Generalized PELVE and applications to risk measures

The continuing evolution of insurance and banking regulation has raised interest in the calibration of different risk measures associated with suitable confidence levels. In particular, Li and Wang (2019) have introduced a probability equivalent level (called PELVE) for the replacement of Value at Risk (VaR) with Conditional Value at Risk (CVaR). Extending their work, we propose two generalizations of PELVE that combine useful theoretical properties with empirical benefits in risk analysis. The former, termed d-PELVE, establishes a correspondence between VaR and suitably parameterized distortion risk measures. The latter, termed g-PELVE, iterates the construction of CVaR starting from VaR to a general coherent risk measure. We state conditions for the existence and uniqueness of the proposed measures and derive additional properties for specific classes of underlying risk functionals. A study of Generalized Pareto Distributions reveals an interesting correspondence between PELVE and g-PELVE, and explores their relationship with the tail index. An empirical application illustrates the usefulness of (g-)PELVE in characterizing tail behavior not only for individual asset returns, but also for possible portfolio combinations.