Optimal Reinsurance Contracts Research Articles

Static Risk Measures in a Frequency-Severity Framework with Systematic Risk: Application in Reinsurance

This article presents the concept of static risk measures as an approach to assessing risk by focusing on loss severity within a frequency-severity framework that encompasses systematic and common shocks. This article will discuss a set of important properties of static risk measures and compare them with the ruin-based risk measures. It will also give a robust representation of static risk measures and discuss the implications of such representation to designing optimal reinsurance contracts. By introducing a flexible framework, the model accommodates additional elements such as systematic and common shocks, enhancing its applicability in the field of reinsurance.

Bowley solution under the reinsurer's default risk

In this paper, we examine how a monopolistic reinsurer designs a Bowley reinsurance contract, under the assumption that the reinsurer will default on payment if the compensated loss exceeds the sum of the initial capital and the premium charged from the contract. The problem is divided into two subproblems faced by the insurer and the reinsurer in turn. The optimal reinsurance contract is analyzed when both the insurer and the reinsurer minimize their retained risks, as quantified by the VaR measure, and the optimal ceded loss function and the optimal pricing function are provided. Explicit expressions are then derived when the reinsurer adopts either VaR- or TVaR-based regulation capital and charges premiums by the expected-value premium principle. Numerical examples using exponential and Pareto distributions are provided to illustrate the sensitivity effect generated by the confidence levels of the VaR for both parties, as well as those for the initial capitals on the set of Bowley reinsurance contracts.

Optimal reinsurance contract and investment strategy for multiple competitive-cooperative insurers and a reinsurer

Abstract Accepted by: Giorgio Consigli In this article, we consider a reinsurance contract design by taking into account the joint interests of multiple insurers and a reinsurer. The reinsurance contract consists of optimal reinsurance strategies and reinsurance prices. The former are chosen by the competitive–cooperative insurers and the latter are determined by a reinsurer. Both insurers and the reinsurer can invest in the common risk-free asset and one different risky asset. The optimal time-consistent reinsurance–investment strategies of the insurers and the optimal reinsurance prices and investment strategy of the reinsurer are derived analytically. Numerical experiments are carried out to illustrate the influences of model parameters on the optimal reinsurance contract and optimal investment strategies. We find that the establishment of the reinsurance contract is affected by the correlation between the claim sizes of insurers as well as that of claim numbers. The results reveal that competition and cooperation will lead to a decrease and increase of the reinsurance price, respectively, showing the importance of opting for cooperation among insurers. Both competition and cooperation are beneficial to insurers, especially for those with high risk aversion.

Optimal reinsurance contract in a Stackelberg game framework: a view of social planner

In this paper, we consider an optimal reinsurance contract under a mean-variance criterion in a Stackelberg game theoretical framework. The reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal per-loss reinsurance to purchase. The objective of the insurer is to maximize a given mean-variance criterion, while the reinsurer adopts the role of social planner balancing its own interests with those of the insurer. That is, we assume that the reinsurer determines the reinsurance premium by maximizing a weighted sum of the insurer's and reinsurer's mean-variance criteria. Under the general mean-variance premium principle, we derive the optimal reinsurance contract by solving the extended Hamilton–Jacobi–Bellman (HJB) systems. Moreover, we provide an intuitive way to set the weight of each party in the reinsurer's objective. Finally, we consider some special cases to illustrate our main results.

Designing of Optimal Reinsurance Indemnity

This paper contributes to relevant research in the area of optimal reinsurance indemnity and deals with the risk measures that are used in reinsurance. The research aims at finding optimal reinsurance contracts under different risk levels. The paper has demonstrated that the method of calculating the indemnity of the reinsurance contract discussed in the aforementioned article—the reduction of the square of excess of loss—can be generalised and is valid in all instances where p ∈ (0; 1) ∪ (1; +∞). The results could be useful for the insurance companies calculating indemnities for different cases, as they could state the degrees that fit their needs most.

Optimal reinsurance pricing, risk sharing and investment strategies in a joint reinsurer-insurer framework

Abstract We investigate the reinsurance contract and investment strategy problem between an insurer and a reinsurer under the continuous-time framework. For the reinsurance contract design, the joint interests of the insurer and the reinsurer are considered. That is, the insurer determines the claim risk sharing strategy, and the reinsurer determines the reinsurance price. The insurer and the reinsurer can invest in the common risk-free asset and different risky assets to increase their respective wealths. The competition between the insurer and the reinsurer is quantified through the relative performance. Both the insurer and the reinsurer aim at maximizing the expected value of the terminal pay-off while minimizing its variance. By using the stochastic optimal control technique, we derive analytically the optimal time-consistent investment strategy and obtain the optimal reinsurance contract. Finally, the influences of model parameters on the optimal reinsurance contract and time-consistent investment strategy are examined through numerical experiments.

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 Reciprocal Reinsurance under VaR or TVaR Constraint

Abstract Most existing researches on optimal reinsurance contract are based on an insurer’s viewpoint. However, the optimal reinsurance contract for an insurer is not necessarily to be optimal for a reinsurer. Hence, this study aims to develop the optimal reciprocal reinsurance which satisfies the benefits of both the insurer and reinsurer. Additionally, due to legislative restriction or risk management requirement, the wealth of insurer and reinsurer are frequently imposed upon a VaR (Value-at-Risk) or TVaR (Tail Value-at-Risk) constraint. Therefore, this study develops an optimal reciprocal reinsurance contract which maximizes the common benefits (evaluated by weighted addition of expected utilities) of the insurer and reinsurer subject to their VaR or TVaR constraints. Furthermore, for avoiding moral hazard problem, the developed contract is additionally restricted to a regular form or incentive compatibility (both indemnity schedule and retained loss schedule are continuously nondecreasing).

Optimal reinsurance with model uncertainty and Stackelberg game

In this paper, we obtain an optimal reinsurance contract explicitly in the form of excess-of-loss with a limit when the risk measure is Range-Value-at-Risk. Then we study optimal reinsurance with model uncertainty, where the uncertainty set contains a greatest element in the sense of stochastic order. Furthermore, we study the Stackelberg game in reinsurance with model uncertainty. In order to illustrate how our findings can be applied in practice to determine the optimal reinsurance contract, we perform an empirical study using the Wisconsin Local Government Property Insurance Fund dataset. Tweedie distributions are fit to the building and contents loss amounts from the property fund, and used to determine the optimal loading factor, and reinsurance contract corresponding to the loading factor. From the analysis, we discover that some policies have more recoveries from the optimal reinsurance contract than the premium paid, while others have smaller recoveries than the premium.

OPTIMAL REINSURANCE FROM THE VIEWPOINTS OF BOTH AN INSURER AND A REINSURER UNDER THE CVAR RISK MEASURE AND VAJDA CONDITION

ABSTRACTIn this paper, we study the optimal reinsurance contracts that minimize the convex combination of the Conditional Value-at-Risk (CVaR) of the insurer’s loss and the reinsurer’s loss over the class of ceded loss functions such that the retained loss function is increasing and the ceded loss function satisfies Vajda condition. Among a general class of reinsurance premium principles that satisfy the properties of risk loading and convex order preserving, the optimal solutions are obtained. Our results show that the optimal ceded loss functions are in the form of five interconnected segments for general reinsurance premium principles, and they can be further simplified to four interconnected segments if more properties are added to reinsurance premium principles. Finally, we derive optimal parameters for the expected value premium principle and give a numerical study to analyze the impact of the weighting factor on the optimal reinsurance.

Optimal reinsurance for both an insurer and a reinsurer under general premium principles

A reinsurance contract should consider the conflicting interests of the insurer and the reinsurer. An optimal reinsurance contract for one party may not be optimal for another party and it might be unacceptable for another party. Therefore, in this paper, we study the optimal reinsurance models from the perspective of both the insurer and the reinsurer by minimizing their total costs under the criteria of loss function which is defined by the joint value-at-risk, assuming that the reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. We derive the optimal reinsurance policies over three ceded loss function sets, the change-loss reinsurance is optimal among the class of increasing convex ceded loss functions; when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the layer reinsurance is shown to be optimal; the quota-share reinsurance with a limit is always optimal when the ceded loss functions are in the class of increasing concave functions. We further use the expectation premium principle and Dutch premium principle to illustrate the application of our results by deriving the optimal parameters.

Robust reinsurance contracts with risk constraint

ABSTRACTThis paper aims to investigate optimal reinsurance contracts in a continuous-time modelling framework from the perspective of a principal-agent problem. The reinsurer plays the role of the principal and aims to determine an optimal reinsurance premium to maximize the expected utility on terminal wealth. It is supposed that the reinsurer faces ambiguity about the insurance claim process. The insurer acts as the agent whose objective is to determine an optimal retention level in a proportional reinsurance to maximize the expected utility on terminal wealth. It is postulated that the insurer is subject to a dynamic Value-at-Risk constraint, which may be attributed to capital requirements specified by Solvency II. The Hamilton-Jacobi-Bellman (HJB) dynamic programming is adopted to discuss the optimization problems of the reinsurer and insurer. Explicit expressions for the optimal solutions of the problems are obtained in the case of exponential utility functions. Numerical examples are provided to illustrate economic intuition and insights.

On the existence of a representative reinsurer under heterogeneous beliefs

This paper studies a one-period optimal reinsurance design model with n reinsurers and an insurer. The reinsurers are endowed with expected-value premium principles and with heterogeneous beliefs regarding the underlying distribution of the insurer’s risk. Under general preferences for the insurer, a representative reinsurer is characterized. This means that all reinsurers can be treated collectively by means of a hypothetical premium principle in order to determine the optimal total risk that is ceded to all reinsurers. The optimal total ceded risk is then allocated to the reinsurers by means of an explicit solution. This is shown both in the general case and under the no-sabotage condition that avoids possible ex post moral hazard on the side of the insurer, thereby extending the results of Boonen et al. (2016). We subsequently derive closed-form optimal reinsurance contracts in case the insurer maximizes expected net wealth. Moreover, under the no-sabotage condition, we derive optimal reinsurance contracts in case the insurer maximizes dual utility, or in case the insurer maximizes a generic objective that preserves second-order stochastic dominance under the assumption of a monotone hazard ratio.

Dynamic Set-Up for Designing Optimal Reinsurance Contracts

In this paper, we consider a dynamic set-up for designing optimal reinsurance contracts. The main objective is to maximise the lifetime dividends of an insurance company. We study three problems. First, we consider a general dividend maximisation problem with just budget constraints; second, we add solvency conditions to the previous problem; and finally, we consider a problem with solvency constraints and a particular type of dividend functions. We show that the optimal solutions of the first and second problems do not evolve over time and can be found by solving a static problem, whereas the third problem's optimal solution evolves over time and depends on the problem specification and the state variables. We find the dual problem of the third problem and show that the optimal reinsurance contract always has a multilayer structure. Last, we consider some examples and show that in many cases, the contract is indeed a two-layer policy.

Optimal reinsurance contracts under the reinsurer’s risk constraint with VaR risk measures

Most researchers in insurance field have not been considering the reinsurer’s risk. Theoretically, the quantity of reinsurer’s risk is not limited. However, in the real-world insurance industry, reinsurer always put a limit on coverage, otherwise the reinsurer will experience a heavy financial burden when the insurer bears an unexpected large loss. In this paper, we review the optimal reinsurance problem under VaR risk measures framework when the limitation for reinsurer’s risk exposure are limited by a constant. By analyzing the VaR of the insurer’s total risk exposure, optimality can be fulfilled and end up in obtaining the expected reinsurance policy. Through the study, it is reveal that the stop-loss reinsurance with a policy limit is constantly the optimal reinsurance policy under VaR risk measures. Determination of optimal quantity of ceded risk depends on the confidence level, the safety loading and the limitation of the constant. A numerical analysis on company claims data is carried out in order to find the optimal ceded loss function.

Golden options in financial mathematics

This paper deals with the construction of “smooth good deals” (SGD), i.e., sequences of self-financing strategies whose global risk diverges to minus infinity and such that every security in every strategy of the sequence is a “smooth” derivative with a bounded delta. Since delta is bounded, digital options are excluded. In fact, the pay-off of every option in the sequence is continuos (and therefore jump-free) with respect to the underlying asset price. If the selected risk measure is the value at risk, then these sequences exist under quite weak conditions, since one can involve risks with both bounded and unbounded expectation, as well as non-friction-free pricing rules. Moreover, every strategy in the sequence is composed of a short European option plus a position in a riskless asset. If the chosen risk measure is a coherent one, then the general setting is more limited. Indeed, though frictions are still accepted, expectations and variances must remain finite. The existence of SGDs will be characterized, and computational issues will be properly addressed. It will be shown that SGDs often exist, and for the conditional value at risk, they are composed of the riskless asset plus easily replicable short European puts. The ideas presented may also apply in some actuarial problems such as the selection of an optimal reinsurance contract.

On the Existence of a Representative Reinsurer under Heterogeneous Beliefs

This paper studies a one-period optimal reinsurance design model with n reinsurers and an insurer. The reinsurers are endowed with expected-value premium principles and with heterogeneous beliefs regarding the underlying distribution of the insurer’s risk. Under general preferences for the insurer, a representative reinsurer is characterized. This means that all reinsurers can be treated collectively by means of a hypothetical premium principle in order to determine the optimal total risk that is ceded to all reinsurers. The optimal total ceded risk is then allocated to the reinsurers by means of an explicit solution. This is shown both in the general case and under the no-sabotage condition that avoids possible ex post moral hazard on the side of the insurer, thereby extending the results of Boonen et al. (2016). We subsequently derive closed-form optimal reinsurance contracts in case the insurer maximizes expected net wealth. Moreover, under the no-sabotage condition, we derive optimal reinsurance contracts in case the insurer maximizes dual utility, or in case the insurer maximizes a generic objective that preserves second-order stochastic dominance under the assumption of a monotone hazard ratio.

Optimal Reinsurance from the Perspectives of Both Insurers and Reinsurers Under General Distortion Risk Measures

In this article, we propose new methodologies to tackle and the problems studied in [Cai, J., Lemieux, C. and Liu, F., 2016, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, ASTIN Bulletin, 46(3), 815-849]. Optimal reinsurance contracts that minimize the convex combination of the risk measurements of the insurer's loss and the reinsurer's loss are characterized under different types of constraints. The results in Cai et al. (2016) are considerably extended to the general framework of distortion risk measures and distortion premium principle.

Robust reinsurance contracts in continuous time

We investigate reinsurance contract problems in a continuous-time principal-agent framework, where the reinsurer (principal) is concerned about potential model ambiguity in the claims process, but the insurer (agent) trusts the claims process, or vice versa. The reinsurer designs a robust reinsurance contract that maximizes his exponential utility of terminal wealth under the worst-case distribution, subject to the insurer’s incentive constraint. Optimal reinsurance contracts are explicitly derived in different ambiguity situations. We first show that the reinsurer’s robustness preference makes him become more conservative, which induces him to raise the reinsurance price, which then decreases the demand for reinsurance. However, the insurer’s robustness preference increases both the reinsurance price and the demand. Furthermore, the reinsurer continuously adjusts the reinsurance price, leading the insurer to always purchase a constant proportion of reinsurance, no matter who faces ambiguity, or whether ambiguity exists. Finally, the economic implications of model ambiguity are illustrated using numerical examples.

Optimal Reinsurance with Heterogeneous Reference Probabilities

This paper studies the problem of optimal reinsurance contract design. We let the insurer use dual utility, and the premium is an extended Wang’s premium principle. The novel contribution is that we allow for heterogeneity in the beliefs regarding the underlying probability distribution. We characterize layer-reinsurance as an optimal reinsurance contract. Moreover, we characterize layer-reinsurance as optimal contracts when the insurer faces costs of holding regulatory capital. We illustrate this in cases where both firms use the Value-at-Risk or the conditional Value-at-Risk.