PELVE from a regulatory perspective

TL;DR

This paper explores the PELVE measure from a regulatory perspective, proposing methods to determine the appropriate Expected Shortfall level under Solvency II, analyzing mathematical properties, and providing practical recommendations for insurers.

Contribution

It introduces PELVE-inspired methods for multiple insurers, analyzing their mathematical properties and applicability in regulatory contexts.

Findings

Existence and uniqueness of ES levels are established.

Expressions for elliptically distributed payoffs are derived.

Method choice significantly impacts risk assessment in different distribution scenarios.

Abstract

Under Solvency II, the Value-at-Risk (VaR) is applied, although there is broad consensus that the Expected Shortfall (ES) constitutes a more appropriate risk measure. Moving towards ES would necessitate specifying the corresponding ES level. The recently introduced Probability Equivalent Level of VaR and ES (PELVE) determines this by requiring that ES equals the prescribed VaR for a given future payoff, reflecting the situation of an individual insurer. We incorporate the regulator's perspective by proposing PELVE-inspired methods for multiple insurers. We analyze existence and uniqueness of the resulting ES levels, derive expressions for elliptically distributed payoffs and establish limit results for multivariate regularly distributed payoffs. A case study highlights that the choice of method is crucial when payoffs arise from different distribution families. We provide recommendations which of our PELVE-inspired methods are most appropriate in certain scenarios.