On peculiarities of CoVaR-based portfolio selection
We consider the portfolio optimization problem with the risk measured by the conditional value-at-risk (as introduced by Adrian and Brunnermeier, i.e. based on the stress event of the chosen asset being equal to the opposite of its value-at-risk level) under the normality assumption. We show that well established drawbacks of conditional value-at-risk, such as the lack of dependence consistency, are not eradicated in the proposed approach. Furthermore, the sign of a certain factor determines the solvability. It depends on the input data: the matrix of correlations, the vector of expected returns and the significance levels for the portfolio as a whole and for the chosen asset. The whimsical behavior of the factor affects not only the existence of the solution, but the set of efficient portfolios as well. Switching to some of conditional value-at-risk modifications brings little improvement. The examples provided illustrate different subcases which may arise. The essential conclusion is that practical users are strongly advised against using conditional value-at-risk in portfolio analysis (for stress-testing or otherwise).