Portfolio Insurance under a risk-measure constraint

TL;DR

This paper addresses portfolio insurance with a risk-measure constraint, providing explicit solutions in complete markets and highlighting the importance of the risk measure choice for the existence of optimal portfolios.

Contribution

It offers a full solution to a nonconvex portfolio optimization problem under risk constraints, emphasizing the impact of different risk measures on portfolio existence.

Findings

Optimal portfolio always exists under entropic risk measure

Existence of optimal portfolio depends on the risk measure used

Explicit solutions are derived for spectral risk measures

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).