Optimal proportional reinsurance with common shock dependence

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with multiple dependent classes of insurance business, which extends the work of Liang and Yuen (2014) to the case with the reinsurance premium calculated under the expected value principle and to the model with two or more classes of dependent risks. Under the criterion of maximizing the expected exponential utility, closed-form expressions for the optimal strategies and value function are derived not only for the compound Poisson risk model but also for the diffusion approximation risk model. In particular, we find that the optimal reinsurance strategies under the expected value premium principle are very different from those under the variance premium principle in the diffusion risk model. The former depends not only on the safety loading, time and interest rate, but also on the claim size distributions and the counting processes, while the latter depends only on the safety loading, time and interest rate. Finally, numerical examples are presented to show the impact of model parameters on the optimal strategiespostprin

Reinsurance, ruin and solvency issues: some pitfalls

In this paper, we consider optimal reinsurance from an insurer's point of view. Given a (low) ruin probability target, insurers want to find the optimal risk transfer mechanism, i.e. either a proportional or a nonproportional reinsurance treaty. Since it is usually admitted that reinsurance should lower ruin probabilities, it should be easy to derive an efficient Monte Carlo algorithm to link ruin probability and reinsurance parameter. Unfortunately, if it is possible for proportional reinsurance, this is no longer the case in nonproportional reinsurance. Some examples where reinsurance might increase ruin probabilities are given at the end, when claim arrival and claim size are not independent.Dependence; Reinsurance; Ruin probability; Solvency requirements

Excess of loss reinsurance under joint survival optimality

Explicit expressions for the probability of joint survival up to time x of the cedent and the reinsurer, under an excess of loss reinsurance contract with a limiting and a retention level are obtained, under the reasonably general assumptions of any non-decreasing premium income function, Poisson claim arrivals and continuous claim amounts, modelled by any joint distribution. By stating appropriate optimality problems, we show that these results can be used to set the limiting and the retention levels in an optimal way with respect to the probability of joint survival. Alternatively, for fixed retention and limiting levels, the results yield an optimal split of the total premium income between the two parties in the excess of loss contract. This methodology is illustrated numerically on several examples of independent and dependent claim severities. The latter are modelled by a copula function. The effect of varying its dependence parameter and the marginals, on the solutions of the optimality problems and the joint survival probability, has also been explored

A BSDE-based approach for the optimal reinsurance problem under partial information

We investigate the optimal reinsurance problem under the criterion of maximizing the expected utility of terminal wealth when the insurance company has restricted information on the loss process. We propose a risk model with claim arrival intensity and claim sizes distribution affected by an unobservable environmental stochastic factor. By filtering techniques (with marked point process observations), we reduce the original problem to an equivalent stochastic control problem under full information. Since the classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite dimensionality of the filter, we choose an alternative approach based on Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize the value process and the optimal reinsurance strategy in terms of the unique solution to a BSDE driven by a marked point process.Comment: 30 pages, 3 figure

Risk Selection in Natural Disaster Insurance – the Case of France

It is widely recognized that “market failure” prevents efficient risk sharing in natural disaster insurance. As a consequence, many countries adopted institutional frameworks presenting public sector participation, often praised as public-private partnerships. We define risk selection as a situation where private companies pass insurance of high risk agents on to the public “partner”, arguing that this is a potentially important issue in such situations. In order to illustrate our concerns we look at the case of France. We build a simple model that incorporates the main features of the system, such as the uniform premium rate in both high and low risk regions and the existence of a state reinsurer. We show that in our model, risk selection is likely to be present at equilibrium and discuss the policy options available. When comparing with the actual situation in France we find that the “stylized facts” of the system correspond to our results. Additionally, the policies implemented by the government correspond to policies characterized to reduce the potential of risk selection.risk selection, property insurance, reinsurance, France

On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model

We consider a risk model where deficits after ruin are covered by a new type of reinsurance contract that provides capital injections. To allow the insurance company's survival after ruin, the reinsurer injects capital only at ruin times caused by jumps larger than a chosen retention level. Otherwise capital must be raised from the shareholders for small deficits. The problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014) on successive ruin events, we show that an explicit formula for these reinsurance premiums exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. Here ruin events are due either to Brownian oscillations or jumps and reinsurance capital injections only apply in the latter case. The results are illustrated explicitly for two specific risk models and in some numerical examples.Comment: 23 pages, 3 figure

Risk Selection in Natural Disaster Insurance - the Case of France

It is widely recognized that “market failure” prevents e¢ cient risk sharing in natural disaster insurance. As a consequence, many countries adopted institutional frameworks involving public-private partnerships”. We de…ne risk selection as a situation where private companies pass insurance of high risk agents on to the public sector. We argue that this is a potentially important issue in such partnerships, illustrating our concerns with the case of France. We build a simple model that incorporates the main features of the system, such as the risk independent premium rate and the existence of a state reinsurer. We show that in our model, risk selection is likely to be present in equilibrium and discuss the policy options available. We …nd that the "stylized facts" of the French system correspond to our results. Additionally, the policies implemented by the gvernment correspond to policies characterized to reduce the potential of risk selection.Risk selection, property insurance, reinsurance, Franc

Risk Selection in Natural Disaster Insurance -The Case of France

It is widely recognized that "market failure" prevents efficient risk sharing in natural disaster insurance. As a consequence, many countries adopted institutional frameworks presenting public sector participation, often praised as public-private partnerships. We define risk selection as a situation where private companies pass insurance of high risk agents on to the public "partner", arguing that this is a potentially important issue in such situations. In order to illustrate our concerns we look at the case of France. We build a simple model that incorporates the main features of the system, such as the uniform premium rate in both high and low risk regions and the existence of a state reinsurer. We show that in our model, risk selection is likely to be present at equilibrium and discuss the policy options available. When comparing with the actual situation in France we find that the "stylized facts" of the system correspond to our results. Additionally, the policies implemented by the government correspond to policies characterized to reduce the potential of risk selection.risk selection; property insurance; reinsurance; France

Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

The final publication is available at Elsevier via https://doi.org/10.1016/j.insmatheco.2018.11.007 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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.Natural Science Foundation of China [11571189, 11871219, 11871220]111 Project [B14019]Natural Sciences and Engineering Research Council [RGPIN-2016-03975