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

Some optimization and decision problems in proportional reinsurance

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs. We consider the classical risk theory model assuming a Poisson process and an individual claim amount phase-type distributed, modified with a proportional reinsurance with a retention level that is not constant and depends on the level of the surplus. Depending on whether the initial surplus is below or above a threshold level, the discounted penalty function behaves differently. General expressions for this discounted penalty function are obtained, as well as interesting theoretical results and explicit expressions for phase-type 2 distribution. These results are applied in numerical examples of decision problems based on the probability of ruin and on different risk measures of the deficit at ruin if ruin occurs (the expectation, the Value at Risk and the Tail Value at Risk)

Some optimization and decision problems in proportional reinsurance [WP]

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function (Gerber & Shiu, 1998) is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs

Tail approximation for reinsurance portfolios of Gaussian-like risks

We consider two different portfolios of proportional reinsurance of the same pool of risks. This contribution is concerned with Gaussian-like risks, which means that for large values the survival function of such risks is, up to a multiplier, the same as that of a standard Gaussian risk. We establish the tail asymptotic behavior of the total loss of each of the reinsurance portfolios and determine also the relation between randomly scaled Gaussian-like portfolios and unscaled ones. Further we show that jointly two portfolios of Gaussian-like risks exhibit asymptotic independence and their weak tail dependence coefficient is non-negative.Comment: In press, Scandinavian Actuarial Journa

Dynamic Financial Analysis - Understanding Risk and Value Creation in Insurance

The changing business environment in non-life insurance and reinsurance has raised the need for new quantitative methods to analyze the impact of various types of strategic decisions on a company’s bottom line. Dynamic Financial Analysis («DFA») has become popular among practitioners as a means of addressing these new requirements. It is a systematic approach based on large-scale computer simulations for the integrated financial modeling of non-life insurance and reinsurance companies aimed at assessing the risks and the benefits associated with strategic decisions. DFA allows decision makers to understand and quantify the impact and interplay of the various risks that their company is exposed to, and – ultimately – to make better informed strategic decisions. In this brochure, we provide an overview and assessment of the state of the industry related to DFA. We investigate the DFA value proposition, we explain its elements and we explore its potential and limitations.reinsurance, dynamic financial analysis, insurance

Optimal moral-hazard-free reinsurance under extended distortion premium principles

We study an optimal reinsurance problem under a diffusion risk model for an insurer who aims to minimize the probability of lifetime ruin. To rule out moral hazard issues, we only consider moral-hazard-free reinsurance contracts by imposing the incentive compatibility constraint on indemnity functions. The reinsurance premium is calculated under an extended distortion premium principle, in which the distortion function is not necessarily concave. We first show that an optimal reinsurance contract always exists and then derive two sufficient and necessary conditions to characterize it. Due to the presence of the incentive compatibility constraint and the nonconcavity of the distortion, the optimal contract is obtained as a solution to a double obstacle problem. At last, we apply the general result to study three examples and obtain the optimal contract in (semi)closed form

Optimal reinsurance of dependent risks

Mestrado em Actuarial ScienceEsta Tese foca-se no problema do resseguro ótimo para dois riscos dependentes, do ponto de vista da seguradora que cede o risco. A dependência entre os dois riscos é modelada através de cópulas. O problema de otimização a resolver consiste em encontrar a combinação de tratados de quota-share e stop-loss, para cada risco, que maximiza a utilidade esperada ou o coeficiente de ajustamento do lucro total da seguradora. Sabe-se que estes dois critérios estão ligados e que o coeficiente de ajustamento está relacionado com a probabilidade da seguradora ficar insolvente em tempo finito, através da desigualdade de Lundberg. Os resultados foram obtidos numericamente, usando o software Mathematica. A sensibilidade da estratégia de resseguro ótimo a vários valores do parâmetro de dependência, a diferentes distribuições dos riscos subjacentes e a diversos princípios de cálculo de prémios de resseguro foi analisada para três famílias diferentes de cópulas, descrevendo diferentes comportamentos da cauda da distribuição conjunta. Os resultados mostram que as dependências alteram o tratado de resseguro ótimo. Diferentes estruturas de dependência, i.e. diferentes cópulas, produzem diferentes valores para os níveis ótimos de retenção. No caso do princípio do valor esperado calculado sobre o risco total cedido, o tratado stop-loss puro é sempre ótimo, mas isso não acontece para os restantes princípios de cálculo de prémios. Em geral, o nível ótimo de retenção do tratado de quota-share decresce quando a dependência entre os riscos aumenta. Para todos os casos considerados, o coeficiente de ajustamento máximo diminui quando a dependência aumenta.This Thesis focuses on optimal reinsurance problem for two dependent risks, from the point of view of the ceding insurance company. We assume that the two risks are dependent by means of a copula structure. By risk we mean a line of business, a portfolio of policies or a policy. The problem consists in finding the optimal combination of quota-share and stop loss treaties, for each risk, that maximizes the expected utility or the adjustment coefficient of the total wealth of the insurer. It is known that these two criteria are connected and moreover the adjustment coefficient is related to the ultimate probability of ruin of the insurer through the Lundberg inequality. Results are obtained numerically, using the software Mathematica. Sensitivity of the optimal reinsurance strategy to several values of the dependence parameter, to different distributions of the underlying risks and to a variety of reinsurance premium calculation principles are performed in three families of copulas describing different tail behaviours of the joint distribution function. Results show that dependencies alter the optimal treaty. Different dependence structures, i.e. different copulas, provide different values for the optimal retention levels. In the case of the expected value principle computed on the total ceded risk, the pure stop loss contract is always optimal, but that is not the case for the remaining premium computation principles. In general, the QS retention level decreases when dependence between the risks increases. For all cases considered, the maximum adjustment coefficient decreases when dependence increases.info:eu-repo/semantics/publishedVersio

Optimal joint survival reinsurance: An efficient frontier approach

The problem of optimal excess of loss reinsurance with a limiting and a retention level is considered. It is demonstrated that this problem can be solved, combining specific risk and performance measures, under some relatively general assumptions for the risk model, under which the premium income is modelled by any non-negative, non-decreasing function, claim arrivals follow a Poisson process and claim amounts are modelled by any continuous joint distribution. As a performance measure, we define the expected profits at time x of the direct insurer and the reinsurer, given their joint survival up to x, and derive explicit expressions for their numerical evaluation. The probability of joint survival of the direct insurer and the reinsurer up to the finite time horizon x is employed as a risk measure. An efficient frontier type approach to setting the limiting and the retention levels, based on the probability of joint survival considered as a risk measure and on the expected profit given joint survival, considered as a performance measure is introduced. Several optimality problems are defined and their solutions are illustrated numerically on several examples of appropriate claim amount distributions, both for the case of dependent and independent claim severitie