Robust non-zero-sum investment and reinsurance game with default risk

This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, who are concerned about the potential model ambiguity and aim to seek the robust optimal reinsurance and investment strategies. The ambiguity-averse insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks; and can invest in a financial market consisting of one risk-free asset, one risky asset and one defaultable corporate bond. The objective of each insurer is to maximize the expected exponential utility of his terminal surplus relative to that of his competitor under the worst-case scenario of the alternative measures. Applying the techniques of stochastic dynamic programming, we derive the robust Nash equilibrium reinsurance and investment policies explicitly and present the corresponding verification theorem. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.     

Pricing and hedging of variable annuities with state-dependent fees

We investigate the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder’s account depends on the account value. It is believed that state-dependent fees are beneficial to policyholders and insurers since they reduce policyholders’ incentives to lapse the policies and match the costs incurred by policyholders with the pay-offs received from embedded guarantees. We consider an incomplete financial market which consists of two risky assets modelled with a two-dimensional Lévy process. One of the assets is a security which can be traded by the insurer, and the second asset is a security which is the underlying fund for the variable annuity contract. In our model we derive an equation from which the fee for the guaranteed benefit can be calculated and we characterize a strategy which allows the insurer to hedge the benefit. To solve the pricing and hedging problem in an incomplete financial market we apply a quadratic objective.     

Long-tail longitudinal modeling of insurance company expenses

The insurance industry is known to have high operating expenses in the financial services sector. Insurers, investors and regulators are interested in models to understand the behavior of expenses. However, the current practice ignores skewness, occasional negative values as well as their temporal dependence.Addressing these three features, this paper develops a longitudinal model of insurance company expenses that can be used for prediction, to identify unusual behavior, and to measure firm efficiency. Specifically, we use a three-parameter asymmetric Laplace density for the marginal distribution of insurers’ expenses in each year. Copula functions are employed to accommodate their temporal dependence. As a function of explanatory variables, the location parameter allows us to analyze an insurer’s expenses in light of the firm’s characteristics. Our model can be interpreted as a longitudinal quantile regression.The analysis is performed using property–casualty insurance company data from the National Association of Insurance Commissioners of years 2001–2006. Due to the long-tailed nature of insurers’ expenses, two alternative approaches are proposed to improve the performance of the longitudinal quantile regression model: rescaling and transformation. Predictive densities are derived that allow one to compare the predictions for individual insurers in a hold-out-sample. Both predictive models are shown to be reasonable with the rescaling method outperforming the transformation method. Compared with standard longitudinal models, our model is shown to be superior in identifying insurers’ unusual behavior.     

Non-zero-sum stochastic differential reinsurance and investment games with default risk

This paper investigates the implications of strategic interaction (i.e., competition) between two CARA insurers on their reinsurance-investment policies. The two insurers are concerned about their terminal wealth and the relative performance measured by the difference in their terminal wealth. The problem of finding optimal policies for both insurers is modelled as a non-zero-sum stochastic differential game. The reinsurance premium is calculated using the variance premium principle and the insurers can invest in a risk-free asset, a risky asset with Heston’s stochastic volatility and a defaultable corporate bond. We derive the Nash equilibrium reinsurance policy and investment policy explicitly for the game and prove the corresponding verification theorem. The equilibrium strategy indicates that the best response of each insurer to the competition is to mimic the strategy of its opponent. Consequently, either the reinsurance strategy or the investment strategy of an insurer with the relative performance concern is riskier than that without the concern. Numerical examples are provided to demonstrate the findings of this study.     

Equilibrium relationships between insurance and self-protection activity

When an insured understakes some costly self-protection activity that reduces the probability of loss, a competitive insurer will increase the insurance coverage, given a fixed premium per dollar of coverage, to reflect the lower insurance risk.However, an imperfectly informed insurer cannot correctly adjust the coverage; while he can observe the self-protection activity of the insured, the insurer cannot determine the cost to the insured of such activity, nor can the insurer determine the reduction in the loss probability of the insured due to the self-protection activity.This paper demonstrates in an equilibrium model that insurers may be able to use the amount of self-protection activity by an insured as a screen to indicate to the insurer what the loss probability of the insured is, thus allowing the insurer to provide correctly priced insurance to all individuals. The model points out that insurers operating in a market with moral hazard may be able to overcome the adverse incentives of insureds by selectively offering certain insurance contracts contingent upon the insured meeting certain screening requirements; in the model here, self-protection activity is the screen.     

Partial splitting of longevity and financial risks: The longevity nominal choosing swaptions

The previous attempts to launch liquid and standardized longevity derivatives in the market failed because banks do not seem to be ready to take longevity risk. Therefore, instead of trying to transfer longevity risk to investors, it could be interesting for financial institutions to propose interest rate hedges adapted to longevity portfolios, in the spirit of liability driven investments. In this paper, we introduce a new structured financial product: the so-called Longevity Nominal Chooser Swaption. Thanks to such a contract, insurers could keep pure longevity risk and transfer to financial markets a great part of interest rate risk underlying annuity portfolios.We use a population dynamics longevity model and a classical two-factor interest rate model to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology.     

The utility of deductibles from the insurer's point of view

The relationship between the premiums for deductible cover and for full cover are analyzed with respect to the utility for the insurer. Theorem 1 shows that within natural bounds for the premiums neither form of coverage is generally preferred by all insurers. Theorems 2–4 indicate that in many realistic cases a deductible cover requires a higher risk loading factor by the insurer than a full insurance cover. Theorem 2(b) also contains an exception to this rule.     

Optimal investment and risk control for an insurer under inside information

This paper is devoted to the study of the optimal investment and risk control strategy for an insurer who has some inside information on the financial market and the insurance business. The insurer’s risk process and the risky asset process in the financial market are assumed to be very general jump diffusion processes. The two processes are supposed to be correlated. Under the criterion of logarithmic utility maximization of the terminal wealth, we solve our problem by using forward integral approach. Some interesting particular cases are studied in which the explicit expressions of the optimal strategy are derived by using enlargement of filtration techniques.     

具有共同冲击相依性的跳扩散金融市场中有着延迟和违约风险的鲁棒最优再保险和投资策略

在本文中,我们考虑跳扩散模型下具有延迟和违约风险的鲁棒最优再保险和投资问题,保险人可以投资无风险资产,可违约的债券和两个风险资产,其中两个风险资产遵循跳跃扩散模型且受到同种因素带来共同影响而相互关联.假设允许保险人购买比例再保险,特别地再保险保费利用均值方差保费原则来计算.在考虑与绩效相关的资本流入/流出下,保险公司的...     

Robust optimal investment and reinsurance problem for a general insurance company under Heston model

In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.     

Applying simulation optimization to the asset allocation of a property–casualty insurer

Proper asset allocations are vital for property–casualty insurers to be competitive and solvent. Theories of finance offer little practical guidance in constructing such asset allocations however. This research integrates simulation models with a newly developed evolutionary algorithm for the multi-period asset allocation problem of a property–casualty insurer. We first construct a simulation model to simulate operations of a property–casualty insurer. Then we develop multi-phase evolution strategies (MPES) to be used with the simulation model to search for promising asset allocations for the insurer. A thorough experiment is conducted to evaluate the performance of our simulation optimization approach. Computational results show that MPES is an effective search algorithm. It dominates the grid search method by a significant margin. The re-allocation strategy resulting from MPES outperforms re-balancing strategies significantly. This research further demonstrates that the simulation optimization approach can be used to study economic issues related to multi-period asset allocation problems in practical settings.     

Optimal investment,consumption and proportional reinsurance for an insurer with option type payoff

This paper is devoted to the study of optimization of investment, consumption and proportional reinsurance for an insurer with option type payoff at the terminal time under the criterion of exponential utility maximization. The surplus process of the insurer and the financial risky asset process are assumed to be diffusion processes driven by Brownian motions which are non-Markovian in general. Very general constraints are imposed on the investment and the proportional reinsurance processes. Based on the martingale optimization principle, we use BSDE and BMO martingale techniques to derive the optimal strategy and the optimal value function. Some interesting particular cases are studied in which the explicit expressions for the optimal strategy are given by using the Malliavin calculus.     

Exponential utility maximization for an insurer with time-inconsistent preferences

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.     

Constant elasticity of variance model for proportional reinsurance and investment strategies

In our model, the insurer is allowed to buy reinsurance and invest in a risk-free asset and a risky asset. The claim process is assumed to follow a Brownian motion with drift, while the price process of the risky asset is described by the constant elasticity of variance (CEV) model. The Hamilton-Jacobi-Bellman (HJB) equation associated with the optimal reinsurance and investment strategies is established, and solutions are found for insurers with CRRA or CARRA utility.     

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.     

Options on tontines: An innovative way of combining tontines and annuities

Increases in the life expectancy, the low interest rate environment and the tightening solvency regulation have led to the rebirth of tontines. Compared to annuities, where insurers bear all the longevity risk, policyholders bear most of the longevity risk in a tontine. Following Donnelly and Young (2017), we come up with an innovative retirement product which contains the annuity and the tontine as special cases: a tontine with a minimum guaranteed payment. The payoff of this product consists of a guaranteed payoff and a call option written on a tontine. Extending Donnelly and Young (2017), we consider the tontine design described in Milevsky and Salisbury (2015) for designing the new product and find that it is able to achieve a better risk sharing between policyholders and insurers than annuities and tontines. For the majority of risk-averse policyholders, the new product can generate a higher expected lifetime utility than annuities and tontines. For the insurer, the new product is able to reduce the (conditional) expected loss drastically compared to an annuity, while the loss probability remains fairly the same. In addition, by varying the guaranteed payments, the insurer is able to provide a variety of products to policyholders with different degrees of risk aversion and liquidity needs.     

Optimal investment and risk control policies for an insurer: Expected utility maximization

Motivated by the AIG bailout case in the financial crisis of 2007–2008, we consider an insurer who wants to maximize his/her expected utility of terminal wealth by selecting optimal investment and risk control strategies. The insurer’s risk process is modeled by a jump-diffusion process and is negatively correlated with the capital gains in the financial market. We obtain explicit solutions of optimal strategies for various utility functions.     

A family of premium principles based on mixtures of TVaRs

Risk-adjusted distributions are commonly used in actuarial science to define premium principles. In this paper, we claim that an appropriate risk-adjusted distribution, besides satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted distributions, we introduce a family of premium principles that gradually incorporate the degree of risk-aversion of the insurer in the risk loading. Members of this family are particular distortion premium principles that can be represented as mixtures of TVaRs, where the weights in the mixture reflect the attitude toward risk of the insurer. We make a systematic study of this family of premium principles.     

Optimal proportional reinsurance and investment under partial information

In this paper, we study the optimal proportional reinsurance and investment strategy for an insurer that only has partial information at its disposal, under the criterion of maximizing the expected utility of the terminal wealth. We assume that the surplus of the insurer is governed by a jump diffusion process, and that reinsurance is used by the insurer to reduce risk. In addition, the insurer can invest in financial markets. We give a characterization for the optimal strategy within a non-Markovian setting. Malliavin calculus for Lévy processes is used for the analysis.     

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.