Portfolio insurance under a risk-measure constraint |
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| Institution: | 1. LPMA, Paris VII University, France;2. CMAP, Ecole Polytechnique, France;1. Institute of Contemporary Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475000, PR China;2. School of Mathematics and LPMC, Nankai University, Tianjin 300071, PR China;1. IPAG Business School, 184 boulevard Saint-Germain, 75006 Paris, France;2. Department of Economics, Ecole Polytechnique, Paris, France;3. CREA, Université du Luxembourg, 162A avenue de la Faïencerie, L-1511 Luxembourg, Luxembourg;1. LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France;2. LPTM, CNRS, Université Cergy-Pontoise, 95302 Cergy-Pontoise, France;1. Tippie College of Business, The University of Iowa, Iowa City, IA 52245, United States;2. Stern School of Business, New York University, New York, NY 10012, United States;3. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, United States |
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| Abstract: | We study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor’s utility function subject to the risk-measure constraint. We give a full solution to this non-convex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists) and for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases). |
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