Breakeven Determination of Loan Limits for Reverse Mortgages under Information Asymmetry

Transcription

1 IRES IRES Working Paper Series Breakeven Deerminaion of Loan Limis for Reverse Morgages under Informaion Asymmery Ming Pu Gang-Zhi Fan Yongheng Deng December, 01

2 Breakeven Deerminaion of Loan Limis for Reverse Morgages under Informaion Asymmery Ming Pu * Gang-Zhi Fan Yongheng Deng December 01 Absrac Since he loan limi of a reverse morgage is a major concern for he borrower as well as he lender, his paper aemps o develop an opion-based model o evaluae he loan limis of reverse morgages. Our model can idenify several crucial deerminans for reverse morgage loan limis, such as iniial housing price, expeced housing price growh, house price volailiy, moraliy disribuion, and ineres raes. We also pay special aenion o he imporan implicaion of morgage lenders informaional advanage over reverse morgage borrowers concerning housing marke risk. In reverse morgage markes, he elderly borrowers ypically hold far less, relaive o he lenders, or no informaion abou he lenders underlying morgage pools. Such informaion asymmery leads hese wo caegories of marke paricipans o generae differen perspecives on he risk of he collaeralized properies, which can be idenified o be imporan in deermining he maximum loan amouns of reverse morgages. We furher find ha he maximum loan amoun of a reverse morgage decreases in he correlaion beween he sysemaic reurn and he reurn on is underlying housing propery bu increases wih he number of he pooled morgages. Key Words: Reverse Morgages, Heerogeneous Beliefs, Informaion Asymmery, Loan Limi JEL: G1, G, J14, R * School of Insurance, Souhwesern Universiy of Finance and Economics, 55 Guanghuacun Sree, Chengdu, China. [email protected]. Corresponding Auhor. Deparmen of Real Esae, Konkuk Universiy, 10 Neungdong-ro, Gwangjin-gu, Seoul , Korea. [email protected]. Insiue of Real Esae Sudies, Naional Universiy of Singapore, 1 Heng Mui Keng errace, #04-0, Singapore [email protected]. We would like o hank Chun-seob Lee, Seven Li, Dogan iriroglu, Seow Eng Ong, Kwong Wing Chau, Zhi Dong, an anonymous referee and paricipans a he 011 Asian Real Esae Sociey-American Real Esae and Urban Economics Sociey Join Inernaional Conference and he 011 Asia-Pacific Real Esae Research Symposium for helpful commens. Any errors are our own. 1

3 Breakeven Deerminaion of Loan Limis for Reverse Morgages under Informaion Asymmery 1. Inroducion Growing number of economies such as Japan, U.S., China, Korea, EU counries are moving rapidly owards aging socieies, which have gradually produced a series of social and economic issues in hese economies (Weizsäcker, 1996; Faruqee and Mühleisen, 003; Li, 005; Davis, 1997). Reverse morgages provide house-rich bu cash-poor elderly homeowners wih a caegory of promising financial producs o obain loans from morgage lenders using heir homes as loan collaeral wihou losing he enure of hese premises, and herefore represens a poenial channel ha allows hem o wihdraw heir subsanial home equiies for improving heir consumpion and reiremen securiy. In he U.S., he Federal Housing Adminisraion (FHA) launched he Home Equiy Conversion Morgage (HECM) program in 1989 afer he passage of he Naional Housing Ac of Under he HECM program, cash-poor bu house-rich homeowners aged 6 and above can wihdraw he equiy of heir homes in he form of a lump sum paymen, a line of credi, monhly paymens, or any combinaion of hese, which is also insured by he FHA [see, e.g., Mayer (1994) and Kuy (1998)]. HECM loans are he mos widely used caegory of reverse morgages and accoun for more han 90% share of he U.S. reverse morgage marke. 1 Alhough here were only abou several hundred HECM loans originaed each year during he whole 1990s (Bishop and Shan, 008), he number of HECM loans originaed each year has significanly increased over he pas decade while experiencing he adverse influence of he recen finance urmoil. Due o he grea poenial of reverse morgage programs in addressing he social and economic problems discussed above in rapidly aging sociey, here is a growing 1 See Hammond (1993) for he deails abou oher caegories of reverse morgages in he U.S.

4 volume of lieraure invesigaing he feasibiliy and poenial of developing he reverse morgage markes in differen counries. For example, Michell and Piggo (004) show ha since elderly dependency raio in Japan, already quie high among he developed economies, is projeced o reach 45 percen by 030 due o increasing longeviy and declining feriliy, releasing home equiy in his counry via reverse morgages will faciliae boosing elderly consumpion, enhancing reiremen securiy and miigaing public pension liabiliy [see also Hayashida and Sasaki (1986)]. Addae-Dapaah and Leong (1996) invesigae he applicabiliy of he Home Equiy Conversion Scheme (HECS) o he elderly in Singapore, and heir findings show ha selling home equiy in exchange for a life enancy will be a subsanial poenial form of HECS program for elderly Singaporeans. Chou e al. (006) explore he viabiliy of culivaing he reverse morgage marke in Hong Kong. Since here is no reiremen proecion scheme for is elderly aduls, who are also one of he larges povery groups in his ciy, heir findings demonsrae ha he reverse morgage program has conribued o he improvemen of he economic saus and well-being of he elderly and o he miigaion of he financial burden of is governmen. For hese reasons, we migh expec ha he reverse morgage program will become more and more popular in many pars of he world in he nex decades in ha boh he absolue number and percenage of elderly people are growing rapidly in hese economies. Alhough he exising lieraure has examined he poenial demand for various reverse morgage programs, lile research has addressed he loan limi issue of reverse morgages. his sudy fills he gap o sudy he valuaion of he loan limis of reverse morgages, which is a major concern for heir borrowers in addiion o heir lenders (Sawyer, 1996). A large body of lieraure has explored he valuaion issue of he convenional morgages subjec o defaul risk or prepaymen risk. he overwhelming majoriy of he lieraure has developed opion-based models, boh For more discussions on he poenial demand of reverse morgages, see for example Levion (001), Rasmussen e al. (1995), Merrill e al.(1994), Mayer and Simons (1994), and Shan (011). 3

5 heoreical and empirical, o he valuaion issue of hese morgages. 3 However, compared wih he convenional ones, we can idenify from he relevan lieraure several remarkable feaures associaed wih reverse morgages. Firs, reverse morgages are ypically a caegory of non-recourse loans such ha a lender can only realize he repaymen of is loan principle and ineres using he sales proceeds of he collaeralized home. If he value of he home is insufficien o reclaim he ousanding balance of he reverse morgage a he ime of sale, he lender has o suffer a loss due o no recourse o oher asses. Chinloy and Megbolugbe (1994) define he risk of he ousanding loan balance exceeding he house equiy value as cross-over risk, which is usually our major concern in he valuaion of reverse morgages. Also, since he reverse morgage borrower is no required o repay any principal and ineres prior o his deah or permanen move-ou from he house, his ousanding loan balance will gradually increase over ime due o accrued ineress and possible monhly paymens. 4 Second, since under he reverse morgage program an elderly borrower can sill coninue o say in his home unil he dies or moves ou of his home permanenly, he morgage erminaion is usually designed on he basis of published acuarial moraliy ables. If he borrower lives longer han he life expecancy esimaed from he ables, hen he lender is exposed o longeviy risk. 5 In conras, a convenional morgage debor is usually only vulnerable o defaul risk and prepaymen risk. Given ha longeviy risk has imporan implicaion for he deerminaion of reverse morgage loan limis, a simple applicaion of he sandard opion pricing model o reserve morgages wihou consideraion of uncerain moraliy is likely o resul in an inaccurae evaluaion of maximum morgage loan amoun. As a resul, his paper develops a breakeven framework for evaluaing morgage loan limis ha aemps o inegrae longeviy risk as well as cross-over risk ino he opion-based valuaion 3 See for example Kau e al. (199, 1995), Hilliard e al. (1998), Epperson e al. (1985), Brunson e al. (001), Deng (1997), Deng e al. (000), Ciochei and Vandell (1999), and Archer e al. (00). Kau and Keenan (1995) provide a comprehensive survey on such caegory of earlier lieraure. 4 Ma and Deng (006) sudy he insurance premium srucure of reverse morgage loans in Korea. hey find ha o he relaively younger borrowers, he graduaed monhly paymen approach is more efficien, while he consan monhly paymen approach is more efficien for he elderly borrowers. 5 We can also idenify from he relevan lieraure oher risk facors, such as home-value risk and mainenance risk (Syzmanoski, 1994; Miceli and Sirmans, 1994). he cross-over risk is acually mainly deermined by boh longeviy risk and home value risk. 4

6 model. In he breakeven poin, he acuarial presen value of he loan amouns paid by he lender o he borrower mus equal he acuarial presen value of he payoff which he lender will receive a he mauriy dae. Furhermore, anoher imporan conribuion of our sudy is he consideraion of morgage lenders informaional advanage over reverse morgage borrowers abou heir home values in he secor of reverse morgages. In he convenional morgage markes, we can usually idenify he informaion asymmery concerning a morgage borrower s credi characerisics beween he borrower and he primary lender, or beween hese wo paries and he ulimae invesors in he secondary morgage marke (Kau e al., 011). In conras, such informaion asymmery is no a major concern in reverse morgage markes, in ha for a reverse morgage, he appraisal value of is underlying home insead of he credi characerisics is usually a major consideraion bu is also common knowledge for boh he elderly borrower and he morgage lender. However, i is noeworhy ha morgage lenders usually provide reverse morgage loans o a grea number of elderly borrowers and manage heir morgage pools, such ha idiosyncraic risk in he underlying morgage pools can, o a large exen, be diversified away, while hose borrowers know very lile abou he deails of he underlying morgage pools. We can show here ha such informaion asymmery migh lead hese wo caegories of marke paricipans o produce heerogeneous beliefs on he volailiy in he housing marke. In effec, morgage lenders informaion advanages have been idenified o have imporan implicaions for morgage markes in several recen sudies. For example, Bond e al., (009) highligh ha morgage lenders informaion advanages over morgage borrowers abou collaeralized home values can provide an ineresing explanaion on he occurrence of predaory morgage lending (Bond e al., 009). Gan and Riddiough (008) provide empirical evidence ha morgage lenders can exploi heir informaion advanages o deer he poenial compeiors from enering he U.S. residenial morgage marke. 5

7 Basak (000) furher emphasizes ha heerogeneous informaion can make marke paricipans generae heerogeneous beliefs or inferences on he sochasic evoluion of asse prices. We allow for he imporan implicaion of heerogeneous beliefs caused by morgage lenders informaional advanage abou house prices for he loan limis of reverse morgages. Several imporan sudies such as Deemple and Murhy (1997) and Basak and Croioru (000, 006) have demonsraed ha heerogeneous beliefs maer in generaing rades in financial markes. Oher financial lieraure has also focused aenion on he significan effecs of heerogeneous beliefs on asse pricing (e.g., Scheinkman and Xiong, 003; Li, 007). Since reverse morgage markes can be characerized by asymmeric or heerogeneous informaion as discussed above, our analysis pays aenion o he key role of morgage lenders informaional advanage in generaing heerogeneous beliefs. We assume ha marke paricipans are of homogeneous or common prior belief on he price process of an underlying individual house, which is a ypical assumpion in game heory and he economics of informaion (e.g., Bonanno and Nehring, 1999). More specifically, boh he morgage lender and he reverse morgage borrowers are specified o have he same or homogeneous belief on he expeced percenage growh and volailiy of he underlying house price. However, since he lender holds complee informaion abou is morgage pools, i can make housing marke inferences based on modern porfolio heory, while he borrowers only have very lile or no informaion on hese pools. Such informaion asymmery can make hese wo caegories of marke paricipans produce he heerogeneous beliefs concerning he volailiy of he pooled underlying houses. his is because, given ha propery price volailiy can be decomposed ino sysemaic volailiy and idiosyncraic volailiy (Miao and Wang, 007), he idiosyncraic risk of he underlying propery price can be diversified away by adoping a porfolio sraegy. As a consequence, we can demonsrae ha he lender informaion advanage can play an imporan role in deermining he loan limis for reverse morgages and herefore he lender economic profi size. 6

8 On he oher hand, we also show ha for he reverse morgage borrowers, heir informaion disadvanage migh lower he amouns of morgage loans hey can obain from he lender. During he pas wo decades, in effec, he marke share of diversified morgage lenders have experienced a furher obvious increase, while he share of concenraed lenders ha provide mos of morgage loans in one local marke has dropped o around 4% five years ago (Louskina and Srahan, 011). his suggess ha he porfolio managemen of morgage loans has become very popular and been widely adoped as one imporan sraegy of risk diversificaion in he secor of morgage lending. Alhough he size of he underlying reverse morgage pool migh be very small a he sage of reverse morgage originaions, he lender can sill realize he diversificaion of morgage porfolios across he differen regions or caegories of morgages, in ha morgage porfolios migh be effecively diversified across he differen regions and caegories (Eichholz e al. 1995). In his case, he lender can gaher he relevan informaion from he morgage porfolios, while he borrowers suffer an informaion disadvanage relaive o he lender and can only acquire he less amouns of morgage loans. As a resul, such informaion asymmery cerainly maers no only for he borrowers who choose he lifeime annuiy-like paymen plan or he line of credi paymen plan, bu also for he borrowers who choose o receive an upfron lump-sum morgage paymen a he ime of reverse morgage originaion. In effec, i can also be shown ha our main findings are likewise applicable o he credi analysis of he radiional house morgage secor and herefore provide new insighs ino he loan limis of radiional house morgages, which are closely relaed wih he recen real esae crisis. he remainder of his aricle is organized as follows. Secion develops an opion-based model for deermining he maximum loan amouns of reverse morgages. Secion 3 proposes hree caegories of models o measure longeviy risk. Secion 4 applies he proposed model o numerical analysis. Secion 5 summarizes his aricle and draws conclusions. 7

9 . Reverse Morgage Models.1 Basic Assumpions Suppose ha a house-rich bu cash-poor elderly homeowner is considering wheher or no o ener ino a reverse morgage conrac in order o resolve his consumpion problem. According o his conrac, he can conver his home equiy ino a lump sum paymen or a life annuiy for he res of his life. He does no need o repay he borrowed morgage loan and accrued ineress unil he dies or moves ou from his dwelling. he ousanding balance owed o a lender will be repaid using he sales proceeds of he house uni. If he proceeds are insufficien o repay he ousanding balance, he lender has o suffer a morgage loan loss. hree major risk facors longeviy, housing prices, and ineres raes are usually aken ino consideraion in he valuaion of reverse morgages [see, e.g., se (1995)]. For he purpose of his sudy, however, we mainly focus on he imporan implicaion of uncerainies on longeviy and housing price in he valuaion of reverse morgages. 6 In exising morgage-pricing lieraure, housing prices are ypically assumed o follow a geomeric Brownian moion process so ha he lieraure can model uncerainy on hese prices. FHA also usually assumes ha housing prices evolve following a geomeric Brownian moion process in is reverse morgage pricing model (Quercia, 1997). o explore he impac of he housing price uncerainy, we likewise assume ha he housing price follows a geomeric Brownian moion process as follows: dp = µ Pd + σpdw, (1) where µ is he expeced percenage growh in house price, σ is he volailiy parameer of house price, and W is a sandard Brownian moion. 6 However, we will exend our model o allow for he effecs of changes in wo caegories of ineres raes in he numerical analysis, such as loan ineres rae and risk-free ineres rae. his will provide addiional insighs ino he usefulness of our valuaion model. 8

10 Longeviy risk is anoher crucial deerminan of he cross-over risk of reverse morgages. o capure he impac of his risk facor, le be he ime insan a which he deah of he homeowner occurs. For any given ime > 0, he cumulaive disribuion funcion and he probabiliy densiy funcion of he random deah ime can be defined, respecively, as follows F ( ) = Pr( ), () f() = F'(). (3) Correspondingly, he survival funcion specifies he probabiliy a which he homeowner can live beyond ime S ( ) = 1 F ( ) = Pr( > ). (4) In addiion, o describe he insananeous deah rae, acuaries define he force of moraliy by S'( ) λ () =. (5) S () In his analysis we allow he homeowner o wihdraw his house equiy by choosing eiher of wo ypical caegories of reverse morgage schemes, namely a lump sum paymen and a lifeime annuiy. In hese wo schemes, we will develop a breakeven framework o deermine boh he maximum morgage loan amouns offered by he lender and he minimum loan amouns accepable for he homeowner. Our model differeniaes wo caegories of ineres raes. he firs one, R, is he rae a which he lender is going o charge he elderly homeowner, and which is used o calculae he ousanding loan balance. he second one, r, is he risk-free ineres rae 9

11 or he opporuniy cos, which is uilized o calculae he presen value of he morgage loan. Since his model is developed in he coninuous ime conex, he ineres raes are compounded coninuously. 7. Porfolio Diversificaion he operaion of insurance indusry largely relies on he Law of Large Numbers and he Cenral Limi heorem,and his principle is likewise applicable o he reverse morgage secor. According o hese heories and modern porfolio heory, if he number of pooled reverse morgages is sufficienly large, he house price risk in he morgage pool can be, o a large exen, diversified away. In order o allow for he imporan implicaion of morgage porfolios in his analysis, we relax he previous assumpion by allowing he lender o provide reverse morgage loans o N elderly homeowners. For simpliciy, suppose ha all he morgage borrowers have he same age x, and ha boh he lender and hese borrowers hold he homogeneous belief on he expeced growh and volailiy of he price of any individual housing uni. As a resul, he prices of N underlying houses can be expressed as an N-dimensional vecor ha follows a sysem of geomeric Brownian processes dp = µ P d + σp dw for i = 1,,3... N (6) () i () i () i () i, where ( i) W represens a sandard Brownian moion driving he sochasic evoluion of underlying propery i price. Miao and Wang (007) show ha he oal volailiy of real esae value can be decomposed ino sysemaic volailiy and idiosyncraic volailiy. his suggess ha he value of a propery is influenced by idiosyncraic risk 7 For exposiional convenience, boh R and r are specified o be deerminisic variables in our analysis and are herefore independen of each oher. An exension of allowing hese wo variables o be correlaed wih each oher will complicae our analysis subsanially, while i does no provide addiional insighs ino our research quesion. Despie such specificaion, however, we will show in Figure 4 ha when hese wo raes are becoming closer o each oher, he annuiy paymen will decline. 10

12 as well as sysemaic risk. Accordingly, we may decompose each W according o () i he following expression dw = ρdz + 1 ρ db, i = 1,,3,..., N (7) () i () i i i where ρ i can be explained as he correlaion coefficien beween he sysemaic reurn and he reurn on underlying propery i, Z is a sandard Brownian moion sanding for he sysemaic shock, B is a sandard Brownian moion represening () i he idiosyncraic shock and is independen of Z and B for j i and j = 1,,3,..., N. In addiion, ρσ i can be explained as he sysemaic componen of ( j ) house price volailiy, while ρ σ represens he idiosyncraic componen. I is 1 i noeworhy ha even hough he parameer ρ i is of he same value across he underlying propery pool, he Brownian moions W migh be differen in ha () i are independen of he sysemaic shock and of each oher. () i B Le P N = P be he oal value of hese N homes. By adding he sochasic i= 1 () i differenial equaions by each P () i, we obain N N N () i () i () i () i dp = µ P d + σ P dw i= 1 i= 1 i= 1. (8) o simplify he analysis, we assume ha he correlaion coefficien ρ i = ρ is consan across he underlying propery pool. his is similar o considering he equally weighed porfolio in modern porfolio heory, and such assumpion can, o a large exen, simplify our model in order o shed new ligh on he deerminaion of he maximum loan amouns. An exension of relaxing his assumpion will complicae our analysis and resul in he inracabiliy in mahemaical reasoning, and he resulan model can only be solved numerically using Mone-Carlo simulaion. Also, such 11

13 relaxaion canno produce addiional insighs ino our main issues of ineres. As a resul, subsiuing (7) ino (8), we direcly obain P dp Pd P ( dz 1 db ). (9a) N () i () i = µ + σ ρ + ρ i= 1 P P where P () i can be inerpreed as he weigh of house i value in he underlying house porfolio. Wihou loss of generaliy, suppose ha he weigh can be expressed as π = i P P () i, where N i i π = 1 and 0< π < 1 for each i. Subsiuing his weigh ino i= 1 (9a) produces he following equaion N i () i = µ + σ ( ρ + π 1 ρ ) i= 1 dp Pd P dz db. (9b) Define N 1 i () i W = ρz π 1 ρ B θ + (10) i= 1 θ = ρ + (1 ρ ) π. I is easy o verify ha W is also a sandard i where ( ) N i= 1 Brownian moion. herefore, from a porfolio perspecive, he oal value of hese pooled properies saisfies he following sochasic differenial equaion N () i 0 0 i= 1 dp = µ Pd + σpdw, P = P, (11) 1

14 where σ = θσ, and P is he iniial price of underlying propery i. Given () i 0 N N i i ( π ) < π = 1, one may easily find ha when N >1, 1 i= 1 i= 1 θ < always holds. his indicaes ha θ is sricly less han uniy such ha he volailiy of he underlying propery porfolio is less han ha of any individual house due o he diversificaion impac of propery porfolio. For exposiional convenience, suppose ha all he underlying homes have he same i 1 weigh a a given ime, ha is π =. In his case, equaion (10) can rewrien as N N 1 1 () i W = ρz 1 ρ B θ + N (1) i= 1 1 whereθ = ρ + (1 ρ ). By equaions (11) and (1), when he lender provides N reverse morgage loans o more han one elderly homeowners, he expeced price growh of he underlying propery pool is sill kep he same as ha of any individual home, bu is volailiy would be reduced compared wih hose of any individual home. his implies ha boh he lender and he elderly borrowers always have a common expecaion on he percenage price growh of he underlying houses bu could have heerogeneous perspecives on heir volailiy magniudes. As he number of reverse morgage loans in he morgage pool increases, we find lim θ = ρ N. his means ha boh he number N and he correlaion coefficien ρ could have imporan implicaion for he deerminaion of reverse morgage loan limis. If here are a sufficienly grea number of elderly homeowners who have obained reverse morgage loans, he lender would be only exposed o sysemaic risk in he housing 13

15 marke due o he diversificaion effec of propery porfolio. However, he individual elderly homeowners are sill exposed o he idiosyncraic risk as well as he sysemaic risk in he marke if hey plan o sell ou heir homes, in ha hey canno diversify away he idiosyncraic risks of heir houses via developing a similar porfolio sraegy. his is analogous o he insurance indusry. Alhough an individual policy holder has a considerable probabiliy o experience a large damage even, insurance companies can largely decrease he uncerainy via selling a grea number of insurance policies according o he Law of Large Numbers. hese companies make full use of his advanage o charge he insurance premiums which are profiable for hem while being also accepable o each policy holder. Furher, we also allow for wo exreme cases: ρ = 1 and ρ = 0. If ρ = 1, one may easily find ha θ = 1 and σ = σ. 8 his indicaes ha if he reurns on he pooled properies are compleely correlaed and heir sochasic evoluions will be driven only by a single sysemaic Brownian moion. In his case, any risk in he underlying propery pool canno be diversified away, and boh he lender and he homeowners herefore have he idenical perspecive on he house volailiy. As a resul, here are no bid-ask spreads of morgage loan amouns generaed by heir marke beliefs on house risk (see Figure 10). Similarly, in he insurance indusry insurance companies are usually no willing o insure hose damages ha are highly correlaed. For example, propery insurance policies usually exemp he insurance companies from he coverage for damages caused by earhquakes. On he oher hand, if ρ = 0, he reurns on he pooled properies are pair wise independen of each oher. In his scenario, we have θ = 1 N, which is he minimum 8 When ρ = 1 and ρ = 1, we can readily find ha boh of hem will produce he same effec on he loan limis from he lender s perspecive. If ρ = 1, we only need o replace moion Z, and hen we can obain he same resul on he limis as in he case of ρ = 1. Z by a new sandard Brownian 14

16 θ value for fixed N, and he idiosyncraic risk of he underlying propery porfolio can be diversified o he larges exen. he lump sum or he annuiy paymen offered by he lender is herefore highes in his case, and he bid-ask spread of morgage loan amouns is larges (see Figure 10). One may also noice ha if N approximaes infiniy in his case, we have lim θ = 0. ha is, as he number of he pooled morgages N approaches infiniy, he price of he underlying propery porfolio in he lender s viewpoin evolves in a deerminisic manner, while hose morgage borrowers sill hink ha housing prices are governed by he sochasic processes specified above..3 Lump Sum Paymen and Bid/Ask Amouns In his secion, we discuss he deerminaion of he breakeven paymens of reverse morgage loans if he morgage borrowers choose o receive a lump sum paymen a = 0. Since all he elderly morgage borrowers are homogeneous, for exposiional convenience we look a his issue from an individual morgage borrower s and a lender s perspecive. As we have menioned above, hese wo marke paricipans hold he homogeneous belief on he expeced growh of he housing prices, whereas hey could have differen beliefs on housing volailiy due o he diversificaion impac of propery porfolio. Under our model framework, we specify ha he acuarial presen value of he loan amoun provided by he lender o he elderly borrower should be equal o he acuarial presen value of he payoff received by he lender a he mauriy dae such ha he breakeven loan amoun can be deermined endogenously. We can find ha because of he heerogeneous beliefs on housing volailiy, hese wo paricipans can hold differen viewpoins abou he breakeven loan amoun such ha a bid-ask amoun spread is generaed. Suppose ha he lender makes a lump sum paymen o he homeowner a he beginning of he conrac, = 0. We iniially fix he homeowner s deah ime a ime = and only allow for uncerainy on housing price. Condiional on his assumpion, he breakeven lump-sum paymen of he loan can be deermined according o he 15

17 following breakeven condiion ( ) max,0 r r R L = e E P e E P Le (13) where he lef-hand side erm represens he breakeven loan amoun received by he homeowner a ime 0, while he righ-hand side represens he presen value of he payoff which he lender will receive a he mauriy dae. Since he soluion o equaion (1) can be expressed as 1 P = P0 exp[( µ σ ) + σw], (14) We can obain he following expression regarding he firs erm of he righ-hand side in (13) r r 1 e E[ P ] = Pe 0 E exp ( µ σ ) + σw ( µ r ) 1 = Pe 0 E exp σ + σw. (15) 1 According o Io s Lemma, we may readily verify ha Z = exp( σ + σw) is a maringale, so is expecaion is EZ = EZ0 = 1. Hence, we have ( ) [ ] r r e E P Pe µ 0 = (16) he second erm of he righ-hand side in (13) may be viewed as he payoff of a European call opion, while is srike price R Le is a funcion of he given deah ime. As a consequence, for he given deah ime we may calculae he second erm using he echnique similar o ha of obaining he value of he European call opion [see Shreve (004, p.19-0)]. he proof deails are given in Appendix A 16

18 while he resulan expression is direcly given by E e P Le Pe Nb L Le Nb L r R ( m r ) ( R r ) max(,0) = 0 ( 1( )) ( ( )) (17) where 1 P 0 1 b1( L) = b( L) + σ = log ( r σ ) R + + σ Le Proposiion 1: Given ha he price of he collaeralized home is governed by a geomeric Brownian moion process and he homeowner s deah ime is fixed a, he breakeven lump-sum paymen of he reverse morgage loan saisfies he following equaion ( µ r ) ( µ r ) r R L = Pe 0 Pe 0 N( b1( L)) + e Le N( b( L)) (18) Proof: his proof is sraighforward by subsiuing (16) and (17) ino (13). I is worh noing ha he derivaion of equaions (17) and (18) is differen from he process of obaining he classic Black-Scholes opion formula. We do no assume he compleeness of he marke such ha he convenional no-arbirage argumen does no work here anymore. Insead, we calculae he breakeven paymen by equaing he acuarial presen value of he cash flow from he lender o he borrower wih ha from he borrower o he lender. Based on he principle, we can derive equaions (17) and (18). he soluion of (18) wih regard o L does no have a closed-form expression, bu we may solve i numerically. I is also noeworhy ha o derive L we have assumed ha he deah of he morgage borrower occurs a a fixed ime, while he individual life span could change a random in acuarial pracice. If he uncerainy of moraliy is aken ino consideraion, hen he breakeven lump-sum paymen is given 17

19 by L = L f () d. (19) 0 We may uilize (18) and (19) o deermine he breakeven loan amoun. I can be shown ha he breakeven paymen always decreases in he housing volailiy. If we exend his model o allow for a grea number of elderly morgage borrowers, his suggess ha he volailiy for he lender be smaller han any one of he elderly borrowers believes due o he diversificaion effec of propery porfolio. As a resul, he lender is acually willing o offer higher breakeven paymens han he elderly borrowers expec. Proposiion : Suppose ha all he N collaeralized homes are homogenous, while heir prices are governed by he respecive geomeric Brownian processes. Given i 1 N > 1and π = a a ime, he bid and ask amouns of lump-sum paymen of a N reverse morgage are deermined by (18), whereas heir volailiy parameers are σ andσ, respecively. Proof: his proof is sraighforward by subsiuing (6) and (1) ino (18). his proposiion suggess ha he volailiy σ of he underlying propery pool can, o a cerain exen, be diversified hrough he porfolio sraegy such ha he bid and ask amouns of lump-sum paymen of a reverse morgage migh be disinc for he lender and he elderly borrower. his is because σ = θσ and θ < 1for N >1, such ha σ < σ holds. Also, one can readily find ha θ ρ > 0, herefore implying ha an drop in he absolue value of he correlaion ρ reduces he value of θ and in urn he value of σ. We may also idenify θ N < 0, which suggess ha an increase in he number N of he pooled reverse morgages decreases he value of σ. 18

20 For he reason, his proposiion can provide imporan implicaion for he bargaining process beween he lender and he elderly borrowers. his proposiion also means ha he lender s marke power plays an imporan role in deermining he marke-clearing amoun of lump-sum paymen of a reverse morgage. Le Q [ 0,1] denoe he lender s deerminisic marke power, ha is, her abiliy o deermine he reverse morgage loan amoun independenly. hen he marke-clearing reverse morgage amoun is given by L = QL + (1 Q) L clear bid ask where bid L respecively. and ask L represen he bid and ask amouns of lump-sum paymens,.4 Annuiy Paymen In his secion, we assume ha insead of a lump-sum paymen, he elderly borrower chooses a lifeime annuiy-like scheme wih coninuous paymens. We will invesigae his caegory of reverse morgages by he similar argumen: we firs fix he elderly borrower s deah ime a and allow for uncerainy in he housing price. Nex, we furher ake longeviy risk ino consideraion. Under his assumpion, upon he deah of he elderly borrower he ousanding loan balance is given by 0 R ( e 1) Rs B( A, R, ) = e Ads = A (0) R where R is he coninuously compounded rae, and A sands for he coninuous rae of morgage loan paymen he lender makes. If he value of he collaeralized house exceeds he ousanding loan balance, his loan canno be repaid unil he deah or move-ou of he homeowner; oherwise, he lender can only receive he proceeds from 19

21 he sale of he residenial propery. As a resul, upon he deah of he homeowner he lender can receive he following payoff X [ P B AR ] = min, (,, ) [ ] = B( AR,, ) max B( AR,, ) P,0 [ ] = P max P B( AR,, ),0. (1) his suggess ha he payoff of he reverse morgage conrac a he mauriy dae is equal o ha of a porfolio consising of a long posiion in he collaeralized house and a shor posiion in a European call opion wih he srike price depending on he homeowner s deah ime. Accordingly, he ne presen value of he lender s cash flows can be wrien as r rs NPV = E e X e Ads 0 ( ) r r r = E e BAR (,, ) e max BAR (,, ) P,0 e BAr (,, ), () where he erm r e X denoes he presen value of he payoff he lender will receive rs a he ime of deah or move-ou, and he erm e Ads is he presen value of he 0 morgage loan. In order for he lender o have an incenive o offer he reverse morgage loan, he NPV should be greaer han zero. As a resul, he following inequaliy should be saisfied ( ) r r r e BAR (,, ) e BAr (,, ) > e E max BAR (,, ) P,0. (3) r Given ha max [ (,, ),0] R e B AR P is no smaller han zero, his implies ha > ralways holds. Since he lender is exposed o boh he longeviy risk and he house value risk, i requires charging he elderly borrower in providing he reverse 0

22 morgage by a higher ineres rae raher han he risk-free ineres rae. In oher words, R should reflec he compensaion for he wo caegories of risks. In conras, he risk-free ineres rae only reflecs he reurn level of a risk-free invesmen. As a consequence, R should be always greaer han r; oherwise, here will be arbirage opporuniy in he financial marke. Since deah ime is a crucial deerminan of he loan paymen, we make use of A o denoe he breakeven coninuous paymen rae if he deah occurs a ime. hen we can deermine A based on our breakeven definiion [ ] r r r NPV= e BAR (,, ) e BAr (,, ) e max EBAR (,, ) P,0 = 0. (4) Subsiuing (0) ino (4), we can obain he following equaion r R R ( 1 ) ( 1) ( 1) e A e A r e A r = e e max E P,0, (5) r R R where he lef-hand side erm is he presen value of he annuiy paymens made by he lender o he homeowner, while he righ-hand side represens he presen value of he payoff which he lender will receive a he mauriy dae. Since he second erm of he righ-hand side of (5) can acually be viewed as a shor posiion on a European pu opion wih he srike price being he ousanding loan balance, i is easy o show ha keeping oher parameers unchanged, a rise in he expeced house price growh µ leads o a fall in he value of he pu opion. his herefore makes he presen value of he payoff received by he lender han ha of he lef-hand side of (5). o achieve he breakeven paymen level, A should be increased as µ rises, herefore implying A µ > 0. A similar argumen can also 1

23 be applied here o inerpre he posiive effec of he rising iniial house price on A, ha is, A P0 > 0. In conras, according o he sandard Black-Scholes opion heory, a higher volailiy σ raises he value of he pu opion, and herefore reduces he presen value of he righ-hand side payoff of (5). As a resul, holding oher hings consan, A is reduced, as σ escalaes, so ha he breakeven paymen level is mainained, hence suggesing A σ < 0. However, he effecs of changes in he elderly borrower s remaining lifeime on A are ambiguous. Alhough a longer migh raise he fuure price of he underlying propery and hence reduce he value of he pu opion, i also implies more uncerainy on he fuure price, herefore raising he value of he opion. As a consequence, while an exension in also causes he presen value of he annuiy paymens o rise, he effec of longer remaining lifeime could be opposie and we hence have A > < 0. Alernaively, according o (1) we can also deermine A based on he following breakeven definiion [ ] r r r NPV= e EP ( ) e BA (, r, ) e max P BA (, R, ),0 = 0. (6) his equaion can be rewrien as r ( ) R ( ) 1 e A e 1 A r r = e E( P) e max E P,0, (7) r R where he righ-hand side sill can be explained as he presen value of he payoff which he lender will receive a he mauriy dae.

24 Since he second erm of he righ-hand side of (7) may be considered o be a shor posiion on a European call opion wih he srike price being he ousanding loan balance, we can readily find ha a higher loan ineres rae R raises he srike price and in urn reduces he value of he call opion. As a resul, keeping oher hings consan, an increase in he ineres rae should increase A so ha he breakeven paymen level is susained, herefore implying A R> 0. In addiion, as he risk-free ineres rae r escalaes, he values of he wo sides of (7) decline due o he higher discoun rae. However, we noice ha he increasing discoun rae has a greaer impac of he value of he righ-hand side of (7) in ha he lender will receive a lump sum paymen a he mauriy dae while on he lef-hand side is a coninuous cash paymen. hus, holding oher hings unchanged, A is decreasing in he risk-free ineres rae, ha is, A r < 0. We will show ha all he comparaive saic resuls above are consisen wih our numerical resuls. I is also noiceable ha he value of he European call opion in (7) may be wrien as E e P Ae Pe Nd A e BA rnd A r R ( m r ) r max(,0) = 0 ( 1( )) (,, ) ( ( )) (8) where 1 P 0 1 d1( A) = d( A) + σ = log + ( r+ σ ). σ B( A, R, ) hus, we can derive he following proposiion. Proposiion 3: Given ha he underlying housing price is governed by a geomeric Brownian moion process, he breakeven annuiy paymen A of a reverse morgage loan saisfies he following equaion 3

25 r ( µ r ) ( µ r ) r 1 e Pe 0 Pe 0 N( d1( A)) + e B( A, R, ) N( d( A)) A = 0. (9) r Proof: his proof is sraighforward by subsiuing (8) and (16) ino (6). Since he analyical soluion o (9) is no available due o he nonlineariy of d 1 and d in A, we use a recursive algorihm o solve his equaion. o be specific, (9) can be rearranged as r ( µ r ) ( µ r ) r A = [ Pe 0 Pe 0 N( d1( A)) + e B( A, R, ) N( d( A))] r 1 e (30) Define A... (0) R = Pe R 0 e 1 µ ( n) r ( µ r) ( n 1) r ( n 1) ( n 1) A = [ Pe 0 PNd 0 ( 1( A )) + e BA (, RNd, ) ( ( A ))] r 1 e for n = 1,,3,.... If we ignore he random erm in (1), he housing price a ime is Pe µ 0, and he fuure value of he coninuous annuiy wih rae (0) A should be exacly equal o his price. Our numerical simulaion has shown ha if we choose his price as he iniial poin of our recursive algorihm, ( n) { A } n = 1 obained using he his algorihm will converge quickly o he breakeven paymen rae. o obain (30), he deah of he elderly borrower has been fixed a ime. We need o furher relax his assumpion and allow for uncerainy in moraliy. As a resul, he coninuous paymen rae A can be wrien as 4

26 A = 0 A f () d = A λ( )exp( λ( s) ds) d. 0 0 (31) he validiy of his inegral depends on he assumpion ha he randomness of he deah ime is independen of he sochasic evoluion of he house price{ P } 0. In addiion, we also have he following proposiion Proposiion 4: Suppose ha all he N collaeralized homes are homogenous, while heir prices are governed by he respecive geomeric Brownian processes. Given i 1 N > 1and π = a a ime, he bid and ask amouns of annuiy paymen of a N reverse morgage are deermined by (30), whereas heir volailiy parameers are σ andσ, respecively. 9 Proof: his proof is sraighforward by subsiuing (6) and (1) ino (30). Similar o he case of lump-sum paymen, if he volailiy of he underlying propery pool is smaller han ha of each individual home, his proposiion suggess ha here is a bid-ask amoun spread of he reverse morgage loan for he lender and he elderly borrower. Also, as discussed earlier, i also implies ha a smaller correlaion ρ can furher reduce he volailiy of he underlying propery pool, while a greaer number of he pooled reverse morgages may, o a larger exen, diversify he uncerainy of he pool. Furhermore, le bid A denoe he loan limi of annuiy paymen of a reverse morgage 9 In Secion 4, we will furher discuss he main implicaions of proposiions 3 and 4. We will invesigae how he key parameers affec he maximum loan amouns in he numerical analysis based on hese proposiions. he effecs of changes in he key parameers on he annuiy paymens will be able o be direcly observed in figures 1 hrough 10. 5

27 len by he lender, and ask A be he lowes morgage amoun accepable by he homeowner. Hence, we have A > A. o deermine he marke-clearing amoun bid ask of annuiy paymen of a reverse morgage, we also require aking ino accoun he lender s marke power, denoed by Q. We can obain he following marke-clearing amoun A = QA + (1 Q) A clear bid ask ( µ r ) ( µ r ) r bid R = P0e QP0e N( d1) Qe A e N( d) ( µ ) ( ) ( ) r r ask R 1 QPe 0 Nd ( 1θ) 1 Qe A e Nd ( θ) where 1 P 1 d1 d θσ θ θ θσ µ θ σ = + = + + log ( ). ask R A e 3. Longeviy Risk Longeviy risk is usually referred o as uncerainy in human lifeime, and is one of he main risks of reverse morgage producs. o measure longeviy risk, we commonly require modeling uncerainy in fuure moraliy rend. here have been muliple probabilisic models developed o describe he evoluion of moraliy [see, e.g., Lee and Miller (001) and Currie e al. (004)]. In his analysis we only allow for wo widely used model specificaions: consan moraliy rae λ() = λ, and uniformly disribued deahs over [0, ω ][see, e.g., Brocke(1991)]. By varying λ andω, we can invesigae he impacs of changes in moraliy rae and remaining lifeime on he breakeven loan amouns of reverse morgages. In addiion, we also ake ino accoun a moraliy able, which is usually uilized o calculae he remaining life expecancy of people a difference ages in acuarial pracice. 3.1 Consan Force of Moraliy 6

28 If he insananeous rae of moraliy is assumed o be consan, we can direcly derive he following resuls from () and (3) F () = 1 e (3a) f() = λe λ. (3b) Under his scenario, he life expecancy is given by λ 1 E[ ] = e d =. λ 0 his suggess ha he expeced residual life is consan, which is independen of he homeowner s age. his independence resuls from he specificaion of he exponenial disribuion. Subsiuing (3b) ino (31) produces he following breakeven loan paymen λ A A λe d 0 =. Differeniaing his equaion wih respec o λ, we have λ A (1 λ ) e d. 0 da = dλ his implies ha A increases in λ when λ 1 bu decreases when λ > 1. As λ 1 increases, he homeowner may decease a an earlier ime because of E( ) =. his λ resul is no surprising in ha he impac of varying on he breakeven loan amoun is ambiguous. In pracice, we usually choose λ 1, and he breakeven loan paymen herefore increases in he force of moraliy. 7

29 3. Uniformly Disribued Deahs Alernaively, we can assume ha he deah of he homeowner is uniformly disribued in he value range[0, ω ]. his is also anoher widely used assumpion in moraliy analysis (Brocke, 1991). Under his assumpion, le x be he homeowner s age and condiional on > x, i is can be shown ha F () = ω x f() 1 = ω x (33a) (33b) where saisfies 0 ω x. Subsiuing (33b) ino (31), we find ha A = = 0 0 ω x A f () d 1 A d. ω x his indicaes ha A is he simple arihmeic average of he funcion A for [0, ω x], while ω x can measure he life expecancy for he homeowner. We can readily find ha an incremen in ω can produce wo opposie effecs on he breakeven loan paymen. 3.3 Moraliy able Anoher alernaive o modeling he moraliy rae is o uilize a moraliy able. In acuarial science, a moraliy able is he one which abulaes he probabiliy of a person dying before his/her nex birhday for each age. I is usually esablished according o he laes demographic daa, and herefore employed o compue he remaining life expecancy of a person. Specifically, suppose ha he homeowner is a he age of x and his morgage loan will be repaid hrough he sale of his house a he end of he year of his deah. Le K be he ime insan a which he house is sold, 8

30 and hen we use Pr( K = k) = Pr( k 1 < k), for k = 0,1,... N, o denoe he probabiliy ha he homeowner can live for k 1 more years bu does no survive in he k -h year, where x+ N is his maximum age. Under his seing, we may obain he following breakeven loan paymen N A= Pr( K= ka ) k (34) k = 1 where Pr( K = k) may be inferred from a moraliy able, and A k is he numerical soluion o (9) using he recursive algorihm. o infer Pr( K = k) based on he moraliy able, suppose ha l x sands for he number of people who survive unil he age of x. Since a moraliy able usually records he number of people wih a beginning of 100,000, we have l 0 = 100,000 and l0 > l1 > l >... If he able shows ha all he people die prior o he age of 111, his means l 111 = 0. Since lx k 1 l + x+ k denoes he number of people dying in he year of x+k, he probabiliy ha he homeowner has survived k -1 more years bu dies before year k can be expressed as for k = 1,,3,...(111 x). x+ k 1 x+ k Pr( K = k) = l l l x 4. Numerical Analysis o illusrae he usefulness of he opion-based model developed above, we apply his model o he valuaion of reverse morgage loan limis in a hypoheical morgage loan. In his loan, a lender designs a reverse morgage produc o provide an elderly homeowner wih he loan in he form of a life annuiy. Our numerical analysis will focus on he breakeven deerminaion of maximum loan amouns which he lender is willing o offer. We assume ha he housing price process follows a geomeric 9

31 Brownian moion process. All he inpu parameer values are se o reflec as closely as possible he marke pracice. Specifically, he basic parameer values are se a P0 = 00, 000, µ = 0.06, σ = 0.1, R= 0.05, r = 0.04, = 15. We firs examine he sensiiviy of annuiy paymens o changes in iniial housing price. Figure 1 shows ha he annuiy paymen is a linear increasing funcion of iniial housing price as one migh expec (see also Appendix B). his is because higher iniial housing price increases he value of he collaeral, herefore raising he paymen he lender is willing o make. Also, he dependence of he paymen on iniial housing price is influenced by he relaive magniude of µ versusσ. When µ is relaively small ( µ = 0.05, σ = 1% ), he paymen increases from abou $7,000 o $34,000 as iniial price rises from $100,000 o $500,000. However, when µ is a relaively large value ( µ = 0.06, σ = 10% ), he paymen rises from abou $8,800 o $45,000 as iniial price escalaes in he same range. [Inser Figure 1] We also invesigae how he expeced percenage growh in housing price affecs annuiy paymens. Figure shows ha holding everyhing else unchanged, here exiss a posiive relaionship beween he expeced price growh and he annuiy paymen, while his relaionship is nonlinear. A a givenσ = 10%, he paymen is enhanced from $13,000 o $175,000 as he expeced price growh rises from 4% o 18%. An increase in he expeced price growh raises he value of he collaeralized house, herefore leading o he incremen in he paymen. [Inser Figure ] Figure 3 displays he impacs of varying house volailiy on annuiy paymens. I is shown ha he annuiy paymen is highly sensiive o changes in housing volailiy. A 30

32 a given µ = 0.06,, he paymen falls from abou $1,500 o $,800 when he volailiy increases from 6% o 35%. his is consisen wih our inuiion in ha increasing house marke uncerainy reduces he lender s willingness o offer he morgage loan, herefore lowering is breakeven loan amoun. [Inser Figure 3] We also explore numerically he implicaion of increases in he homeowner s remaining lifeime on annuiy paymens. Figure 4 demonsraes ha given R = 0.05 and r = 0.04, he annuiy paymen decreases from abou $38,800 o $15,100 as he remaining lifeime lenghens from 5 o 30 years. However, if he remaining lifeime coninues o lenghen, he paymen sars o rise slighly. When R = 0.06 and r = 0.03,, we can also find a similar effec of varying remaining lifeime on annuiy paymens, while he higher difference beween hese wo raes implies higher loan reurns and herefore more loan amouns he lender is willing o offer. hese imply ha longer remaining lifeime can produce wo opposing effecs on he lender s annuiy paymens. his resul is somewha surprising. Longer remaining lifeime suggess more annuiy paymens and larger uncerainy in he housing marke, herefore decreasing he paymens which he lender is willing o make. However, since here is an expeced growh in housing price, he decreasing effec will be weakened, as he remaining lifeime lenghens and he housing price rises coninuously. When he expeced growh in housing price dominaes his effec evenually, he lender will be willing o make slighly higher annuiy paymens. [Inser Figure 4] Furhermore, we find ha annuiy paymens are sensiive o changes in loan ineres rae. Specifically, a a given r = 0.04, if he loan ineres rae rises from 4.5% o 10%, 31

33 Figure 5 shows ha he paymen increases by abou $7700. his resul is consisen wih our expecaion. Since a higher loan ineres rae increases he financial cos he homeowner bears, he lender is willing o provide more annuiy paymens o he homeowner in order o enhance he aracion of he reverse morgage scheme. his also suggess ha uncerainy in he ineres rae could be an imporan deerminan for he annuiy paymen amoun which a bank is willing o provide. [Inser Figure 5] We invesigae he effecs of changes in he risk-free ineres rae on annuiy paymens as well. Figure 6 shows clearly ha a a given R = 0.05, when he risk-free ineres rae rises from 0.1% o 4.5%, he annuiy paymen decreases from abou $8,000 o $15,000. his resul is no surprising in ha higher risk-free rae raises he aracion of invesing in risk-free projecs and herefore reduces he lender s willingness o provide he reverse morgage scheme. In addiion, we also find ha he loan ineres rae effec on he nonlinear negaive relaionship becomes more significan as he risk-free ineres rae rises. [Inser Figure 6] We also examine how he rae (force) of moraliy and maximum life span affec annuiy paymens. Figure 7 shows ha he annuiy paymen is an increasing funcion of he force of moraliy as one migh expec. Holding everyhing else unchanged, when he force of moraliy rises from % o 0%, he paymen is enhanced by abou $115,000. However, Figure 8 shows ha here exiss a nonlinear negaive relaionship beween he maximum life span and he paymen. When he maximum life span lenghens from 70 o 110, he paymen falls from abou $44,00 o $9,800. [Inser Figure 7] [Inser Figure 8] 3

34 We ake ino accoun he usefulness of a moraliy able in deermining annuiy paymens as well. We make use of he 006 period life able for he U.S. Social Securiy area populaion. his able shows ha all he 100,000 people die before he age of 111. We also analyze he pas 35 years of he U.S. naional housing price index daa, which are compiled and mainained by he U.S. Federal Housing Finance Agency (FHFA). Based on he daa, we use he Generalized Mehod of Momens (GMM) o derive he esimaes of wo crucial parameers: µ = and σ = Finally, we can derive he annuiy paymens using recursive algorihm (30) and equaion (34). We summarize he resuls in able 1 for several ineres raes. [Inser able 1] Finally, we also look a he effecs of changes in boh he correlaion coefficien ρ and he number of reverse morgages N on annuiy paymens. he numerical resuls are evaluaed on he basis of Proposiion 4 and demonsraed in Figures 9 and 10. When he force of moraliy is fixed a λ = 0.05, Figure 9 shows ha he annuiy paymen drops as he correlaion coefficien escalaes, due o he increase in sysemaic risk. A N=100, when he correlaion coefficien increases from 0% o 100%, he oal annuiy paymen falls from abou $195,000 o $157,500. A N=5, his coefficien increase reduces he annuiy paymen from abou $187,500 o $157,500. On he oher hand, if deah is uniformly disribued, we can sill observe, from Figure 10, a similar relaionship beween he annuiy paymen and he correlaion coefficien. Given N=100, x=40, andω = 80, if he correlaion coefficien rises from 0% o 100%, he oal annuiy paymen drops from abou $4,400 o $34,750. However, when N=5, he annuiy paymen will falls from abou $40,600 o $34, 750 as he correlaion coefficien increases in his value range. hese resuls are consisen wih wha one migh expec. As menioned earlier, ρ measure he correlaion beween he sysemaic reurn and he reurn on an underlying propery. A a given house volailiy and when 33

35 ρ > 0, an increase in he value of ρ can reduce he idiosyncraic componen of house volailiy bu increase he sysemaic componen, herefore increasing he volailiy of he underlying propery pool. Higher pool volailiy means he less effec of porfolio diversificaion and herefore decreases he maximum loan amouns of he reverse morgages he lender is willing o offer. Of course, if ρ < 0, we can readily find ha a larger value of ρ raises he idiosyncraic componen, herefore increasing he maximum loan amouns. Our resuls also sugges ha no only correlaion coefficien bu also he number of loans in he morgage pool are found o play imporan roles in deermining he maximum loan amouns of he reverse morgages. As he number of he pooled loans increases, he morgage pool will become larger and he value risks of underlying properies will be more diversified. In his case he lender is willing o offer more morgage amouns for hose elderly homeowners. However, if ρ = 1, his means ha he sochasic evoluions of all he house prices are driven only by he sysemaic shocks and herefore he risks of house values are compleely undiversifiable. In his case, here is no informaion asymmery beween he lender and he borrowers, and he number of loans, herefore, does no produce any impacs on he maximum loan amouns as shown in Figures 9 and 10. Las bu no leas, hese resuls also imply ha he disribuion of deah is anoher crucial deerminan for he annuiy paymen amoun. [Inser Figure 9] 5. Conclusions [Inser Figure 10] As a growing number of economies are moving rapidly owards aging socieies, he reverse morgage program is aracing more and more aenion in hese counries. his program provides a poenial channel ha allows cash-poor elderly homeowners 34

36 o wihdraw heir home equiies for improving heir consumpion and reiremen securiy, whereas hey do no have o suffer a subsanial loss caused by moving away from heir homes wih hose memories, neighbors, resaurans, and so on. his paper herefore develops a breakeven opion-based model for invesigaing he loan limis of reverse morgages. Our model aemps o allow for he effecs of heerogeneous beliefs, caused by morgage lenders informaional advanage, on he risk of he collaeralized houses and inegrae longeviy risk ino he opion-based valuaion approach. o our knowledge, his paper is he firs o look a he deerminaion of maximum loan amouns of reverse morgages, which are main concerns o boh he lenders and he borrowers of reverse morgages. Alhough our main resuls seem o be sraighforward, we have provided an exac approach on he deerminaion of he loan amouns, herefore also implying he obvious conribuion o real esae lieraure. We have idenified several crucial deerminans of maximum morgage loan amouns, which include iniial housing price, expeced housing price growh, housing price volailiy, remaining lifeime, loan ineres rae and risk-free ineres rae. I is shown ha annuiy paymens increase in iniial housing price, expeced price growh and loan ineres rae, bu falls in housing price volailiy, remaining lifeime and risk-free ineres rae. We also find ha boh he correlaion coefficien beween he sysemaic reurn on and he reurn on an underlying propery and he number of pooled reverse morgages have imporan implicaion for he deerminaion of he maximum loan amouns. Higher correlaion reduces he loan amouns due o he increase in sysemaic risk, while a rise in he number of pooled morgages raises he loan amouns because of he diversificaion role of morgage pools. However, since morgage lenders and reverse morgage borrowers hold asymmeric informaion abou he diversificaion impac, heir heerogeneous beliefs on he risk of he collaeralized houses are generaed and idenified o maer in inerpreing he maximum loan amouns. 35

37 Furhermore, we also allow for hree alernaives approaches o measure longeviy risk for compuing he remaining life expecancy of an elderly homeowner. Our numerical resuls show ha boh he force of moraliy and he maximum life span produce significan impacs on he maximum loan amouns, while heir impac direcions are opposie. We also calculae a breakeven loan amoun able based on he 006 U.S. period life able and he pas 35 years of he U.S. naional housing price indexes. hese resuls sugges he usefulness of our model in he morgage pricing pracice as well as he real esae academic field. Finally, for he purpose of his paper, our model focuses on he effec of porfolio diversificaion upon he pool of reverse morgages, and finds ha a rise in he number of pooled morgages migh raise he loan amouns for he elderly borrowers. his is consisen wih he predicion of modern porfolio heory. Modern porfolio heory aemps o inerpre he favorable effec of diversificaion on he expeced reurns of invesmen porfolios. According o his heory, porfolio diversificaion can lower he risk of porfolio reurns across various asses, and he effec of diversificaion migh even become more significan across he broader asse classes. As a resul, i will be appealing o furher exend our model o invesigae he loan limi issue associaed wih he broader caegories of loans. Of course, his also implies ha if here were an increasing number of reverse morgage borrowers in aging sociey, he banking indusry would be likely o be exposed o more sysemaic uncerainy in he house marke. As a consequen, o hedge agains his caegory of uncerainy, he lenders probably lower he amouns of loans offered o oher propery-relaed business. Appendix A: Proof of (17) he soluion for (1) is given by σ P = Pexp[( µ ) + σw]. A he expiraion dae, we have 36

38 1 P = P µ σ + σw 1 = P µ σ σ Y exp[( ) ] exp[( ) ], W where Y = is a sandard normal variable. Nex, we calculae he value of he call opion r R e max E[ P Le,0] r 1 e 1 y R = max P exp σ y ( m σ ) L e,0 e dy. p + he inegrand R max Pexp σ y+ ( m σ ) Le,0 vanishes if and only if 1 1 P 1 > = µ σ σ + L. y b log ( R ) As a resul, r R e max E[ P Le,0] ( r ) b 1 1 y R r e = max P exp σ y + ( m σ ) Le,0 e dy p b 1 1 y R r e = [ P exp σ y + ( m σ ) Le ] e dy p b 1 ( R r ) exp σ y ( m σ ) dy Le N( d) r e P y = + p Pe m = p b 1 ( ) exp ( ) R r y + σ dy Le N ( d ) b + σ 1 ( R r ) exp z dz Le N( d) ( m r ) Pe = p By leing z = b1 = b + σ, we obain (17). Q.E.D Appendix B: Proof of lineariy From equaion (4), we have 37

39 r ( ) R ( ) 1 e A e 1 A r r = e E( P) e max E P,0. r R Since only he housing price P is random in he second erm of he righ-hand side, his equaion can be rewrien as r ( ) R ( ) 1 e A e 1 A r r = e E( P) e max EP ( ),0. r R We noice ha he lef-hand side of his equaion is linear in eiher EP ( ) or A. As he housing price evolves in a geomeric Brownian moion, is soluion is given by 1 P = P + W 0 exp[( µ σ ) σ ] From his equaion, we can readily find ha EP ( ) is linear in he curren house price P 0. As a resul, given ha he iniial housing price is only muliplied by. consans in he equaion, we can conclude ha A is linear in P 0. References Q.E.D Addae-Dapaah, K. and Leong, K. M. (1996). Housing finance for he ageing Singapore populaion: he poenial of he home equiy conversion scheme, Habia Inernaional, 1996, 0, Archer, W. R., Elmer, P. J., Harrison, D. M. and Ling, D. C., Deerminans of mulifamily morgage defaul, Real Esae Economics, 00, 30(3), Basak, S., A model of dynamic equilibrium asse pricing wih heerogeneous beliefs and exraneous risk, Journal of Economic Dynamics & Conrol, 000, 4, Basak, S., and Croioru, B., Equilibrium mispricing in a capial marke wih porfolio consrains, Review of Financial Sudies, 000, 13, Basak, S., and Croioru, B., On he role of arbirageurs in raional markes, Journal of Financial Economics, 006, 81, Bishop,. B. and Shan, H., Reverse Morgages: A Closer Look a HECM Loans, Naional Bureau of Economic Research, 008, Available from hp:// 38

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44 Figure 1 Annuiy Paymen vs. Iniial House Price 4.5 x mu=0.06,sigma= mu=0.05,sigma=0.1 Annuiy Paymen Iniial House Price x 10 5 Figure Annuiy Paymen vs. Expeced House Price Growh 1 x 104 sigma= sigma=0.1 Annuiy Paymen Expeced House Price Growh 43

45 Figure 3 Annuiy Paymen vs. House Price Volailiy. x mu=0.05 mu= Annuiy Paymen House Price Volailiy Figure 4 Annuiy Paymen vs. Remaining Lifeime 4.5 x R=0.06, r=0.03 R=0.05,r=0.04 Annuiy Paymen Remaining Lifeime 44

46 Figure 5 Annuiy Paymen vs. Loan Ineres Rae.6 x r=0.03 r=0.04 Annuiy Paymen Loan Ineres Rae Figure 6 Annuiy Paymen vs. Risk-Free Ineres Rae 3 x 104 R=0.05 R=0.06 Annuiy Paymen Risk-free Ineres Rae 45

47 Figure 7 Annuiy Paymen vs. Force of Moraliy 16 x Annuiy Paymen Force of Moraliy Figure 8 Annuiy Paymen vs. Maximum life Span 4.6 x Annuiy Paymen Maximum Life Span 46

48 Figure 9 Annuiy Paymen vs. Correlaion Coefficien 1.9 x 105 N=1 N=5 N=100 Breakeven Annuiy Paymen Correlaion Coefficien Figure 10 Annuiy Paymen vs. Correlaion Coefficien Breakeven Annuiy Paymen 4.3 x N=1 N=5 N= Correlaion Coefficien 47

49 able 1 Annuiy Paymen able R=0.06 R=0.06 R=0.05 R=0.05 r=0.03 r=0.04 r=0.03 r=0.04 Age x= Age x= Age x= Age x= Age x= Age x=