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Every insurance companies have a capacity limit related to the maximum claim that can be borne. Therefore, insurance companies need to reinsure risks to reinsurance companies. Besides of quota-share, types of reinsurance contracts that commonly used is stop-loss. The quota-share reinsurance premium is proportional based on the amount claim that is covered, but not safe against a large claim. While for stop-loss, the reinsurance premium is relatively large but safe for a large claim. So, this paper combines both types of reinsurance to cover the shortcomings with their respective strengths. After being combined, it is necessary to determine the optimal quota-share proportion and stop-loss retention. One criterion of determines optimal proportion and retention is based on Value-at-Risk (VaR) optimization. With the reinsurance premium as a constraint, this optimization problem is solved for each type of reinsurance combination, be it quota-share before stop-loss or stop-loss before quota-share. From each of these types combinations, the result is optimal quota-share proportion and stop-loss retention, so as produce a minimum VaR value from the borne risk by insurance companies. by comparing the results of VaR optimization of these combinations, stop-loss before quota-share is obtained resulting in a more minimum VaR value.

'Catastrophe Risk Financing in Developing Countries' provides a detailed analysis of the imperfections and inefficiencies that impede the emergence of competitive catastrophe risk markets in developing countries. The book demonstrates how donors and international financial institutions can assist governments in middle- and low-income countries in promoting effective and affordable catastrophe risk financing solutions. The authors present guiding principles on how and when governments, with assistance from donors and international financial institutions, should intervene in catastrophe insurance markets. They also identify key activities to be undertaken by donors and institutions that would allow middle- and low-income countries to develop competitive and cost-effective catastrophe risk financing strategies at both the macro (government) and micro (household) levels. These principles and activities are expected to inform good practices and ensure desirable results in catastrophe insurance projects. 'Catastrophe Risk Financing in Developing Countries' offers valuable advice and guidelines to policy makers and insurance practitioners involved in the development of catastrophe insurance programs in developing countries.

It is well-known that reinsurance can be an effective risk management solution for financial institutions such as the insurance companies. The optimal reinsurance solution depends on a number of factors including the criterion of optimization and the premium principle adopted by the reinsurer. In this paper, we analyze the Value-at-Risk based optimal risk management solution using reinsurance under a class of premium principles that is monotonic and piecewise. The monotonic piecewise premium principles include not only those which preserve stop-loss ordering, but also the piecewise premium principles which are monotonic and constructed by concatenating a series of premium principles. By adopting the monotonic piecewise premium principle, our proposed optimal reinsurance model has a number of advantages. In particular, our model has the flexibility of allowing the reinsurer to use different risk loading factors for a given premium principle or use entirely different premium principles depending on the layers of risk. Our proposed model can also analyze the optimal reinsurance strategy in the context of multiple reinsurers that may use different premium principles (as attributed to the difference in risk attitude and/or imperfect information). Furthermore, by artfully imposing certain constraints on the ceded loss functions, the resulting model can be used to capture the reinsurer’s willingness and/or capacity to accept risk or to control counterparty risk from the perspective of the insurer. Under some technical assumptions, we derive explicitly the optimal form of the reinsurance strategies in all the above cases. In particular, we show that a truncated stop-loss reinsurance treaty or a limited stop-loss reinsurance treaty can be optimal depending on the constraint imposed on the retained and/or ceded loss functions. Some numerical examples are provided to further compare and contrast our proposed models to the existing models.

In this paper, we analyze the relationship between the equilibrium reinsurance strategy and the tail of the distribution of the risk. Since Mean Residual Life (MRL) has a close relationship with the tail of the distribution, we consider two classes of risk distributions, Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) distributions, which can be used to classify light-tailed and heavy-tailed distributions, respectively. We assume that the underlying risk process is modelled by the classical Cramér-Lundberg model process. Under the mean-variance criterion, by solving the extended Hamilton-Jacobi-Bellman equation, we derive the equilibrium reinsurance strategy for the insurer and the reinsurer under DMRL and IMRL, respectively. Furthermore, we analyze how to choose the reinsurance premium to make the insurer and the reinsurer agree with the same reinsurance strategy. We find that under the case of DMRL, if the distribution and the risk aversions satisfy certain conditions, the insurer and the reinsurer can adopt a reinsurance premium to agree on a reinsurance strategy, and under the case of IMRL, the insurer and the reinsurer can only agree with each other that the insurer do not purchase the reinsurance.

Optimal reinsurance from an insurer's point of view or from a reinsurer's point of view has been studied extensively in the literature. However, as two parties of a reinsurance contract, an insurer and a reinsurer have conflicting interests. An optimal form of reinsurance from one party's point of view may be not acceptable to the other party. In this paper, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the Value-at-Risk (VaR) risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interests of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance forms in a wide class of reinsurance policies and under the expected value reinsurance premium principle. These optimal reinsurance forms are more complicated than the optimal reinsurance contracts from the perspective of one party only. The proposed models can also be reduced to the problems of minimizing the VaR of one party's loss under the constraints on the interests of both the insurer and the reinsurer.

We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors.

We consider an optimal robust investment and reinsurance problem for a general insurance company which holds shares of an insurance company and a reinsurance company. It is assumed that the decision-maker is ambiguity-averse and does not have perfect information in drift terms of the investment and insurance risks. To capture the ambiguity aversion in the objective function, the criterion of this paper is to minimize a robust value involving the probability of drawdown and a penalization of model uncertainty. By using the technique of stochastic control theory and solving the corresponding boundary-value problems, the closed-form expressions of the optimal strategies are derived explicitly, and a new verification theorem is proved to show that a non-increasing solution to the Hamilton–Jacobi–Bellman equation is indeed our value function. Moreover, we examine theoretically how the level of ambiguity aversion affects the value function and optimal drift distortion. In the end, some numerical examples are exhibited to illustrate the influence of the different investment patterns on our optimal results.

In this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.

We consider the optimization problem of a large insurance company that wants to maximize the expected utility of its surplus through the optimal control of the proportional reinsurance. In addition, the insurer is exposed to the risk of default of its reinsurer at the worst possible time, a setting that is closely related to a scenario of the Swiss Solvency Test.

This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common goal of the reinsurer and the insurer is to maximize the expectation of the weighted sum of their wealth processes and minimize the corresponding variance. Based on the game method, we obtain the Nash equilibrium investment-reinsurance strategies for the above-proposed model and find out the equilibrium strategies when unilateral interest is considered. In addition, the Nash equilibrium investment-reinsurance strategies are deduced under two special premium calculated principles (i.e., the expected value premium principle and the variance value premium principle). We theoretically study the effect of the intertemporal restrictions on Nash equilibrium investment-reinsurance strategies and find the effect depends on the value of some parameters, which differs from the previous researches that generally believed that intertemporal restrictions would make investors avoid risks. Finally, we perform corresponding numerical analyses to verify our theoretical results.