Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model

We consider an optimal time-consistent reinsurance-investment strategy selection problem for an insurer whose surplus is governed by a compound Poisson risk 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 stock. The dynamics of the risky stock is governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity as well as the feedback effect of an asset’s price on its volatility. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategies and the corresponding value functions are obtained in both the compound Poisson risk model and its diffusion approximation. Numerical examples are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary.     

Budget-constrained optimal reinsurance design under coherent risk measures

ABSTRACT

Reinsurance is a versatile risk management strategy commonly employed by insurers to optimize their risk profile. In this paper, we study an optimal reinsurance design problem minimizing a general law-invariant coherent risk measure of the net risk exposure of a generic insurer, in conjunction with a general law-invariant comonotonic additive convex reinsurance premium principle and a premium budget constraint. Due to its intrinsic generality, this contract design problem encompasses a wide body of optimal reinsurance models commonly encountered in practice. A three-step solution scheme is presented. Firstly, the objective and constraint functions are exhibited in the so-called Kusuoka's integral representations. Secondly, the mini-max theorem for infinite dimensional spaces is applied to interchange the infimum on the space of indemnities and the supremum on the space of probability measures. Thirdly, the recently developed Neyman–Pearson methodology due to Lo (2017a) is adopted to solve the resulting infimum problem. Analytic and transparent expressions for the optimal reinsurance policy are provided, followed by illustrative examples.     

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

The quest for optimal reinsurance design has remained an interesting problem among insurers, reinsurers, and academicians. An appropriate use of reinsurance could reduce the underwriting risk of an insurer and thereby enhance its value. This paper complements the existing research on optimal reinsurance by proposing another model for the determination of the optimal reinsurance design. The problem is formulated as a constrained optimization problem with the objective of minimizing the value-at-risk of the net risk of the insurer while subjecting to a profitability constraint. The proposed optimal reinsurance model, therefore, has the advantage of exploiting the classical tradeoff between risk and reward. Under the additional assumptions that the reinsurance premium is determined by the expectation premium principle and the ceded loss function is confined to a class of increasing and convex functions, explicit solutions are derived. Depending on the risk measure's level of confidence, the safety loading for the reinsurance premium, and the expected profit guaranteed for the insurer, we establish conditions for the existence of reinsurance. When it is optimal to cede the insurer's risk, the optimal reinsurance design could be in the form of pure stop-loss reinsurance, quota-share reinsurance, or a combination of stop-loss and quota-share reinsurance.     

Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria

In this paper, an ambiguity-averse insurer (AAI) whose surplus process is approximated by a Brownian motion with drift, hopes to manage risk by both investing in a Black–Scholes financial market and transferring some risk to a reinsurer, but worries about uncertainty in model parameters. She chooses to find investment and reinsurance strategies that are robust with respect to this uncertainty, and to optimize her decisions in a mean-variance framework. By the stochastic dynamic programming approach, we derive closed-form expressions for a robust optimal benchmark strategy and its corresponding value function, in the sense of viscosity solutions, which allows us to find a mean-variance efficient strategy and the efficient frontier. Furthermore, economic implications are analyzed via numerical examples. In particular, our conclusion in the mean-variance framework differs qualitatively, for certain parameter ranges, with model-uncertainty robustness conclusions in the framework of utility functions: model uncertainty does not always result in an agent deciding to reduce risk exposure under mean-variance criteria, opposite to the conclusions for utility functions in Maenhout and Liu. Our conclusion can be interpreted as saying that the mean-variance problem for the AAI explains certain counter-intuitive investor behaviors, by which the attitude to risk exposure, for an AAI facing model uncertainty, depends on positive past experience.     

Time-consistent reinsurance and investment strategies for an AAI under smooth ambiguity utility

This paper investigates time-consistent reinsurance(excess-of-loss, proportional) and investment strategies for an ambiguity averse insurer(abbr. AAI). The AAI is ambiguous towards the insurance and financial markets. In the AAI's attitude, the intensity of the insurance claims' number and the market price of risk of a stock can not be estimated accurately. This formulation of ambiguity is similar to the uncertainty of different equivalent probability measures. The AAI can purchase excess-of-loss or proportional reinsurance to hedge the insurance risk and invest in a financial market with cash and an ambiguous stock. We investigate the optimization goal under smooth ambiguity given in Klibanoff, P., Marinacci, M., & Mukerji, S. [(2005). A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892], which aims to search the optimal strategies under average case. The utility function does not satisfy the Bellman's principle and we employ the extended HJB equation proposed in Björk, T. & Murgoci, A. [(2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics 18(3), 545–592] to solve this problem. In the end of this paper, we derive the equilibrium reinsurance and investment strategies under smooth ambiguity and present the sensitivity analysis to show the AAI's economic behaviors.     

Optimal insurance in the presence of reinsurance

This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution.     

Dynamic asset–liability management in a Markov market with stochastic cash flows

This paper provides a general model to investigate an asset–liability management (ALM) problem in a Markov regime-switching market in a multi-period mean–variance (M–V) framework. Emphasis is placed on the stochastic cash flows in both wealth and liability dynamic processes, and the optimal investment and liquidity management strategies in achieving the M–V bi-objective of terminal surplus are evaluated. In this model, not only the asset returns and liability returns, but also the cash flows depend on the stochastic market states, which are assumed to follow a discrete-time Markov chain. Adopting the dynamic programming approach, the matrix theory and the Lagrange dual principle, we obtain closed-form expressions for the efficient investment strategy. Our proposed model is examined through empirical studies of a defined contribution pension fund. In-sample results show that, given the same risk level, an ALM investor (a) starting in a bear market can expect a higher return compared to beginning in a bull market and (b) has a lower expected return when there are major cash flow problems. The effects of the investment horizon and state-switching probability on the efficient frontier are also discussed. Out-of-sample analyses show the dynamic optimal liquidity management process. An ALM investor using our model can achieve his or her surplus objective in advance and with a minimum variance close to zero.     

Reinsurance decisions in life insurance: An empirical test of the risk–return criterion

Recent studies have analyzed optimal reinsurance contracts within the framework of profit maximization and/or risk minimization. This type of framework, however, does not consider reinsurance as a tool for capital management and financing. In the present paper, we consider different proportional reinsurance contracts used in life insurance (viz., quota-share, surplus, and combinations of quota-share and surplus) while taking into account the insurer's capital constraints. The objective is to determine how different reinsurance transactions affect the risk/reward profile of the insurer and whether factors, such as claims severity, premiums, and insurer's risk appetite, influence the choice of a proportional reinsurance coverage. We compare each reinsurance structure based on actual insurance company data, using the risk–return criterion. This criterion determines the type of reinsurance that enables insurer to retain the largest underwriting profits and/or minimize the risk of the retained claims while keeping the insurer's risk appetite constant, assuming a given capital constraint. The results of this study confirm that the choice of reinsurance arrangement depends on many factors, including risk retention levels, premiums, and the variance of the sum insured values (and therefore claims). As such, under heterogeneous insurance portfolio single type of reinsurance arrangement cannot maximize insurer's returns and/or minimize the risk, therefore a combination of different reinsurance coverages should be employed. Hence, future research on optimal risk management choices should consider heterogeneous portfolios while determining the effects of different financial and risk management tools on companies' risk–return profiles.     

A note on optimal expected utility of dividend payments with proportional reinsurance

In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company that controls risk exposure by purchasing proportional reinsurance. We assume the preference of the insurer is of CRRA form. By solving the corresponding Hamilton–Jacobi–Bellman equation, we identify the value function and the corresponding optimal strategy. We also analyze the asymptotic behavior of the value function for large initial reserves. Finally, we provide some numerical examples to illustrate the results and analyze the sensitivity of the parameters.     

Optimal reinsurance and portfolio selection: Comparison between partial and complete information models

We consider partial and complete information models to investigate how partial information has a unique quality over complete information for insurers. We find that optimal reinsurance and investment strategies for the partially informed insurer depend on prior beliefs, whereas those for the completely informed insurer do not. In addition, information quality can affect insurer behaviour, mainly through the relative difference between risk-adjusted market premium and risk-adjusted insurance premium projected on the financial markets. Numerical results indicate that partial information increases the conservativeness of insurer strategies.     

Expected exponential utility maximization of insurers with a Linear Gaussian stochastic factor model

In this paper, we consider the problem of optimal investment by an insurer. The wealth of the insurer is described by a Cramér–Lundberg process. The insurer invests in a market consisting of a bank account and m risky assets. The mean returns and volatilities of the risky assets depend linearly on economic factors that are formulated as the solutions of linear stochastic differential equations. Moreover, the insurer preferences are exponential. With this setting, a Hamilton–Jacobi–Bellman equation that is derived via a dynamic programming approach has an explicit solution found by solving the matrix Riccati equation. Hence, the optimal strategy can be constructed explicitly. Finally, we present some numerical results related to the value function and the ruin probability using the optimal strategy.     

Optimal Reinsurance Under the Risk-Adjusted Value of an Insurer’s Liability and an Economic Reinsurance Premium Principle

In this article, an optimal reinsurance problem is formulated from the perspective of an insurer, with the objective of minimizing the risk-adjusted value of its liability where the valuation is carried out by a cost-of-capital approach and the capital at risk is calculated by either the value-at-risk (VaR) or conditional value-at-risk (CVaR). In our reinsurance arrangement, we also assume that both insurer and reinsurer are obligated to pay more for a larger realization of loss as a way of reducing ex post moral hazard. A key contribution of this article is to expand the research on optimal reinsurance by deriving explicit optimal reinsurance solutions under an economic premium principle. It is a rather general class of premium principles that includes many weighted premium principles as special cases. The advantage of adopting such a premium principle is that the resulting reinsurance premium depends not only on the risk ceded but also on a market economic factor that reflects the market environment or the risk the reinsurer is facing. This feature appears to be more consistent with the reinsurance market. We show that the optimal reinsurance policies are piecewise linear under both VaR and CVaR risk measures. While the structures of optimal reinsurance solutions are the same for both risk measures, we also formally show that there are some significant differences, particularly on the managing tail risk. Because of the integration of the market factor (via the reinsurance pricing) into the optimal reinsurance model, some new insights on the optimal reinsurance design could be gleaned, which would otherwise be impossible for many of the existing models. For example, the market factor has a nontrivial effect on the optimal reinsurance, which is greatly influenced by the changes of the joint distribution of the market factor and the loss. Finally, under an additional assumption that the market factor and the loss have a copula with quadratic sections, we demonstrate that the optimal reinsurance policies admit relatively simple forms to foster the applicability of our theoretical results, and a numerical example is presented to further highlight our results.     

Optimal insurance and reinsurance policies chosen jointly in the individual risk model

The paper studies the so-called individual risk model where both a policy of per-claim insurance and a policy of reinsurance are chosen jointly by the insurer in order to maximize his/her expected utility. The insurance and reinsurance premiums are defined by the expected value principle. The problem is solved under additional constraints on the reinsurer’s risk and the residual risk of the insured. It is shown that the solution to the problem is the following: The optimal reinsurance is a modification of stop-loss reinsurance policy, so-called stop-loss reinsurance with an upper limit; the optimal insurer’s indemnity is a combination of stop-loss- and deductible policies. The results are illustrated by a numerical example for the case of exponential utility function. The effects of changing model parameters on optimal insurance and reinsurance policies are considered.     

Optimal investment of an insurer with regime-switching and risk constraint

We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly.     

Optimal Reinsurance Arrangements Under Tail Risk Measures

Regulatory authorities demand insurance companies control their risk exposure by imposing stringent risk management policies. This article investigates the optimal risk management strategy of an insurance company subject to regulatory constraints. We provide optimal reinsurance contracts under different tail risk measures and analyze the impact of regulators' requirements on risk sharing in the reinsurance market. Our results underpin adverse incentives for the insurer when compulsory Value-at-Risk risk management requirements are imposed. But economic effects may vary when regulatory constraints involve other risk measures. Finally, we compare the obtained optimal designs to existing reinsurance contracts and alternative risk transfer mechanisms on the capital market.     

Optimal asset allocation for participating contracts under the VaR and PI constraint

ABSTRACT

Participating contracts provide a maturity guarantee for the policyholder. However, the terminal payoff to the policyholder should be related to financial risks of participating insurance contracts. We investigate an optimal investment problem under a joint value-at-risk and portfolio insurance constraint faced by the insurer who offers participating contracts. The insurer aims to maximize the expected utility of the terminal payoff to the insurer. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the joint value-at-risk and the portfolio insurance constraint impacts the optimal terminal wealth.     

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

Abstract

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.     

Optimal excess-of-loss reinsurance contract with ambiguity aversion in the principal-agent model

ABSTRACT

We discuss an optimal excess-of-loss reinsurance contract in a continuous-time principal-agent framework where the surplus of the insurer (agent/he) is described by a classical Cramér-Lundberg (C-L) model. In addition to reinsurance, the insurer and the reinsurer (principal/she) are both allowed to invest their surpluses into a financial market containing one risk-free asset (e.g. a short-rate account) and one risky asset (e.g. a market index). In this paper, the insurer and the reinsurer are ambiguity averse and have specific modeling risk aversion preferences for the insurance claims (this relates to the jump term in the stochastic models) and the financial market's risk (this encompasses the models' diffusion term). The reinsurer designs a reinsurance contract that maximizes the exponential utility of her terminal wealth under a worst-case scenario which depends on the retention level of the insurer. By employing the dynamic programming approach, we derive the optimal robust reinsurance contract, and the value functions for the reinsurer and the insurer under this contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we discuss the case when the claims follow an exponential distribution; it is then possible to show explicitly the impact of ambiguity aversion on the optimal reinsurance.     

Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps

This paper considers a robust optimal excess-of-loss reinsurance-investment problem in a model with jumps for an ambiguity-averse insurer (AAI), who worries about ambiguity and aims to develop a robust optimal reinsurance-investment strategy. The AAI’s surplus process is assumed to follow a diffusion model, which is an approximation of the classical risk model. The AAI is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset whose price is described by a jump-diffusion model. Under the criterion for maximizing the expected exponential utility of terminal wealth, optimal strategy and optimal value function are derived by applying the stochastic dynamic programming approach. Our model and results extend some of the existing results in the literature, and the economic implications of our findings are illustrated. Numerical examples show that considering ambiguity and reinsurance brings utility enhancements.     

基于随机微分博弈的保险公司最优决策模型

本文研究了基于保险公司与自然之间二人-零和随机微分博弈的最优投资及再保险问题。假设保险公司具有指数效用,自然是博弈的虚拟对手,通过求解最优控制问题对应的HJB I方程,得到了保险公司的最优投资和再保险策略以及最优值函数的闭式解。结果显示,在完全分保时(即自留比例为零),保险公司应该将全部财富购买无风险资产,即风险资产投资额为零;在不完全分保时保险公司将卖空风险资产,且卖空数量及保险自留比例都随保险公司盈余过程与风险资产间的相关性的提高而增大,随终止时刻T的临近而增加,但随市场中无风险资产回报率的增加而减少。