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1、1CatBondPricingUsingProbabilityTransformsbyShaunS.Wang1.IntroductionCatastrophe(cat)bondsareinsurance-linkedsecuritiesthattranslatereferencedcatrisks(naturalcatastrophesorman-madecats)intocreditrisks.Intheeventofacatloss,investors,asholdersofcatbond,maylosepar
2、toralloffuturecouponpaymentsandprincipalofthebond.Catlossescoveredunderthecatbondareoftencharacterisedbyalossexceedancecurve,S(x)=Pr{X>x},thatis,theprobabilitythatthecatlossXwillexceedamount$x.Thelossexceedancecurveisrelatedtothecumulativedistributionfunction(
3、CDF)byF(x)=1−S(x).Foracatbond,investorsaregenerallyprovidedwithaloss-exceedancecurveS(x)thatisobtainedeither:•byrunningcompanyexposuredatathroughcommerciallyavailablecatmodellingsoftware,or•bydesigningpayoutfunctionsalongsomeparametricindicators(e.g.theRichter
4、Scaleofanearthquakeataspecifiedlocation,anaggregateindustrylossindex,etc).EmbeddedinalossexceedancecurveS(x)includeinformationof•theexpectedfrequencyofdefault,and•therecoveryrate,givendefault.Ascompensationforcreditrisks,justlikecorporatebonds,catbondsnormally
5、offerinvestorsyieldsthatarehigherthantherisk-freeinterestrate(e.g.theLIBOR).Theexcessyieldsspreadovertherisk-freerateshouldmorethancompensatefortheexpecteddefaultrate,sinceitshouldalsocontainriskloadforuncertaintiesassociatedwiththedefaultrisk.Forinvestors,iti
6、sdesirabletocomparetherelativeattractivenessoftheyieldsspreadsbetweencatbondsandcorporatebonds.Inordertocomparerisk-adjustedperformanceofvariousassetclasses,wewouldneedacommonyardstickthatisapplicabletoalltypesofrisks.Formutualfunds,apopularmeasureofrisk-adjus
7、tedperformanceistheSharperatio,namelytheexcessreturnperunitofvolatility.TheSharperatioworkswellforassetswhosereturnsfollownormaldistributions.However,forasinglecatbondissue,wecannotreadilyapplythetraditionalSharperatioconceptsincetheassetreturnisskewedandwithj
8、umps:mostoftheprobabilitymassiscenteredatzeroloss,whilethereisasmallprobabilityofpotentiallylargenegativereturns.Inthispaper,IwilluseprobabilitytransformstoextendtheSharperatioconc
