Portfolio optimization with wealth-dependent risk constraints

Regulatory risk constraints as in the European Solvency II standard formula for insurance companies may lead to wealth-dependent constraints on the investment strategy. We develop two solution approaches for portfolio optimization problems in continuous time with wealth-dependent constraint sets. In the first approach, we reduce the optimization problem to an associate problem with constraints independent of wealth and a different utility function. The associate problem is then solved using known convex duality results. In the second approach, we use a change of control. We apply these results to Solvency II constraint sets and find that even for an investor with HARA utility who inherently reduces risk in times of distress, the constraints help to prevent the investor from taking too much risk in an optimistic market. Furthermore, we measure significant loss in utility and reduction in risk caused by the constraints, and we also evaluate the trade-off between these two effects.