Multiple per-claim reinsurance based on maximizing the Lundberg exponent

Continuous-time optimal reinsurance strategy with nontrivial curved structures

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

ABSTRACTWe discuss classes of risk measures in terms both of their axiomatic definitions and of the economic theories of choice that they can be derived from. More specifically, expected utility theory gives rise to the exponential premium principle, proposed by Gerber (1974), Dhaene et al. (2003), whereas Yaari's (1987) dual theory of choice under risk can be viewed as the source of the distortion premium principle (Denneberg, 1990; Wang, 1996). We argue that the properties of the exponential and distortion premium principles are complementary, without either of the two performing completely satisfactorily as a risk measure. Using generalised expected utility theory (Quiggin, 1993), we derive a new risk measure, which we call the distortion-exponential principle. This risk measure satisfies the axioms of convex measures of risk, proposed by Föllmer & Shied (2002a,b), and its properties lie between those of the exponential and distortion principles, which can be obtained as special cases.

Optimal reinsurance with premium constraint under distortion risk measures

ABSTRACTIn this paper, we consider the ruin probability minimization of an insurance company that buys proportional reinsurance and invests in markets where borrowing is constrained. We use a diffusion approximation model for the surplus process of this company and assume that the company invests its surplus into a riskless and multiple risk assets that are modeled as geometric Brownian motions. To find the results, we introduce an auxiliary market parametrized with fictitious processes to relax the borrowing constraint and apply the techniques of stochastic optimal control. In this way, we find the optimal proportional reinsurance and investment strategy of an insurance company investing into multiple risky assets to minimize its ruin probability under the borrowing constraint. Furthermore, from our solutions, we show how our results connect to economic survival analysis and how investment and reinsurance strategies are related.

This paper considers a non-zero-sum stochastic differential reinsurance and investment game between two competitive insurers under the expected exponential utility. It is assumed that the surplus process of each insurer is characterized by a compound Poisson model and their claim businesses are correlated through thinning dependence structure. Each insurer can purchase proportional reinsurance and is allowed to invest in a risk-free asset and a risky asset. The insurers compete with each other and are concerned with the relative performance. By using the stochastic control technique, not only the existence and uniqueness of the optimal strategies are proved, but also the specific analyses for the constraint reinsurance strategies are provided. We find that (i) the thinning dependence structure has significant impact on optimal reinsurance strategies for both insurers and can help them reduce risks; (ii) the behavior of non-zero-sum game will reduce the proportion of purchasing reinsurance for each insurer.

<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>

The role of a representative reinsurer in optimal reinsurance

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

Christofides (1998) studies the proportional hazards (PH) transform of Wang (1995) and shows that for some parametric families, the PH premium principle reduces to the standard deviation (SD) premium principle. Christofides conjectures that for a parametric family of distributions with constant skewness, the PH premium principle reduces to the SD principle. I will show that this conjecture is false in general but that it is true for location-scale families and for certain other families.Wang's premium principle has been established as a sound measure of risk in Wang (1995, 1996), Wang, Young, and Panjer (1997), and Wang and Young (1998). Determining when the SD premium principle is a special case of Wang's premium principle is important because it will help identify circumstances under which the more easily applied SD premium principle is a reliable measure of risk.First, recall that a distortion g is a non-decreasing function from [0, 1] onto itself. Wang's premium principle, with a fixed distortion g, associates the following certainty equivalent with a random variable X, (Wang, 1996) and (Denneberg, 1994):in which Sx is the decumulative distribution function (ddf) of X, Sx(t) = Pr(X > t), t ∈ R. If g is a power distortion, g(p) = pc, then Hg is the proportional hazards (PH) premium principle (Wang, 1995).Second, recall that a location-scale family of ddfs is , in which Sz is a fixed ddf. Alternatively, if Z has ddf Sz, then {X = μ + σZ: μ∈ R, σ > 0} forms a location-scale family of random variables, and the ddf of . Examples of location-scale families include the normal, Cauchy, logistic, and uniform families (Lehmann, 1991, pp. 20f). In the next proposition, I show that Wang's premium principle reduces to the SD premium principle on a location-scale family. Christofides (1998) observes this phenomenon in several special cases.

This paper considers a proportional reinsurance-investment problem and an excess-of-loss reinsurance-investment problem for an insurer, where price processes of the risky assets and wealth process of the insurer are both described by Markovian regime switching. The target of the insurer is assumed to maximize the expected exponential utility from her terminal wealth with a state-dependent utility function. By employing the dynamic programming approach, the optimal value functions and the optimal reinsurance-investment strategies are derived. In addition, the impact of some parameters on the optimal strategies and the optimal value functions is analyzed, and lots of interesting results are discovered, such as the conclusion that excess-of-loss reinsurance is better than proportional reinsurance is not held in the regime-switching jump-diffusion model.

Optimal reinsurance and investment problem for an insurer with counterparty risk

Optimal reinsurance-investment strategy with thinning dependence and delay factors under mean-variance framework

Optimal Hedging and Reinsurance Strategies under Risk Measures