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Generalized PELVE and applications to risk measures

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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 (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.

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Notes

  1. Note that this integrability condition is automatically fulfilled, for instance, for any \(X \in L^{\infty }\).

  2. It is worth emphasizing that [1, 2] use sign notations that are slightly different from ours. The increasing monotonicity property in our case corresponds to a decreasing monotonicity property in [1, 2].

  3. For similar arguments see the proof of Proposition 2 in Li and Wang [21].

  4. There are several ways to characterize the L function into an explicit expression, in particular, Zhou [32], Sect. 2 , provides a representation in terms of spectral measure and discusses the interpretation of L in relation to the tail dependence structure.

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Acknowledgements

We are grateful to the Editor and two anonymous Referees for useful comments and remarks that contributed to improve the original manuscript. We would like to thank the organizers and participants in OWARS Seminar Series (January 26, 2022), 24th International Congress on Insurance: Mathematics and Economics (July 05–09, 2021), STAR seminars in Oslo (March 2022), Workshop of Quantitative Finance in Rome (April 2022) for insightful suggestions and stimulating discussion on the subject of the paper.

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  1. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano Bicocca, Via Bicocca Degli Arcimboldi 8, 20126, Milan, Italy

    Anna Maria Fiori & Emanuela Rosazza Gianin

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  1. Anna Maria Fiori
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  2. Emanuela Rosazza Gianin
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Correspondence to Anna Maria Fiori.

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Fiori, A.M., Rosazza Gianin, E. Generalized PELVE and applications to risk measures. Eur. Actuar. J. 13, 307–339 (2023). https://doi.org/10.1007/s13385-022-00320-6

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