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Bernard, He, Yan, and Zhou (Mathematical Finance, 25(1), 154-186) studied an optimal insurance design problem where an individual's preference is of the rank-dependent utility (RDU) type, and show that in general an optimal contract covers both large and small losses. However, their results suffer from the unrealistic assumption that the random loss has no atom, as well as a problem of moral hazard that provides incentives for the insured to falsely report the actual loss. This paper addresses these setbacks by removing the nonatomic assumption, and by exogenously imposing the "incentive compatibility" constraint that both indemnity function and insured's retention function are increasing with respect to the loss. We characterize the optimal solutions via calculus of variations, and then apply the result to obtain explicitly expressed contracts for problems with Yaari's dual criterion and general RDU.Finally, we use numerical examples to compare the results between ours and Bernard et al. K E Y W O R D Sincentive compatibility, indemnity function, moral hazard, optimal insurance design, probability weighting function, quantile formulation, rankdependent utility theory, retention function

Optimal reinsurance indemnities have widely been studied in the literature, yet the bargaining for optimal prices has remained relatively unexplored. Therefore, the key objective of this paper is to analyze the price of reinsurance contracts. We use a novel way to model the bargaining powers of the insurer and reinsurer, which allows us to generalize the contracts according to the Nash bargaining solution, indifference pricing and the equilibrium contracts. We illustrate these pricing functions by means of inverse-S shaped distortion functions for the insurer and the Value-at-Risk for the reinsurer.

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This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution. ARTICLE HISTORY