Securitization of Crossover Risk in Reverse Mortgages. ChouWen Wang 1 HongChih Huang 2 YuanChi Miao 3 ABSTRACT


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1 Secuiizaion of Cossove Risk in Revese Mogages ChouWen Wang HongChih Huang YuanChi Miao 3 ABSTRACT When he ousaning balance excees he housing value befoe he loan is sele, he insue suffes an exposue o cossove isk inuce by hee isk facos: inees aes, house pices an moaliy aes. Une he consieaion of housing pice isk, inees ae isk an longeviy isk, we povie a heeimensional laice meho which simulaneously capues he evoluion of housing pice an shoem inees ae o numeically calculae he fai valuaion of evese mogages. Fo a mogage evese insue, he pemium sucue of evese mogage insuance is eemine by seing he pesen value of oal expece claim losses equal o he pesen value of he pemium chages. Howeve, when he acual loss is highe han he expece loss, he insue will incu an unexpece loss. To offse he poenial loss, we also esign a cossove bon o ansfe he unexpece loss ino he bon invesos. Theefoe, hough he cossove bons, he evese mogage insues can paly ansfe he cossove isk ino he bon holes. Keywo: evese mogages, cossove isk, longeviy isk, cossove bons. Associae Pofesso, Depamen of Risk Managemen an Insuance, Naional Kaohsiung Fis Univesiy of Science an Technology, Kaohsiung, Taiwan. *Coesponing auho. Pofesso, Depamen of Risk Managemen an Insuance, Naional Chengchi Univesiy, Taipei, Taiwan. 3 Depamen of Risk Managemen an Insuance, Naional Chengchi Univesiy, Taipei, Taiwan.
2 . Inoucion Demogaphic aging offes one of he mos seious challenges fo he evelope an eveloping counies. The ens of moaliy impovemen coninue o heaen he secuiy of social secuiy sysems wolwie. The epenen aio, especially ha of he age efine as he aio of he numbe of senio epenens (ove 65 yeas of age) o he oal populaion (age 5 64 yeas) keeps ising in mos counies, which means ha he oveall economy faces a geae buen o suppo an aging populaion. Govenmens an inusies seek o ecease hei financial buen by efeing he eiemen age an/o eucing he benefis people eceive in a efine benefi pension plan. In aiion, people migh aain financial secuiy in eiemen hough savings, puchasing pivae insuance, o coninuing o wok. Howeve, fo many people who ean less fom hei employmen, he aleaive is unealisic an impacical. As emogaphic shifs incease he popoion of elely homeownes who ae ofen houseich an cashpoo, evese mogages migh povie a moe easonable soluion fo he aging homeowne populaion. Revese mogages ae financial conacs ha allow eiees o conve hei home equiies ino eihe a lump sum o annuiy income bu sill mainain owneship an esience unil hey ie, sell o vacae hei homes o live elsewhee. Thee ae hee basic paymen foms fo evese mogages: () enue, () em, o (3) line of cei. The isinguishing chaaceisic of he enue mogage is ha i povies a monhly paymen o he boowe as he boowe occupies he house. In conas, a em loan povies monhly paymens fo only a fixe peio. The line of cei pemis he boowe o make aws a any ime up o some maximum pespecifie amoun; he mogage is no ue an payable unil he boowe sells he popey, moves ou pemanenly, o ies. A he mogage's ue ae, he loan ges
3 epai wih accumulae inees hough he sale of he popey. Fuhemoe, he lene can only eceive he minimum of he enie eb o he ne value of he popey, which pevens he boowe fom owing moe han he value of he popey. This nonecouse clause makes he evese mogage ifficul o pice. The evese mogage conacs involve a ange of isks fom he insue s pespecive. The ousaning balance usually accumulaes a a fase ae han he appeciaing ae of he housing value; heefoe, if he ousaning balance excees he housing value befoe he loan is sele, he lene sas o incu a lossnamely cossove isk, one of he cucial isks o manage in evese mogages. Appaenly, his cossove isk is inuce by hee isk facos: inees aes, house pices an moaliy aes. As poine ou by Phillips an Gwin (99), he lening feaue of evese mogages subjec loan povies o muliple isks. An incease of he lifespan of he loan esule fom moaliy impovemen o euce mobiliy aes will impose a highe cossove isk. A ise in inees aes will spee up he ae a which he loan accumulaes, an will possibly hi he cossove poin ealie. Besies, a epesse eal esae make will wosen he value of he home. Szymanoski (994) analyzes he isks involve wih evese mogage insuance an explains a picing moel evelope fo he Home Equiy Convesion Mogage (HECM) a publicly guaanee evese mogage offee by he U.S. Depamen of Housing an Uban Developmen. Chinloy an Megbolugbe (994) evelop an alenaive picing moel fo a evese mogage in which he boowe eceives paymens as eihe a lump sum o an annuiy. Boh suies invesigae evese mogages wih a consan inees ae assumpion. Boehm an Ehha (994) povie analysis of he isks associae wih evese mogage loans an pesene picing moels fo evese mogages une inees ae isk inheen o fixeae evese mogages. They fin ha he inees ae isk of a evese mogage is geae 3
4 han ha of eihe a ypical coupon bon o a egula mogage. Michell an Piggo (004) also exploe he feasibiliy of eveloping he evese mogage make in Japan an conclue ha i has he poenial o elieve he fiscal buen on aiional, saefune eiemen povisions. Roa, Lam, an Youn (004) analyze he HECM pogam using a simulaion moel in which inees aes an house pices vay accoing o hisoically accuae ansiion pobabiliies followe by he yea Teasuy since Apil 953 an he OFHEO house pice inex since 975. Ma, Kim an Lew (007) analyze he isk of govenmen insue evese mogages in Koea, using he MoneCalo simulaion meho consieing he housing pices following geomeic Bownian moion an inees aes following Vasicek pocess. Employing he LeeCae moel wih pemanen jump effecs an an ARIMAGARCH housing picing moel, Chen, Cox an Wang (00) pice he nonecouse povision of evese mogages an compae i wih calculae mogage insuance pemiums. The loan balance of evese mogage may gow o excee he popey value a he ime of eminaion because of muliple isks: eminaion isk (longeviy isk), inees ae isk an housing pice isk. Howeve, mos of he exising lieaue on isk moeling in he HECM pogam oes no simulaneously consie he ynamics of moaliy aes, inees ae an housing pice. Theefoe, une he consieaion of housing pice isk, inees ae isk an longeviy isk, we pice evese mogages wih upoae mehos. Specifically, we consie a evese mogage sucue like an HECM scheme an assume ha housing pices follow a Geomeic Bownian moion (GBM) pocess (Cunningham an Henesho, 984; Kau, J. B., e al., 99; Chinloy an Megbolugbe, 994; Hillia, J. E., e al., 998; Szymanoski, 994; Yang, T. T., e al., 998), while he inees ae eflecs he lognomal pocess of Black, Deman, an Toy (990) an he moaliyelae isks is assesse by he LeeCae moel. 4
5 Howeve, when he pocess fo he shoem inees ae follows he lognomal pocess, he closefom soluions of evese mogages ae no available. In his aicle, we use a heeimensional laice meho which can simulaneously capue he evoluion of housing pice an shoem inees ae o calculae he fai valuaion of evese mogages. Fo he housing pice pocess, we use he CoxRossRubinsein (CRR, 979) moel o geneae he possible saes of fuue housing pice. Fo he pocess of shoem inees aes, we use he BlackDemanToy (BDT, 990) moel o geneae he possible saes of fuue spo aes. Secuiizaion is a financial innovaion ha emege in he 970 s in he US financial make. Accoing o Cummins (004), secuiizaion involves he isolaion of a pool of asses o ighs o a se of cash flows an he epackaging of he asses o cash flows ino secuiies ha ae ae in capial makes. The iea of secuiizing moaliy an/o longeviy isks is inouce (see Blake an Buows (00) an Blake (003)). Thee is an incease inees o moel hese ypes of moaliybase secuiies; hence, he ieas of moaliy bons an moaliy swaps ae popose in he lieaue (Lin an Cox, 005) an effecively pu ino pacice. Following a simila appoach use by Lin an Cox (005), Wang, Valez an Piggo (007) popose a secuiizaion meho o hege he longeviy isk by using suvivo bons an suvivo swaps fo evese mogage poucs. Following a simila appoach use by Lin an Cox (005), Denui e al. (007) an Wang, Valez an Piggo (007), his pape poposes a secuiizaion mehocossove bonso hege he cossove isk inheen in evese mogage poucs. Fo a mogage evese insue, he pemium sucue of evese mogage insuance is eemine by seing he pesen value of oal expece claim losses equal o he pesen value of he pemium chages. Howeve, when he acual loss is highe han 5
6 he expece loss, he insue will incu an unexpece loss. To offse he poenial loss, we esign a cossove bon o ansfe he cossove isk ino he bon invesos. The payoff sucue of cossove bons is elae o he acual losses an expece losses of evese mogages. A each paymen ae, if he acual loss of evese mogage is less han he expece loss, he bon invesos can obain a highe level of coupon ae; ohewise, hey can only eceive a lowe level of coupon ae. Theefoe, hough he cossove bons, he evese mogage insues can ansfe he unexpece loss ino he bonholes. In his aicle, using he heeimensional laice meho, we will numeically calculae he fai coupon aes of cossove bons.. Picing Moel of Revese Mogage Insuance Conacs In his secion, we fis escibe he conac sucue of evese mogages, which povies he basis fo ou valuaion. We hen moel he ynamics of he spo inees aes, he house pices, an he moaliy aes sequenially... Revese mogage conacs In U.S. HECM pogam, boowes ae equie o pay % of housing values as an upfon mogage insuance pemium ( UP 0 ) an a monhly mogage insuance pemium ( MIP ) accoing o he annual ae of 0.5% of he ousaning loan balances. Using his peeemine insuance pemium sucue, we evaluae he pesen values of expece claim losses an ha of insuance pemiums, eemining he maximum levels of consan monhly paymens une he coniion saisfying he pesen values of expece claim losses ae equal o ha of insuance pemiums. 6
7 We invesigae evese mogage wih a lump sum paymen, analogous o he U.S. HECM pogam (Szymanoski, 994). The iniial popey value, enoe H 0, enables us o eemine he lump sum paymen. We assume ha he loan becomes ue an payable only a he boowe s eah. The boowe eceives a lump sum paymen, BAL 0, an oes nohing else, because he house is his o he pincipal esience. BAL, he ousaning balance a ime, is eemine by he ousaning balance a ime  plus he pemium chage wih inees accue. follows: BAL can be calculae as BAL BAL MIP y () whee BAL The ousaning loan balance a ime BAL M H UP M H 0.0 H ( M 0.0) H 0 0 M Maximum Level of Mogages UP0 Upfon mogage insuance pemium a incepion MIP Yealy mogage insuance pemiums a ime ; MIP BAL y Mogage inees ae a ime We can ecusively obain he expessions fo MIP an BAL as follows:.05 BAL ( M 0.0)( y) H, () 0 MIP =0.005 BAL ( M 0.0)( y) H. (3) The evese mogage conacs involve a ange of isks fom he insue s pespecive. The ousaning balance usually accumulaes a a fase ae han he appeciaing ae of he housing value; heefoe, if he ousaning balance excees 7
8 he housing value befoe he loan is sele, he lene sas o incu a lossnamely cossove isk, one of he cucial isks o manage in evese mogages. Appaenly, his cossove isk is inuce by hee isk facos: inees aes, house pices an moaliy. In he following secion, we escibe hei ynamics, especively... Inees ae pocess Geneally speaking, une he oinay economic envionmen, we can assume he shoem iskfee inees ae is consan. Howeve, in some cicumsances, he shoem inees ae changes amaically. Fo example, when he cenal bank suenly changes moneay policies o oil shocks occu, he shoem inees ae will flucuae ove ime. Thus, insea of using consan inees ae, we mus assume he sho em iskfee inees ae o be sochasic. Thee ae a numbe of moels of he local pocess fo he shoem inees ae a nomal pocess (Vasicek, 977; Jamshiian, 989), a lognomal pocess (Dohan, 978; Black, Deman, an Toy (990); Black an Kaasinski 99) an a squaeoo pocess (Cox, Ingesoll, an Ross (985)) o ohes. Among hem, lognomal moels keep he ae away fom zeo eniely, while he nomal pocess may fall below zeo an some squaeoo moels make zeo ino a "eflecing baie." In his aicle, we use he lognomal moel o escibe he evoluion of shoem inees aes as follows: ln ln ln W, (4) whee is he insananeous spo ae a ime ; is longem inees ae paamee; is he volailiy veco of he spo ae a ime an sasifies 8
9 ,0, ha is, ; enoes he Eucliean nom in R ; an W epesens a imensional sana Bownian moion une a Riskneual pobabiliy measue Q an saisfies W W, W H. Noe ha Equaion (4) is a coninuous ime limi of he BlackDemanToy onefaco moel (BDT moel), incopoaing wo inepenen funcions of ime, θ() an σ(), chosen so ha he moel fis he em sucue of spo inees aes an he em sucue of spo ae volailiies. Assuming a iffeen lognomal sho ae isibuion fo each fuue ime allows boh mean an vaiance o epen on ime. In conas o he Vasicek moel, in he lognomal epesenaion he sho aes ae lognomally isibue; wih he esuling avanage ha inees aes canno become negaive..3. House pice moel Fo mogage valuaion, i ypically assumes ha housing pice follows a sochasic Geomeic Bownian moion (GBM) pocess (Cunningham an Henesho, 984; Kau, J. B., e al., 99; Chinloy an Megbolugbe, 994; Hillia, J. E., e al., 998; Szymanoski, 994; Yang, T. T., e al., 998). This pocess is also known as a coninuous ime limi of anom walk wih if fo he ynamics of insananeous ae of euns. Theefoe, une he iskneual measue Q, we assume ha he house pice pocess is govene by H H W, (5) H 9
10 whee is a mainenance yiel (o enal ae) fo he house; H is he volailiy veco of he housing pice an sasifies H H H, H, ha is, H H ; an H is he coelaion coefficien beween he inees ae pocess an he house pice pocess..4. Moaliy moel Eve since Lee an Cae pesene hei oiginal wok in 99, he LeeCae moel has been wiely use in moaliy en fiing an pojecion. The Census Bueau populaion foecas has use i as a benchmak fo he longun foecas of U.S. life expecancy. The wo mos ecen Social Secuiy Technical Avisoy Panels have suggese he Tusees o aop his meho o ohe mehos consisen wih i (Lee an Mille, 00). In his aicle, we use he LeeCae moel o assess he moaliyelae isks. Le mx, be he cenal eah ae fo age x a ime. The LeeCae moel assumes: ln m k e, (6) x, x x x, whee x epesens he age paen of eah aes, x escibes he paen of eviaions fom he age x pofile when he paamee k vaies, k explains he change of moaliy ove ime, an ex, escibes he eo em, which is expece o be whie noise wih zeo mean an a elaively small vaiance (Lee, 000). The LeeCae moel canno be fie by he oinay leas squae appoach, because all vaiables on he igh sie of he moel ae unobsevable. Moeove, his moel is obviously ovepaameeize. We use he singula value ecomposiion appoximaion (Lee an Cae, 99) o fi he soluions of he paamees. To obain 0
11 a unique soluion, a nomalizaion coniions is impose such ha he x ems sum o uniy an he k ems sum o zeo, i.e., k 0 an x. (7) x Then x becomes he aveage value of lnm x,. Fo each age goup x, we can obain ˆx by egessing log m x ˆ x on k ˆ wihou a consan em. Following Lee an Cae (99), we foecas fuue values of k wih k k z, (8) whee z is he if paamee, an is a sequence of inepenen an ienically nomal isibuions wih mean 0 an vaiance. We assume ha he values k,, k 0 ae known bu ha k ae unknown an mus be foecas, whee j 0 j, fo j any naual numbe j. By viue of Equaion (8), we have j 0 i i j k k jz, (9) Moeove, coniional on 0, k j is nomal isibue wih mean k 0 jz an vaiance j. Le p ( ) enoe he oneyea suvival pobabiliy ha an x0 age peson in x0 0 calena yea eaches age 0 x 0 +. We assume ha he agespecific moaliy aes ae consan wihin bans of age an ime bu may vay fom one ban o he nex. Specifically, given any inege age x0 an calena yea 0, we suppose ha m m fo 0,. (0) x0, 0 x0, 0
12 Thus, he oneyea suvival pobabiliy can be calculae as p ( ) = exp( m ). x0 0 x0, 0 Le p enoe he nyea suvival pobabiliy ha an x0 age peson in calena n x0, 0 yea eaches age 0 x0 n, which is p exp m exp exp k n n j. () n x0, 0 x, 0 j x0 j x0 j 0 j j0 j0.5. Picing moel fo evese mogage insuance We eemine he lump sum paymen BAL0 when he pesen value of he insuance pemiums coves he pesen value of expece losses fom fuue claims. Though he picing pocess, i is moe convenien o se he valuaion ae 0 valuaion ae 0 (=0), he money make accoun is efine by exp 0 uu B o 0. Thus, a he. () We also assume ha he make is complee an wihou abiage. Base on he abiage picing heoy, he value of he evese mogage insuance, which is also he pesen value of he expece losses fom fuue claims, equals he expecaion of iscoune fuue cash flows une iskneual measue Q. Le x0 be he age of he boowe a ime 0 ; hen, he value of he evese mogage insuance is of he fom: whee N MIP j PVMIP UP 0 EQ p j x0, 0 j B( j ) N E Q px p x PVEL, j B( j ) PVMIP Max BAL H,0 j j j 0, 0 j 0, 0 (3) Pesen value of oal mogage insuance pemiums a incepion (ime ) 0 PVEL Pesen value of oal claim losses a incepion j N x0, 0 p The numbe of yeas ha boowes wih age x will live unil hey each 0 yeas of age The pobabiliy ha a boowe of age x a incepion will suvive a age x+j
13 Une he assumpion ha moaliy ae an financial isk ae inepenen, we can ewie Equaion (3) as follows: whee p(, T ) N j j ( 0.0) 0.05 ( ) ( 0, j ) x0 j, (4) j PVMIP UP M H y p S, N x0 j x0 j j (5) j PVEL S S C enoe he pice of a zeocoupon bon issue a ime ha pays $ a ime T, T ; S E p ; an C j is of he fom: x0 n Q n x0, 0 j = C E Q Max BAL H j B( j ) j,0. (6) Howeve, when he pocess fo he shoem inees ae follows he lognomal pocess efine in Equaion (4), he closefom soluions of j C ae no available. To numeically obain he values of j C, hee ae many echniques in numeical meho of opion picing, such as finie iffeence mehos, laice o ee mehos, an Mone Calo mehos. In his aicle, we use a heeimensional laice meho which can simulaneously capue he evoluion of housing pice an shoem inees ae. Fo he housing pice pocess, we use he CoxRossRubinsein (CRR) moel o geneae he possible saes of fuue housing pice. This means ha, if H = 0 H is he asse value a ime 0, hen, afe one peio, a ime, i can ise o uh o ecease o H, whee u an epesen, especively, he magniue of one up sep an he magniue of one own sep. Consequenly, he unelying asse pice evoluion can be epesene by a binomial ee whee each noe coespons o one possible value of he asse pice. Noe ha, in he CRR moel, since he logchange of housing pice is moele by a anom walk wih if, hen by aking he limi of he numbe of 3
14 peios equal o infiniy, he coninuousime limi of anom walk wih if moel fo logchange of housing pice will become he GBM pocess efine in Equaion (5). [Inse Figue ] Fo he pocess of shoem inees aes, we use he BlackDemanToy moel o geneae he possible saes of fuue spo aes. The BDT moel is a onefaco shoae (noabiage) moel all secuiy pices an aes epen only on a single faco, he sho ae he annualize onepeio inees ae. The cuen sucue of long aes (yiels on zeocoupon Teasuy bons) fo vaious mauiies an hei esimae volailiies ae use o consuc a ee of possible fuue sho aes (fo eails, please see Black, Deman an Toy (990)). This ee can be use o value ineesaesensiive secuiies such as evese mogage conacs. Similaly, when he numbe of peios appoaches o infiniy, he pocess of shoem inees aes govene by BDT moel become a lognomal pocess efine in Equaion (4). Combining he CRR moel an BDT moel, Figue epics he heepeio laice moel o numeically obain he values of j C. The iniial avance BAL0 can be eemine by seing he pesen value of oal expece claim losses equal o he pesen value of he pemium chages, namely PVMIP PVEL. 3. Secuiizaion fo Revese Mogage Insuance Suppose he lene hols a pofolio of L loans. A ime 0, all he boowes ae of he iffeen ages anging fom age 6 o age 00. Each boow a lump sum agains hei home popey. When he ousaning balance excees he housing value befoe he loan is sele, he insue sas o incu a loss. Theefoe, fo a mogage 4
15 evese insue, he pemium sucue of evese mogage insuance is eemine by seing he pesen value of oal expece claim losses equal o he pesen value of he pemium chages. Howeve, when he acual loss is highe han he expece loss, he insue will incu an unexpece loss. To offse he poenial loss, we esign a pincipalguaanee cossove bon o ansfe he unexpece losses inuce by cossove isk ino he bon invesos. The payoff sucue of he cossove bon is elae o he acual loss an he expece loss. Simila o Teasuy bons, he cossove bons pay inees a each coupon paymen ae an he pincipal a mauiy. Unlike he Teasuy bons, when he acual loss is less han he expece loss a he coupon paymen ae, he bon invesos will eceive a highe level of coupon ae han ha of he Teasuy bons wih he same mauiy. ohewise, he bon invesos will eceive a lowe level of coupon ae. Since he close fom soluion of he cossove bons ae ha o eive, using he heeimensional laice meho, we can calculae he fai valuaion of he cossove bons. 4. Numeical analysis Fo numeically analyze he impacs of longeviy isk, inees ae isk an housing picing isk on he picing evese mogage, we fis escibe he paamees fo he ynamics of inees ae, housing pice an moaliy ae, hen we pesen he numeical esuls fo he loanovalue (LTV) aios as well as hei sensiiviy analysis. Finally, we compue he fai coupon aes of he cossove bons wih mauiy up o hiy yeas. 5
16 Fis, as shown in Figue, we employ he Teasuy zeo aes fom GeTai Secuiies Make in Taiwan o calibae he paamees of BDT moel. Using he zeo ae cuve, he minimum, meium an maximum of he spo ae ee ae epice in Figue. The highe he spo ae volailiy, he lage he iffeence beween he minimum an maximum of he fuue spo aes. Wihou loss of genealiy, we apply he meiums a each ime peio of he BDT moel as he coesponing mogage aes of evese mogage. As shown in Figue 3, he cuves of mogage aes fo iffeen spo ae volailiies ae vey close o each ohe. [Inse Figue ] [Inse Figue 3] We employ Taiwan moaliy aa fom he Depamen of Saisics, Minisy of he Ineio in Taiwan o calibae he paamees of he LeeCae moel. Final age all lives ae assume o en is 0. The paen of empiical moaliy aes appea in Figue 4. Applying he LeeCae moel, we epic he suvival pobabiliies Sx 0 n fo 0 x 65, 70, 75 an 80 in Figue 5. The highe is he age ceeis paibus he lowe is he suvival pobabiliy. [Inse Figue 4] [Inse Figue 5] Using he heeimensional laice meho, we fis pesen he numeical esuls fo a epesenaive base case. Fo he paamees of he base case, he iniial housing value is assume o be $5,000,000, i.e., H 0 =5,000,000; he housing pice volailiy is 45%; an he inees ae volailiy is %. Figue 6 epics he LTV aios fo iffeen ages. The lowe is he age ceeis paibus he lowe is he LTV aio. Fo is economic implicaion, he pesen value of he house is he sum of he pesen value of fuue enal incomes. Accoing o he evese mogage mechanism, he 6
17 boowe uses he enal income afe his o he eah in exchange fo he lump sum paymen a he incepion. An ole boowe can boow moe money since his o he expece eah comes soone an he pesen value of he enal income afe eah is geae. [Inse Figue 6] We nex examine he sensiiviy of he LTV aios by vaying he level of housing pice volailiy an inees ae volailiy in Table. Fom Table, i can be seen ha he highe he inees ae volailiy, he lowe he LTV aio. Fo is economic implicaion, he pesen value of he house is he sum of he pesen value of fuue enal incomes. The highe inees ae volailiy may lea o a highe level of inees ae an hence esuls in a lowe pesen value of he house as well as a lowe LTV aio. Similaly, since he highe housing pice volailiy may conibue o a highe level of housing pice, he highe is he housing pice volailiy, he highe is he LTV aio. Noe ha, fom Table, he impac of housing pice volailiy is moe significan han ha of inees ae volailiy on LTV aio, which inicaes ha fo picing evese mogage i is cucial o esimae he volailiy of housing pice pecisely. [Inse Table ] Fo he secuiizaion of evese mogage, we assume he payoff sucue of he pincipalguaanee cossove bon is elae o he acual losses an he expece losses. The acual loss a each peio is calculae accoing o he isibuion of boowe age an gene as follows: 00 g, x g x gf o m x6, (6) AL w w AL whee AL is he acual loss a ime ; wg is he gene weigh; wage is he age weigh; an AL ( AL f, x m, x ) is he acual loss fo a female (male) wih age x an 7
18 saisfies: AL = g, x0 j j q g x0 Max BAL g, x0 j B( ) j H j,0, (7) whee j q p p enoes an iniviual live fo j g g g x0 j x0, 0 j x0, 0 yeas an will ie in one yea. Using he heeimensional laice meho, we can calculae he acual losses, ogehe he expece losses, fo iffeen ages a each peio. Fo example, he expece losses fo he nex hiy yeas fo age x=65, 70, 75 an 80 ae epice in Figue 7. Since he expece loss a ime is eemine by he pobabiliy q x an he elaion beween q x an ime peio in Figue 8 also exhibis a humpe cuve, i is easonable ha he elaionship beween he expece losses an ime peio exhibis a humpe cuve. [Inse Figue 7] [Inse Figue 8] The payoff sucue of he cossove bon is efine as follows. Simila o he Teasuy bons, he bon invesos will eceive he inees a each coupon paymen ae an he pincipal a mauiy. Howeve, unlike he Teasuy bons, when he acual loss is lage han he expece loss a he coupon paymen ae, he bon invesos will eceive coupon ae equal o 0.5%. Ohewise, he bon invesos will eceive a highe level of coupon ae han ha of he Teasuy bons wih he same mauiy. In aiion, accoing o he gene an age isibuion of HECM loan boowes (Bishop an Shan, 008), he female weigh is assume o be 0.6 an he age weigh wx ae given in Figue 9. Using he heeimensional laice meho, in Table we calculae he fai coupon ae of he pincipalguaanee cossove bon wih iffeen ime o mauiies. As he ime o mauiy inceases, boh he coupon ae an he makup incease. Theefoe, when he acual loss is lage han he 8
19 expece loss a he coupon paymen ae, he issue of cossove bon can paly hege he unexpece loss an he bon holes also eceive a minimum coupon ae an pincipal a mauiy. Ohewise, he cossove bon holes can eceive a highe coupon han he Teasuy bon wih he same ime o mauiy. The winwin siuaion pobably makes he bon aacive an hence he issues can successfully ansfe he cossove isk o he bon make. 5. Conclusion In aiion o govenmensponsoe social secuiy sysems an employesponsoe eiemen plans, evese mogages ae becoming emakably popula in he las few yeas since i allows eiees o conve hei subsanial home equiies ino eihe a lump sum o annuiy income an o emain in hei homes unil hey ie, sell o vacae hei homes o live elsewhee. Fom he insue s pespecive, he evese mogages involve a ange of isks, incluing housing pice isk, inees ae isk an longeviy isk. In his pape, une he consieaion of housing pice isk, inees ae isk an longeviy isk, we povie a heeimensional laice meho which simulaneously capues he evoluion of housing pice an shoem inees ae an numeically calculae he fai valuaion of evese mogages. Fo a mogage evese insue, he pemium sucue of evese mogage insuance is eemine by seing he pesen value of oal expece claim losses equal o he pesen value of he pemium chages. Howeve, when he acual loss is highe han he expece loss, he insue will incu an unexpece loss. To paly hege he unexpece loss, we esign a pincipalguaanee cossove bon, he payoff sucue of which is elae o he acual losses an expece losses. Theefoe, hough he cossove bons, he evese mogage insues can paly ansfe he 9
20 cossove isk ino he bon invesos. Fo fuhe eseaches, since he wok of Clak (973), i has been ecognize ha he ynamics of asse euns pesen some commonly obseve saisical popeies, known as sylize empiical facs in he financial economeics (Con, 00). Theefoe, he ynamics of housing euns may no be aequaely escibe by geomeic Bownian moion wih consan if an volailiy. Fom he numeical analysis, we fin ha he impac of housing pice volailiy is moe impoan han ha of inees ae volailiy on picing evese mogage; heefoe, i is an ineesing opic o incopoae he sylize facs wih he housing pice pocess such as exponenial Lèvy pocesses o GARCH Lèvy moels fo picing evese mogages. 0
21 Inees Rae v=0.0 v=0.0 v=0.03 Inees Rae v=0.0 v=0.0 v=0.03 Inees Rae v=0.0 v=0.0 v=0.03 Volailiy Volailiy Volailiy Housing Pice vs=0.40 vs=0.40 vs=0.40 Housing Pice vs=0.40 vs=0.45 vs=0.45 Housing Pice vs=0.50 vs=0.50 vs=0.50 Table LoanoValue Raios Age Gene Female Male Female Male Female Male Age Gene Female Male Female Male Female Male Age Gene Female Male Female Male Female Male Time o Mauiy Table Fai Coupon Raes of Cossove Bons Coupon Rae The Acual Loss Acual Loss >Expece Loss Expece Loss Coesponing Makup Zeo Rae %.0%.66% +44bps %.6%.88% +74bps % 3.09%.00% +09bps
22 =0 u e P u e P u e P u e P 4 3 u e P u e P u e P u e P u e P u e P u e P u e P 0 9 = = =3 Figue The TheePeio Laice meho fo Picing Revese Mogage Insuance Conacs H u H u Huu Hu H u uu Huu u Huu Hu H uuu uu u P P P3 P4 P5 P6 P7 P8 P9 P0 P P, H
23 Figue Zeo Rae Cuve an he Range of BDT moel Figue 3 Mogage Raes fo iffeen Spo Rae Volailiies 3
24 Panel A Female Moaliy Raes Panel B Male Moaliy Raes Figue 4 Taiwan Moaliy Raes fo Diffeen Ages 4
25 Figue 5 Suvival Pobabiliies fo Diffeen Ages Figue 6 LoanoValue Raios fo Diffeen Ages wih H 45% an % 5
26 Figue 7 Expece Losses fo Diffeen Ages Figue 8 The Relaionship Beween q x an Time Peio Figue 9 Age weighs fo he Cossove Bon 6
27 Refeence Blake, D., 003. Reply o suvivo bons: a commen on Blake an Buows. Jounal of Risk an Insuance, 70,, Blake, D., Buows, W., 00. Suvivo bons: helping o hege moaliy isk. Jounal of Risk an Insuance, 68,, Black, F., Deman, E., Toy, W., 990. A onefaco moel of inees aes an is applicaion o easuy bon opions. Financial Analyss Jounal, 46,, Boehm, T. P., Ehha, M. C., 994. Revese mogages an inees ae isk. Jounal of he Ameican Real Esae an Uban Economics Associaion,,, Black, F., Kaasinski P., 99. Bon an opion picing when shoem aes ae lognomal. Financial Analyss Jounal, 47, 4, Chinloy P., Megbolugbe, I. F., 994. Revese Mogage: Conacing an Cossove Risk. Jounal of Ameican Real Esae an Uban Economics Associaion,,, Clak, P.K., 973. A suboinae sochasic pocess moel wih finie vaiance fo speculaive pices. Economeica 4, Con, R., 00. Empiical popeies of asse euns: Sylize facs an saisical issues, Quaniaive Finance,, pp. 4. Cox, J. C., Ingesoll, J., Ross, S. A., 985. A heoy of he em sucue of inees aes. Economeica, 53,, Cox, J. C., Ross, S. A., Rubinsein, M., 979. Opion picing: a simplifie appoach. Jounal of Financial Economics, 7, Cummins, J. D., 004. Secuiizaion of life insuance asses an liabiliies.submie o TIAACREF Insiue. Cunningham, D., Henesho, P., 984. Picing FHA Mogage Defaul Insuance. Housing Finance Review, 3, 4, Denui, M., Devole, P., Goeniaux, A.C., 007. Secuiizaion of longeviy isk: picing suvivo bons wih wang ansfom in he Leecae famewok. The Jounal of Risk an Insuance, 74,, Dohanm, M., 978. On he Tem Sucue of Inees Raes. Jounal of financial economics, 7, Hillia, J. E., Kau, J. B., Slawson V. C. J., 998. Valuing pepaymen an efaul in a fixeae mogage: A bivaiae binomial opions picing echnique. Real Esae Economics, 6, 3, Jamshiian, F., 989. An exac bon opion fomula. Jounal of Finance, 44,,
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