Optimal Quota-Share and Excess-of-Loss Reinsurance and Investment with Heston’s Stochastic Volatility Model

An optimal quota-share and excess-of-loss reinsurance and investment problem is studied for an insurer who is allowed to invest in a risk-free asset and a risky asset.Especially the price process of the risky asset is governed by Heston's stochastic volatility(SV)model.With the objective of maximizing the expected index utility of the terminal wealth of the insurance company,by using the classical tools of stochastic optimal control,the explicit expressions for optimal strategies and optimal value functions are derived.An interesting conclusion is found that it is better to buy one reinsurance than two under the assumption of this paper.Moreover,some numerical simulations and sensitivity analysis are provided.

THE OPTIMAL REINSURANCE-INVESTMENT PROBLEM CONSIDERING THE JOINT INTERESTS OF AN INSURER AND A REINSURER UNDER HARA UTILITY

This paper focuses on an optimal reinsurance and investment problem for an insurance corporation which holds the shares of an insurer and a reinsurer.Assume that the insurer can purchase reinsurance from the reinsurer,and that both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset which are governed by the Heston model and are distinct from one another.We aim to find the optimal reinsuranceinvestment strategy by maximizing the expected Hyperbolic Absolute Risk Aversion(HARA)utility of the insurance corporation’s terminal wealth,which is the weighted sum of the insurer’s and the reinsurer’s terminal wealth.The Hamilton-Jacobi-Bellman(HJB)equation is first established.However,this equation is non-linear and is difficult to solve directly by any ordinary method found in the existing literature,because the structure of this HJB equation is more complex under HARA utility.In the present paper,the Legendre transform is applied to change this HJB equation into a linear dual one such that the explicit expressions of optimal investment-reinsurance strategies for-1≤ρi≤1 are obtained.We also discuss some special cases in a little bit more detail.Finally,numerical analyses are provided.

Pareto-Optimal Reinsurance Based on TVaR Premium Principle and Vajda Condition

Reinsurance is an effective risk management tool for insurers to stabilize their profitability. In a typical reinsurance treaty, an insurer cedes part of the loss to a reinsurer. As the insurer faces an increasing number of total losses in the insurance market, the insurer might expect the reinsurer to bear an increasing proportion of the total loss, that is the insurer might expect the reinsurer to pay an increasing proportion of the total claim amount when he faces an increasing number of total claims in the insurance market. Motivated by this, we study the optimal reinsurance problem under the Vajda condition. To prevent moral hazard and reflect the spirit of reinsurance, we assume that the retained loss function is increasing and the ceded loss function satisfies the Vajda condition. We derive the explicit expression of the optimal reinsurance under the TVaR risk measure and TVaR premium principle from the perspective of both an insurer and a reinsurer. Our results show that the explicit expression of the optimal reinsurance is in the form of two or three interconnected line segments. Under an additional mild constraint, we get the optimal parameters and find the optimal reinsurance strategy is full reinsurance, no reinsurance, stop loss reinsurance, or quota-share reinsurance. Finally, we gave an example to analyze the impact of the weighting factor on optimal reinsurance.

OPTIMAL PROPORTIONAL REINSURANCE AND INVESTMENT FOR A CONSTANT ELASTICITY OF VARIANCE MODEL UNDER VARIANCE PRINCIPLE

This article studies the optimal proportional reinsurance and investment problem under a constant elasticity of variance （CEV） model. Assume that the insurer＇s surplus process follows a jump-diffusion process, the insurer can purchase proportional reinsurance from the reinsurer via the variance principle and invest in a risk-free asset and a risky asset whose price is modeled by a CEV model. The diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term＇s explanation. In all cases, by using techniques of stochastic control theory, closed-form expressions for the value functions and optimal strategies are obtained.

Optimal proportional reinsurance and dividend payments with transaction costs and internal competition

We study the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside company and transaction costs when dividends occur. The management of the company controls the reinsurance rate, the timing and the amount of dividends paid out to maximize the expected total dividends paid out to the shareholders until ruin time. By solving the corresponding quasi-variational inequality, we obtain the optimal return function and the optimal strategy.

Empirical study on optimal reinsurance for crop insurance in China from an insurer's perspective

This study investigates the optimal reinsurance for crop insurance in China in an insurer＇s perspective using the data from Inner Mongolia, Jilin, and Liaoning, China. On the basis of the loss ratio distributions modeled by An Hua Crop Risk Evaluation System, we use the empirical model developed by Tan and Weng（2014） to study the optimal reinsurance design for crop insurance in China. We find that, when the primary insurer＇s loss function, the principle of the reinsurance premium calculation, and the risk measure are given, the level of risk tolerance of the primary insurer, the safety loading coefficient of the reinsurer, and the constraint on reinsurance premium budget affect the optimal reinsurance design. When a strict constraint on reinsurance premium budget is implemented, which often occurs in reality, the limited stop loss reinsurance is optimal, consistent with the common practice in reality. This study provides suggestions for decision making regarding the crop reinsurance in China. It also provides empirical evidence for the literature on optimal reinsurance from the insurance market of China. This evidence undoubtedly has an important practical significance for the development of China＇s crop insurance.

Martingale method for optimal investment and proportional reinsurance

Numerous researchers have applied the martingale approach for models driven by L¶evy processes to study optimal investment problems.This paper considers an insurer who wants to maximize the expected utility of terminal wealth by selecting optimal investment and proportional reinsurance strategies.The insurer's risk process is modeled by a L¶evy process and the capital can be invested in a security market described by the standard Black-Scholes model.By the martingale approach,the closed-form solutions to the problems of expected utility maximization are derived.Numerical examples are presented to show the impact of model parameters on the optimal strategies.

CONDITIONAL RECURSIVE EQUATIONS ON EXCESS-OF-LOSS REINSURANCE

The rharginal recursive equations on excess-of-loss reinsurance treaty are investignted, under the assumption that the number of claims belongs to the family consisting of Poisson, binomial and negative binomial, and that the severity distribution has bounded continuous density function. On conditional of the numbers of claims associated with the reinsurer and the cedent, some recursive equations are obtained for the marginal distributions of the total payments of the reinsurer and the cedent.

VAR AND CTE BASED OPTIMAL REINSURANCE FROM A REINSURER'S PERSPECTIVE

In this article,we study optimal reinsurance design.By employing the increasing convex functions as the admissible ceded loss functions and the distortion premium principle,we study and obtain the optimal reinsurance treaty by minimizing the VaR(value at risk)of the reinsurer's total risk exposure.When the distortion premium principle is specified to be the expectation premium principle,we also obtain the optimal reinsurance treaty by minimizing the CTE(conditional tail expectation)of the reinsurer's total risk exposure.The present study can be considered as a complement of that of Cai et al.[5].

Risk Measure and Premium Distribution on Catastrophe Reinsurance

In this paper, we propose a new risk measure which is based on the Or- licz premium principle to characterize catastrophe risk premium. The intention is to develop a formulation strategy for Catastrophe Fund. The logarithm equivalent form of reinsurance premium is regarded as the retention of reinsurer, and the differential earnings between the reinsurance premium and the reinsurer＇s retention is accumu- lated as a part of Catastrophe Fund. We demonstrate that the aforementioned risk measure has some good properties, which are further confirmed by numerical simu- lations in R environment.

OPTIMAL REINSURANCE UNDER EXPECTED VALUE PRINCIPLE

The paper concerns the problem how to purchase the reinsurance in order to make the insurer and the reinsurance company＇s total risk to be least under the expected value principle. When the insurer and reinsurance company take arbitrary risk measures, sufficient con- ditions for optimality of reinsurance contract are given within the restricted class of admissible contracts. Further, the explicit forms of optimal reinsurance contract under several special risk measures are given, and the method to decide parameters as well.

Optimal Investment-Reinsurance Strategies for Insurers with Mean-Reversion and Mispricing under Variance Premium Principle

This paper considers a robust optimal reinsurance-investment problem for an insurer with mispricing and model ambiguity. The surplus process is described by a classical Cramér-Lunderg model and the financial market contains a market index, a risk-free asset and a pair of mispriced stocks, where the expected return rate of the stocks and the mispricing follow mean reverting processes which take into account liquidity constraints. In particular, both the insurance and reinsurance premium are assumed to be calculated via the variance premium principle. By employing the dynamic programming approach, we derive the explicit optimal robust reinsurance-investment strategy and the optimal value function.

Equilibrium Reinsurance Strategy and Mean Residual Life Function

In this paper,we analyze the relationship between the equilibrium reinsurance strategy and the tail of the distribution of the risk.Since Mean Residual Life(MRL)has a close relationship with the tail of the distribution,we consider two classes of risk distributions,Decreasing Mean Residual Life(DMRL)and Increasing Mean Residual Life(IMRL)distributions,which can be used to classify light-tailed and heavy-tailed distributions,respectively.We assume that the underlying risk process is modelled by the classical CramérLundberg model process.Under the mean-variance criterion,by solving the extended Hamilton-Jacobi-Bellman equation,we derive the equilibrium reinsurance strategy for the insurer and the reinsurer under DMRL and IMRL,respectively.Furthermore,we analyze how to choose the reinsurance premium to make the insurer and the reinsurer agree with the same reinsurance strategy.We find that under the case of DMRL,if the distribution and the risk aversions satisfy certain conditions,the insurer and the reinsurer can adopt a reinsurance premium to agree on a reinsurance strategy,and under the case of IMRL,the insurer and the reinsurer can only agree with each other that the insurer do not purchase the reinsurance.

Optimal Reinsurance and Dividend Under Model Uncertainty

In this paper,the authors analyze the optimal reinsurance and dividend problem with model uncertainty for an insurer.Here the model uncertainty represents possible deviations between the real market and the assumed model.In addition to the incorporation of model uncertainty into the traditional diffusion surplus process,the authors include a penalty function in the objective function.The proposed goal is to find the optimal reinsurance and dividend strategy that maximizes the expected discounted dividend before ruin in the worst case of all possible scenarios,namely,the worst market.Using a dynamic programming approach,the problem is reduced to solving a Hamilton-Jacob-Bellman-Isaac(HJBI)equation with singular control.This problem is more difficult than the traditional robust control or singular control problem.Here,the authors prove that the value function is the unique solution to this HJBI equation with singular control.Moreover,the authors present a verification theorem when a smooth solution can be found,and derive closed-form solution when the function in the objective function is specified.

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

This study considers an optimal investment and reinsurance problem involving a defaultable security for an insurer in an ambiguous environment.In other words,the insurer is ambiguous about the insurance claim that is exponentially distributed with an uncertain rate parameter.The insurer can purchase proportional reinsurance and invest its wealth in three assets:a risk-free asset,a risky asset,the price process of which satisfies the Heston local-stochastic volatility model,and a defaultable corporate bond.For the optimal investment–reinsurance objective with a smooth ambiguity utility proposed by Klibanoff,P.,Marinacci,M.,and Mukerji,S.[A smooth model of decision making under ambiguity,Econometrica,2005,73(6):1849-1892],the equilibrium strategy is introduced and the extended Hamilton–Jacobi–Bellman equation is established through a stochastic control approach.However,the analytical solution of the strategy under the Heston local-stochastic volatility model cannot be obtained because of the complicated nonlinearity of the partial differential equation.In this study,we employ a perturbation method to derive an asymptotic solution for the post-and pre-default cases.In addition,we present a sensitivity analysis to explain the impact of model parameters on the equilibrium investment–reinsurance strategy.

Optimal reinsurance designs based on risk measures:a review

Reinsurance is an effective way for an insurance company to control its risk.How to design an optimal reinsurance contract is not only a key topic in actuarial science,but also an interesting research question in mathematics and statistics.Optimal reinsurance design problems can be proposed from different perspectives.Risk measures as tools of quantitative risk management have been extensively used in insurance and finance.Optimal reinsurance designs based on risk measures have been widely studied in the literature of insurance and become an active research topic.Different research approaches have been developed and many interesting results have been obtained in this area.These approaches and results have potential applications in future research.In this article,we review the recent advances in optimal reinsurance designs based on risk measures in static models and discuss some interesting problems on this topic for future research.

Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection

In this paper, the surplus process of the insurance company is described by a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and purchase excess-of-loss reinsurance. Under short-selling prohibition, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. We first show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions. Then, by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risky-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson's longstanding conjecture about the relation between the two problems.

Optimal Proportional Reinsurance for Controlled Risk Process which is Perturbed by Diffusion

In this paper, we study optimal proportional reinsurance policy of an insurer with a risk process which is perturbed by a diffusion. We derive closed-form expressions for the policy and the value function, which are optimal in the sense of maximizing the expected utility in the jump-diffusion framework. We also obtain explicit expressions for the policy and the value function, which are optimal in the sense of maximizing the expected utility or maximizing the survival probability in the diffusion approximation case. Some numerical examples are presented, which show the impact of model parameters on the policy. We also compare the results under the different criteria and different cases.

On Optimal Proportional Reinsurance and Investment in a Markovian Regime-Switching Economy

In this paper, the surplus of an insurance company is modeled by a Markovian regime- switching diffusion process. The insurer decides the proportional reinsurance and investment so as to increase revenue. The regime-switching economy consists of a fixed interest security and several risky shares. The optimal proportional reinsurance and investment strategies with no short-selling constraints for maximizing an exponential utility on terminal wealth are obtained.

Optimal Time-consistent Investment and Reinsurance Strategy for Mean-variance Insurers Under the Inside Information

In this paper, we consider the problem of the optimal time-consistent investment and proportional reinsurance strategy under the mean-variance criterion, in which the insurer has some inside information at her disposal concerning the future realizations of her claims process. It is assumed that the surplus of the insurer is governed by a Brownian motion with drift, and the insurer has the possibility to reduce the risk by purchasing proportional reinsurance and investing in financial markets. We first formulate the problem and provide a verification theorem on the extended Hamilton-Jacobi-Bellman equations. Then, the closed-form expression is obtained for the optimal strategy of the optimization problem.