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This paper studies an optimal insurance problem with finitely many potential policyholders. A monopolistic, risk-neutral insurer applies linear pricing, and cannot discriminate in the insurance premium rate. The individuals are endowed with exponential expected utility preferences, and there is heterogeneity in the risk-aversion parameters. We study two models. In the first model the individuals can self-select their insurance coverage given the market premium rate. We find that partial or no insurance is generally optimal, and the premium optimization can be reduced to a piecewise concave problem. In the second model, the insurer offers only one insurance contract and individuals can either buy it or not. We show that it is optimal for the insurer to offer a full insurance contract. The premium optimization problem is reduced to a discrete problem, where the premium is an indifference premium of one individual in the market. Since the risk-aversion parameters of individuals are generally unobserved, we also present a simulation-based framework in which we simulate the risk-aversion parameters of the individuals. We show that the model with finitely many policyholders converges to the model with a continuum of potential policyholders when the number of potential individuals increases.

Inspired by the recent developments in risk sharing problems for the Value-at-Risk (VaR), the Expected Shortfall (ES), or the Range-Value-at-Risk (RVaR), we study the optimization of risk sharing for general tail risk measures. Explicit formulas of the inf-convolution and Pareto-optimal allocations are obtained in the case of a mixed collection of left and right VaRs, and in that of a VaR and another tail risk measure. The inf-convolution of tail risk measures is shown to be a tail risk measure with an aggregated tail parameter, a phenomenon very similar to the cases of VaR , ES and RVaR. Optimal allocations are obtained in the setting of elliptical models, and several results are established for tail risk measures and risk sharing problems in the presence of model uncertainty. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk sharing.

In the optimal insurance literature one typically studies optimal risk sharing between one insurer (or reinsurer) and one policyholder. However, the insurance business is based on diversification benefits that arise when pooling many insurance policies. In this paper, we first show that results on optimal insurance that are valid in the case of a single policyholder extend to the case of multiple policyholders, provided their insurance claims are independent. However, due to natural catastrophes, increasing life expectancy and terrorism events, insurance claims show tendency to be correlated. Interestingly, in the case of interdependent insurance policies, it may become optimal for the insurer to refuse selling insurance to some prospects, based on their attitude towards risk or due to their risk exposure characteristics. This finding calls for government policies to ensure that insurance stays available and affordable to everyone.

It is natural to connect reinsurance problems with risk measures since a reinsurance contract is an efficient risk management tool for an insurer and the reinsurance premium can also be viewed as a measure of a reinsurer's risk. In this paper, we assume that the insurer uses a law-invariant convex risk measure, while reinsurers use a Wang's premium principle to determine their premiums. We study an optimal reinsurance policy design from an insurer's perspective in a market of multiple reinsurers. Both the insurer's risk measure and the reinsurer's premium principle represent broad families of risk measures with considerable generality. We provide a general formula for the optimal solution which recovers existing results if particular law-invariant convex measures, such as the AVaR, and particular premium principles are assigned.

The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, value at risk (VaR) and expected shortfall, are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.