A new portfolio choice model in continuous time is formulated for both complete and incomplete markets, where the quantile function of the terminal cash flow, instead of the cash flow itself, is taken as the decision variable. This formulation covers a wide body of existing and new models with law-invariant preference measures, including expected utility maximisation, mean-variance, goal reaching, Yaari's dual model, Lopes' SP/A model, behavioural model under prospect theory, as well as those explicitly involving VaR and CVaR in objectives and/or constraints. A solution scheme to this quantile model is proposed, and then demonstrated by solving analytically the goal-reaching model and Yaari's dual model. A general property derived for the quantile model is that the optimal terminal payment is anti-comonotonic with the pricing kernel (or with the minimal pricing kernel in the case of an incomplete market) if the investment opportunity set is deterministic. As a consequence, the mutual fund theorem still holds in a market where rational and irrational agents co-exist.

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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. KEY WORDS: optimal insurance design, rank-dependent expected utility, inverse-S shaped probability distortion, indemnity, quantile formulation, deductible.

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We formulate and carry out an analytical treatment of a single-period portfolio choice model featuring a reference point in wealth, S-shaped utility (value) functions with loss aversion, and probability weighting under Kahneman and Tversky's cumulative prospect theory (CPT). We introduce a new measure of loss aversion for large payoffs, called the large-loss aversion degree (LLAD), and show that it is a critical determinant of the well-posedness of the model. The sensitivity of the CPT value function with respect to the stock allocation is then investigated, which, as a by-product, demonstrates that this function is neither concave nor convex. We finally derive optimal solutions explicitly for the cases in which the reference point is the risk-free return and those in which it is not (while the utility function is piecewise linear), and we employ these results to investigate comparative statics of optimal risky exposures with respect to the reference point, the LLAD, and the curvature of the probability weighting. This paper was accepted by Wei Xiong, finance.portfolio choice, single period, cumulative prospect theory, reference point, loss aversion, S-shaped utility function, probability weighting, well-posedness