Abstract
Risk measures are defined as functionals of the portfolio loss distribution, thus implicitly assuming the knowledge of such a distribution. However, in practical applications, the need for estimation arises and with it the need to study the effects of mis-specification errors, as well as estimation errors on the final conclusion. In this paper we focus on the qualitative robustness of a sequence of estimators for set-valued risk measures. These properties are studied in detail for two well-known examples of set-valued risk measures: the value-at-risk and the maximum average value-at-risk. Our results illustrate, in particular, that estimation of set-valued value-at-risk can be given in terms of random sets. Moreover, we observe that historical set-valued value-at-risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as the maximum likelihood average value at-risk.
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The authors wish to thank the referees for their detailed reports, that helped to improve the research, and, hopefully, will lead to further developments in the future.
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Crespi, G.P., Mastrogiacomo, E. Qualitative robustness of set-valued value-at-risk. Math Meth Oper Res 91, 25–54 (2020). https://doi.org/10.1007/s00186-020-00707-9
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DOI: https://doi.org/10.1007/s00186-020-00707-9