• We are grateful for comments from participants at the 2011 Conference on Mathematical Finance and Partial Differential Equations at Rutgers and the Perspectives in Analysis and Probability Conference in Honor of Freddy Delbaen at ETH Zurich. We also thank the Associate Editor and an anonymous referee for their valuable suggestions. The first author acknowledges support from the Natural Sciences and Engineering Research Council of Canada; the second author acknowledges support from a start-up fund at Columbia University; the third author acknowledges support from the National Basic Research Program of China (973 Program) (No. 2007CB814902), the Key Laboratory of Random Complex Structures and Data Science, CAS (No. 2008DP173182), and the Science Fund for Creative Research Groups of NNSF (No. 11021161); and the last author acknowledges support from a GRF grant (No. CUHK419511), and research grants from both CUHK and Oxford.


We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank-dependent expected utility (RDEU) theory with a concave utility function and an inverse-S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.