Optimal Investment Reinsurance Strategy Research Articles

On penalized goal-reaching probability minimization under borrowing and short-selling constraints

Abstract We consider a robust optimal investment–reinsurance problem to minimize the goal-reaching probability that the value of the wealth process reaches a low barrier before a high goal for an ambiguity-averse insurer. The insurer invests its surplus in a constrained financial market, where the proportion of borrowed amount to the current wealth level is no more than a given constant, and short-selling is prohibited. We assume that the insurer purchases per-claim reinsurance to transfer its risk exposure to a reinsurer whose premium is computed according to the mean–variance premium principle. Using the stochastic control approach based on the Hamilton–Jacobi–Bellman equation, we derive robust optimal investment–reinsurance strategies and the corresponding value functions. We conclude that the behavior of borrowing typically occurs with a lower wealth level. Finally, numerical examples are given to illustrate our results.

Minimizing the penalized goal-reaching probability with multiple dependent risks

We consider a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who wishes to minimize the probability that the value of the wealth process reaches a low barrier before a high goal. We assume that the insurer can purchase per-loss reinsurance for every class of insurance business and invest its surplus in a risk-free asset and a risky asset. Using the technique of stochastic control theory and solving the associated Hamilton-Jacobi-Bellman (HJB) equation, we derive the robust optimal investment-reinsurance strategy and the associated value function. We conclude that the robust optimal investment-reinsurance strategy coincides with the one without model ambiguity, but the value function differs. We also illustrate our results by numerical examples.

Optimal Investment-reinsurance Strategies for an Insurer with Options Trading Under Model Ambiguity

Optimal Investment-reinsurance Strategies for an Insurer with Options Trading Under Model Ambiguity

Non-zero-sum investment-reinsurance game with delay and ambiguity aversion

In this paper, we investigate a investment-reinsurance game problem with delay and ambiguity aversion for two insurers under mean–variance preference. Each insurer can purchase proportional reinsurance whose surplus process is a diffusion approximation process, and invest oneself’s wealth in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a jump–diffusion process. Based on the introduction of delay, we obtain wealth dynamics for each insurer depicted by a stochastic delay differential equation. By the stochastic control theory in the framework of game theory, the robust time-consistent Nash equilibrium investment-reinsurance strategies and the corresponding robust equilibrium value functions for each insurer are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment-reinsurance strategy.

Optimal investment and reinsurance strategies for an insurer with stochastic economic factor

This work considers optimal investment and reinsurance strategies for an insurer with stochastic economic factor. In our mathematical model, a risk-free asset and a risky asset are assumed to rely on a stochastic economic factor which is described by a diffusion process. We generalize the claim process to a compound Poisson process with the stochastic economic factor. Using expected utility maximization, we characterize the optimal strategy of investment-reinsurance under the power utility function. We use dynamic programming principle to derive the Hamilton–Jacobi–Bellman (HJB) equation. Then, by analysing the solution of the HJB equation, the optimal investment-reinsurance strategy is obtained and given in the verification theorem. Finally, sensitivity analysis is given to show the economic behavior of the optimal investment and reinsurance strategies.

Non-zero-sum reinsurance and investment game with correlation between insurance market and financial market under CEV model

<p style='text-indent:20px;'>In this paper, we study a non-zero-sum investment and reinsurance game for two insurers. Each insurer's surplus process is described by a Brownian motion with drift. Both insurers are allowed to purchase proportional reinsurance and invest in a risk-free asset and a risky asset. The price process of the risky asset follows the constant elasticity of variance (CEV) model, and the correlation between the risky asset's price process and the claim process is considered. Each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. By applying stochastic control theory, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation and derive optimal investment-reinsurance strategies for two insurers under exponential utility function. Furthermore, we consider the insurer's optimal investment-reinsurance strategies without competition. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies.</p>

The optimal investment-reinsurance strategies for ambiguity aversion insurer in uncertain environment

<p style='text-indent:20px;'>We study the optimal investment-reinsurance strategies for insurer, who is assumed to be ambiguous about factors related to the stock process and surplus process. In the financial market, the stock and derivatives are traded freely, the optimal investment-reinsurance strategies are obtained under the worst-case scenario with or without derivative trading. We find the evidence that the optimal exposures to different risks of the stock are significantly affected by the ambiguity aversion to the corresponding risk factors. Insurer who ignores model uncertainty always incur welfare losses, which can be proved to be strictly positive. Furthermore, we find that volatility ambiguity has a smaller impact in incomplete markets and surplus process ambiguity has a great impact on reinsurance strategies. Numerical experiments are provided to demonstrate the impact of the optimal exposure and utility loss with different parameters.</p>

A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

Robust optimal investment and reinsurance to minimize a goal-reaching probability with constrained control variables

This paper considers a robust optimal investment and reinsurance problem with constraints for an Ambiguity-Averse Insurer (AAI). The criterion is to minimize the goal-reaching probability, namely, the probability that the value of the wealth process reaches a low barrier before a high goal. The robust optimal investment-reinsurance strategy and closed-form expression of the associated value function are derived explicitly by applying stochastic dynamic programming and solving the corresponding Hamiliton-Jacobi-Bellman (HJB) equation. It is extremely interesting that the sum of our value function and the value function of Luo et al. [23] is equal to 1 in two cases of ambiguity and ambiguity-neutral. Finally, numerical examples are given to illustrate the influence of typical parameters on our results.

Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model

<abstract><p>This paper considers the problem of optimal investment-reinsurance for the insurer and reinsurer under the constant elasticity of variance (CEV) model. It is assumed that the net claims process is approximated by a diffusion process, both the insurer and reinsurer can invest in risk-free assets and risky assets. We use the variance premium principle to calculate the premiums of the insurer and reinsurer, and the reinsurance proportion is constrained by the net profit condition. Our objective is to maximize the joint exponential utility of the insurer and reinsurer's terminal wealth for a fixed time. By solving the HJB equation, we obtain the explicit expressions of the optimal investment-reinsurance strategy and value function. We find that the optimal reinsurance strategy can be divided into many cases and is related to the risk aversion coefficient of the insurer and reinsurer, but independent of the price of risky assets. Furthermore, we give the proof of the verification theorem. Finally, we demonstrate a numerical analysis to explain the results.</p></abstract>

Optimal investment-reinsurance strategy with derivatives trading under the joint interests of an insurer and a reinsurer

<abstract> <p>Considering the common interests of an insurer and a reinsurer, the optimal investment-reinsurance problem with derivatives trading is studied. Suppose that both parties would invest a stock and a risk-free asset for capital appreciation, the insurer could purchase reinsurance and trade derivatives, the optimization problem is formulated by maximizing the expected exponential utility of two parties' wealth processes. The corresponding HJB equations are built for optimal strategy through the dynamic programming principle. In addition, derivatives trading is evaluated based on the certainty-equivalence principle. A numerical study directly illustrates how model parameters influence optimal strategies.</p> </abstract>

Optimal investment-reinsurance strategy in the correlated insurance and financial markets

<p style='text-indent:20px;'>Within the correlated insurance and financial markets, we consider the optimal reinsurance and asset allocation strategies. In this paper, the risk asset is assumed to follow a general continuous diffusion process driven by a Brownian motion, which correlates to the insurer's surplus process. We propose a novel approach to derive the optimal investment-reinsurance strategy and value function for an exponential utility function. To illustrate this, we show how to derive the explicit closed strategies and value functions when the risk asset is the CEV model, 3/2 model and Merton's IR model respectively.</p>

Robust optimal investment-reinsurance strategies with the preferred reinsurance level of reinsurer

<abstract><p>This paper investigates robust equilibrium investment-reinsurance strategy for a mean variance insurer. With a larger market share, a reinsurer has a greater say in negotiating reinsurance contracts and makes the decision to propose the preferred level of reinsurance and charges extra fees as a penalty for losses that deviate from the preferred level of reinsurance. Once the insurer receives a decision from the reinsurer, the insurer weighs its risk-bearing capacity against the cost of reinsurance in order to find the optimal investment-reinsurance strategy under the mean-variance criterion. The insurer who is ambiguity averse to jump risk and diffusion risk obtains a robust optimal investment-reinsurance strategy by dynamic programming principle. Moreover, the reinsurance strategy is no longer excess-of-loss reinsurance or proportional reinsurance. In particular, the insurer may purchase proportional reinsurance for different ranges of loss and the proportion depends on the extra charge rate, which is more consistent with market practice than the standard excess-of-loss reinsurance and proportional reinsurance. The optimal reinsurance policy depends on the degree of ambiguous aversion to jump risk and not on the degree of ambiguous aversion to diffusion risk. Finally, several numerical experiments are presented to illustrate the impacts of key parameters on the value function and the optimal reinsurance contracts.</p></abstract>

Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks

This article investigates the optimal investment-reinsurance problem for state-dependent risk aversion with delay and common shock dependence. We suppose that the insurer manages insurance risk which is characterized by a two-dimensional dependent claim process by purchasing proportional reinsurance, and the insurer can invest his wealth in a financial market that consists of a risk-free asset and a risky asset. On the basis of the consideration of the performance-related capital inflow/outflow, the wealth process of the insurer is modeled by a stochastic delay differential equation. Then we formulate the optimal investment and reinsurance mean-variance problem within a game-theoretic framework and then applying the stochastic control theory with delay, we derive the corresponding extended HJB equations with delay. Next by solving the extended HJB equations and constructing the exponential martingale process, the closed-form solution to the optimal investment-reinsurance strategy and the corresponding equilibrium value function are derived. Besides, we use a numerical example to analyze the influence of the model parameters on the optimal time-consistent strategy and give an economic explanation.

Mean-variance problem for an insurer with dependent risks and stochastic interest rate in a jump-diffusion market

In this paper, we study an optimal investment and reinsurance problem in which the interest rate is driven by the Vasicek process and two dependent classes of insurance business are correlated through a common shock component. The goal of insurer is to find an optimal investment-reinsurance strategy to minimize the variance of terminal net wealth for a given expected terminal net wealth. By using the linear-quadratic optimal control theory and the corresponding Hamilton-Jacobi-Bellman (HJB) equation, the closed-form expressions for the value function and optimal strategies are obtained under the special case of perfect correlation between the bond and stock processes. We present that the solution of the HJB equation is no longer a classical solution, but a viscosity solution due to the non-negativity constraint of the reinsurance strategy. Furthermore, the efficient strategies and efficient frontier are derived explicitly. Finally, we explore some numerical examples to show the influence of model parameters on the optimal investment and reinsurance strategies.

Robust optimal investment and reinsurance for an insurer with inside information

This paper studies a robust optimal investment–reinsurance problem for an insurer who possesses inside information on the financial market and the insurance business under model uncertainty. The insurer’s surplus process and the risky asset process in the financial market are assumed to be correlated jump diffusion processes with random coefficients. The inside information is modelled by a general random variable related to the future realizations of the surplus process and the risky asset process. Under the criterion of expected utility maximization, the optimal investment–reinsurance problem with inside information is transformed into an adapted stochastic control problem. By incorporating Knightian uncertainty into the model, we establish an adapted stochastic differential game problem with a nonstandard performance functional, which aims to select the robust optimal investment–reinsurance strategy for the insurer with inside information. The stochastic maximum principles are adopted to derive a characterization of the optimal strategy. Some interesting special cases are discussed and the corresponding optimal strategies are derived.

Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform

The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

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.

Martingale and duality methods for optimal investment and reinsurance problem in a Lévy model

In this paper, we employ the martingale and duality methods to study the optimal investment and proportional reinsurance problem for an insurer. The insurer’s risk process is modeled by a Lévy process and the capital can be invested in a security market described by a geometric Lévy process. The objective of the insurer is to maximize the expected utility of her terminal wealth. We derive the expression for optimal investment-reinsurance strategies for various utility functions. Furthermore, an example is considered, and numerical simulations are presented to illustrate the effect of the parameters on the optimal strategies.

Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles

This paper considers the optimal investment-reinsurance strategy in a risk model with two dependent classes of insurance business under two kinds of premium principles, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the expected value premium principle and the variance premium principle, we use the stochastic optimal control theory to derive the optimal strategy and the value function for the compound Poisson risk model as well as for the Brownian motion diffusion risk model. In particular, we find that the optimal investment strategy on the risky asset is independent to the reinsurance strategy and the reinsurance strategy for the compound Poisson risk model are very different from those for the diffusion model under both two kinds of premium principles, but the investment strategies are the same in this two risk models. Finally, numerical examples are presented to show the impact of model parameters in the optimal strategies.