Insurance with multiple insurers: A game-theoretic approach

This paper studies the set of Pareto optimal insurance contracts and the core of an insurance game. Our setting allows multiple insurers with translation invariant preferences. We characterise the Pareto optimal contracts, which determines the shape of the indemnities. Closed-form and numerical solutions are found for various preferences that the insurance players might have. Determining associated premiums with any given optimal Pareto contract is another problem for which economic-based arguments are further discussed. We also explain how one may link the recent fast growing literature on risk-based optimality criteria to the Pareto optimality criterion and we show that the latter is much more general than the former one, which according to our knowledge, has not been pointed out by now. Further, we extend some of our results when model risk is included, i.e. there is some uncertainty with the risk model and/or the insurance players make decisions based on divergent beliefs about the underlying risk. These robust optimal contracts are investigated and we show how one may find robust and Pareto efficient contracts, which is a key decision-making problem under uncertainty

Some Aspects of the Economics of Catastrophe Risk Insurance

The ability to share risk efficiently in the economy is essential to welfare and growth. However, the increased frequency of natural catastrophes over the last decade has raised once again questions associated to the limits of insurability in a free-market economy, and to the relevance of public interventions on risk-sharing markets. In this paper, we explore the potential reasons for the lack of insurance specifically associated to catastrophe environmental risks. Our final aim is to link each source of possible market inefficiency to its possible remedies.

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

On the Pricing of Intermediated Risks: Theory and Application to Catastrophe Reinsurance

We model the equilibrium price and quantity of risk transfer between firms and financial intermediaries. Value-maximizing firms have downward sloping demands to cede risk, while intermediaries, who assume risk, provide less-than-fully-elastic supply. We show that equilibrium required returns will be "high" in the presence of financing imperfections that make intermediary capital costly. Moreover, financing imperfections can give rise to intermediary market power, so that small changes in financial imperfections can give rise to large changes in price. We develop tests of this alternative against the null that the supply of intermediary capital is perfectly elastic. We take the US catastrophe reinsurance market as an example, using detailed data from Guy Carpenter & Co., covering a large fraction of the catastrophe risks exchanged during 1970-94. Our results suggest that the price of reinsurance generally exceeds "fair" values, particularly in the aftermath of large events, that market power of reinsurers is not a complete explanation for such pricing, and that reinsurers' high costs of capital appear to play an important role.

HEDGING CROP RISK WITH YIELD INSURANCE FUTURES AND OPTIONS

This paper analyses the optimal hedging decisions for risk-averse producers facing crop risk, assuming crop yield insurance futures and options can be used. The first-best optimal hedge requires a futures position or an option position proportionate to the individual beta depending on whether the financial markets are perceived unbiased or biased. Using yield data for a sample of wheat producers in France, the producers' hedge ratios are derived. These new hedging instruments are more effective to reduce farm yield variability than the individual yield contracts, except if the individual yield guarantee is at least equal to the individual average yield.crop insurance, hedging position, incomplete markets, Marketing, Risk and Uncertainty,

The Design of Optimal Insurance Decisions in the Presence of Catastrophic Risks

This paper deals with the development of decision making tools for managing catastrophic (low probability - high consequences) risks. Catastrophes produce rare and highly correlated claims, which depend on various decision variables, i.e., coverages at different locations, mitigation measures and reinsurance agreements. Joint probability distributions of these claims depicting their complex spatial and temporal interactions and effects of decision variables are analytically intractable. Spatial stochastic models of catastrophes can bypass these difficulties. Catastrophic models combine the simulation of realistic and geographically explicit catastrophic events with the differentiation of property values and insurance coverages in different locations of the region. Catastrophic models can be combined with stochastic optimization techniques to aid decision making on the spatial diversification of contracts, insurance premiums, reinsurance requirements, effects of mitigation measures, and the use of other financial mechanisms. The aim of this paper is to extend a two-stage spatial catastrophic model to dynamic cases reflecting dependencies of risk accumulation processes in time. This extension is important since it can be used for the analysis of decisions under changing frequencies of events and values of properties. It is also possible to incorporate catastrophes caused by the clustering in time of such events as rains and droughts due to persistence in climate. The model can be used by individual insurers, pools of insurers or regulatory authorities

Optimal reinsurance of dependent risks

We analyse the problem of nding the optimal combination of quota-share and stop loss treaties, maximizing the expected utility or the adjustment coecient of the ce- dent, for each of two risks dependent through a copula structure. By risk we mean a line of business or a portfolio of policies. Results are obtained numerically, using the software Mathematica. Sensitivity of the optimal reinsurance strategy to several factors are investigated, including: i) the dependence level, by means of the Kendall's tau and the dependence parameter; ii) the type of dependence, using dierent copulas describing dierent tail behaviour; iii) the reinsurance calculation principles, where expected value, variance and standard deviation principles are considered. Results show that dierent dependence structures, yield signicantly dierent optimal solu- tions. The optimal treaty is also very sensible to the reinsurance premium calculation principle. Namely, for variance related premiums the optimal solution is not the pure stop loss. In general, the maximum adjustment coecient decreases when dependence increases..info:eu-repo/semantics/publishedVersio