Ambiguity Aversion, Generalized Esscher Transform, And Catastrophe Risk Pricing

Transcription

1 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing Ambiguiy Avrsion, Gnralizd Esschr ransform, And Caasroph Risk Pricing Wng ZHU School of Financ Shanghai Univrsiy of Financ and Economics Shanghai, , P. R. China Phon: Fax: ABSRAC oivad by h obsrvaion ha h sprad prmium of CA loss bonds is vry high rlaiv o h xpcd loss of h bond principal, and ha h prmium sprad is much mor pronouncd for CA bonds wih low probabiliy ha a coningn loss paymn o h bond issurs will b riggrd, w xnd h radiional Esschr ransform o a gnralizd framwork and ra i as a sochasic discoun facor o pric h CA risk. h gnralizd Esschr ransform is drivd from a modifid quilibrium modl by allowing a rprsnaiv agn o ac in a robus conrol framwork agains modl misspcificaion wih rspc o rar vns in h sns of Andrson, Hansn and Sargn h modl is xplicily solvd and h drivd pricing krnl is shown o b xacly h gnralizd Esschr ransform. Using caasroph bonds daa, w xamin h mpirical implicaion of our modl. JE Classificaion: G12, G13 Ky Words: Ambiguiy Avrsion; Esschr ransform; Caasroph Risk Pricing; CA Bonds 1

2 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing 1. Inroducion In rcn yars, h pricing of caasroph CA risks has bn of major inrs among insuranc and acuarial profssionals. his yp of rsarch inrs is mainly du o h ris of h magniud of disasr losss and hir rsuling ffcs on insuranc and rinsuranc indusry in h pas dcads. hs normous incrass in disasr losss hav challngd h abiliy of priva insuranc and rinsuranc indusry as a mchanism o provid covrag agains caasroph risks. wo yps of soluions o h insuranc capaciy gap hav bn proposd and pu ino pracic. On is mandaory public provision of insuranc, which rlis on govrnmn o sprad losss across ciizns. h ohr is hrough CA risk scuriizaion. Sinc h incpion of CA risk scuriizaion in 1992 wih h inroducion of indx-linkd caasroph loss fuurs by h Chicago Board of rad CBO, h mark of insuranc-linkd scuriis has volvd ino a businss of mor han US$ 10 billion issuancs. A major sgmn of h CA scuriis mark has bn caasroph-linkd bonds CA bonds wih hir arlis inroducion by Winrhur R, USAA, and Swiss R in abl 1 displays som CA bonds issud from 1997 hrough 2000 rpord by Goldman-Sachs. Plas insr abl 1 abou hr hr ar wo imporan poins o b mad from abl 1. Firs, h sprad prmium of CA loss scuriis is vry high rlaiv o h xpcd loss of bonds principal. o s his, no ha h raio of h sprad prmium o h xpcd loss of bond principal rangs from 2.28 o 50, wih h avrag of Sinc hisorical daa suggss ha caasroph risk can usually b lookd as uncorrlad wih h capial mark, or mor xacly, amouns o a small fracion of h oal walh in h conomy, hy ough o b pricd a clos o h risk-fr inrs ra. In ohr words, h CA bond prmium should qual acuarial fair losss covrd by h conrac. h fac ha h CA bond sprad is far abov h xpcd loss of bond principal is conradicory o any sandard capial 2

3 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing mark hory such as CAP hory or Arbirag Pricing hory. hr is a scond, mor subl poin o b akn from abl 1. I appars ha h prmium sprad is much mor pronouncd for CA bonds wih low probabiliy ha a coningn loss paymn o h bond issurs will b riggrd. his may gnra a kind of smirk parn in h cross-scional plo of h prmium o E[loss] raio agains h probabiliy of coningn loss s Figur 1. As a comparison, hr is no apparn rlaion bwn h raio and h xpcd loss prcnag condiional on a loss occurrnc, which indicas ha h prmium implici in CA bonds is mainly snsiiv o h rarnss of h caasroph vns. Wih h abov wo mpirical facs in mind, i is worh considrabl inrs o xplor an conomic modl xplaining why sprads in h CA bond mark ar so high and why, moving from h CA bonds wih larg probabiliis of loss occurrnc o lowr ons, h prmium sprads bcom mor pronouncd. Basd on h conomic modl, i will as wll b inrsing o dvlop a mhodology for pricing h CA linkd scuriis and rinsuranc conracs. Plas insr Figur 1 abou hr Prvious liraur on CA bonds and ohr rlad caasroph conracs could b roughly dividd ino wo major groups: h firs group of aricls concnrad on h possibl conomic xplanaions for high sprad puzzl S,.g., Banwal and Kunruhr 2000, Froo For xampl, Banwal and Kunruhr suggsd ha myopic loss avrsion and prospc hory, ambiguiy avrsion, slcion bias and hrshold bhavior, impac of worry and fixd cos of ducaion may accoun for h puzzl. h scond group dvos o dscrib h pricing formula of CA-linkd conracs. S, for xampl, Cummins and Gman 1995, Gman and Yor 1997, and Cummins, wis and Phillips On common problm wih hs rsarchs is ha hy faild o xplain h scond mpirical fac 3

4 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing mniond abov, i.., hy did no ry o xplain why prmium sprads bcom lss pronouncd whn w mov from CA bonds wih low probabiliis of loss occurrnc o largr ons. Anohr problm is ha mos CA risk pricing formulas ar sill basd on a rahr prfc framwork and did no accoun for h imprfc propris of agn bhavior as lisd in Banwal and Kunruhr 2000 ino hir modls. In his papr, w will inroduc a kind of gnralizd Esschr ransform for h pricing of CA risk. h gnralizd ransform will b suppord by an conomic modl dscribing h agn bhavior of ambiguiy avrsion in h sns of Andrson, Hansn and Sargn o xplain h scond mpirical fac mniond abov, w xamin h implicaions of varying ambiguiy avrsion oward rarnss of caasroph vns. Wihou considring ambiguiy avrsion, our gnralizd ransform will b rducd o h Esschr ransform inroducd in Grbr and Shiu From his prspciv, his papr can b viwd as an addiion o h lis of many acuarial and financial rsarchs conribuing o h gnralizaion of h original Grbr-Shiu framwork. h rs of h papr is organizd as follows. Scion 2 proposs a kind of gnralizd Esschr ransform for compound Poisson procss risk pricing and ss up an conomic quilibrium modl o xamin h conomic implicaion of h gnralizd ransform. Scion 3 dmonsras how o us h ransform o pric h caasroph-linkd conracs. his scion also prsns an simaion of h modl using h CA bonds daa and valuas h modl fficincy by xamining CA bonds daa. Scion 4 concluds. chnical dails ar collcd in h Appndix. 2. Gnralizd Esschr ransform and Economic Implicaions 2.1 Sochasic Discoun Facor and Gnralizd Esschr ransform Ovr h pas svral dcads, hr appars a ndncy owards a unificaion of financial conomics and acuarial horis. Boh insuranc and financ ar inrsd in h fair pricing of financial producs and h horical and mpirical dvlopmns for ass pricing in boh filds currnly bcom mphasizing h concp of sochasic discoun facor ha rlas payoffs for 4

5 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing coningn claims o mark prics in an quilibrium mark. h basic quaion for h sochasic discoun facor can b wrin as follows: C = E [ η, Z ], 2.1 whr C is h im- pric of a coningn claim wih random payoff Z a im ; E is h condiional xpcaion opraor condiioning on h informaion availabl up o im, and η, is h so-calld sochasic discoun facor, or SDF. If marks ar compl, hn h sochasic discoun facor is uniqu. Bu compl cas is rar in insuranc and as a consqunc, hr will b infinily many such sochasic sa prics so a naural qusion o com in his cas is which sochasic discoun facor should b applid. A paricularly racabl spcificaion for h sochasic discoun facor is η, = xp δ ζ / ζ, and ζ has h form ζ = ζ ; α = xp αx / E[xp α X ], whr X is a spcifid risk procss; δ is h risk-fr coninuously compoundd inrs ra. α is a ral numbr paramr and E is h xpcaion opraor. his form of SDF is calld Esschr ransform, which can b dad back o h Swdish acuary F. Esschr. Grbr and Shiu 1994 pionrd h us of Esschr ransform as a kind of SDF and applid i in pricing sock opions. Esschr ransform spcifid in Grbr and Shiu 1994 involvs only on fr paramr, which is rlad wih h risk avrsion cofficin, and is hus sill wihin h subjciv xpcd uiliy framwork. As xplaind in h inroducion, h radiional conomic hory basd on subjciv xpcd uiliy funcion has difficuly xplaining h high prmium sprad of CA bonds and sprad discrpancy of bonds wih diffrn loss probabiliis. As a consqunc, h radiional Esschr ransform sms no propr as a pricing principl assigning prmium o caasroph risk. hus in his papr w dvlop a gnralizd Esschr ransform o pric CA risk. 5

6 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing Sinc CA losss is usually dscribd as a compound Poisson procss, h following argumn will b applid o a gnral risk Y which follows a compound Poisson procss; ha is, Y = = N j j 1, 2.2 whr N is a Poisson procss wih innsiy λ>0 and j j=1,2,f, indpndn of ach ohr and N, dnos h random loss amoun. For convninc, w will assum h random loss variabl j can b dscribd by idnical disribuion funcion, Pr x F x,whrf x dnos h disribuion funcion of a random variabl. j = For h risk procss {Y } spcifid abov, w dfin ζ o hav h following gnralizd form: ζ = ζ ; α, β = xp αy + βn / E[xp αy + βn ]. 2.3 Fx; b h disribuion funcion of Y, i.., F x; = Pr Y x, h gnralizd Esschr ransform of Y is hus dfind as a random variabl having h cumulaiv disribuion funcion F x, ; α, β = E[ I Y x ζ ; α, β ],whr I dnos an vn indicaor funcion. In ohr words, for gnralizd Esschr ansform bsids applying Esschr ransform o h original CA loss procss o rprsn risk avrsion, w augmn an Esschr ransform applying only o Poisson procss N o rprsn ambiguiy avrsion of agns. In nx scion, i is shown ha his form of SDF is suppord by an quilibrium conomy wih agns who ar avrs no only o risk bu also o uncrainy wih rspc o loss occurrnc in a robus conrol framwork. Similar o h drivaion in Grbr and Shiu 1994, h corrsponding momn-gnraing funcion of h random variabl Y undr h gnralizd Esschr ransform can hn b calculad as Y z, ; α, β = E[xp α + z Y + βn / E[xp αy + βn ]] 6

7 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing = xp[ λ = xp[ λ α + z α β β 1 ]/xp[ λ α β 1 ] α + z 1 ]. 2.4 α whr dnos h momn gnraing funcion of h random variabl. Hnc h gnralizd Esschr ransform of h compound Poisson procss Y is again a compound Poisson procss, wih modifid Poisson paramr λ β α and loss amoun α + z bcoms a random variabl whos momn-gnraing funcion is. α Exampl 1: Gamma cas G x; a, b dno h Gamma disribuion wih shap paramr a and scal paramr b, a b x a 1 by G x; a, b = y dy, x 0. Γ a 0 If loss amoun follows Gamma disribuion, h momn gnraing funcion of h Y hn bcoms Y b a z, = xp λ [ 1]. b z Hnc h corrsponding momn gnraing funcion wih h modifid probabiliy disribuion is b a b α a Y z, ; α, β = xp λ β [ 1], b α b α z which shows ha h ransformd procss is of h sam yp, wih paramrs λ, a, b rplacd by λ β b b, a, b α b α. 2.2 h Economic Implicaions In his scion w sablish an quilibrium modl and discuss h undrlying conomic implicaion o h gnralizd Esschr ransform inroducd abov. W assum hr xiss an insuranc mark whr h insuranc risk Y is radd. A rprsnaiv agn sars wih an iniial walh w a h iniial im 0 and bsids invsing in h risk-fr bond; 7

8 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing h agn will also invs in CA risk conracs. h risk-fr bond valu is accumulad a risk-fr coninuously compoundd inrs ra δ. W furhr assum a im Ûh agn undrwris a par of h oal risk Y wih dnos a rminal, prspcifid im. h pric of insuranc covring Y is c;h proporion h agn undrwris in h full insuranc is m so h oal prmium h agn rcivs is mc. W assum all loss paymns occur a im. h agn s ass procss W 0 < hnc follows W = W 0 δ wih W 0 =w+mc. As inroducd in h inroducion, in his papr, w dvia from h sandard approach by considring h rprsnaiv agn who, in addiion o bing risk avrs, also xhibis uncrainy o h insuranc risk modl in a robus conrol framwork in h sns of Andrson, Hansn and Sargn In h robus conrol sings, h agn is assumd o dal wih h risk modl as follows. Firs, having noicd h unrliabl aspcs of h modl simaion basd on xising informaion, h valuas alrnaiv modl dscripion. Scond, acknowldging h fac ha h rfrnc modl is indd h bs saisical characrizaion of h availabl informaion, h pnalizs h choic of alrnaiv modl by a disanc funcion masuring how far i dvias from h rfrnc modl. Andrson, Hansn and Sargn 2000 masur discrpancy bwn alrnaiv modl and rfrnc modl by rlaiv nropy, dfind as h xpcd valu of a log-liklihood raio. or xacly, ling P = P b h probabiliy masur associad wih h rfrnc modl, h alrna modl b dscribd by a probabiliy masur ~ ~ P = P, inwhichp dnos h P s of all alrnaiv probabiliy masurs ha may b chosn. Dnos ξ = d P ~ / dp as h Radon-Nikodym drivaiv of P ~ wih rspc o P, h rlaiv nropy is dfind by I ξ 1 E ~ = lim [log + ],whr E ~ 0 ξ is h condiional xpcaion opraor condiioning on h informaion availabl up o im ha is valuad wih rspc o h dnsiy associad wih h 8

9 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing alrnaiv or wisd modl i.., no h rfrnc modl. W now urn o h opimizaion problm according o h spiri of robus conrol hory S,.g., iu, Pan and Wang 2005, w assum h agn sks o maximiz h robus-conrol uiliy funcion U, which is dfind as h soluion o h following sochasic ingral quaion: U = U W,, y, m = inf {E ~ [ u W my ~ P P I s U Ws, s, Ys, m dsw = W, Y = y]}, 2.5 wih boundary condiion U W,, y, m = u W my, whr E ~ is h condiional xpcaion opraor wih rspc o an alrnaiv probabiliy masur in P; u is h xponnial uiliy funcion xprssd as u x x = 1/, and >0 is h risk cofficin; I su Ws, s, Ys, m ds rprsns a pnaly funcion conrolling h dviaion from h rfrnc modl, in which is a paramr masuring h rlaiv imporanc of h rfrnc and alrnaiv modls In ohr words, an agn wih highr xhibis highr avrsion o modl uncrainy. Sinc hr sms no apparn rlaion bwn sprad prmium and loss svriy for CA scuriis, w rsric hr ha h uncrainy avrsion of h agn only applis o h liklihood componn of h loss arrival. ha is, w ffcivly assum h agn only has doub abou h occurrnc probabiliy of loss vns, whil is comforabl wih h loss magniud aspc of h modl. h Radon-Nikodym drivaiv quaion. ξ h h dξ 1 ξ dn 1 λξ d = is hus dfind by h following sochasic diffrnial whr h is a im dpndn funcion conrolling h modl disorion magniud, and whr ξ = 1 0. By consrucion, h procss { ξ, 0 } is a maringal of man on. h ~ ~ masur P = P hus dfind is indd a probabiliy masur, and l P b h nir 9

10 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing collcion of such probabiliy masurs. Givn h alrnaiv modl spcificaion dfind by as: ξ, h disanc masur I can b calculad h I = h λ h S Appndix B in iu, Pan and Wang 2005 for h proof of a mor gnral formula han 2.7. h quaion 2.5 for uiliy funcion U hn bcoms U = U W,, y, m = inf {E ~ [ u W my { h } s λ hs [ 1+ hs 1 ] U Ws, s, Ys, m dsw = W, Y = y]} h corrsponding HJB quaion for U is hn as follows: U h + δ + inf { λ [EU W,, y +, m WU W h λ h U W,, y, m] [1 + h 1 ] U} =0, 2.8 whr U is h drivaiv of U wih rspc o, U W,U WW ar is firs and scond drivaivs 1 wih rspc o W, and h rminal condiion is U W,, y, m = xp W. W conjcur ha h indirc funcion U is of h form my U W,, y, m = V W, f, m δ whr VW, is dfind as V W, xp W =,andf,m is a im-dpndn funcion. Insr h form of U in 2.9 ino h abov HJB quaion and cancling h facor of hn rwri 2.8 as 10 my,wcan

11 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing V f + Vf + δwv W f h λ h + inf { λ E[xp m] Vf Vf [1 + h 1 ] Vf } = h h firs and hird rms cancl and cancling V<0 from h rmaining rms obains h following ordinary diffrnial quaion for f f h + sup{ λ [ m 1] f λ [ 1 + h h 1 ] f } =0, 2.11 h wih boundary condiion f,m=1. h firs ordr condiion for h givs h following quaion [ m 1] h = * Noic is soluion h = [ m 1] is a consan indpndn of im and subsiuing i and h corrsponding soluion of 2.11 ino h indirc uiliy funcion form 2.9, h funcion U a im 0 is hn givn by 1 δ U w + mc,0,0, m = xp{ w + mc + h* h* λ [ m h * 1 ] d} Finally h firs ordr condiion for m in quaion 2.13 givs h following quaion for h opimal insuranc proporion m*: c = δ 1 h* m* δ m* λ E[ ] d λ E[ ] 0 = In quilibrium, h rprsnaiv agn accps full risk in h insuranc mark so m*=1, h soluion o mark quilibrium and h pricing krnl can hn b summarizd by h following 11

12 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing proposiion: Proposiion 1: In quilibrium, h pric of CA risk is givn by 1 δ c = λ E[ ] h quilibrium SDF is hn givn by a gnralizd Esschr ransform o hav h form ζ = ζ ; α, β = xp αy + βn / E[xp αy + βn ], wih α = and β = 1. S Appndix 1 for proof of h proposiion. Rmark 1: Esschr ransforms as candidas of SDF ar usually suppord by risk xchangs quilibrium modls. S, for xampl, Kallsn & Shiryav 2002 for conomic manings of Esschr ransform inroducd in Grbr & Shiu In his papr, w hav shown ha h spcifid class of gnralizd Esschr ransform for CA risk pricing can also b suppord by an quilibrium framwork. h diffrnc from h prvious work is ha, his im h involvd agn is avrs o uncrainy as wll as o risk. h conomic undrpinnings of Esschr ransform mak i uniqu as an insuranc prmium principl and i is also in his sns ha Esschr ransform can b lookd as a bridg pulling radiional acuarial scinc and modrn financial conomics oghr Grbr and Shiu Rmark 2: h abov discussion can b gnralizd o includ sock in h financial mark. Alhough w hav assumd h insuranc risk is radd a h iniial im, h abov argumn for insuranc risk pricing can b applid o any im and h drivd SDF is also givn by a gnralizd Esschr ransform. Noic ha onc h insuranc risk has bn ransfrrd a h iniial im, h insuranc pric a lar im will b adjusd o kp h insuranc fully undrwrin by h rprsnaiv agn. 3. Caasroph Bond Pricing and Empirical Analysis 12

13 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing 3.1. Caasroph Bond Pricing In his scion, w apply h SDF corrsponding o h gnralizd Esschr ransform o pric CA bond. W assum for convninc of discussion ha h form of CA bond is dscribd as follows: h CA bond is pricd a K, which dnos h bond principal. If h loss for any singl CA vn in h priod 0, is lss han a loss riggr A, h agn will g back his principal K, plus risk-fr inrs K δ 1,plusspradprmium K ~ l a h mauriy im, in which l ~ dnos h sprad prmium ra. Onc a CA loss xcds h riggr, h agn will forfi som or all h principal a im. W assum h sprad prmium plus h risk-fr inrs is guarand no mar whhr any principal loss occurs. h aggrga CA losss xcding A in h im priod 0, is assumd o follow a compound Poisson disribuion dscribd as follows, N Y = = j j 1, 2.2 whr CA largr han A occurrncs numbr N is Poisson wih innsiy λ>0 and j j=1,2,f, indpndn of ach ohr and N, dnos h random loss amoun and can b dscribd by idnical disribuion funcion, Pr x F x, whr F x dnos h j = disribuion funcion of a random variabl which is largr han A. h loss fracion f AB of h principal is in proporion o h firs CA loss 1 in h rang bwn h riggr A andacapb, and is givn by f AB = ax[ 0, in 1 A, B A]/ B A = ax[0, 1 A] ax[0, B]/ B A 1 = A B + / B A In ohr words, h cash flow h agn g back a im can b dscribd as a random variabl 13

14 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing δ ~ K + l I N > 0 f AB, whri is h vn indicaor funcion. hrfor h random principal loss fracion a im is I N > 0 f. AB h probabiliy ha a las on caasroph loss occurs in h priod 0, is givn λ by E[ I N > 0] = 1. h avrag loss fracion condiional on h caasroph occurrnc can b calculad as E f proporion is hus givn by l = E[ I N > 0 f ] = E[ I N > 0]E f. AB AB B [1 F x] dx A =. h man of h bond principal loss B A AB B λ 1 [1 F x] dx A =. 3.2 B A Now w apply h sochasic discoun facor η 0, drivd from h gnralizd Esschr ransform spcifid in Scion 2.1 o pric h abov CA bond. h basic quaion can b wrin as: δ ~ K = E[ η 0, K + l I N > 0 f ], AB from which w can g h formula o calcula h sprad prmium ra l ~ as follows, ~ l = E[xp α Y + βn I N > 0 f ]/ E[xp αy + βn ]. 3.3 AB If w furhr assum h caasroph risk is nonsysmaic, ha is, no corrlad wih h mark porfolio of scuriis, h absolu risk avrsion paramr of h rprsnaiv invsor hus gos o zro, i.., α 0. In his cas w can only focus on h ffc of h ambiguiy avrsion. Equaion 3.3 hn bcoms ~ l = E[xp β N I N > 0 f = E f AB AB ]/ E[xp βn E[xp β N I N > 0]/ E[xp βn ] ] 14

15 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing in which E[xp N I N 0]/ E[xp βn ] β > is h modifid probabiliy ha a las on caasroph loss occurs and can b calculad o b β 1 λ. hrfor w β B λ ~ 1 [1 F A hav l = B A loss can b givn by h following proposiion: x] dx, and h raio of h sprad prmium o h man principal Proposiion 2: Assuming h rprsnaiv agn is risk nural o h caasroph risk Y,h raio of sprad prmium ra o h xpcd principl loss proporion i.., h raio of h modifid man principal loss o h xpcd loss of h rlad CA bonds is givn by ~ l l 1 1 λ = λ β λ λ β = β, 3.4 in which h scond approximaion quaion holds whn λ is vry small. In Proposiion 2, w involv a paramr β o dno h ambiguiy avrsion. Sinc h uncrainy rsuls parly from h scarciy of hisorical saisical informaion, i sms naural o assum β = β l is a dcrasing funcion of h man principl loss proporion l In-sampl Fiing and Ou-of-Sampl Prformanc In his scion w xamin h mpirical implicaion of our modl using mpirical caasroph bonds daa. W firs calibra our modl o mach h mpirical daa lisd in abl 1. Sinc w hav ignord h prmium loadings for xpnss in h prvious discussion whil h caasroph scuriis as financial insrumns, usually hav high ransacion coss, w should limina h xpnss ffcs in h simaion of h mulipl of prmium rlaiv o acuarially xpcd losss. Froo 1999 has assumd in his xplanaion of high pric of caasroph rinsuranc ha h brokrag and undrwriing xpnss com o b abou 10 prcn of prmium. h liminaion of hs xpnss will driv down h avrag raio of prmiums o xpcd losss from 9.09 o abou

16 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing o calibra h mpirical daa o our modl spcificaion, w fac h problm of which kind of funcional form for β b chosn o fi h sampl daa. An in-sampl s shows ha h funcion form for β β b1 corrsponding o h powr funcion, ha is: = b0l provids a simpl and good fiing and h paramrs ar simad o b b 0 = and b 1 = Figur 2 shows h calibrad rsul simad by h powr funcion as wll as h mpirical raio of xpns-adjusd prmium o xpcd loss agains h xpcd CA loss basd on daa. I can b sn from h graph ha h calibrad rsul provids an xclln fi o h obsrvd daa. Plas insr Figur 2 abou hr Having spcifid h form of β and rlad paramrs ha bs fi h caasroph bond sampls issud from 1997 o W now urn o xamin h modl s ou-of-sampl pricing prformanc. For his purpos, w rly on h caasroph scuriis daa collcd from 2000 o h daa ar obaind from Sigma journal of Swiss R. abl 2 is a lis of CA scuriis ousanding as of 31 Dcmbr Plas insr abl 2 abou hr Figur 3 shows h calculad rsul basd on h modl and paramrs simad abov, as wll as h mpirical raio of xpns-adjusd prmium o xpcd loss agains h xpcd CA loss as lisd in abl 2. h figur shows ha h simad rsul provids an adqua fi o h obsrvd daa. hrfor, using h gnralizd Esschr ransform basd on h robus conrol hory sms srving as an xclln modl o pric h caasroph risk and CA linkd scuriis. Plas insr Figur 3 abou hr 4. Conclusion 16

17 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing oivad by h obsrvaion ha h sprad prmium of CA loss scuriis is vry high rlaiv o h xpcd loss of principal of bonds, and h prmium sprad is much mor pronouncd for CA bonds wih low probabiliy ha a coningn loss paymn o h bond issurs will b riggrd, w hav xndd h radiional Esschr ransform o a gnralizd framwork and rad i as a sochasic discoun pricing facor o pric h CA risk. h gnralizd Esschr ransform is drivd from a modifid quilibrium modl by allowing h rprsnaiv agn o ac in a robus conrol framwork agains modl misspcificaion wih rspc o rar vns in h sns of Andrson, Hansn and Sargn h modl is xplicily solvd and h drivd pricing krnl is shown o b xacly h gnralizd Esschr ransform. Using caasroph bonds daa, w xamin h mpirical implicaion of our modl. hr ar svral xnsions w may do o our currn invsigaion. Firs, w hav considrd h issu of uncrainy avrsion in a robus conrol framwork and hus nvision h modl misspcificaion as a prmann psychological characrisic of a dcision makr. On h ohr hand, an agn may larn h modl hrough succssiv approximaions. I would b an imporan xnsion o incorpora forms of larning, i.., h crdibiliy hory, ino our framwork. Scond, w rsric h CA loss procss hr o b a compound Poisson procss. Sinc hr is usually sasonaliy in caasroph occurrnc, i will b inrsing o dscrib h CA loss by a im-changd vy procss and in h mos gnral sing, h procss wrin as a smimaringal. Discussion for h corrsponding chang of masur in hs cass can b found in Carr & Wu 2004, and BuKlmann al Finally, jus as Froo poind ou 1999, hr is ypically a shif of covrag window in rinsuranc mark afr a larg CA vn occurs. o xplain his im sris mpirical fac, i migh b hlpful o modl how h prior prformanc of CA vns migh affc h magniud of uncrainy avrsion of dcision makrs in an insuranc mark. For his purpos, w may simula h approach on similar opic in h framwork of walh-basd prospc hory s,.g., h sminal conribuion of Barbris, Huang and Sanos For all hs issus w lav for fuur rsarch. 17

18 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing 18 Appndix 1 Proof of Proposiion 1: According o quaion 2.4, h momn gnraing funcion undr h gnralizd Esschr ransform spcifid in Proposiion 1 can b calculad as ] 1 xp[, ;, z z Y + = α α α λ β α β, wih α = and 1 = β. Hnc h gnralizd Esschr ransform of Y is again a compound Poisson ransform, wih modifid Poisson paramr 1 λ and loss amoun bcoms a random variabl whos momn-gnraing funcion is + z. h modifid man of loss amoun can hn b calculad as ] [ ] [ 1 1 z z z z z dz d = E = E + = =. hrfor h pric of CA risk Y in priod, isgivnby d c = 0 1 ] E[ λ δ ] E[ 1 δ λ =.

19 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing REFERENCES 1. Andrson, E,.P. Hansn and.j.sargn, Risk and Robusnss in Gnral Equilibrium, Working Papr, Univrsiy of Chicago, Banwal, V.J. and H.C. Kunruhr, A Ca Bond Prmium Puzzl, h Journal of Psychology and Financial ark, 11: 76-91, Barbris, N.C.,. Huang and. Sanos, Prospc hory and Ass Prics, h Quarrly Journal of Economics, 116: 1-53, BKhlmann, H., F. Dlban, P. Embrchs, and A. Shiryav, No-arbirag, chang of masur and condiional Esschr ransforms, CWI Quarly, 9: , Carr, P. and. Wu, im-changd vy Procsss and Opion Pricing, Journal of Financial Economics 71: , Cummins, J.D. and H. Gman, Pricing Caasroph Insuranc Fuurs and Call Sprads: An Arbirag Approach, Journal of Fixd Incom 4: 46-57, Cummins, J.D., C.. wis and R.D. Phillips, Pricing Excss-of-oss Rinsuranc Conracs agains Caasrophic oss, in K. Froo, d., h Financing of Caasroph Risk, Univrsiy of Chicago Prss: , Cummins, J.D., D.alond and R.D.Phillips, h Basis Risk of Caasrophic-loss Indx Scuriis, Journal of Financial Economics 71: , Froo, Knnh, Inroducion o K. Froo, d., h Financing of Caasroph Risk, Univrsiy of Chicago Prss: 1-22, Gman, H. and. Yor, Sochasic im Changs in Caasroph Opion Pricing, Insuranc: ahmaics and Economics, 213: , Grbr, H.U. and Shiu, E.S.W., Opion pricing by Esschr ransforms, ransacions of h Sociy of Acuaris, XVI, , Grbr, H.U. and Shiu, E.S.W., Acuarial bridgs o Dynamic Hdging and Opion Pricing, Insuranc: ahmaics and Economics, 18: , Kallsn, J. and A.N. Shiryav, h Cumulan Procss and Esschr s Chang of asur, Financ and Sochasics, 6: , iu J., J. Pan and. Wang, An Equilibrium odl of Rar-Evn Prmia, Rviw of 19

20 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing Financial Sudis 181: , SwissR, Naural Caasrophs and an-mad Disasrs in 2003: any Faaliis, Comparaivly odra Insurd osss, Sigma 1:1-44,

21 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing abl 1: Caasroph bond issus 1997~2000 bonds daa h Sprad Prmium is h annual coupon ra abov on-yar IBOR. h Prob of Firs oss is h probabiliy ha a coningn paymn will b riggd undr h bond. h E[ >0] is h xpcd principal paymn o h issuing insurr, condiional on h occurrnc of a loss ha riggrs paymn undr h bond, xprssd as a prcnag of h principl of h bond. h Expcd oss is h produc of h probabiliy of firs loss and E[ >0]. Prm o E[loss] is h raio of h sprad prmium o h xpcd loss of principl of h bond. Da ransacion Sponsor Sprad Prmium % Prob of Firs oss % E[ >0] % Expcd oss % Prm o E[oss] arch-00 SCOR arch-00 SCOR arch-00 SCOR arch-00 hman R Novmbr-99 Amrican R Novmbr-99 Amrican R Novmbr-99 Amrican R Novmbr-99 Grling Jun-99 Grling Jun-99 USAA July-99 Sorma July-98 Yasuda arch-99 Kmpr arch-99 Kmpr ay-99 Orinal and Fbruary-99 S. Paul/F&G R Fbruary-99 S. Paul/F&G R Dcmbr-98 Cnr Soluions Dcmbr-98 Allianz Augus-98 X/idOcan R Augus-98 X/idOcan R July-98 S. Paul/F&G R July-98 S. Paul/F&G R Jun-98 USAA arch-98 Cnr Soluions Dcmbr-97 okio arin & Fir Dcmbr-97 okio arin & Fir July-97 USAA Augus-97 Swiss R Augus-97 Swiss R Augus-97 Swiss R Augus-97 Swiss R Sourc: Cummins, alond and Phillips 2004 Avrag dian

22 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk V V R O > Û R W Û P H U 3 3UREÛRIÛIUVWÛORVVÛà Figur 1: h smirk curv of mpirical prmium sprads agains loss probabiliis 22

23 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk V V R O > Û R W Û P H U 3 Expcd oss % PSUFDOÛ'DWD &DOEUDWHGÛ'DWD Figur 2 h calibrad and mpirical raios of xpns-adjusd prmium sprads o xpcd loss agains xpcd CA losss In-sampl fiing o CA bonds daa 23

24 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk Pricing abl2 CA scuriis ousanding as of 31 Dcmbr ~2003 bonds daa CA bond sponsor Schduld mauriy Sprad a issuanc % Expcd loss % Sprad o xpcd loss Swiss R Û 15.5Û 4.86Û 3.19Û Swiss R Û 15.25Û 4.86Û 3.14Û Swiss R Û 15Û 4.86Û 3.09Û Swiss R Û 1Û 0.01Û Û SCOR Û 2.38Û 0.05Û 47.60Û SCOR Û 6.75Û 0.9Û 7.50Û Orinal and Û 3.1Û 0.41Û 7.56Û REIP Û 3.45Û 0.53Û 6.51Û Nissa Dowa Û 4Û 0.67Û 5.97Û AGF Û 2.6Û 0.22Û 11.82Û AGF Û 5.85Û 1.16Û 5.04Û Swiss R Û 4.75Û 1.27Û 3.74Û Swiss R Û 5.75Û 1.28Û 4.49Û Swiss R Û 5Û 1.28Û 3.91Û okio arin & Fir Û 4.3Û 0.7Û 6.14Û Znkyorn Û 2.45Û 0.22Û 11.14Û Znkyorn Û 3.5Û 0.49Û 7.14Û Swiss R Û 6Û 1.28Û 4.69Û Swiss R Û 5Û 1.27Û 3.94Û Swiss R Û 1.75Û 0.22Û 7.95Û Swiss R Û 4.25Û 1.29Û 3.29Û Swiss R Û 7.5Û 1.31Û 5.73Û EDF Û 1.5Û 0.02Û 75.00Û EDF Û 3.9Û 0.54Û 7.22Û Swiss R Û 3.85Û 0.52Û 7.40Û Swiss R Û 2.3Û 0.22Û 10.45Û USAA Û 4.99Û 0.68Û 7.34Û USAA Û 4.9Û 0.67Û 7.31Û USAA Û 4.95Û 0.48Û 10.31Û Swiss R Û 4.5Û 1.29Û 3.49Û Swiss R Û 5.75Û 1.28Û 4.49Û Swiss R Û 5.25Û 0.68Û 7.72Û Swiss R Û 5.75Û 0.76Û 7.57Û Synidica Û 6.75Û 1.14Û 5.92Û Zurich R/Convrium Û 8Û 1.11Û 7.21Û Zurich R/Convrium Û 4Û 0.67Û 5.97Û Swiss R Û 1.35Û 0.02Û 67.50Û Sourc: Sigma of Swiss R, No.1,

25 Ambiguiy Avrsion,Gnralizd Esschr ransform and Caasroph Risk V V R O > Û R W Û P H U 3 Expcd ossûà PSUFDOÛ'DWD &DOEUDWHGÛ'DWD Figur 3 h calibrad and mpirical raios of xpns-adjusd prmium sprads o xpcd loss agains xpcd CA losss Ou of sampl fiing o 2000~2003 CA bonds daa using bonds daa basd simaion 25