Portfolio optimization with relative tail risk

This paper proposes analytic forms of portfolio conditional value at risk (CoVaR) and the mean of the portfolio loss conditional on it being in financial distress (CoCVaR) on the normal tempered stable market model. Since CoCVaR captures the relative risk of the portfolio with respect to a benchmark return, we apply it to relative portfolio optimization. Moreover, we derive analytic forms for the marginal contribution to CoVaR and the marginal contribution to CoCVaR. We discuss the Monte-Carlo simulation method for calculating CoCVaR and the marginal contributions of CoVaR and CoCVaR. We provide an empirical illustration to show relative portfolio optimization with 30 stocks included in the Dow Jones Industrial Average under distressed conditions. Finally, we apply the risk budgeting method to reduce the CoVaR and CoCVaR of the portfolio based on the marginal contributions to CoVaR and CoCVaR.