Generalized PELVE and applications to risk measures

Abstract

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 with Conditional Value at Risk. Extend- ing their work, we here propose a new measure (generalized PELVE, or g-PELVE) that permits the calibration between pairs of monotone risk measures, where the latter risk measure is obtained by integrating the former with respect to a suitable probability measure. We state conditions for the existence and uniqueness of g-PELVE, and derive additional properties for specific families. A study of Generalized Pareto Distributions reveals an interesting correspondence between PELVE and g-PELVE, and ex- plores 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.