By Greg N. Gregoriou (eds.)
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Matthys, G. and Beirlant, J. (2003) “Estimating the Extreme Value Index and High Quantiles with Exponential Regression Models”, Statistica Sinica, 13: 853–80. Moscadelli, M. (2004) “The Modelling of Operational Risk: Experience with the Analysis of the Data Collected by the Basel Committee”, Working Paper no. 517, Bank of Italy. Picklands, J. (1975) “Statistical Inference Using Extreme Order Statistics”, The Annals of Statistics, 3: 119–31. V. (1939) “Sür Les Écarts de la Courbe de Distribution Empirique”, Matematiceskii Sbornik, 6: 3–26.
This issue is addressed in the following proposition whose proof is found in Appendix B. 1. A M I Y A T O S H P U R N A N A N D A M E T A L.
4. 26 INCORPORATING DIVERSIFICATION INTO RISK MANAGEMENT Deﬁne a trivial acceptable portfolio ηc consisting of $1 invested only in riskfree capital, in other words, a N + 1 vector with one as the ﬁrst element and zero in the remaining elements. This portfolio has the property Pηc = (1 + r)1 > 0 where 1 is a N + 1 vector of ones. 1, portfolio risk is deﬁned in terms of the l2 norm, N 2 ||x − y||2 equals i=0 (xi − yi ) , on M. Our risk function ρ(η) maps from the domain of portfolio holdings, M, into the non-negative real line, ρ(η): 1.