Portfolio insurance under a risk-measure constraint

• Published in: Insurance, mathematics & economics Vol. 49; no. 3; pp. 361 - 370
• Main Authors: De Franco, Carmine, Tankov, Peter
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.2011; North Holland Publ. Co; Elsevier; Elsevier Sequoia S.A
• Series: Insurance: Mathematics and Economics
• Subjects: Convex risk measures; Entropic risk measure; Fund management; Insurance; Investment advisors; Investors; Mathematics; Optimization; Portfolio analysis; Portfolio insurance; Portfolio insurance Utility maximization Convex risk measures Spectral risk measure Entropic risk measure; Portfolio management; Probability; Risk assessment; Risk management; Spectral risk measure; Studies; Transaction costs; Utility functions; Utility maximization; Utility maximization; Convex risk measures; Portfolio insurance; Entropic risk measure; 91G10; Spectral risk measure
• ISSN: 0167-6687; 1873-5959
• DOI: 10.1016/j.insmatheco.2011.05.009

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). ► We study a portfolio insurance where a fund manager guarantees that the portfolio value will be above a threshold. ► The fund manager pays an initial fee to the bank which refunds, when it is needed, the investor. ► In exchange for this protection, the bank imposes a limit on the risk exposure of the fund manager. ► We give a full solution to this problem. Explicit results are provided for the entropic and spectral risk measures.