A Effectveess of Itegrated Portfolo Bacassurace Taea Karya Research Ceter for Facal Egeerg Isttute of Ecoomc Research Kyoto versty Sayouu Kyoto 606-850 Japa arya@eryoto-uacp Itroducto As s well ow the tradtoal bag busess s to get face va tme depost savg mae loas to frms I that a ba maages credt rs ts loa portfolo or equvaletly credt rs pool to mae a proft from the spread of terests betwee loa rate depost rate O the other h the surace busess surace rs s maaged ts rs pool of the sured to mae a proft from the dfferece betwee the premums collected the loss damage actually caused There the separato or segregato betwee face surace busess s sttutoally made o the bass that the fucto of mag face the fucto of provdg surace agast rss are regarded as dfferet separable at least from the sde of the supplers However as the ew terms bacassurace assureba fassurace are sometmes used a tred toward covergece of bag surace busess s clearly observed may facal commodtes busess corporatos Ths tred wll correspod to the eeds of the frms cosumers who have to separably maage facal rs surace rs ther value developmet lfe cycles Merto 994 Karya 000 dscussed the problem vew of fuctoal approach to face surace provded clues to uderstg the tred as atural Gora997 provded may actual examples of bacassurace assureba Europe Caada SA Accordg to the author bacassurace s the case a ba sells surace commodtes developed by a surace compay the ba ows assurebag s the reversed case The author s basc vewpot stll les the sttutoal approach he dscussed about the merts demerts of the busess schemes of bacassurace assurebag I ths paper from a vewpot of the tradtoal busess schemes bag surace we cosder the effectveess of the covergece of bag busess surace busess Specfcally regardg bag busess surace busess as the busesses whch respectvely mae profts by poolg maagg the rss ther loa portfolo polcy portfolo we cosder the effectveess of combg the
two portfolos or two busesses compared to each portfolo The effectveess s evaluated terms of asymptotc default probablty whe each portfolo sze s large Accordg to ths crtero the asymptotc default probablty of a tegrated portfolo s smaller tha that of each portfolo uder a certa codto The cocept o whch the surace busess reles for poolg maagg rs ts portfolo s a law of large umbers probablty theory I a large homogeeous pool of polces the probablty of the occurrece of a accdet s costat the frequecy of accdets relatve to the sze of polces s regarded as close to costat Hece so log as the rate of premum s set more tha the probablty of the occurrece the probablty that the collecto of premums receved s bgger tha the loss pad goes to oe as the portfolo sze gets large The prcple of the law of large umbers s fact used together wth the dversfcato prcple of rs the maagemet of a loa portfolo the tradtoal bag Whe the default probablty of each loa s costats the default evets are depedet the default probablty of a ba becomes smaller as the portfolo seze gets larger so log as a default-adusted spread of loa terest rate depost terest rate s postve Secto 3 I ths sese rs maagemet bag busess s smlar to that surace busess the evaluato crtero based o the asymptotc default probablty s reasoable Ths paper oly treats oe-perod model for smplcty of argumets A dyamc exteso s left ope for complcato of modelg Our results state that uder a codto o default probablty of loas depost rate loa rate premum rate accdet rate retur rate for premums a tegrated portfolo s more effectve asymptotc default probablty tha ay of a ba portfolo a surace portfolo Cosequetly the covergece of bag surace wll be theoretcally ustfed I addto the codto ca be used to fd a strategc posto for a fassuracce sttuto wth loa rate depost rate surace premum as cotrol varables The cotet of the paper s as follows Secto Prcple of Tradtoal Isurace Busess:Law of Large Numbers Secto 3 Effect of Poolg Credt Rs Isurace Rs o Portfolos Secto 4 Effectveess of a Fassurace Portfolo: Geeral Case Prcple of Tradtoal Isurace Busess: Law of Large Numbers As s dscussed Secto the tradtoal surace busess a pool of surace rss a portfolo s maaged relatve to ts reserve of premums so that the busess ca be cotued wthout default Needles to say the busess scheme provdes a
mportat fucto of rs maagemet agast surable rs our socety The theoretcal cocept o whch the scheme of creatg the fucto reles s the law of large umbers probablty theory To develop our argumets fassurace let us brefly state the relevat pot troduce some otato we wll use below Let the preset tme be deoted by 0 cosder oe perod model over [0] Supposg a homogeeous surace populato for a specfc rs let be the sze of polces or equvaletly the umber of the obects persos to be sured a sured portfolo let X ~ deote the clam amout whe the -th sured obect gets a accdet Here we assume that each maxmum clam for X s bouded by polcy as 0 < X < A The the total loss possbly caused a surace portfolo s expressed as X X L X ~ = ~ + L + L s the dcator fucto of the surace evet of the th sured defed by f the th sured gets a accdet L = { 0 otherwse We call L the accdet evet geerato fucto of the th obect Here the ~ ~ losses X X are assumed to be depedetly detcally dstrbuted d the evet geerato fuctos L L are also assumed to be d ~ It s ofte the case that X } L } are also assumed But here we do ot assume t Put X ~ = X L { deote respectvely the total loss ts mea loss per obect by { X = X + + X X = X Further assumg that the expectato of = E [ ] µ X X s fte let The the wea law of large umbers states that for ay Q X µ > ε 0 whle the strog law of large umbers states 3
Qlm X = = µ Here we use the wea law for smplcty the sequel Sce by q µ + ε Q X > µ + ε = Q X > µ + ε 0 so log as the premum s set as µ + ε the default probablty goes to zero as that the default evet { X > µ + ε} of the surace compay occurs But realty s fte hece the premum probablty less tha or equal to a certa level γ ; µ + ε s eeded to be set to eep the default q µ + ε γ The assumg that the varace of X s fte by the cetral lmt theorem the default probablty 7 s approxmately evaluated by the ormal probablty ε q µ + ε Φ γ Φ s the cumulatve stard ormal dstrbuto fucto Let α γ be the γ -percetle pot of the stard ormal dstrbuto The the equalty 8 holds f oly f ε = αγ Hece order for 8 to be guarateed the surace premum eeds to be set at least more tha or equal to x = µ +α 0 γ Actually a surace compay usually regards the premum as Premum x = Pure Premum + Safety Loadg Expese I other words the mea µ of X correspods to the pure premum α γ to the safety loadg the mmum premum to x = µ + α γ µ µ + 0 ξ The the expese s set as a proporto sayη of the premum x from whch the premum s gve by x = x 0 + ηx or equvaletly 4
x = µ + ξ η The umerator of ths formula s µ + ε It s oted that the above argumet holds eve whe X ~ ~ s costat X A ~ X = X L Ths s because L s rom the cetral lmt theorem holds for the sum of L s Effect of Poolg Credt Rs Isurace Rs o Portfolo To treat a cetral part of our problem we frst cosder the followg smplfed oe perod surace problem Though t s deleted later we assume that whe a accdet occurs the perod a fxed amout A s pad wthout respect to the sze of ts damage or loss Ths assumpto correspods to the assumpto of 00% damageablty surace The the above settg the problem of cotrollg the probablty that a surace portfolo or compay gets defaulted agast a homogeeous surace populato s regarded as the problem of cotrollg the probablty that = AY x r A + s postve Here x s a premum r s a rate of terest for the premum reserve L L Y = L + + L s the total umber of accdets possbly occurrg oe perod for the portfolo The frst term AY of the rght sde 3 s the total loss amout the surace compay has to pay at tme The secod term s the premum reveue xa receved at tme 0 plus terest receved at tme uder the rate r Therefore > 0 a meas default for the surace portfolo Now to cosder a covergece problem of face surace we frst defe the default of a loa portfolo bag for a homogeeous credt rs populato each loa sze s assumed to be A Correspodg to the default problem surace the default of the loa portfolo s defed as the stuato that = A + r = A + r Y A + r A r r Y s postve r s a loa terest rate r s a savg rate 5
Y = L + L + + L s the total umber of defaulted loas the total sze of the loa portfolo Here t s assumed that the recovery rate of a defaulted loa s zero the default geerato fuctos L are depedet The frst term of the rght sde of the frst equalty s the amout the ba has to pay bac wth terest at tme the secod term s the remag amout whe Y frms get barupted durg the perod The secod equalty of 33 meas the dfferece betwee the total loss the reveue from the terest spread r r Hece the default problem s regarded as the problem of cotrollg the probablty that > 0 I the argumet below we vew the covergece problem as the problem of fdg a codto for a effectveess of the combed portfolo of the surace rs portfolo the credt rs portfolo To treat ths problem frst ote that Y s are depedetly dstrbuted as bomal; Y ~ B p p = E[ L ] = p s the default probablty of dvdual frm = or sured obect = oe perod Hece as s well ow they are approxmated as ormal ; Z Y p = p p ~ N 0 wth sg ths fact we evaluate the default probabltes of dvdual portfolos the tegrated portfolo approxmately Default probablty of a ba portfolo Expressg terms of Z 34 as = + A{ + r m Z + m [ r r p + ]} r the default evet 0 of a loa portfolo s equvalet to the evet a > Z > + 6
a m = = m u u = r r ν + p + r ν = + r Here u s the spread of loa savg rates whe the default probablty p of a dvdual frm s tae to accout whch we shall call the default-adusted spread or smply adusted ba spread Here u > 0 s equvalet to r r + r > p whch s satsfed whe p s small We assume u > 0 Hece whe s large the default probablty of a ba portfolo s approxmately evaluated as Q > 0 Φ a It s easy to see that whe u s egatve a s egatve so the default probablty s larger tha / Ths mples that a ba caot form a loa portfolo uless the dscouted spread of loa savg rates s bgger tha the default rate p Default of a surace portfolo The default evet 0 of a surace portfolo s equvalet to the evet > Z > + a sce s expressed as { } 0 + A m Z + m + r x p = a u m = m u = = + r x p + Here u s the dfferece betwee the surace premum + r x vewed at tme the default rate p of a sured obect whch we shall call the surace spread Aga whe u 0 the default probablty of a surace portfolo becomes greater tha or equal to / Hece we assume u > 0 For a large the default probablty s expressed as Q > 0 Φ + a 7
Default of the tegrated portfolo I the tegrated portfolo of a loa portfolo a surace portfolo the default evet becomes + > 0 For otato settg ν + s expressed terms of Z Z as { } 3 + = + A m Z + m ν Z + [ m u + m ] ν u v s gve 36 Here lettg 4 W = [ m + ν νz + mν Z] mν m + 0 s equvalet to > W + > b 5 b = m u + ν + mu mν m Therefore whe are large satsfy 6 m δ costat m the default probablty of the tegrated portfolo s approxmated as 7 Q + > 0 Φ + b Now let us cosder a codto for whch the tegrated portfolo s more effectve tha ay of the ba portfolo the surace portfolo terms of default probablty Clearly from 7 a codto s 8 b > a = The codtob > a s expressed as 9 z > z β 0 z = mu mu [ ν mν m ν ] β = = m + Also the codtob > a s expressed as 8
z β > z β Note that z > z s always satsfed Theorem Assume 6 The a ecessary suffcet codto for the tegrated portfolo to be asymptotcally more effectve tha ay of the portfolos s z > > or equvaletly mu m u z m mu m + r < < m + r + m m u + m u m + r + m u = r r p + r u = + r x p m = + are respectvely the ba spread the surace spread the proporto of the dvdual portfolo szes the tegrated portfolo The codto s of course a codto o the set r r p p r x m m Hece whe the portfolo szes default rates p bag p surace are gve ths codto gves a restrcto o the set { r r r x } each of whch s the compoet of the prces {loa rate retur rate for premums surace premum} of the fassurace busess A strategc choce of a pot ths set wll gve a fasssurace sttuto a better portfolo posto Whe = = the codto gves o restrcto o whle p = p = p t gves a lear restrcto o p As a specal case let us cosder the case the portfolo szes are equal = the dvdual default rates are equal p = p = p The we have Corollary Suppose = p = p = p The the codto 3 s equvalet to the codto of 33 34; 3 r r [ < + r r r + + r ] + r + r [ + r + + ] + r + r < x 9
4 a < p < a Here β β[ r r + + r x] r r a = β + r + r a β [ r r + + r x] + r x = β + r β [ + + ] = = r β Proof The codto z > mu m u s equvalet to 5 β r r + r x β > p [ β + r ] β Here sce the rght sde s easly show to be postve the left sde eed to be postve whch gves 6 < x < r r [+ + + r ] + r + 0 r Ths s the rght equalty of O the other h from m u m u > z we obta 7 p [ + r β ] > r r β β r x + β Here sce the sde of [ ] s egatve the rght sde must be egatve whch leads to the left equalty of 33 Therefore whe 3 s satsfed combg 5 7 yelds the result 4 Now we delete the assumpto of 00% damageablty I the case of o-lfe damage casualty surace whe a accdet occurs the full paymet of the surace s ot pad but oly the loss caused by the damage s pad wth a maxmum level A To treat ths geeral stuato let 8 w X A = Also the case of loa whe a frm gets barupted some porto of the loa made to the frm s recovered Hece let the loss rate be w or equvaletly let X ~ = w A 0
3 The ether case expressed as w s a rom varable o [ ] are 9 = A+ r = A = w = L w L A r r x+ r A der the assumpto of the depedece of { L } { w } 30 E[ w L ] = ξ p µ wth E[ w ] = ξ Var w L = E w p p the stardzed varate 3 Z [ w L µ ] = s asymptotcally stard ormal Cosequetly replacg p by µ 37 38 p by µ 30 3 36 by 330 the spreads u u 38 3 by 3 u = r r p ξ r + u = + r x p ξ respectvely the argumet made above holds as t sts Theorem 3 I a geeral case wth a commo level A for loa surace the codto 3 s ecessary suffcet for a tegrated portfolo to be more effectve u are defed by 330 33 Effectveess of a Fasurace Portfolo :Geeral Case I ths secto let us cosder a geeral case there are K homogeeous credt rs populatos wth dfferet dvdual default rate p s for each populato dfferet loa szes A s = K there are K homogeeous surace rs
populatos wth dfferet default rates surace szes = K p for each populato dfferet The default problem bag s the problem of cotrollg the default probablty of a ba portfolo wth loas of dfferet credt ratgs Ths problem s treated the same way as Secto 3 Here we oly treat the case the recovery rate s zero e w = 38 sce the ozero case s a drect exteso of the zero case as has bee see Secto 3 The the default probablty s the probablty that the evet 0 occurs > Y K = A = = L { + r Y r r } + + L Here deotes the -th credt ratg = K r s the loa rate for frms wth the th ratg s ts umber of the loas A s the loa sze Of course the loa sze s ot commo to all the frms of the -th ratg geeral But the argumet below s easly exteded wth some complcato otato to a geeral case there are M classes loa sze for each ratg as Credt Rs Plus997 Hece wthout loss of geeralty we tae ths smplfed settg Rewrte 4 as Also assume that as = A u = K A = a { + r Z u } = r r K = A + r a = A p A m δ 0 > wth = p wth p The bomal varates { Y } are asymptotcally multvarate ormal the sese that { Z } asymptotcally follows Z = Z Z ~ N 0 Λ wth Λ = ρ K ε s the correlato of Z Z Cosequetly N µ ~ γ wth
µ = A γ = A K = a u a a + r + r = ρ O the other h also the case of a surace portfolo wth dfferet surace rss the default probablty s descrbed as the probablty that > 0 Y K = B = = L { Y x + r } + + L Here the retur rate of maagg the reserve of premums s assumed to be r The smlarly to 4 rewrte = B u K = B = b as { Z u } = + r x K = B = p b p p = B B wth The { Z } s are asymptotcally multvarate ormal; Z = Z Z ~ N 0 Λ wth Λ = ρ Here t s also assumed that as K m δ 0 > Cosequetly µ γ ~ = B = B = N µ γ K = b u b b ρ wth Therefore the default problem of the tegrated portfolo of credt rs surace rs s the problem of cotrollg the probablty that + 0 Here we assume > 3
that as m m m + ξ Theorem 4 For the tegrated portfolo to be asymptotcally more effectve tha ay of the ba portfolo the surace portfolo t s ecessary suffcet that 0 β β < β = µ µ < β [ γ γ + γ ] = The proof s the same as the case Secto 3 Note that 40 s equvalet to µ + µ γ + γ > µ γ = Of course Theorem 4 the depedece of credt rs surace rs s assumed 5 Cocluso Ths paper specfcally gves a explct codto for whch a tegrated portfolo or equvaletly fassurace portfolo s better asymptotc default probablty tha ay dvdual portfolo:ba portfolo surace portfolo It ca be used to fd a strategc posto for a fassurace sttuto to mae a complex commodty wth credt rs surace rs Refereces BeardREPetaeT PesoeE996 Rs Theory Chapma Hall Crede Susse Facal Product 966 Credt Rs+ GoraJC997 Bacassurace: Postog for Afflatos LOMA Karya T000 What s Facal Egeerg Iwaamshote Japaese MertoRC994 Facal systems ecoomc performace Joural of Facal Servces Research 4 63-300 4