Financial institutions have to satisfy capital adequacy tests, e.g., Solvency II for insurers or the Basel Accords for banks. At the same time their main concern is to maximize their own benefit, i.e., they aim for maximizing their expected utility. Combining both aspects leads to a portfolio optimization problem under a risk constraint.

In this talk, we show an example of a tight financial situation in which no reallocation of the initial endowment exists such that the capital adequacy test is satisfied. In this case, the classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the scalarized utility-based multi-asset (SUBMA) risk measure which optimizes the hedging costs and the expected utility of an agent simultaneously subject to the capital adequacy test.

We find that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a one-period financial market model we present sufficient conditions for the SUBMA risk measure to be finite-valued and continuous. Under further assumptions on the utility function we obtain existence and uniqueness results for the optimal hedging strategies. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for specific capital adequacy tests.