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Abstract
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We study a continuous-time portfolio selection problem where the agent tries to minimize the law-invariant coherent risk measure of terminal payoffs while achieving a pre-specified expected return level. We extend the representation theorem of law-invariant coherent risk measures to the space of lower-bounded random variables in order to be consistent with tame portfolio setting in continuous-time portfolio selection literature. Using the quantile formulation and a min-max theorem, we solve this portfolio selection problem completely. The sufficient and necessary conditions on the well-posedness of the problem and the existence of optimal solutions are derived. It turns out that the optimal value is independent of the expected return level, which results in a vertical efficient frontier on the return-risk plane. At last, we study this problem with an additional uniform lower-bound on terminal payoffs. We solve it completely when the law-invariant coherent risk measure is comonotonic, and the optimal value does not depend on the expected return level either.
This is a joint work with Han Qing Jin (Oxford) and Xun Yu Zhou (Oxford)
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