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1、15DISTORTIONRISKMEASURESANDECONOMICCAPITALByWernerHürlimann,SwitzerlandAbstract.Toprovideincentiveforactiveriskmanagement,itisarguedthatasoundcoherentdistortionriskmeasureshouldpreservesomehigherdegreestop-lossorders,atleastthedegreethreeconvexorder.Suchriskmeasuresarecalledf
2、reeoftailriskorsimplytail-freeriskmeasures.Itisshownthatundersomecommonaxiomsandotherplausibleconditions,atail-freecoherentdistortionriskmeasureidentifiesnecessarilywiththeWangright-tailmeasureortheexpectedvaluemeasure.Thismainresultisappliedtoderiveanoptimaleconomiccapitalfo
3、rmula.Keywords:coherentriskmeasure,distortionriskmeasure,tail-freeriskmeasure,higherdegreeconvexorder,Bernoullidistribution,Paretodistribution,economiccapital1.Introduction.Theaxiomaticapproachtoriskmeasuresisanimportantandveryactivesubject,whichappliestodifferenttopicsofactu
4、arialandfinancialinterestlikepremiumcalculationandcapitalrequirements.BesidesthecoherentriskmeasuresbyArztneretal.(1997/99),oneisinterestedinthedistortionriskmeasuresbyDenneberg(1990/94),Wang(1995/96)andWangetal.(1997).Undercertaincircumstances,distortionriskmeasuresarecohere
5、ntriskmeasures(e.g.Wangetal.(1997),Theorem3).Forthisreason,theycanbeusedtodeterminethecapitalrequirementsofariskybusiness,assuggestedbyseveralauthorsincludingWirchandHardy(1999),Goovaertsetal.(2002)andWang(2002).However,despiteofbeingcoherent,alotofdistortionriskmeasures,like
6、conditionalvalue-at-risk(identicaltoexpectedshortfall)ortheveryrecentWangtransformriskmeasure,donotalwaysprovideincentiveforriskmanagementbecausetheylackofgivingacapitalreliefinsomesimpletwoscenariossituationsofreducedrisk(Examples3.1and3.2).Topreventtheexistenceofsuchpatholo
7、gicalcounterexamples,oneisinterestedindistortionriskmeasuresthatpreservethehigherdegreestop-lossorders(e.g.Hürlimann(2000b),YoshibaandYamai(2001)).Moreprecisely,itisknownthatacoherentdistortionriskmeasurepreservestheusualstochasticorderandtheusualstop-lossorder(e.g.Hürlimann(
8、1998a)).Thisisadesirablepropertybecauseincreasedriskshouldbepenalize
