Portfolio Optimisation with a Value at Risk Constraint in the Presence of Unhedgeable Risks

Abstract

This paper addresses the type of portfolio optimisation problem that most European insurance companies will face after the introduction of Solvency II (the new regulatory framework in Europe to be introduced in 2013). Solvency II will limit the total Value at Risk of an insurance company. In this paper therefore we derive the optimal portfolio of hedgeable risks when also unhedgeable risks are present and the sum of both risks is constrained by a Value at Risk constraint. This paper extends the current literature on portfolio optimisation by including both a Value at Risk constraint and unhedgeable risks where in the current literature maximally only one of these two is included. To obtain flexibility with respect to assumptions regarding the probability functions of both the hedgeable and unhedgeable risks, the state price density and the utility function used, the problem is optimised numerically. An example shows the importance of a correct specification of the characteristics of the hedgeable risk. The results also show that the optimal portfolio is much less skewed than the optimal portfolio that is obtained when only hedgeable risks are present.