functionals: the first one depends on the whole trajectory of the controlled process and the second one is based on its terminal-time value. These results enable the development of numerical methods for mean–variance problems for a pre-determined risk-aversion coefficient. We apply them to study optimal trading strategies pursued by fund managers in response to various types of compensation schemes. In particular, we examine the effects of continuous monitoring and scheme’s symmetry on trading behavior and fund performance.     

Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump–diffusion model

In this paper, we study an optimal investment problem under the mean–variance criterion for defined contribution pension plans during the accumulation phase. To protect the rights of a plan member who dies before retirement, a clause on the return of premiums for the plan member is adopted. We assume that the manager of the pension plan is allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process is modeled by a jump–diffusion process. The precommitment strategy and the corresponding value function are obtained using the stochastic dynamic programming approach. Under the framework of game theory and the assumption that the manager’s risk aversion coefficient depends on the current wealth, the equilibrium strategy and the corresponding equilibrium value function are also derived. Our results show that with the same level of variance in the terminal wealth, the expected optimal terminal wealth under the precommitment strategy is greater than that under the equilibrium strategy with a constant risk aversion coefficient; the equilibrium strategy with a constant risk aversion coefficient is revealed to be different from that with a state-dependent risk aversion coefficient; and our results can also be degenerated to the results of He and Liang (2013b) and Björk et al. (2014). Finally, some numerical simulations are provided to illustrate our derived results.     

Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility

This paper investigates the open-loop equilibrium reinsurance-investment (RI) strategy under general stochastic volatility (SV) models. We resolve difficulties arising from the unbounded volatility process and the non-negativity constraint on the reinsurance strategy. The resolution enables us to derive the existence and uniqueness result for the time-consistent mean variance RI policy under both situations of constant and state-dependent risk aversions. We apply the general framework to popular SV models including the Heston, the 3/2 and the Hull–White models. Closed-form solutions are obtained for the aforementioned models under constant risk aversion, and the non-leveraged models under state-dependent risk aversion.