Calibrating distribution models from PELVE

TL;DR

This paper addresses the problem of calibrating distribution models to match a specified PELVE value, providing methods for different constraint types and analyzing properties like monotonicity and convergence.

Contribution

It introduces a framework for PELVE calibration from data or expert opinion, including differential equation approaches for curve constraints.

Findings

Developed calibration techniques for one-point, two-point, n-point, and curve constraints.

Applied methods to insurance datasets for estimation and simulation.

Provided new theoretical results on PELVE's monotonicity and convergence.

Abstract

The Value-at-Risk (VaR) and the Expected Shortfall (ES) are the two most popular risk measures in banking and insurance regulation. To bridge between the two regulatory risk measures, the Probability Equivalent Level of VaR-ES (PELVE) was recently proposed to convert a level of VaR to that of ES. It is straightforward to compute the value of PELVE for a given distribution model. In this paper, we study the converse problem of PELVE calibration, that is, to find a distribution model that yields a given PELVE, which may either be obtained from data or from expert opinion. We discuss separately the cases when one-point, two-point, n-point and curve constraints are given. In the most complicated case of a curve constraint, we convert the calibration problem to that of an advanced differential equation. We apply the model calibration techniques to estimation and simulation for datasets used in insurance. We further study some technical properties of PELVE by offering a few new results on monotonicity and convergence.