Calibrating Distribution Models from PELVE

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 article, we study the converse problem of PELVE calibration; that is, to find a distribution model that yields a given PELVE, which may be obtained either 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.

ACKNOWLEDGMENT

The authors thank Xiyue Han for many helpful comments.

Notes

1 In this article, we use the “small α” convention for $\text{VaR}$ and $\text{ES}.$ Hence, “VaR at 1% confidence” and “ES at 2.5% confidence” correspond to ${\text{VaR}}_{99\%}$ and ${\text{ES}}_{97.5\%}$ in BCBS (2019), respectively.

2 Recall that PELVE is location-scale free, and hence we need to pick two free parameters to specify a distribution calibrated from PELVE.

3 Throughout the article, all terms like “increasing” and “decreasing” are in the non-strict sense.

Additional information

Funding

RW is supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590).