Distortion Risk Measures for Nonnegative Multivariate Risks

24 Pages Posted: 12 Jun 2018

See all articles by Montserrat Guillén I Estany

Montserrat Guillén I Estany

University of Barcelona

José María Sarabia

University of Cantabria - Department of Economics

Jaume Belles-Sampera

University of Barcelona - Riskcenter-IREA

Faustino Prieto

University of Cantabria - Department of Economics

Date Written: June 11, 2018

Abstract

We apply distortion functions to bivariate survival functions for nonnegative random variables. This leads to a natural extension of univariate distortion risk measures to the multivariate setting. For Gini’s principle, the proportional hazard transform distortion and the dual power transform distortion, certain families of multivariate distributions lead to a straightforward risk measure. We show that an exact analytical expression can be obtained in some cases. We consider the independence case, the bivariate Pareto distribution and the bivariate exponential distribution. An illustration of the estimation procedure and the interpretation is also included. In the case study, we consider two loss events with a single risk value and monitor the two events together over four different periods. We conclude that the dual power transform gives more weight to the observations of extreme losses, but that the distortion parameter can modulate this influence in all cases. In our example, multivariate risk clearly diminishes over time.

Keywords: distortion functions; multivariate risk, multiperiod risk assessment, dependence, risk aggregation, multivariate loss.

Suggested Citation

Guillén I Estany, Montserrat and Sarabia, José María and Belles-Sampera, Jaume and Prieto, Faustino, Distortion Risk Measures for Nonnegative Multivariate Risks (June 11, 2018). Journal of Operational Risk, Vol. 13, No. 2, 2018. Available at SSRN: https://ssrn.com/abstract=3194245

José María Sarabia

University of Cantabria - Department of Economics ( email )

Av. Los Castros s/n
Santander 39005, Cantabria 39005
Spain

Jaume Belles-Sampera

University of Barcelona - Riskcenter-IREA ( email )

Av. Diagonal, 690
Barcelona, 08034
Spain

Faustino Prieto

University of Cantabria - Department of Economics ( email )

Av. Los Castros s/n
Santander 39005, Cantabria 39005
Spain

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