Insurance Market Effects of Risk Management Metrics

We extend the classical analysis on optimal insurance design to the case when the insurer implements regulatory requirements (Value-at-Risk). Presumably, regulators impose some risk management requirement such as VaR to reduce the insurers’ insolvency risk, as well as to improve the insurance market stability. We show that VaR requirements may better protect the insured and improve economic efficiency, but have stringent negative effects on the insurance market. Our analysis reveals that the insured are better protected in the event of greater loss irrespective of the optimal design from either the insured or the insurer perspective. However, in the presence of the VaR requirement on the insurer, the insurer's insolvency risk might be increased and there are moral hazard issues in the insurance market because the optimal contract is discontinuous.     

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.     

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 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 investment-reinsurance with dynamic risk constraint and regime switching

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.     

Optimal proportional reinsurance with a loss-dependent premium principle

ABSTRACT

Empirical studies suggest that many insurance companies recontract with their clients on premiums by extrapolating past losses: a client is offered a decrease in premium if the monetary amounts of his claims do not exceed some prespecified quantities, otherwise, an increase in premium. In this paper, we formulate the empirical studies and investigate optimal reinsurance problems of a risk-averse insurer by introducing a loss-dependent premium principle, which uses a weighted average of history losses and the expectation of future losses to replace the expectation in the expected premium principle. This premium principle satisfies the bonus-malus and smoothes the insurer's wealth. Explicit expressions for the optimal reinsurance strategies and value functions are derived. If the reinsurer applies the loss-dependent premium principle to continuously adjust his premium, we show that the insurer always needs less reinsurance when he also adopts this premium principle than when he adopts the expected premium principle.     

Insurance Contract Design When the Insurer Has Private Information on Loss Size

This article examines the optimal indemnity contract in an insurance market, when the insurer has private information about the size of an insurable loss. Both parties know whether or not a loss occurred, but only the insurer knows the true value of the loss and/or to what extent the losses are covered under the policy. The insured may verify the insurer's loss estimate for a fixed auditing cost. The optimal contract reimburses the auditing costs in addition to full insurance for losses less than some endogenous limit. For losses exceeding this limit, the contract pays a fixed indemnity and requires no monitoring. The optimal contract is compared with the contracts obtained in cases where it is only the insured who can observe the loss size.
     

Pareto-Optimal Insurance Policies in the Models with a Premium Based on the Actuarial Value

This article analyzes the problem of designing Pareto‐optimal insurance policies when both the insurer and the insured are risk averse and the premium is calculated as a function of the actuarial value of the insurer's risk. Two models are considered: in the first, the set of admissible policies is constrained by a given size of the premium; in the second, the premium size is not constrained so that it varies with the actuarial value of a policy chosen by the agents. For both cases a characterization of the Pareto‐optimal policies is derived. The corresponding optimality equations for the Pareto‐optimal policies are obtained and compared with the results on the classical risk exchange model.     

Optimal insurance contract under a value-at-risk constraint

This study develops an optimal insurance contract endogenously under a value-at-risk (VaR) constraint. Although Wang et al. [2005] had examined this problem, their assumption implied that the insured is risk neutral. Consequently, this study extends Wang et al. [2005] and further considers a more realistic situation where the insured is risk averse. The study derives the optimal insurance contract as a single deductible insurance when the VaR constraint is redundant or as a double deductible insurance when the VaR constraint is binding. Finally, this study discusses the optimal coverage level from common forms of insurances, including deductible insurance, upper-limit insurance, and proportional coinsurance. JEL Classification G22     

Optimal insurance contracts without the non-negativity constraint on indemnities: revisited

In the literature on optimal indemnity schedules, indemnities are usually restricted to be non-negative. Keeler [1974] and Gollier [1987] show that this constraint might well bind: insured could get higher expected utility if insurance contracts would allow payments from the insured to the insurer at some losses. This paper extends Collier’s findings by allowing for negative indemnity payments for a broader class of insurers’ cost functions and argues that the indemnity schedule derived here is more appropriate for practical applications (e.g. in health insurance). JEL Classification D80 · D81 · D89     

Optimal Insurance Design Under a Value-at-Risk Framework

This study designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the Value-at-Risk (VaR) constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Additionally, this study also calculates the optimal insurance levels for these models when we restrict the indemnity to be one of three common forms: a deductible policy, an upper-limit policy, or a policy with proportional coinsurance. JEL Classification No: G22     

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.     

A constraint-free approach to optimal reinsurance

Reinsurance is available for a reinsurance premium that is determined according to a convex premium principle H. The first insurer selects the reinsurance coverage that maximizes its expected utility. No conditions are imposed on the reinsurer's payment. The optimality condition involves the gradient of H. For several combinations of H and the first insurer's utility function, closed-form formulas for the optimal reinsurance are given. If H is a zero utility principle (for example, an exponential principle or an expectile principle), it is shown, by means of Borch's Theorem, that the optimal reinsurer's payment is a function of the total claim amount and that this function satisfies the so-called 1-Lipschitz condition. Frequently, authors impose these two conclusions as hypotheses at the outset.     

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 reinsurance arrangements in the presence of two reinsurers

In this paper, we investigate the optimal form of reinsurance from the perspective of an insurer when he decides to cede part of the loss to two reinsurers, where the first reinsurer calculates the premium by expected value principle while the premium principle adopted by the second reinsurer satisfies three axioms: distribution invariance, risk loading, and preserving stop-loss order. In order to exclude the moral hazard, a typical reinsurance treaty assumes that both the insurer and reinsurers are obligated to pay more for the larger loss. Under the criterion of minimizing value at risk (VaR) or conditional value at risk (CVaR) of the insurer's total risk exposure, we show that an optimal reinsurance policy is to cede two adjacent layers, where the upper layer is distributed to the first reinsurer. To further illustrate the applicability of our results, we derive explicitly the optimal layer reinsurance by assuming a generalized Wang's premium principle to the second reinsurer.     

Optimal insurance under maxmin expected utility

We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the maxmin expected utility model of Gilboa and Schmeidler (J. Math. Econ. 18:141–153, 1989), whereas the insurer is a (risk-averse or risk-neutral) expected-utility maximiser. We characterise optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, of marginal utility of wealth, of belief heterogeneity, as well as of ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how a singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Rényi ambiguity set.

     

Arrow's Theorem of the Deductible with Heterogeneous Beliefs

In Arrow's classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer—or decision maker (DM)—is a deductible contract when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU maximizer. In Arrow's framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This article reexamines Arrow's problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer's and the DM's subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow's classical result is then obtained as a special case.     

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.     

Interplay of insurance and financial risks in a stochastic environment

Consider an insurer who makes risky investments and hence faces both insurance and financial risks. The insurance business is described by a discrete-time risk model modulated by a stochastic environment that poses systemic and systematic impacts on both the insurance and financial markets. This paper endeavors to quantitatively understand the interplay of the two risks in causing ruin of the insurer. Under the bivariate regular variation framework, we obtain an asymptotic formula to describe the impacts on the insurer's solvency of the two risks and of the stochastic environment.     

Utility Functions

Abstract

This article is a self-contained survey of utility functions and some of their applications. Throughout the paper the theory is illustrated by three examples: exponential utility functions, power utility functions of the first kind (such as quadratic utility functions), and power utility functions of the second kind (such as the logarithmic utility function). The postulate of equivalent expected utility can be used to replace a random gain by a fixed amount and to determine a fair premium for claims to be insured, even if the insurer’s wealth without the new contract is a random variable itself. Then n companies (or economic agents) with random wealth are considered. They are interested in exchanging wealth to improve their expected utility. The family of Pareto optimal risk exchanges is characterized by the theorem of Borch. Two specific solutions are proposed. The first, believed to be new, is based on the synergy potential; this is the largest amount that can be withdrawn from the system without hurting any company in terms of expected utility. The second is the economic equilibrium originally proposed by Borch. As by-products, the option-pricing formula of Black-Scholes can be derived and the Esscher method of option pricing can be explained.