Portfolio selection with tail nonlinearly transformed risk measures—a comparison with mean-CVaR analysis

Abstract

We compare portfolio selection under the tail nonlinearly transformed risk measure (TNT) with mean-CVaR analysis for normally distributed returns. TNT arises as a natural extension to CVaR that additionally distorts the portfolio returns' outcomes by means of a concave transformation function. We first address the portfolio setting where no risk-free asset exists. Here, the global minimum CVaR portfolio, also under fairly realistic asset returns, may not exist. TNT is able to resolve this shortcoming: its additional transformation of the portfolio outcomes ensures that the global minimum TNT portfolio and, accordingly, the ()-efficient frontiers, do always exist. Second, we analyze the setting where the risk-free asset is available. In this case, Tobin's theorem holds both for CVaR and TNT, under the condition that the respective investors are sufficiently risk averse. Under TNT, this condition is less restrictive, Tobin's theorem holds more often. We third address the choice of optimal portfolios under CVaR and TNT. Under CVaR, optimal portfolios exhibit plunging: either the investor invests entirely in the risky tangency portfolio or she invests entirely in the risk-free asset; diversification is never optimal. TNT, in contrast, yields more realistic portfolio structures. Below a certain minimum risk premium, we do not find any risky investment. Once this minimum risk premium is passed, the risky investment is continuous and strictly monotonously increasing in the risk premium. This pattern is in line with empirical evidence on the stock market participation puzzle and the equity premium puzzle.