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Portfolio Insurance under a risk-measure constraint

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  • Carmine De Franco
  • Peter Tankov

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 nonconvex 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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  • Carmine De Franco & Peter Tankov, 2011. "Portfolio Insurance under a risk-measure constraint," Papers 1102.4489, arXiv.org.
  • Handle: RePEc:arx:papers:1102.4489
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    References listed on IDEAS

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    1. Rama Cont & Peter Tankov, 2007. "Constant Proportion Portfolio Insurance in presence of Jumps in Asset Prices," Working Papers hal-00129413, HAL.
    2. El Karoui, Nicole & Jeanblanc, Monique & Lacoste, Vincent, 2005. "Optimal portfolio management with American capital guarantee," Journal of Economic Dynamics and Control, Elsevier, vol. 29(3), pages 449-468, March.
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    4. Basak, Suleyman & Shapiro, Alexander, 2001. "Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices," Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 371-405.
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    6. Gundel, Anne & Weber, Stefan, 2007. "Robust utility maximization with limited downside risk in incomplete markets," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1663-1688, November.
    7. M. Kaina & L. Rüschendorf, 2009. "On convex risk measures on L p -spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 475-495, July.
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    9. Susanne Emmer & Claudia Klüppelberg & Ralf Korn, 2001. "Optimal Portfolios with Bounded Capital at Risk," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 365-384.
    10. Rama Cont & Peter Tankov, 2009. "Constant Proportion Portfolio Insurance In The Presence Of Jumps In Asset Prices," Mathematical Finance, Wiley Blackwell, vol. 19(3), pages 379-401.
    11. Black, Fischer & Perold, AndreF., 1992. "Theory of constant proportion portfolio insurance," Journal of Economic Dynamics and Control, Elsevier, vol. 16(3-4), pages 403-426.
    12. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    13. Phelim Boyle & Weidong Tian, 2007. "Portfolio Management With Constraints," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 319-343.
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    Cited by:

    1. Goovaerts, Marc & Linders, Daniël & Van Weert, Koen & Tank, Fatih, 2012. "On the interplay between distortion, mean value and Haezendonck–Goovaerts risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 10-18.
    2. Pézier, Jacques & Scheller, Johanna, 2013. "Best portfolio insurance for long-term investment strategies in realistic conditions," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 263-274.
    3. Catherine Donnelly & Russell Gerrard & Montserrat Guillén & Jens Perch Nielsen, 2015. "Less is more: increasing retirement gains by using an upside terminal wealth constraint," Working Papers 2015-02, Universitat de Barcelona, UB Riskcenter.
    4. Donnelly, Catherine & Gerrard, Russell & Guillén, Montserrat & Nielsen, Jens Perch, 2015. "Less is more: Increasing retirement gains by using an upside terminal wealth constraint," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 259-267.

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