Insurance Premium Structure of Reverse Mortgage Loans in Korea Seungryul Ma and Yongheng Deng. September, 2006

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1 Insurance Premium Srucure of Reverse Morgage Loans in Korea Seungryul Ma and Yongheng Deng Sepember, 2006 Absrac We analyze he insurance premium srucure of reverse morgage loans in Korea. Our analyses provide a comparison beween he reverse morgage loans srucured wih consan monhly paymens o hose srucured wih graduaed monhly paymens which are indexed o he growh rae of consumer prices. Using he oal annual loan cos measure, we find ha, o he relaively younger borrowers, he graduaed monhly paymens approach is more efficien; while he consan monhly paymens approach is more efficien o he older borrowers. Our sensiiviy analyses confirm ha he younger borrowers are more sensiive o he change of loan erms. Therefore, we propose ha insurance premium srucure should be more conservaive o he relaively younger borrowers group. The resuls of his sudy can provide useful guideline o he operaion of reverse morgage sysem in Korea as well as in oher counries. JEL classificaion: G21; G22; G28 Keywords: Insurance premium srucure; Reverse morgage sysem; Graduaed monhly paymens; Morgage insurance premium; Toal annual loan coss raes This work is suppored by he Korea Research Foundaion Gran (KRF B00037). Deparmen of Insurance and Finance, Daegu Universiy, Souh Korea, Tel: +82 (53) School of Policy, Planning, and Developmen, Universiy of Souhern California, U.S.A., Tel: +1 (213) The auhors are graeful for he commens of David Downs, Charles Leung, and paricipans in he 2006 HKU-NUS Symposium on Real Esae Research, and 2006 AsRES/AREUEA Inernaional Conference a Vancouver. Deng acknowledges he financial suppor from he Lusk Cener for Real Esae a Universiy of Souhern California.

2 1. Inroducion Korea is moving rapidly ino an aging sociey. However, social securiy sysems for he elderly in Korea are lagged behind. Reverse morgage sysem is recenly recognized in Korea as an imporan financial vehicle o supplemen curren social securiy sysems for elderly homeowners. A governmen-insured reverse morgage program is under consideraion by he Korean governmen. A reverse morgage allows he senior homeowner o ake ou a loan agains he equiy from house ha he borrower does no have o pay back during he lifeime as long as he (she) lives here. Wih a reverse morgage, he lender makes paymens o he borrower (hence he paymen sream is reversed comparing o hose in he regular morgage) based on a percenage of he value of he house. When he borrower no longer occupies he house, he lender akes over he propery. Reverse morgage herefore allows equiy rich bu cash poor senior homeowners o conver par of heir home equiy ino ax-free cashes wihou having o sell he house. I gives he seniors financial sabiliy and independence while remain conrols of he houses hey say. Reverse morgages in he Unied Saes can be raced back o 1960 s. The mos popular reverse morgage program in he U.S. is he Home Equiy Conversion Morgage (HECM) program, which is he only reverse morgage insured by he U.S. federal governmen. HECM was developed by he Deparmen of Housing and Urban Developmen (HUD) in 1987 o accomplish hree broad objecs: (1) o permi he conversion of home equiy ino liquid asses o mee he special needs of elderly homeowners; (2) o encourage and o increase he paricipaion of he primary and secondary morgage markes in convering home equiy ino liquid asses; and (3) o deermine he exen of demand for home equiy conversion and he ypes of home equiy conversion morgages ha bes serve he needs of elderly households (Szymanoski, 1994). Alhough numerous reverse morgage producs have been developed since hen, here are hree major producs in oday s marke, i.e., HUD s HECM, Fannie Mae s proprieary Home Keeper Morgage, and Financial Freedom s proprieary Cash Accoun. Among hese hree major producs, The HECM program is considered he safes reverse morgage produc curren available in he U.S. because i is insured by he U.S. federal governmen. Over 95 percen of all reverse morgage borrowers choose he HECM produc. Depending on he producs borrowers chose, he governmen or a privae eniy provides guaranee owards reverse morgage producs. For example, he Federal Housing Adminisraion (FHA), a division of HUD, insures he HECM program; Fannie Mae, he larges non-banking financial services company in he world, guaranees he Home Keeper Morgages; and Financial Freedom, currenly he larges privae reverse morgage lender and servicer in he U.S., guaranees he Cash Accoun (Ballman, 2004). Mayer and Simons (1994 a, b), Case and Schnare (1994), Merrill, Finkel, and Kuy (1994), and 1

3 Fraanoni (1999) provided evidence of srong demand for reverse morgages among houserich, cash-poor elderly homeowners in he U.S. In Korea, Cho, Park, and Ma (2004) esimaed ha he poenial demand for he reverse morgage loans could be well over half a million of senior homeowners of age 60 and over in Korea. Lim and Cho (1999), Cho and Ma (2004), and Cho, e al. (2004) provided evidences of poenial demand for reverse morgage loans o he elderly homeowners in he Korean housing marke and assered ha he governmen-insured reverse morgage sysem would perform an imporan role o supplemen curren social securiy sysem for he elderly in Korea. When inroducing he governmen-insured reverse morgage program in Korea, i is desirable o adop a similar sysem o he HECM program in he U.S. because i has been esed and improved since is incepion in Reverse morgage funds can be dispensed in several opions. Borrower may receive an upfron lump sum in cash; an annuiy of pre-deermined amoun of monhly cash paymens as long as he (she) resides in he house as his (her) primary residence (he enure plan); an annuiy of monhly cash paymen for a fixed period of ime deermined by he borrower (he erm plan); a line of credi; or a combinaion of he above. In his paper, we analyze he mehods of wo enure plans: he consan monhly paymen plan and he graduaed monhly paymen plan. Because he reverse morgage enure plan provides a fixed amoun of monhly paymens for as long as he borrower occupying he house as his (her) primary residence, he real value of cash seam migh decline over ime due o he inflaion. Therefore, i is useful o analyze he graduaed monhly paymens plan where he monhly paymens are indexed o he growh rae of consumer prices 1. The reverse morgage program presens cerain risks o he borrowers and he lenders alike. Under he HECM program, hese risks are proeced hrough he FHA morgage insurance program. Reverse morgage insurance premiums consis of an up-fron insurance premium and monhly morgage insurance premiums (MIP) colleced from he borrowers. For he purpose of sable and coninual applicaion of reverse morgage sysem, i is imporan o consruc an appropriae insurance premium srucure which reflecs he expeced claims properly. To do his, we develop an acuarial model of pricing reverse morgage insurance premiums for he governmen-insured reverse morgage program in Korea. Exising lieraure, including Boehm and Ehrhard (1992, 1994), Szymanoski (1994), Chinloy and Megbolugbe (1994), Rodda, Herber, and Lam (2000) and Rodda, Youn, Ly, 1 In HECM program, o provide readily available cash for unexpeced expenses and give borrowers a hedge agains rising prices, i also allows o add a line of credi o a consan monhly advance (The American Associaion of Reired Person, 2005). 2

4 Rodger, and Thompson (2003), and Rodda, Lam, and Youn (2004), provided analysis of he risks associaed wih reverse morgage loans and presened pricing models for reverse morgages. Recenly, several empirical sudies analyzed reverse morgage sysem in Korea. Yoo (2002) analyzed he economic effecs of he reverse morgage sysem. Ma and Cho (2003) analyzed he mehods of long erm forecasing of housing values and ineres raes for building reverse morgage models. Ma and Cho (2004) and Cho, e al. (2004) analyzed he paymen plans of reverse morgage and analyzed reverse morgage relaed risks. Sung (2005) analyzed life insurer s sraegy for operaing he reverse morgage loans in Korea. The main purpose of his paper is o develop a reasonable insurance premium srucure of he reverse morgage sysem and o provide some guideline for he implemenaion of he reverse morgage sysem in Korea. Insurance claim losses are expeced o occur in he even ha he borrower s oal ousanding loan balance exceeds he appreciaed value of his (her) propery a he ime he loan is due and payable. However, i has been well documened ha he exac iming of a loan becoming due and payable is unknown and is difficul o esimae (Rodda, e al., 2000 and Rodda, e al., 2003). In his sudy, develop a framework of he acuarial srucure of he reverse morgage program based on forecased growh raes of fuure housing values and ineres raes, and esimaed loan erminaion raes and coss of reverse morgage. Our research sraegy is described as follows: we firs deermine he maximum levels of consan monhly paymens and graduaed monhly paymens using he maximum levels of Loan o Value Raio (LTV) classified by borrower s ages; secondly, we esimae he presen values of expeced claim losses and expeced reverse morgage insurance premiums and hen, adjus he levels of monhly paymens if here are discords wih he levels of presen values; hirdly, we propose he mehod of paymens ha have he minimum levels of Toal Annual Loan Coss (TALC) raes such ha he values of expeced insurance premiums are equal o he values of expeced claim losses, and finally we conduc sensiiviy analysis of he reverse morgage pricing models based on a se of scenarios wih differen house price appreciaion raes and ineres raes. The remaining of he paper is organized as follows: secion 2 presens a model of reverse morgage focusing on enure plans wih alernaive paymen approach, secion 3 describes mehodology and daa, secion 4 repors he resuls of analysis, and he las secion draws conclusion. 2. A Model of Reverse Morgage 2.1. Alernaive Paymen Plans 3

5 The maximum level of lump sum paymen Following he HECM program, cash advances in our reverse morgage program are limied by he principal limi facor, which is deermined when he expeced losses from fuure claims can be covered by he insurance premium (Szymanoski, 1994). This principal limi facor is also known as he maximum level of Loan-o-Value (LTV) raio of he reverse morgage. In his paper, we adop he concep of life expecancy o deermine he unique level of maximum LTVs corresponding o borrower cohors varying by borrower s age, house price appreciaion raes and ineres raes. Following Ma and Cho (2004), he maximum level of lump sum reverse morgage paymen (LSUM) can be calculaed according o he following formula: ( 1 + g ) H LSUM = (1) 0 = 1 Ta ( 1 + m ) = 1 Ta where LSUM = he lump sum reverse morgage paymen H = he iniial housing equiy 0 T a = he life expecancy of he borrower wih iniial age a g = he house price appreciaion rae a period m = he marke ineres rae a period In equaion (1), he non-linear relaionship of house price appreciaion and discoun facor, 1 + g ) ( 1 + m ), is called he ne discoun raio. If he ime series of ne discoun raio ( is saionary, we can use he mean value of ne discoun raio, ( 1 + g ) ( 1+ m), for calculaing he maximum level of he lump sum reverse morgage paymen 2. In ha case, equaion (1) can be simplified ino following equaion 3 : Ta 1 + g Ta LSUM = H 0 = H 0 NDR = H 0 LTV (2) 1 + m 2 Refer o Haslag, Nieswiadomy, and Sloje (1991) and Ma and Cho (2004). 3 Mayer and Simon (1994b) used he simplified model similar o he Equaion (2), bu hey did no es he saionary of he ne discoun raio series before applying he mean values of he house price appreciaion rae and ha of he morgage rae in heir analysis. 4

6 where Ta NDR = maximum LTV condiioning on he life expecancy of he borrower wih iniial age a NDR = ( 1 + g ) ( 1 + m) = he mean value of ne discoun raio In his analysis, we compue LSUM in equaion (2) as he original principal limi 4. The amoun of LSUM borrowers can ge varies depending on heir ages, curren ineres raes, and heir propery values in he reverse morgage program. However, by using equaion (2), we can find ou he unique level of LTV by borrower s ages regardless of he curren ineres raes flucuaion because we use he mean value of ne discoun raios in equaion (2). Wih HECM producs, borrowers may obain heir reverse morgage funds as a single lump sum, a enure plan, a erm plan, or a line of credi. In addiion, borrowers can choose any combinaion of hese opions. In his analysis, we focus on wo mehods of he enure plan, he consan monhly paymens and he graduaed monhly paymens. Maximum level of consan monhly paymen Under he consan monhly paymen scheme, he annuiy paymen is compued such ha he lump sum paymen afer adjusing he equilibrium facor equals he presen discouned value of he sream of annuiy paymens muliplied by he borrower s survival rae 5. ( ) PMT LSUM ( 1 β) = p ( 1+ i) T a 1 C a, (3) = 0 4 Currenly, he HECM program uses a differen mehod o deermine he level of LTV. Under he single insurance premium srucure and arbirary assumpions of he key variables (i.e., growh rae of housing value, ineres raes, loan erminaions, loan coss, ec.), he HECM calculaes he level of LTV where he presen values of expeced losses equal he expeced insurance premiums using he mehod of rial and error (Szymanoski, 1990, 1994). 5 However, in he HECM model, he value of consan monhly paymens can be calculaed using he (100 a) 12 formula as follows; = LSUM ( 1+ r) PMT (where, a = borrower s age). As we can see in = 0 he HECM model, borrowers of he enure plan in he HECM model were assumed o live unil hey are 100 years old (Szymanoski,1990, and Rodda, e al., 2000). Therefore, he calculaion in HECM is more conservaive compared o equaion (4) which is based on expeced survival rae, ( p a, ). In his paper, we assume he paymens of an annuiy are made a he beginning of ineres conversion periods (Muksian, 2003). Ma and Cho (2004) used monhly paymens model similar o his paper. However, hey simply used he concep of life expecancy by borrower s age insead of using borrower s survival rae as we do in his paper. 5

7 where β = he equilibrium facor for balancing income and ougo T ( a) = he number of monhs ha borrowers wih age a will coninue o receive he annuiy paymens unil hey reach 100 years of age PMT C = he annuiy paymen (consan monhly paymen) i = he annuiy rae p a, = he probabiliy ha a borrower of age a will survive a age a+ In equaion (3), he value of he equilibrium facor, β, for balancing he presen value of expeced claims and ha of expeced morgage insurance premium will be changed according o he assumpions of insurance premium srucure, he levels of housing price appreciaion raes, and expeced ineres raes in he acuarial model of reverse morgage. Solving he equaion (3) for he monhly annuiy paymen gives: PMT C = ( ) T a 1 = 0 ( 1+ i) ( ) LSUM 1- β 1 p a, (4) We can calculae he value of equilibrium facor, β, using he rial-and-error approach. Maximum level of graduaed monhly paymens The graduaed monhly paymens scheme can alleviae he risk of decrease in purchasing power over ime. In his paper, we analyze he graduaed monhly paymens scheme which are indexed o he growh rae of consumer prices. Provided ha he ime series of growh rae of consumer prices ( c ) is saionary, we can calculae he base saring amouns ( PMT G ) of graduaed monhly paymens using he following relaionship. T( a) 1 PMTC LSUM ( 1 β) = p = 0 ( 1+ i) =PMT G a, T( a) 1 1+ c p = 0 1+ i a, (5) 6

8 where PMT = he base saring amoun of graduaed monhly paymen G c = he mean value of growh rae of consumer prices Using he above relaionship, we can evaluae he maximum level of base saring amoun of graduaed monhly paymens as follows: PMT G = ( ) T a 1 = 0 ( ) LSUM 1- β 1+ c p 1+ i a, (6) 2.2. Insurance Premium Srucure The main purpose of his sudy is o develop an appropriae insurance premium srucure of reverse morgage sysem in Korea. To do his, we conduc an analysis based on he acuarial srucure of he reverse morgage. We assume ha he combinaion of up-fron cos and monhly morgage insurance premium (MIP) begins wih an up-fron cos of 5% of housing values and a monhly MIP according o he annual rae of 0.5% of he loan s ousanding balance 6. We firs evaluae he presen values of expeced claim losses and expeced insurance premiums under predeermined insurance premium srucure. Nex, we modify he levels of monhly paymens if here is a discord wih he levels of presen values. We hen find he mehods of paymens ha have he minimum levels of oal annual loan cos (TALC) raes under he presen value condiions ha he expeced insurance premiums are equal o he expeced claim losses. Presen Value of Fuure Premiums and Fuure Claim Losses A he ime of loan closing, following Rodda, e al. (2000; 2003), presen value of oal 6 In he HECM, afer loan closing, every borrowers are required o pay 2 percen of he maximum claim amoun (adjused propery value) as an up-fron (iniial) insurance premium and a monhly MIP according o he annual rae of 0.5% of he loan s ousanding balance (Rodda, e al., 2000, 2003). The iemized coss of a HECM loan include an originaion fee, hird-pary closing coss, an up-fron morgage insurance premium, a monhly MIP, a servicing fee, and ineres (AAPR, 2005). In 1999, he ypical HECM borrowers paid $1,800 in originaion fees, $1,500 in closing coss, $2,100 in up-fron insurance premium or a oal of $5,400 as he iniial cos of a HECM (Rodda e al., 2004). In his analysis, we assume ha he up-fron cos is composed of he originaion fee and hird-pary closing coss (3% of housing values) and up-fron insurance premium (2% of housing values). These high iniial ransacion coss creae a large hurdle ha has essenially blocked refinancing of reverse morgages. Therefore, prepaymen raes in reverse morgage loans could be conrolled by he up-fron coss (Szymanoski, 1994, and Rodda, e al., 2004). 7

9 expeced claim losses ( PVEC ) can be calculaed as follows: PVEC EC q (7) ( 1+ i) T(a) a, = = 1 where PVEC = presen value of oal expeced claim losses a =0 EC = { 0, [( OLB H ) q ]} max a+ = expeced claim losses a OLB = expeced ousanding loan balance 7 H = expeced house price a ( ( ) H = H 1 + g ) 8 q = probabiliy ha he loan will be erminaed a age a + a, i = discoun rae 0 A he ime of loan closing, following Rodda, e al. (2000, 2003), presen value of oal projeced morgage insurance premium ( PVMIP ) can be calculaed according o he following discouning formula. T(a) MIP q a, PVMIP = UP0 + (8) = 1 ( 1+ i) where PVMIP = presen value of oal projeced morgage insurance premiums a =0 UP 0 = up-fron insurance premium a =0 MIP = projeced monhly morgage insurance premiums Ne Expeced Insurance Liabiliy The ne expeced insurance liabiliy ( NEL ) is calculaed as follows. 7 The ousanding loan balance ( OLB ) is esimaed as he previous period s loan balance plus he projeced amoun of cash paymens o borrowers, monhly MIP (morgage insurance premiums), service fees, and ineres charges accrued during ha period. The HECM acuarial model assumes ha here is no parial prepaymen before he loan is due and payable because hey do no appear o happen frequenly. (Rodda, e al., 2003). 8 A he ime when he loan becomes due and payable, he fuure value of house price (FV) can be T compued according o he following formula, FV = H 0 = ( 1 + g ) 1. As we can see equaion (2), if he ime series of he house price appreciaion rae ( g ) is also confirmed saionary, we can simplify formula ino he equaion which using he mean value of he house price appreciaion rae ( g ). 8

10 ( RESV PVMIP) NEL = PVEC - + (9) where NEL = ne expeced insurance liabiliy n [ ] RESV k = ( TC )( 1 + i) = 1 TIP = ne reserve 9 TIP = oal amoun of insurance premiums including up-fron premium TC = oal amoun of claim disbursemens k = loan duraion n = oal number of monhs beween loan closing and he cuoff dae Under he presen condiions ha he values of expeced premiums are equal o he values of expeced claim losses, we finally search for he mehod of paymens ha shows he minimum levels of TALC raes Toal Annual Loan Coss (TALC ) raes The TALC rae is he annual average rae ha includes all he coss of a reverse morgage bu do no considers he survival probabiliies of he loan. The advanage of he TALC mehod is ha i presens us he oal cos of he loan by a single rae a a specific period ( = n ) during he borrowers become aged 100 (Scholen, 1996). Consan Monhly paymen Under he consan monhly paymens scheme, he oal ousanding loan balance a period = n can be expressed as follows: (10) n ( ) ( ) n OLB = PMT n 1 + TALCR C where TALCR = oal annual loan coss (TALC) raes In equaion (10), we can calculae TALC raes a period = n as follows. 9 A he ime of loan closing, RESV will be only an up-fron premium (2% of he house price). 9

11 (11) 1 TALCR n OLB n 1 = PMTC n Graduaed Monhly Paymen In he case of graduaed monhly paymens, he oal ousanding loan balance a period = n can be expressed as follows: n OLB n = PMTG + = 1 ( 1 + c) ( 1 TALCR) n (12) In equaion (12), we can calculae TALC raes a period = n as follows. OLB TALCR = PMTG = 1 n n ( 1 + c) 1 n 1 (13) 3. The Daa House price appreciaion, Marke ineres raes, and Ne discoun raios The ime series daa used in he analysis include he monhly daa of aparmen house price index (APT) mainained by Kookmin Bank from January 1986 o December , he monhly daa of yields on naional housing bonds wih 5-year mauriy (HB5) of Korea Naional Saisical Office (KNSO), and he consumer price index (CPI) of Korea Naional Saisical Office. We generae he ime series of growh rae of consumer price by log differencing he ime series of CPI and he growh rae of APT by log differencing he ime series of APT. As discussed in previous secion ha equaion (2) is valid only if he discoun raios series is saionary. A ime series is said o be saionary if he generaing funcion for he series does no 11 iself change hrough ime. We conduc he uni roo es in order o confirm he saionary of ime series. We primarily conduc OLS (ordinary leas squares) esimaion wih DF model and 10 The aparmen housing in Korea is he mos popular housing ype. In Korea, he aparmen housing is similar o he condominium secor of he housing marke in he Unied Saes. 11 See Appendix for a discussion of he uni roos es. 10

12 modify -saisics of δ, coefficien from he AR(1) regression, o accoun he serial correlaion in u. The ime series of he ne discoun raio, he growh rae of consumer price are confirmed saionary, respecively. We hen conclude ha he mean values of he ime series can be used on calculaing equaion (2) o (6) in his research. Following equaion (1), we use he ne discoun raio (NDR ) and he borrower s life expecancy for calculaing he value of lump sum reverse morgage paymen ( LSUM ). We generae he ime series of monhly discoun raes ( m ) based on a marke ineres rae (i.e. HB5). We hen generae he ime series of he ne discoun raio (NDR ) by using he growh rae of APT and he ime series of he marke ineres raes ( m ). Figure 1 show he ne discoun raio (NDR ) from January 1986 o December Mean: NDR Figure 1. Trend of he ne discoun raio (January 1986 o December 2004) The loan erminaion raes Because here was no previous experience on reverse morgage erminaion raes (he probabiliy ha he loan will become due and payable), U.S. HUD used unesed assumpions for erminaion raes. In HECM erminaion model, he loan erminaion probabiliy was assumed o be 1.3 imes he moraliy rae of he younges borrower in he family 12. In HECM erminaion model, we can find he relaionship beween monhly loan survival probabiliies and ha of erminaion probabiliies as below. 12 HECM loans erminae when he borrower leaves her home permanenly, simply chooses o pay off he loan, or borrower dies. However, no erminaion experience was available when he premiums were originally se, so an assumpion of erminaions a 1.3 imes moraliy was made (Rodda e al. (2004)). We assume ha he younger co-borrower in he family is a wife (i.e. female). Therefore, insead of using he co-borrower s join age sex moraliy rae, we simply use he female s moraliy rae in his analysis. Meanwhile, if considering he single saus, i is necessary o use male s moraliy rae. 11

13 da, = pa. qa, = pa, p a, + 1 (14) where d = he probabiliy ha he loan will be erminaed wihin a monh afer a, age a + p = he probabiliy ha a borrower of age a will survive a age a + a, q = he probabiliy ha he loan will be erminaed a age a + a, To conver annual borrower survival probabiliies, S i, j, ino monhly loan survival probabiliies, p a,, he HECM used he following equaion which boh inerpolaed geomerically and adjused for loan erminaions for reasons oher han deah of he borrower (Szymanoski (1990)). p a, r 12 S i, j+ 1 = S i, j S i, j 1+ m (15) where i = iniial age in years = {62, 63,, 99} j = aained age in full years = { i, i +1,, 100} a = iniial age in monhs = 12 i = aained age minus iniial age in monhs = 12( j - 1)+ r r = monhs beween aained ages j and j + 1 = {0,1,, 11} m = Move-ou rae expressed as a decimal = 0.3 Bu, according o Chow, Szymanoski, and DiVeni (2000) and Rodda, e al. (2003), he 1.3 imes moraliy assumpion under-predics erminaion probabiliies a young ages, and overpredics erminaion probabiliies a lae ages based on he hisorical daa. To alleviae his problem, we use male s moraliy rae as female s erminaion rae in his analysis. 13 In Korea, 13 According o his moraliy assumpion, we use he male s life expecancy from he daa of 2002 Complee Life Tables in he Korea Naional Saisical Office (KNSO) when we calculae he values of LTV in equaion (2). 12

14 we use he moraliy rae (he probabiliy of dying) from he daa of 2002 Complee Life Tables in he Korea Naional Saisical Office (KNSO) and generae monhly moraliy rae of aged 60 and above o aged For he purpose of inerpolaing, we used he smoohed Hodrick- Presco (HP) rends as follows Moaliy raes QF QFHP QM QMHP [Figure 2] Smoohed Moraliy Raes by Age In Figure 2, QF is female s moraliy raes and QM is male s moraliy raes. QFHP is HP rend of female s moraliy raes and QMHP is HP rend of male s moraliy raes QM. We can see in Figure 2 ha boh HP rends penerae he median poin of each age inerval. This is jus he reason why we used HP rends for inerpolaing he value of moraliy raes in his analysis because he annual moraliy raes of KNSO represen he median value of each age inerval. If we simply follow he mehod of HECM, he monhly loan survival raes (erminaion raes) will be a lile higher (lower) han ha of our analysis 16. In Figure 3, we compared QFHP, QMHP, and 1.3 imes of female s moraliy rae (QFHP 1.3). 14 The KNSO assumes ha he moraliy rae of aged 100 and above equal o 1.0 (if age 100 hen, q + =1.0). a 15 See Appendix for a discussion of he Hodrick-Presco Filering approach. 16 In acual analysis, we divided he values of HP rend by 12 monhs o generae he monhly moraliy rae of aged 60 and above. 13

15 .4.3 Moraliy raes.2.1 QFHP QFHP*1.3 QMHP Figure 3. Moraliy raes. The annuiy rae, discoun rae, expeced ineres rae When applying equaion (4) o (8), proxy he annuiy rae, discoun rae, and expeced ineres rae which adoped o deermine he ousanding loan balance by using he mean value of life insurer s raio of inves asses profi (RIAP) 17 ime series from April, 2000 o November, The Resuls The descripive saisics of each ime series in his paper are as follows. Table 1 The descripive saisics of each monhly ime series (January, 1986 o December, 2004) 17 Ma and Park (2004) confirmed ha here was a srucural change in he ime series of RIAP before and afer he poin of April, 2000 and he wo sub-period ime series of life insurer s RIAP were saionary before and afer April, 2001, respecively. Therefore, o alleviae he problem of ineres rae risk resuling from he difficulies of forecasing long erm ineres raes, hey suggesed ha he mean value of RIAP afer April, 2000 should be used a reference ineres rae o calculae he life insurer s assumed ineres rae in Korea. Reverse morgages have wo ineres raes, one for deermining he annuiy and he oher for accumulaing he loan (i.e. he oal ousanding loan balance). In he reverse morgage program, he fixed ineres rae deermines he annuiy paymen o avoid flucuaions in he borrower s cash flow while he adjusable rae compounds he loan (Chinloy and Megbolugbe, 1994). 14

16 growh rae Marke rae growh rae of ne RIAP of APT ( HB5) consumer price discoun raio Mean Sd. Dev Noe: The resuls of RIAP are from April, 2000 o November, We conduc uni roo es o es he saionary of each ime series. Table 2 repors he resuls of ADF es and PP es. 18 Table 2 The resul of uni roo es ( o ) growh rae of APT Marke rae ( HB5) growh rae of consumer price Ne discoun raio RIAP ADF PP Noe: 1. The resuls from including boh inercep and rend in es equaion. 2. The resuls of RIAP are from April, 2001 o November, Criical values: Period (January, 1986 December, 2004): 1%: , 5%: , 10%: ; Period (April, 2000 November, 2004): 1%: , 5%: , 10%: In Table 2, boh ADF and PP ess show ha he ime series of growh rae of APT, growh rae of consumer price, ne discoun raio, and RIAP are saionary a a 1 % level of significance, respecively. Bu, he marke ineres rae (HB5) does no rejec he null hypohesis ha here exiss a uni roo a a 10 % level of significance. Alhough he marke ineres is non-saionary ime series, we can sill use he mean value of ne discoun raio for calculaing he lump sum reverse morgage paymen (LSUM) in equaion (2) because he ime series of ne discoun raio is saionary. The lump sum reverse morgage paymen (LSUM) Based upon he mean value of ne discoun raio (NDR = ) and male s life expecancy, we can deermine LTV and calculae he lump sum reverse morgage paymen (LSUM). We assume ha he housing value is 200 million won 19 and he up-fron cos is 5% of housing value 20. The maximum values of LTV and LSUM in equaion (2) are repored in Table See Appendix for a discussion of ADF and PP ess. 19 USD $1 is abou 1,000 won. Therefore, 200million won is abou USD $200, As a base case, we assume ha he value of up-fron cos is 5 % of housing value and he up-fron cos involves 2 % of housing value as an up-fron insurance premium according o he insurance premium srucure of HECM program. 15

17 Table 3 Maximum value of LTV and LSUM (uni: Korean 1,000 won) Borrower s age Life expecancy 222 monh 179 monh 140 monh 106 monh 80 monh 59 monh LTV LSUM 74,000 90, , , , ,000 Noe: LSUM = 200 million Won * LTV We focus on wo paymen mehods of he enure plan in his paper. One is he consan monhly paymens and he oher is he graduaed monhly paymens which index o he growh rae of consumer prices. The consan monhly paymens We deermine he maximum level of annuiy rae in equaion (3) and (4). The monhly CDF (cumulaive discoun facor), he presen values of oal loan coss, and he maximum level of consan monhly paymens ( PMT ) are repored in Table 4. C Table 4 CDF and maximum value of PMT C (uni: Korean won) Borrower s age CDF C β 23.5% 31% 37.5% 44% 48.2% 50.4% PMT C 513, , , ,555 1,246,561 1,638,812 T ( a ) Noe: 1. [( ) ] 1 CDFC = 1 1+ i = 0 pa 2. he annuiy rae: i = per annum (RIAP rae 7% + MIP 0.5%): The mean value of RIAP ime series from April, 2000 o November, 2004 was 7.35 % per annum. We assume 7.0 % per annum as he RIAP rae. 3. We can calculae he values of β using rial-and-error approach such ha he values of LSUM afer adjusing β equals o he presen discouned values of annuiy paymens muliplied by he borrower s survival rae. In his case, he presen values of expeced claims equal o ha of expeced morgage insurance premiums. The graduaed monhly paymens In his paper, we analyze a mehod of graduaed monhly paymens which index o he growh rae of consumer prices. Based upon he mean value of monhly growh rae of consumer price ( c = ) and 7.5 % of annuiy rae, we can calculae he maximum level of base saring amouns of graduaed monhly paymens in equaion (6). Table 5 16

18 CDF and Maximum base saring amouns ( PMT G ) (uni: Korean won) Borrower s age CDF G β 23.5% 31% 37.5% 44% 48.2% 50.4% PMT 343, , , ,504 1,016,182 1,389,760 G [ p ] T ( a ) Noe: 1. 1 CDFG = ( + c 1+ i) 1 = 0 a 2. inflaion rae: c = 4.67% per annum 3. he annuiy rae: i =7.5% per annum. 3. We calculae he values of PMTG using he same values of β in consan paymen mehod (Table 4). The presen Value of Fuure Claim Losses and Fuure Insurance Premiums We calculae he presen value of fuure claim losses and fuure insurance premiums using he deermined values of monhly paymens. Afer calculaing he presen values, we evaluae he ne expeced insurance liabiliy of reverse morgage loans. We apply several assumpions during he analysis. Table 6 repors he resuls of ne expeced insurance liabiliy of reverse morgage loans. Under equal insurance premium srucure, all he raios of PVEC o PVMIP of consan paymens show similar o 1.0 because here were a pre-adjusmen of β in he acuarial model of reverse morgage. However, he raios of graduaed paymens show greaer han 1.30 because he loan erminaion raes show a paern of exponenial increase in he ail period and he magniude of monhly paymens increase by passage of ime. We adjus he level of monhly paymens of graduaed paymen mehods o improve he raios of PVEC o PVMIP. Table 7 repors he resuls of revised paymens plan which represen he raios of PVEC o PVMIP of graduaed paymens approach o he value of 1.0. Evaluaing oal annual loan coss (TALC) raes Firs of all, we evaluae he values of TALC raes under he condiion ha he values of β are equal o boh consan paymen and graduaed paymen mehods and hen calculae he values of TALC raes under he condiion ha he raios of PVEC o PVMIP are equal o 1.0 in he graduaed paymen mehod. Table 8 repors he TALC raes evaluaed using he resuls of Table 6 and 7 a various imes ( = n) by borrower s age. 17

19 Table 6 Ne Expeced Insurance Liabiliies (uni: Korean won) AGE Paymen LSUM β LSUM(1- β ) PMT mehod PVEC PVMIP NEL PVEC/PVMIP 60 consan 513,616 8,807,347 8,822,409-15, ,000, % 56,610,000 graduaed 343,299 11,544,074 8,473,348 3,070, consan 631,763 8,288,931 8,234,802 54, ,000,000 31% 62,100,000 graduaed 445,149 11,165,067 7,975,857 3,189, consan 779,889 7,551,454 7,573,311-21, ,000, % 66,250,000 graduaed 578,823 10,387,778 7,396,846 2,990, consan 975,555 6,864,932 6,927,860-62, ,000,000 44% 69,440,000 graduaed 760,504 9,493,598 6,815,776 2,677, consan 1,246,561 6,364,146 6,365,416-1, ,000, % 72,520,000 graduaed 1,016,182 8,649,222 6,296,976 2,352, consan 1,638,812 5,874,415 5,906,753-32, ,000, % 76,384,000 graduaed 1,389,760 7,659,491 5,863,936 1,795, Noe: 1. The growh rae of APT is 3/12 % (i.e. H H ( 1+ ( 0.03 )) = : For calculaing he presen values of fuure claim losses, we assume ha he 0 12 growh rae of APT is 3 % per annum. U.S. house price appreciaion raes based on he OFHEO index have averaged 5.7% since Bu, for he conservaive assumpion, he HECM acuarial model assumed house prices follow a 3 percen annual appreciaion rae from he loan originaion unil loan erminaion. This assumpion abou house price appreciaion rae has been kep conservaive o allow for under-mainenance given he exreme age of many HECM borrowers (Rodda, e al., 2003). In Korea, he mean value of annual growh rae of APT from January, 1986 o December, 2004 was 5.64% (= *12). 2. The value of discoun rae which used on calculaing he presen values of expeced claims and ha of MIP, he expeced ineres rae which adoped o deermine he ousanding loan balance (OLB) is 7.5 % per annum including margin and MIP (i.e. 0.5 % per annum) 3. Up-fron coss is 5 % of he iniial house price ( H 0 *5 %) and we assume ha up-fron coss includes 2 % of up-fron insurance premium (i.e. H 0 *2 %) 4. The iniial house price is 200 million Won. 5. The probabiliy ha he loan will become due and payable is male s moraliy rae. 18

20 Table 7 Modified Ne Expeced Insurance Liabiliies: Graduaed monhly paymens (uni: Korean won) AGE Paymen LSUM β LSUM(1- β ) PMT mehod PVEC PVMIP NEL PVEC/PVMIP 60 before 23.5% 56,610, ,299 11,544,074 8,473,348 3,070, ,000,000 afer 32.6% 49,876, ,463 8,111,320 8,051,544 59, before 31% 62,100, ,149 11,165,067 7,975,857 3,189, ,000,000 afer 39.5% 54,450, ,312 7,581,886 7,578,428 3, before 37.5% 66,250, ,823 10,387,778 7,396,846 2,990, ,000,000 afer 44.8% 58,512, ,216 7,076,847 7,068,936 7, before 44% 69,440, ,504 9,493,598 6,815,776 2,677, ,000,000 afer 50% 62,000, ,021 6,580,341 6,562,552 17, before 48.2% 72,520,000 1,016,182 8,649,222 6,296,976 2,352, ,000,000 afer 53.2% 65,520, ,095 6,121,451 6,108,246 13, before 50.4% 76,384,000 1,389,760 7,659,491 5,863,936 1,795, ,000,000 afer 54.1% 70,686,000 1,286,089 5,756,549 5,743,958 12,

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