An Attractive Financial Tool in Aging Populations: Determining an Optimal Principal Limit Factor for Reverse Mortgages under the Economic-Based Model

Transcription

1 An Attractive Financial Tool in Aging Population: Determining an Optimal Principal Limit Factor for Revere Mortgage under te Economic-Baed Model Ciang, Su Ling Correponding autor. Profeor. Department of Buine Management National Kaoiung Normal Univerity Kaoiung, Taiwan Tel : Tai, Ming Sann Aociate Profeor Department of Finance National Univerity of Kaoiung Kaoiung, Taiwan mtai@nuk.edu.tw Tel :

2 An Attractive Financial Tool for Aging Population: Determining an Optimal Principal Limit Factor for Revere Mortgage Under an Economic-Baed Model Abtract Te revere mortgage (RM) i an attractive financial tool government can ue to improve te economic tatu of teir aging population, becaue it give elder greater peronal reponibility in covering teir living expene. Determining an optimal principal limit factor (PLF) i very important, ince it influence elder willingne to participate in te RM program, wic in turn influence not only te future development of te RM market but alo te effect of portfolio diverification on te collateral rik and te longevity rik. Te goal of ti tudy wa to develop a general economic model for calculating an optimal PLF tat include te longevity rik, te collateral rik and te interet rik. We provide numerical analye tat illutrate ti calculation for bot uninured and inured RM under two general economic aumption: te lender break even and te lender acieve maximum profit. Our enitivity analye provide ueful information to elp policymaker and market participant modulate a reaonable PLF reponive to cange in te economic ituation and te relevant rik. Our model and reult provide ueful information tat can elp tem determine a reaonable PLF and manage rik. Keyword: Principal Limit Factor, Revere Mortgage, Croover Rik, Longevity Rik, Collateral Rik, Interet Rik

3 1. Introduction Large-cale demograpic cange are taking place in many countrie, including Cina, Japan, Korea, te U.S. and member of te European Union. Tee countrie ave rapidly aging population, wic a gradually created ocial and economic problem (Von Weizacker, 1996; Davi, 1997; Faruqee and Muleien, 2003; Li, 2005). Becaue oue wealt i often te major component of aving for elderly omeowner, 1 effectively putting ti wealt to ue i important for improving teir quality of life. However, elderly omeowner ave a trong ditate for moving out of teir oue (Venti and Wie, 2000; Davidoff and Welke, 2007). Revere mortgage (ereafter RM) allow elderly omeowner to obtain loan from lender (i.e., bank) uing teir oue a collateral. An added bonu i tat RM borrower can till tay in teir oue. Tu, te RM i an attractive financial tool for borrower. Moreover, developing te RM market i a good way for government to olve te problem tey face wit aging population, becaue elder can ue ti product to take more peronal reponibility in covering teir living expene and unexpected bill. RM borrower enjoy te following benefit. Firt, a noted above, RM allow tem to convert teir ome equity into ca witout aving to move out of te oue. Second, omeowner ave no obligation to repay te loan a long a tey tay in te oue. Finally, te loan repayment are capped at te proceed from te ale of te property, te o-called non-recoure proviion. Economit believe tat RM ave te potential to improve te wellbeing of tee oue-ric but ca-poor elderly omeowner 2 by paying for teir long-term 1 For many elderly omeowner, oue wealt i te larget non-penion component of teir total wealt. For example, according to te 2004 Survey of Conumer Finance (SCF), oue wealt contitute at leat 80% of te total wealt of 27.8% of omeowner age 62 or above. In addition, 13.3% of uc omeowner ave a oue-value-to-income ratio of at leat 10. For more detail, ee te Federal Reerve Bulletin. 2 Uing data from te Engli Longitudinal Study of Ageing , Soda (2005) etimated tat 10.2 percent of retiree in Britain ad an income below te "modet but adequate" tandard ( 157 per week before ouing cot), 1

4 care, auring teir financial independence and maintaining teir tandard of living wile allowing tem to continue living in teir ome. Several tudie ave demontrated tat RM indeed improve elder financial ituation (Speare, 1992; Mayer and Simon, 1994b; Ramuen et al. 1995; Morgan, Megbolugbe and Ramuen, 1996; Venti and Wie, 2000). Altoug RM ave been in exitence for everal dozen year in te U.S., te growt of te RM market a not been a great a expected. Only about 400,000 of te ten of million of eligible omeowner in te U.S. took out loan troug te Home Equity Converion Mortgage (ereafter HECM) program 3 from 1989 (te inception of te RM program) to te end of 2007 (Biop and Hui, 2008). Only about 1.4 percent of omeowner ad active RM in 2009 (Merrill et al., 1994). Academic tudie ave uggeted a number of explanation for te mall ize of te RM market: te ig cot of participation in te RM program (e.g., origination fee, mortgage inurance premium, interet), regulatory and legal barrier, moral azard and advere election, financial awarene and literacy, te perception of oue equity a a afety for large medical expene, bequet motive and te difficultie aociated wit RM ecuritization (Szymanoki et al., 2007; Biop and Hui, 2008). Some autor ave argued tat te ig cot of financing RM, a well a te low montly payment reulting from te conervatively low principal limit factor (ereafter PLF) et by te Department of Houing and Urban Development (ereafter HUD), 4 could be major factor adverely affecting elder demand for RM (Mayer and Simon, 1994; Redfoot, Scolen and Brown, 2007; Pu, Fan and Deng, 2013). Some tudie ow tat te PLF may not be a reaonable value for an RM contract (Weinrobe, 1987). Several tudie indicate tat even toug teir oue wealt exceeded 100, Te mot common type of revere mortgage loan i te HECM, inured by te Federal Houing Adminitration. HECM account for over 90% of all revere mortgage loan originated in te U.S. market. 4 Te HUD webite provide more information about te RM program. It can be found at: ttp://earc.ua.gov/. 2

5 te mortgage inurance premium for HECM loan i too ig, jutifying a reduction in te front-end carge (Cen, Cox and Wang, 2010; Ji, 2012). In oter word, tee autor are uggeting tat te PLF ould be increaed under te current inurance ytem. An optimal PLF i important for te development of te RM market. Too low a PLF i unfavorable for RM borrower, becaue tey ten need relatively ig collateral to obtain te ame payment form an RM. For example, a PLF le tan 0.5 mean tat te actual oue value i le tan alf it market value for te purpoe of calculating te RM payment. Ti decreae elder willingne to participate in te RM program and retrict te development of te RM market. In ti paper, we ue economic-baed model to illutrate te calculation of an optimal PLF. Our numerical data prove tat te current PLF i indeed too low. In decribing ow to etimate an optimal PLF, it i neceary firt to note te function of a PLF. Since RM contract ave non-recoure proviion, te collateral (te oue) i te only ource tat lender ave for reclaiming te non-recoure debt. Te lender may incur a lo from te croover rik, defined a te rik for lender wen te RM balance exceed te value of te property at termination of te RM contract. Analyi of ti rik i particularly important if elderly borrower cooe a tenure-payment plan, becaue in ti cae tey receive montly payment from te lender for a long a tey live wile continuing to occupy te property a teir primary reidence. 5 Lender uually ue two way to protect temelve againt loe reulting from croover rik: te PLF and mortgage inurance. From te actuarial viewpoint, te preent value of all 5 Borrower option can generally be claified a follow: (1) receive a lump um ca payment up front; (2) receive predetermined montly ca payment from an annuity a long a te oue i te primary reidence (tenure plan); (3) receive predetermined montly ca payment from an annuity for a fixed period of time determined by te borrower (term plan); (4) receive mortgage proceed, eiter in unceduled payment or intallment, at time and in amount of te borrower cooing until te line of credit i exauted (line of credit plan); or (5) a combination of ome or all of te above. 3

6 payment ould equal an upper limit of principal at te origination time of te RM. Ti upper limit i calculated by multiplying te initial value of te oue by te PLF value. Ti mean tat lender can ue te PLF to control te payment for an RM contract. According to te teory of rik management, limit on te payment advance can retrict future loan balance, tereby limiting te rik of a future lo. To mitigate te rik of te property depreciating more tan i forecat, lender uually decide te maximum initial loan balance baed on a conervative PLF. Doing o make tem more confident tat te property value will remain greater tan te accreting loan amount for a long period of time, even if te depreciation of property i ig. Terefore, one may well ay tat te PLF for lender i a firt buffer againt lo from te croover rik. If te conervative PLF i not ufficient to defend againt ti lo, lender mut reort to te econd buffer, mortgage inurance. Ti inurance cover any ortfall in lender recovery from te ale of property, in teory eliminating teir collateral rik. Nowaday, PLF value in te U.S. can be obtained from a table publied by HUD. Szymanoki (1994) advocated a model for determining a fair value for a PLF defined a te iget poible initial loan-to-value (LTV) ratio. Te calculation of ti PLF i baed on te aumption tat te preent value of te total received inurance premium cover te current value of te inurer expected loe. PLF calculated uing te model in Szymanoki (1994) are nearly equal to te value own in te HUD table. Many reearcer ave referred to Szymanoki (1994) in dicuing ti iue a related to RM (Te, 1995; Piggott and Mitcell, 2004; Cen, Cox and Wang, 2010; Ma, Zang and Kannan, 2011; Fan and Deng, 2013). PLF etimate baed on Szymanoki (1994) rely on a fully edged croover rik for te lender. In oter word, te lender can ift all it poible loe reulting from croover rik to te inurer, tu incurring no lo from te RM contract. However, in ti cae, a PLF calculated 4

7 from te metod of Szymanoki (1994) i too conervative, becaue it conider only te edge for te inurer. A noted above, te PLF i alo te buffer againt lo from te croover rik. We argue tat lender need not in fact fully edge te croover rik. Terefore, tey can eek greater profit by etting a iger PLF. A ig PLF a everal advantage. Firt, a i well known, te increaed loan amount reulting from a large PLF can earn te lender greater profit from te interet rate pread. Second, if te PLF increae, te payment tat borrower receive are alo increaed, tereby cauing elder participation rate in te RM program to rie. Tird, ome autor ave own ow muc benefit i obtained from portfolio diverification depend on ow muc te number of pooled RM rie (Pu, Fan and Deng, 2013). Suc an increae can be expected to raie te amount te elderly borrow, becaue troug diverification lender can coniderably attenuate teir collateral and longevity rik. Tu, altoug lender may increae teir probability of lo on individual loan by raiing te PLF, teir overall lo can be effectively diverified troug a larger portfolio if tey make more of tee loan. Te determination of an optimal PLF i important for participant in te RM market. Te goal of ti tudy wa to come up wit a way to determine an optimal PLF baed on two general economic aumption: one tat aume te lender break even and one tat aume te lender make maximum profit. To determine te breakeven point, te optimal PLF i calculated baed on te aumption tat te expected income from te interet rate pread equal te expected average lo from te croover rik. If ti aumption i met, lender get zero profit; in oter word, tey break even. On te oter and, if te PLF i et to le tan ti optimal level, lender will likely earn a profit by providing RM. Even if te lender jut break even, PLF determined by ti metod can elp te government develop te RM market, becaue uc PLF 5

8 offer te maximum benefit to te borrower, provided tere i no lo for te lender. Tu, if te lender i a government-ponored organization, for wic profit-making i not an objective, it may be more appropriate to cap te probability of a real lo rater tan to focu on profit (Diventi and Herzong, 1991). Te optimal PLF i better to be determined at te breakeven point. In te cenario of te model of maximum profit, te optimal PLF i determined under te aumption tat te marginal expected profit equal te marginal expected lo. From te general economic teory, one can aure tat te lender i getting te maximum profit from it RM wen uing ti model. Wen determining an optimal PLF, it i neceary to etimate te expected average lo from te croover rik. Several tudie ave demontrated tat maximum loan amount and maximum annuity payment are influenced by te following crucial factor: oue price volatility, te remaining life of te elder, te loan interet rate, and te rik-free interet rate (Lee, Wang and Huang, 2012; Pu, Fan and Deng, 2013). Tee factor are all related to te interet rik, te collateral rik and te longevity rik. Studie on te pricing of RM contract or croover rik ave eiter ued periodic life table, tu neglecting te dynamic of mortality rate (Weinrobe, 1988; Szymanoki, 1994; Cinloy and Megbolugbe, 1994; Te, 1995; Zai, 2000), or tey ave ignored te inerent dynamic of interet rate (Cinloy and Megbolugbe, 1994; Szymanoki, 1994; Cen et al. 2010; Li et al., 2010; Pu, Fan and Deng, 2013). Only a few autor ave dicued te effect of te dynamic of mortality rate and interet rate on te amount of te loan, te annuity payment, or te PLF (Diventi and Herzong, 1991; Szymanoki, 1994; Pu, Fan and Deng, 2013). Recently, economic variable uc a interet rate and oue price ave become more and 6

9 more variable. In addition, elder are living longer tan ever before becaue of advance in medicine. Tu te interet rik, te collateral rik and te longevity rik ave been increaing more and more. In ti paper, we decribe a well-deigned model for calculating an appropriate PLF tat incorporate imultaneouly te tree main rik: te interet rik, te collateral rik and te longevity rik. We aume tat te interet rate follow te extended Vaicek model and tat te oue price follow te geometric Brownian motion proce tat i uually applied by te Federal Houing Adminitration (FHA). Te longevity rik i meaured uing a Gompertz-type urvival probability denity function. To evaluate te effect of tee rik factor, and to furter decribe ow cange in te optimal PLF repond to tem, we alo report enitivity analye. Our model provide valuable information tat can elp policy implementer and market participant determine an optimal PLF and manage RM rik. Ti paper i organized a follow. In Section 2, we pecify te valuation framework for calculating te expected lo from te croover rik, including pecification for te interet rik, te collateral rik and te mortality rik. In ti ection, we alo ow ow to determine te optimal PLF baed on te traditional model of Szymanoki (1994) and te economic model, auming a breakeven point and maximum profit. Reult of a numerical example applying te model are preented in Section 3. We firt decribe ow to calculate te optimal PLF for an uninured RM, in wic cae te lender ue te PLF a te only buffer againt loe; ten we decribe te calculation for an inured RM. Section 4 give te reult of enitivity analye aimed at aeing ow enitive PLF are to cange in te parameter related to te longevity rik, te collateral rik and te interet rik. Finally, in Section 5 we offer concluding remark. 7

10 2. Decription of te Model Ti ection include four ubection. In Subection 2.1, we introduce te baic framework of te current RM ytem. In Subection 2.2, we preent a formula for calculating te expected lo from te croover rik, including pecification for te mortality rik, te interet rate proce and te oue price proce. Our model for determining te PLF following from te traditional model of Szymanoki (1994) and te economic model are repectively own in Subection 2.3 and Baic RM model baed on te current ytem Let te elder age wen joining te RM program be t 0 year; in te current RM ytem, t Aume te elder own a oue atifying all FHA property tandard and floor requirement. Let te oue value be (0) at te time of RM origination. To participate in an RM program, te elder mut pay te initial cot ( D (0) ), wic include te organization cot, tird-party carge and te upfront mortgage inurance premium. According to te non-recoure proviion, te elder need not repay te interet and principal during te RM period; owever, te elder i carged a montly ervicing fee. Aume te lender et aide, from te borrower principal limit, te preent value of te total montly ervicing fee ( S (0) ), wic i calculated from cloing until te borrower would reac age 100. We denote te mortgage interet rate a r C and te mortgage inurance premium a r I. Letting r RM be te compounding rate for an RM, we ave r RM r r. I C Te net principal limit ( NPL (0) ) i defined a: 6 See te HUD webite: ttp://earc.ua.gov/. 8

11 NPL( 0) PL(0) S(0) D(0), (1) were PL (0) i te principal limit for te RM; (0) PL min( (0), H), were i te PLF value provided by te HUD table; H i te maximum HECM loan amount allowed by te HUD regulation (defined a te mortgage limit); min( (0), H) i te maximum claim amount. Te firt term on te rigt ide of Equation (1) i te oue value at te time te borrower igned up for te RM program. Te lat two term are te participation cot for te borrower. We let PL (0) (0) H, were min(1, ), an exogenou variable. Ti give u: (0) NPL (0) ( 0)( ), (2) were i te total cot ratio of te RM. Let be a percentage of te initial oue value. Tu, we ave S(0) D(0). (0) We denote te total amount of payment in te borrower account at time a (). Tu, we ave: t ( ) V ( t)(1 ), (3) t0 r RM were r RM i te contract rate of te RM; and V (t) i te payment from te RM at time t. Since RM borrower will receive different ca payment according to wic RM payment plan tey cooe, V (t) can be pecified in different way in te application. Te borrower receive predetermined montly ca payment from te annuity, te amount depending on te type of tenure payment tey cooe, until te RM i terminated; tu, V( t) V for t 0, were V i te fixed montly ca payment from te RM. If te plan i a term plan, te borrower receive predetermined montly ca payment from te annuity for a fixed period of 9

12 time (ere t 1 ) determined by te borrower; tu, we ave V V( t) if 0 t t1 and V ( t) 0 if t t 1. If te plan i a lump-um plan, te borrower receive a lump um of ca up front; tu, we ave V (0) NPL (0) and V (t) 0 for t 0. A for te payment line of te credit plan, te borrower receive mortgage proceed eiter a unceduled payment or in intallment, at time and in amount of te borrower cooing until te line of credit i exauted. Tu, V (t) i te actual witdrawal amount at time t, provided tat te accumulated loan amount doe not exceed te net principle limit (i.e., V (t) NPL (0) ). Wen te borrower elect te type of RM payment V (t), te debt amount () at time can be calculated. In te current RM market, te lump-um payment plan i te mot popular. However, if te borrower elect a tenure-payment plan, te lender probably incur a larger lo tan if te borrower elect a different type of payment plan. We terefore aume in our model tat te elder cooe te tenure-payment plan. Te RM payment depend on te age of te borrower at te origination of te RM, te borrower iget attainable age, te PLF, te initial oue value, te total cot of joining te RM program and te RM interet rate. Wit te tenure-payment plan, te lender ue te net principal limit to decide te maximum payment at te time of origination of te contract. Let T be te aumed total number of payment year, T T t0 in ti cae; T i te iget attainable age (100 in te current ytem). 7 From te actuarial viewpoint, we ave: NPL(0), (4) V a T, r RM were i te preent value of te annuity per monetary unit from time 0 to time a T, r RM T, 7 Ti pecification i own on te HUD webite: ttp://earc.ua.gov/. 10

13 wit interet rate r RM and T being te total number of year until maturity. 8 From financial teory, one can obtain te following formula for te preent value of te annuity wit a contant contract rate and for a contant period : a, ((1 ) 1). (1 ) 2.2 Te expected lo from te croover rik In determining an optimal PLF, it i neceary to conider te factor tat influence te etimate of te lo expected from te croover rik. Here we decribe a model for accurately calculating ti croover rik, wic a tree main component: te interet rik, te collateral rik and te longevity rik. Our model aume tat te dynamic of te interet rate are pecified by te extended Vaciek model, te beavior of te oue price follow te log-normal ditribution and te longevity rik for te elder i pecified by te Gompertz urvival probability denity function. We illutrate tee tree aumption in detail in te following ubection. Aumption 1: Dynamic of te interet rate following te extended Vaicek model We propoe tat te beavioral function of te interet rate and te oue price are located in probability pace (, F, Q), were i te tate; te filtration F i generated by te information proce F, F F t ) 0tT ( ; Q i te rik-neutral probability. We ue te extended Vaicek model to decribe cange in te interet rate r (t). Te evolution of te term tructure i decribed by te dynamic of te ort-term interet rate (Vaicek, 1977): dr( t) a( r( t) r( t)) dt r dzr ( t), (5) were 8 Wit te current tenure plan, te RM diburement i paid montly. For implicity, we pecify ti payment a yearly. 11

14 a i te adjuted peed of interet rate (a poitive contant); r i te volatility of te ort interet rate (a poitive contant); r (t) i te long-run ort interet rate (a determinitic function of t ); and Z r (t) repreent a tandard Brownian motion of interet rate under te rik-neutral meaure Q. Baed on Equation (5), one can obtain a cloed-form pricing formula for a zero coupon bond wit maturity date at time t. Letting P ( t, ), t, be te price of ti bond, we ave: P ( t, ) E[exp( r( u) du) ] A( t, )exp( B( t, ) r( t)), (6) t F were E [] i te expected operator; A( t, ) ( exp( a ( t) 1 e B( t, ). a B( t, ) t)( a 2 a r( t) 0.5 r ) r B( t, ) 4a 2 ) ; and Aumption 2: Dynamic of te oue price following a log-normal ditribution Te oue price proce (t) pecified under te pyical meaure i defined a: d( t) P ( t) dt dz ( t), (7) ( t) were (t) i te intantaneouly expected oue return, i te intantaneou volatility of te oue return and Z P (t) repreent te tandard Brownian motion of te oue return under 12

15 pyical meaure. To evaluate te croover rik, we tranfer te probability from te pyical meaure to te rik-neutral meaure. Uing te rik-neutral meaure, te oue price proce (t) i defined a: d( t) ( ( t) ( t) ) dt dz ( t), (8) ( t) were (t) i te rik premium and Z (t) repreent te tandard Brownian motion of te oue return under te rik-neutral meaure Q. In ti rik-neutral valuation framework, we ave ( t) r( t) ( t) under te aumption of no arbitrage and a complete market. Tu, in our valuation model, we ave te following: d( t) r( t) dt dz( t). (9) ( t) Wen te pecification of te oue price proce include te ervice flow rate, wic act a a dividend and i uually aumed to be a percentage of te oue value in te literature (Kau, Keenan, Muller Ⅲ, and Epperon 1992, 1993; Kau, Keenan and Muller Ⅲ, 1993; Kau and Keenan, 1996; Ambroe, Buttimer and Capone, 1997; Bardan, Karapandza and Uroevic, 2006), we ave: d( t) ( r( t) ) dt dz( t), (10) ( t) were i te ervice flow rate a a percentage of te oue price. We let Z (t) be correlated wit Z r (t), denoted a E[ dz ( t) dz ( t)] dt, were E [] i te expected operator, and denote te correlation between te ort-term interet rate and te oue return. r Aumption 3: Specification of te mortality rik uing te Gompertz urvival probability 13

16 function Becaue we are addreing mortality rik, we let te probability pace be (,, Q). We pecify te deat time a a non-negative random variable; t mean tat deat occurred at or before time t. We let te jump proce aociated wit be t I{ t}, were I i an indicator function: I 1 for tate (e.g., deat) and I 0 for oter tate. We let l ( ) t0 t tand for te filtration generated by and we let G ( 0, ) denote te probability of urvival until time, given tat te omeowner i alive at time 0. We ten ave G( 0, ) E[1 ] P( 0). Up to te preent, everal well-known mortality function ave commonly been ued to etimate mortality rate. Example are te Weibull-type function for young cildren, te invere-weibull type for teenager and te Gompertz type for adult (Brillinger, 1961; Lee and Carter, 1992; Carriere, 1992, 1994; Jaiulewicz, 1997). Becaue our dicuion i focued on elder, we ue te Gompertz function to meaure te urvival probability. We let G (u) be te urvival function at u year of age and F (u) te deat probability at age u, t u 0 T ; tu, F( u) 1G( u) and f (u) i te probability denity function for deat at time t, were F( u) f ( u). Te Gompertz function i decribed a follow: u m u m G( u) exp(exp( ) exp( )), (11) were m and are te location and diperion parameter repectively. Terefore, te deat probability denity function can be decribed a: 14

17 (1 G( u)) 1 u m f ( u) G( u)( exp( )). (12) u Auming te RM borrower i alive at time t, te conditional urvival probability and conditional deat probability denity function are decribed repectively a: G( t ) G( t, ) ; and (13) G( t) f ( t ) f ( t, ). (14) G( t) Given te previou tree aumption, te croover rik can be obtained under te condition of no arbitrage. In te following ection, we firt ue te dicrete-time approximation to compute te preent value of te expected lo from te croover rik (Broadie and Glaerman, 1997). Ten we extend te model to te continuou-time framework. We give a reference filtration of te probability pace (, G, Q), were G i an enlarged filtration generated a G F. We propoe tat te valuation proceed in ti pace. Te debt (accumulated outtanding balance) for te RM i () at time 0,1,, T. According to Equation (3) and (4), we ten ave: () = V ( 1 r RM ) a, r. (15) RM Te lender receive an amount equivalent to te leer of te debt value ( ) and te oue value ( ) wen te borrower terminate te RM contract (e.g., becaue of deat) at time, [1,, T ] : 15

18 min( ( ), ( )) ( ) max( 0, ( ) ( )). (16) Becaue ( ) i te amount tat ould be repaid to te lender, a own in Equation (16), te lender potential lo (croover rik) i: max( 0, ( ) ( )). (17) Equation (17) ow tat te expected lo from te croover rik i conidered a put option, were te exercie price i te accumulated debt ( ) and te underlying aet value i te oue value at time. Ti put option i influenced by tree underlying factor: te random deat time, te accumulated debt ( ) and te oue price ( ) (Szymanoki, 1994). Given determinitic deat time (i.e., ), te value of te put option i B( 0, ). If B(0, ) (0, ) i te expected lo per $1 of te initial oue value at time, (0) we ave: 9 ( 0, ) ( ) A(, T, ) P 0, ) N( d (0, )) e N( d (0, )), (18) r RM ( 2 1 were A(, T, r RM e ) T rrm e T ( e rrm RM 1) ; 1 ln( ) ln( A(, T d (0, ) 1 d (0, ) d (0, ) v(0, ); 2 r 1 1, rrm )) ln( P(0, )) v 2 v(0, ) 2 (0, ) ; 2 2 v ( 0, ) ( u) b(0, u) du ; 0 9 See Appendix A. 16

19 r au ( u) u ; and b ( 0, u) (1 e ). a Te above pecification mean tat ( 0, ) i obtained under te aumption of a determinitic deat time. To extend te above framework to a model wit a tocatic deat time, we aume te elder will live to a maximum age of 100 year. Tere i a probability of deat at eac time point before ti maximum deat age. Let T be te tocatically ditributed number of year tat te elder urvive, tarting wit te age at joining te RM program and ending wit te maximum age of 100 year. For eac deat time we can obtain an expected average lo ( 0, ) ; te et i defined a ( 0, 1), ( 0, 2),, to (0, T ). If (0) i te expected lo given a random deat time, we ave: (0) E[ (0, )]. (19) We can obtain te value for (0) if te conditional probability denity function for te deat time i pecified. A own by Aumption 3, f ( 0, ) i te conditional probability denity function at te time of deat. Accordingly, we ave: (0) T 0 (0, ) f (0, ). (20) 2.3 Principal limit factor calculated from te traditional model of Szymanoki (1994) To reduce te croover rik, te lender dicount te elder oue value by te value of te PLF in calculating te maximum loan balance at te origination time. We now illutrate te traditional model from Szymanoki (1994) tat i ued to calculate te prevailing PLF. Szymanoki (1994) define te PLF a te iget initial loan-to-value ratio for wic te inurance premium cover te expected lo from te future claim. Calculation of te PLF i baed on te following aumption: te preent value of te expected lo from te pooled RM 17

20 equal te preent value of te expected mortgage premium to be collected from te pooled RM: T 0 E[ L( )(1 r( )) ] T 0 E [ MIP( )(1 r( )) ], (21) were L () i te lo from te croover rik incurred at time and MIP () repreent te mortgage inurance premium ceduled to be collected at time. Becaue (0) i te expected lo per $1 of te initial oue value, we ave T 0 E [ L( )(1 r( )) ] ( 0) (0) (22) Tere are two type of MIP: (1) te upfront mortgage inurance premium (UMIP ) at te initial time and (2) te inurance premium paid for eac period until te RM contract i terminated. We ten ave MIP( 0) UMIP and MIP( ) r ( ) for 1,, T. Ti lead to: I T 0 T E [ MIP( )(1 r( )) ] UMIP r G(0, ) ( ) P(0, ) I (23) 1 According to Szymanoki (1994), te equilibrium PLF (denoted a ) can be calculated baed on te following aumption: T (0 ) r ( ) G(0, ) A(, T, r ) P(0, ), (24) I I 0 RM were (0 ) i te value of (0) given, and I i te upfront mortgage inurance premium per $1 of te initial oue value. For a lender, ti calculation of i baed on a fully edged etimate of te croover rik, becaue te inurer incur all te poible loe from te croover rik. However, a 18

21 previouly mentioned, bot te oue collateral and te mortgage inurance protect againt ig potential loe due to te croover rik. If te etimate i fully edged, may be too conervative to encourage development of te RM market. Tu, we propoe a model for calculating te lender optimal tat i baed on two general economic model of te lender ituation: breaking even and maximum profit. 2.4 Determination of an optimal PLF baed on economic concept We argue tat determination of an optimal PLF ould be baed on two economic concept: te lender earning and lo. We aume te lender make a ceduled payment at te origination time for iuing te RM and pay te capital cot at a fixed capital cot rate ( r ); tu, r r( 0) r. Te accumulated capital cot of te RM program up to time i () for I 1,, T. Ti can be repreented a: () = V ( 1 r a ), r. (25) Given te above pecification, if te elder pae away at time, te lender receive te value of te ( ) from te elder and ue it to pay te total cot of te capital ( ( ) ) to te depoitor. Terefore, te lender earning obtained from te interet rate pread of te RM at te time te RM contract i terminated are denoted a ( ) ( ) tee earning i:. Te expected preent value of E [( ( ) ( ))(1 r ( )) ] ( 0) R(0 ), (26) were R ( 0 ) repreent te expected average earning per $1 of te initial oue value given. We ten ave: 19

22 T R( 0 ) ( ) ( A(, T, r ) B(, T, r, r )) f (0, ) P(0, ), (27) 0 RM RM were B(, T, r RM, r ) a a, T, r RM ( 1 r ). In Equation (26), R ( 0 ) i calculated witout taking account of te poible lo from te croover rik. If te RM i not inured, te lender face a poible lo, a te market value of te oue i le tan te accumulated debt. Tu, we treat ti poible lo from te croover rik a te implied cot of te RM for te lender. We need to take ti cot into account in calculating te profit from te RM. If we decribe te lender expected lo a ( 0) C(0 ), were C ( 0 ) i te expected average lo per $1 of te initial oue value, te lender profit from te RM it iued i: ( 0 ) R( 0 ) C( 0 ), (28) were ( 0 ) i te expected profit per $1 of te initial oue value. For an uninured RM, it i te lender wo oulder te expected average lo. Tu, we ave C ( 0 ) ( 0 ). If te government want to encourage development of te RM market, it ould et te PLF a ig a poible. On te oter and, if te lender i a government-ponored organization, for wic profit-making i not an objective, according to economic teory te optimal PLF ould be et under te aumption tat te lender break even (i.e., zero profit), tat i, te expected average earning equal te expected lo: R ( 0 ) C ( 0 ). (29) If te RM lender i not a government-ponored organization, it can maximize it profit by 20

23 determining te PLF baed on te aumption tat te marginal earning equal te marginal cot (i.e., te marginal poible lo from te croover rik): MR ( 0 ) MC ( 0 ), (30) were dr(0 ) MR( 0 ) and d dc(0 ) MC( 0 ) repreent te lender marginal d expected revenue and marginal expected cot repectively, bot baed on. If te RM contract i inured, te lender expected lo may differ from te above. We tu ugget an approac for calculating te optimal PLF of an inured RM from te lender tandpoint. In term of ti framework, if a lender want to maximize it profit or encourage te greatet development of te RM market, it mut be willing to incur ome of te lo from te croover rik. In oter word, te inurer can only oulder te lo tat correpond to te received mortgage inurance premium. Te ratio of te inurer burden on lo can be defined a te total collected inurance premium divided by te total expected lo from croover rik. From te inurer tandpoint, if an increae in te PLF caue a rie in te loan amount tat an RM borrower receive, bot te preent value of te expected lo and te preent value of te expected to-be-received premium increae. In general, te growt rate i larger for te former tan for te latter (a own in Figure 3). Terefore, te preent value of te expected lo i larger tan wat te inurer mut bear if an increae in te PLF, calculated baed on economic concept. For a new PLF, te difference between te expected lo and te received premium i: (0) T 0 E[ L( )(1 r( )) ] T 0 E [ MIP( )(1 r( )) ]. (31) If te lender intend to raie te PLF above, it mut be prepared to bear any potential 21

24 lo (0) in exce of wat te inurer mut bear. Terefore, for an inured RM, te lender expected lo i denoted a max( 0, (0)). We ten ave ( 0) C(0 ) max( 0, (0)). According to Equation (22), (24) and (31), ti give u: T C ( 0 ) max( 0, (0 ) ( r ( ) G(0, ) A(, T, r ) P(0, ) )). (32) I I Wen C ( 0 ) i determined by Equation (32), one can ue Equation (29) and (30) to determine te optimal PLF taking account repectively of te maximum development of te RM market (te lender breakeven ituation) and te lender maximum profit. 0 RM 3. Numerical Analye Following i a numerical example to illutrate our model. 10 We let te elder participation age t 0 = 70, te initial oue value P 0 = $100,000, 1, r C 10% and r 0.5% ; 11 tu, te montly interet rate for te RM ( r ) 10% 0.5%. In addition, we aume tere i no RM ervicing fee ( S ( 0) 0 ), te initial MIP i $2,000, and te origination fee i $2,500; 12 tu, I and are 0.02 and repectively. I Jarrow, Lando and Yu (2005) empaize tat te equivalence of te pyical and rik-neutral meaure old under te aumption of a well-diverified portfolio. Becaue te croover rik can be well diverified by aving a large number of pooled RM, we calculate te market price of an RM under te aumption tat te pyical meaure i equivalent to te rik-neutral meaure. All te rik can be calculated from market information. We ue parameter 10 All value are taken from figure publied by te U.S. Department of Houing and Urban Development, available at: ttp://earc.ua.gov/earc?affiliate=ouingandurbandevelopment&query= In 2015, te initial MIP wa 0.5% or 2.5%, depending on te borrower diburement. Over te life of te loan, te annual MIP i 1.25% of te mortgage balance. Our aumption are only for comparative purpoe. 12 Te origination fee tat HUD permit for HECM range from $2,500 to $6,

25 etimated from actual data to perform te numerical analye. To meaure te expected average lo from te croover rik, we etimate te parameter of te interet rate and te oue price procee uing a 3-mont U.S. Treaury bond and te U.S. Houing Price Index (HPI) repectively. Te ampling frequency i montly and te ample period i from January 1987 to September 2010, yielding 260 data point. Table 1 preent decriptive tatitic (mean, tandard deviation, median, and maximum and minimum value) for te ort-term interet rate and te HPI. < Inert Table 1 Here > We ue te maximum likeliood metod to etimate te parameter of te ort-term interet rate and oue price procee baed on te data. 13 Te reult are own in Table 2. Te etimated parameter are: r = , a = , r = , , = and = < Inert Table 2 Here > Te mortality rik i calculated from market information. Te parameter in te conditional probability denity function of te deat time are calculated baed on te U.S. Life-Table for According to ti table, te oldet urvival age for an RM borrower i 100 year; tu, te urvival period ( T ), defined a from te elder age at participation onet (70) to te maximum attainable age (100), i 30 year. Te etimated parameter value for te Gompertz urvival probability denity function are m = and = We ue tee two parameter to etimate te conditional expected deat probability for an RM borrower wo i older tan See Appendix B. 14 Te etimation metod i own in Appendix B. 23

26 year wen joining te RM program. < Table 3 Inert Here > Baed on te previou pecification, te baic parameter value are a follow: t 0 70 year, 1, r 10%, r 0.5%, , 0. 02, r ( 0) 0. 04, ( 0) $100, 000, C I I r = , a = , r = , , = and =0.2105, T 30 year, m =87.46 and = In Figure 1, we ow te relationip between te expected average lo from te croover rik and te determinitic deat age baed on te above parameter. A own in Figure 1, te expected average lo i an increaing convex function of te borrower deat age. Tat i, te longer te elder live, te greater te lender potential lo, wic i attributable to te increaing croover rik. < Inert Figure 1 Here > Figure 2 ow ow to determine te optimal PLF for an uninured RM contract, conditional on eiter te lender making maximum profit or zero profit (te breakeven point). It diplay te expected average earning obtained from te interet rate pread ( R ( 0 ) ), te lender expected average lo from te croover rik ( C ( 0 ) ) and te expected profit ( ( 0 ) ). Te y-axi repreent te lender expected average earning or expected average loe, and te x-axi repreent te PLF value ( ). Te dotted line repreent te expected average earning calculated from Equation (27); te unbroken line give te lender expected average loe calculated from Equation (24); te daed line repreent te expected average profit, defined a te difference between te lender expected average earning and expected 24

27 average loe. If te lender cooe to bae te optimal value of te PLF on it maximum profit, Equation (30) can be ued to determine ti optimal value. Given te equilibrium point a te iget expected profit, we calculate te optimal PLF in our example to be A own in Table 4, it follow from ti value tat te lender expected average earning, expected average lo, and expected profit are , , and repectively. On te oter and, if te lender cooe to bae te optimal PLF for an uninured RM contract on te aumption of zero profit (te breakeven ituation), ti optimal value i located at te breakeven point for te lender ( R ( 0 ) C ( 0 ) ); in our example, ti value i A own in Table 4, it follow from ti value tat te lender expected profit i zero, and it expected average earning and expected average lo are bot < Inert Figure 2 Here > < Inert Table 4 Here > We now ow ow to calculate te optimal PLF for an inured RM. Figure 3 illutrate, for different PLF value, ow to obtain te expected average lo from te croover rik and te expected premium collected for eac $1 of te initial oue value. Note tat te expected average lo i a convex function tat i poitively related to te PLF value; in oter word, te expected collected premium increae a te PLF increae. Tee increae are caued by te increaing croover rik, wic in turn increae te expected premium. According to te model in Szymanoki (1994), te fair PLF i te point at wic te line for te expected average lo and te expected inurance premium cro in Figure 3; in our example, ti value i If te actual PLF i le tan te fair PLF, te inurer may reap a profit. Moreover, Figure 3 tell u tat te increae in te expected premium i le tan te increae in te expected average lo, becaue te PLF value exceed te value at te croover point in te figure. Tu, one can 25

28 conclude tat te inurer may incur a lo if te actual PLF i larger tan te fair PLF. Finally, Table 4 ow tat wen te PLF i et at in our example, te lender expected average earning, expected average lo and expected profit are , 0 and repectively. < Inert Figure 3 Here > Figure 4 ow ow to determine te optimal PLF for an inured RM contract auming maximum profit and zero profit (breakeven ituation) for te lender. If te PLF i larger tan (te fair PLF), te expected average lo, calculated from Equation (32), ould be greater tan zero. If te lender cooe te PLF value baed on maximum profit, te optimal PLF in our example i Ti value i larger tan te fair PLF determined by te traditional metod, meaning tat te lender ould lend te RM borrower more money. According to Table 4, te PLF for an inured RM (0.4576) i iger tan te PLF for an uninured RM (0.4451). Ti i reaonable, becaue te inurer a aborbed part of te lo. Table 4 ow tat if te lender et te optimal PLF at , te expected average earning, expected average lo and expected profit are , and repectively. Altoug te expected average lo increae due to te rie in te PLF, becaue te interet rate pread increae te earning more tan it increae te lo, te expected profit alo increae. Te optimal PLF determined by our model repreent a win-win ituation, becaue te increae in te maximum loan caued by te iger PLF increae bot elder rate of participation in te RM program and te lender profit. Finally, if te PLF i et at in our example, te expected average lo incurred from te croover rik i and te expected premium received by te lender i only In oter word, te inurer burden ratio of lo i 54.18% (0.0506/0.0934). Tee reult can elp lender and inurer et a reaonable PLF for ti win-win ituation. < Inert Figure 4 Here > 26

29 Figure 4 alo ow te optimal PLF for an inured RM wen te lender profit i zero. In our example, te optimal PLF i at ti breakeven point. Even toug te lender a no profit tat can be ued to et te PLF value, te borrower a te iget PLF and tu can get more payment from te RM program, tereby greatly encouraging te advancement of te RM market. Table 4 ow tat at ti breakeven point te lender expected average earning and expected average lo are bot Tu, te inurer burden ratio i 11.67% (0.0866/0.7418). It follow from te above dicuion tat a PLF et at a conervatively low value not only reduce te borrower maximal loan balance but alo reduce te lender profit. Alo, our numerical reult imply tat raiing te PLF value i a practical alternative. Addreing ti quetion i eential to undertanding ow lender are protected from tee loe, and it anwer provide evidence tat regulator can ue to conduct cot-benefit analye and deign more efficient policie. 4. Senitivity Analye Becaue of advance in medicine, elder are living longer tan ever before. In addition, economic factor uc a interet rate and oue price are becoming more and more variable. How muc ould PLF cange in repone to tee oter cange? In ti ection, we report enitivity analye in wic we calculate te fair PLF by te traditional metod and te optimal PLF baed on te cange in te lender maximum profit in repone to tee variou factor. Te government can ue te reult of tee analye to determine te mot uitable policy for developing te RM market. In Figure 5 we ow te relationip between te PLF and te different RM contract rate. 27

30 Te figure demontrate tat bot te fair PLF and te optimal PLF are negatively correlated wit te RM contract rate. In addition, te optimal PLF i alway larger tan te fair PLF. However, te value of tee two PLF are almot te ame if te RM contract rate i low. A cange in te RM contract rate a a larger effect on te fair PLF tan on te optimal PLF. < Inert Figure 5 Here> Figure 6 diplay te relationip between te PLF and variou parameter related to te interet rate, namely, te initial interet rate, it adjutment peed and it volatility. A own in Figure 6a, te fair PLF and te optimal PLF are poitively related wit te initial interet rate. Te lope of tee two curve reveal tat te influence of a cange in te initial interet rate i greater on te fair PLF tan on te optimal PLF. Figure 6b ow tat cange in te adjutment peed do not affect tee two PLF to a tatitically ignificant degree. Figure 6c ow tat te fair PLF and te optimal PLF are bot negatively related wit interet rate volatility. Collectively, tee reult imply tat a cange in te initial interet rate a te greater effect on te fair PLF and te optimal PLF. < Inert Figure 6 Here> Figure 7 give te reult of te enitivity analye of ow variou parameter related to oue price, uc a te initial oue price, te oue ervice flow rate, te volatility of oue return, and te correlation between te interet rate and te oue return, influence te PLF. Becaue te lope of tee curve are negative in Figure 7a-7c, we infer tat te fair PLF and te optimal PLF are negatively correlated wit te initial oue price, te oue ervice flow rate and te volatility of oue return. To te contrary, te lope of te curve are poitive in Figure 7d. Terefore, te two PLF are poitively related wit te correlation between te oue return 28

31 and te interet rate, altoug not to a tatitically ignificant degree. Te reult reveal te factor tat a te greatet influence on PLF i te oue ervice flow rate. <Inert Figure 7 Here> Finally, Figure 8a and 8b ow tat all te relationip between te two PLF (fair and optimal) and te parameter of te urvival probability denity function (location and diperion) are negative. Tee reult tell u tat in te current climate, te longer te elder i expected to live, te lower te value of te PLF ould be et. Ti appearance i conitent wit te current tatu of te RM market, wit te longevity rik being negatively related wit te maximum loan amount. < Inert Figure 8 Here> We argue tat if te PLF increae, te payment tat borrower receive alo increae, tereby cauing elder participation rate in te RM program to rie. Te lender can ave te benefit becaue of te effect of diverification reulting from an increae in te number of pooled RM. Prior reearc demontrate tat te oue price rik aociated wit mortgage portfolio can be effectively diverified acro te different region and categorie (Eicoltz et al., 1995). Tu, altoug lender may increae teir probability of lo on individual loan by raiing te PLF, teir overall loe in a large portfolio can be effectively diverified if tey make more of tee loan. Here we illutrate a numerical example to ow ti argument. In our example, we ue tree PLF (0.35, 0.5 and 0.65) and tree value of te volatility of oue return (0.5, 0.25 and 0.125) to dicu te different level of effectivene of te diverification of te RM portfolio. We give te following numerical example to clarify ow te overall lo from an RM ( (0) ) can be decreaed troug portfolio diverification. 29

32 < Inert Table 5 Here > In Table 5, wen 0. 5 te lender expected loe (0) are , and repectively for PLF value ( ) 0.35, 0.5 and Tat i, te lender expected lo from individual loan i likely to increae a te PLF value increae, auming contant volatility of oue return. Ti implie tat lender rik increaing teir loe on uc loan by raiing te PLF wen te oue price rik cannot be effectively diverified troug a larger RM portfolio. Next, we ue te numerical reult to illutrate te following inference: if te volatility of oue return decreae, te lender expected loe ould be decreaed. In view of ti fact, Table 5 ow tat if i 0.35, te lender expected loe (0) are , and repectively wen te correponding volatilitie of te oue return ( ) are 0.5, 0.25 and Ti mean tat lender can reduce teir expected loe if teir oue price rik can be decreaed. If an increae in te PLF increae elder participation rate in te RM program, te number of pooled revere mortgage may become ufficiently large. Tu, te volatility of te underlying property portfolio i le tan tat of any individual oue due to te conequence of diverifying te property portfolio. Next, we clearly illutrate tat aving a large RM portfolio may reduce te expected lo if te oue price rik can be effectively diverified. We aume tat volatility of oue return decreae wen te number of pooled revere mortgage become larger and larger becaue of an increae in te PLF. For example, if =0.35, 0.5 and 0.65, ten become 0.5, 0.25 and 0.125, repectively. In Table 5, (0) = for =0.35 and 0.5; (0) = for =0.5 and 0.25; and (0) = for =0.65 and 30

33 Tee value reveal tat even toug te PLF increae, te expected lo i till reduced becaue of te effect of RM portfolio diverification. Collectively, all te above numerical example tell u tat altoug lender may increae teir probability of lo on individual loan by raiing te PLF, teir overall loe can be decreaed by effectively diverification wit a larger portfolio if tey make more of tee loan. 5. Concluion Becaue te RM contract include a non-recoure proviion, lender can incur a croover rik if te RM balance exceed te value of te property at termination. In te current RM ytem, lender ue a conervatively low PLF, determined by mortgage inurance, to protect temelve from poible loe. However, etting te PLF value o low tat it adverely influence elder participation rate to a ignificant degree decreae te development of te RM market, becaue te montly payment te elderly borrower receive migt be too low to ufficiently improve teir financial ituation. In ti paper, we ave decribed a model to determine an optimal PLF. A i well known, RM lender uually protect temelve from loe due to te croover rik by utilizing PLF and mortgage inurance. Traditional reearcer ave alway calculated te fair PLF value uing te model in Szymanoki (1994). However, te PLF obtained by ti metod i too conervative to be optimal, becaue it ue only mortgage inurance to fully edge te lender croover rik. We calculate te optimal PLF under two general economic aumption: te lender breaking even and te lender obtaining maximum profit. To te bet of our knowledge, ti i te firt paper to decribe determination of te optimal PLF under tee two economic circumtance. To determine te optimal PLF, we mut firt reaonably model te croover rik. We 31

34 incorporate in our model te tree main rik aociated wit RM: te longevity rik, te collateral rik and te interet rik. Previou model ave rarely incorporated tee imultaneouly. In our model, we ue a Gompertz-type meaure for te longevity rik; te interet rate and te oue price are aumed to follow te extended Vaicek model and te geometric Brownian motion proce, repectively. Our numerical analye and enitivity analye upport te concluion tat our model a te following four ueful policy implication for market participant and government policymaker. For te government, RM can be a good way to improve te financial circumtance of an aging population and alleviate te government financial burden. In view of ti, if te government primary policy goal i to develop te RM market, we ugget tat te optimal PLF be calculated under te aumption of te lender breaking even. In our numerical example, ti optimal PLF i Altoug te lender make no profit in ti cae, te increae in te montly payment tat elder receive increae teir deire to participate in te RM program, tereby greatly encouraging te development of te RM market. If te participation rate rie, te beneficial effect of portfolio diverification on te collateral rik and te longevity rik can coniderably attenuate lender and inurer potential loe from a croover rik. Moreover, lender alo can iue RM mortgage-backed ecuritie (e.g., HECM mortgage-backed ecuritie, HMBS) if te RM portfolio become large enoug. Tee ecuritie not only can provide capital and liquidity for lender and inurer, but tey alo can increae teir profit. Referring to tee dicuion ould elp government determine optimal policie for popularizing RM. A for lender, determining PLF on te bai of a fully edged rik, wic i currently ow it i done, i too conervative to increae profit in te RM market. Determining te optimal PLF 32

35 under te aumption of maximum profit i better tan te current metod if te lender policy i to eek maximum profit. Altoug te expected average lo increae due to te rie in te PLF, te interet income increae te earning more tan it increae te lo, and te expected profit alo increae. For ti reaon, te optimal PLF determined by our model (a own in our numerical example) lead to an increae in bot te RM borrower maximum loan balance and te lender profit. Suc an optimal PLF i a win-win for te elder and te lender. Tirdly, according to our previou dicuion, a PLF et at a conervatively low value not only reduce te lender profit but alo decreae te borrower maximal loan balance, tereby lowering elder deire to participate in te RM program. Our numerical reult reveal tat altoug lender may increae teir expected loe on individual loan by raiing te PLF, teir overall loe can be effectively diverified troug larger portfolio if tey make more of tee loan. Ti implie tat raiing te PLF value i a practical alternative. Our reult regarding te optimal PLF wen lender break even or acieve maximum profit ould elp policymaker et reaonable PLF tat are beneficial to lender, RM borrower and inurer. Finally, te information provided by te reult of our enitivity analye can elp policymaker and market participant modulate a reaonable PLF reponive to cange in te economic ituation and te relevant rik. For example, tee analye demontrate tat PLF value are negatively correlated wit te volatility of te oue return and te volatility of interet rate. Policymaker ould increae (decreae) te PLF wen te interet rate rik and te oue price rik decreae (increae). Modulating PLF by te economic ituation and te relevant rik i more reaonable tan te current PLF obtained from a table publied by HUD and ould elp increae participation in te RM market. 33

36 Reference Ambroe, B. W., R. J. Buttimer, C. A. Capone, Jr Pricing Mortgage Default and Forecloure Delay. Journal of Money, Credit and Banking 29(3): Bardan, A., R. Karapandza and B. Uroevic Valuing Mortgage Inurance Contract in Emerging Market Economie. Journal of Real Etate Finance and Economic 32(1): Biop, T. B. and S. Hui Revere Mortgage: A Cloer Look at HECM Loan. Working Paper. NBER Paper on Retirement Reearc Center Project. Brillinger, D. R A Jutification of Some Common Law of Mortality. TSA, XIII: Broadie, M. and P. Glaerman Pricing American-Style Securitie Uing Simulation. Journal of Economic Dynamic and Control 21, Carriere, J. F Parametric Model for Life Table. Tranaction of te Society of Actuarie 44: Carriere, J. F A Select and Ultimate Parametric Model. Tranaction of te Society of Actuarie 46: Cen, H., S. H. Cox, and S. S. Wang I te Home Equity Converion Mortgage in te United State Sutainable? Evidence from Pricing Mortgage Inurance Premium and Non-Recoure Proviion Uing te Conditional Ecer Tranform. Inurance: Matematic and Economic 46: Cinloy, P. and I. F. Megbolugbe Revere Mortgage: Contracting and Croover Rik, Journal of te American Real Etate and Urban Economic Aociation 22: Davidoff, T., and G. Welke Selection and Moral Hazard in te Revere Mortgage Market. Working Paper. Davi, E. P Can Penion Sytem Cope? European Buine Review 98: Diventi, T. R., and T. N. Herzog Modeling Home Equity Converion Mortgage. Tranaction of te Society of Actuarie 43: Eicoltz, P., M. Hoeli, B. MacGregor and N. Nantakumaran Real etate portfolio diverification by property type and region. Journal of Property Finance 6(3): Faruqee, H., and M. Muleien Population Aging in Japan: Demograpic Sock and 34

37 Fical Sutainability. Japan and te World Economy 15: Jarrow, R. A., D. Lando, and F. Yu Default Rik and Diverification: Teory and Empirical Implication. Matematical Finance 15: Jaiulewicz, H Application of Mixture Model to Approximation of Age-at-Deat Ditribution. Inurance: Matematic and Economic 19: Ji, Min, A Semi-Markov Multiple State Model for Revere Mortgage Termination. Annal of Actuarial Science 6: Kau, J. B. and D. C. Keenan An Option-Teoretic Model of Catatrope Applied to Mortgage Inurance. Journal of Rik and Inurance 63(4): Kau, J. B., Keenan, D. C. and W. J. Muller Ⅲ An Option-Baed Pricing Model of Private Mortgage Inurance. Journal of Rik and Inurance 60(2): Kau, J. B., D. C. Keenan, W. J. Muller Ⅲ and J. F. Epperon A Generalized Valuation Model for Fixed-Rate Reidential Mortgage. Journal of Money, Credit, and Banking 24(3): Kau, J. B., D. C. Keenan, W. J. Muller Ⅲ and J. F. Epperon Option Teory and Floating-Rate Securitie wit a Comparion of Adjutable- and Fixed-Rate Mortgage. Journal of Buine 66: Lee, R. D. and L. R. Carter Modeling and Forecating U.S. Mortality. Journal of te American Statitical Aociation 87: Lee, Y. T., C. W. Wang and H. C. Huang On te Valuation of Revere Mortgage wit Regular Tenure Payment. Inurance: Matematic and Economic 51(2): Li, Y. S Te Callenge of Aging toward Cinee Society. Public Adminitration and Management: An Interactive Journal 10: Li, J. S. H., M. R. Hardy and K. S. Tan On Pricing and Hedging te No-Negative- Euity Guarantee in Equity Releae Mecanim. Te Journal of Rik and Inurance 77 (2): Ma, L., J. X. Zang and D. Kannan A Markov Proce Modeling and Analyi of Indifference Pricing of Inurance Contract for Home Reverion Plan for a Pair of Inured. Stocatic Analyi and Application 29: Mayer, C. J. and K. Simon Revere Mortgage and te Liquidity of Houing Wealt. 35

38 Journal of te American Real Etate and Urban Economic Aociation 22: Mayer, C. J. and K. Simon. 1994b. A New Look at Revere Mortgage: Potential Market and Intitutional Contraint. Federal Reerve Bank of Boton: New England Economic Review (iue Mar): Merrill, S. R., M. Finkel and N. K. Kutty Potential Beneficiarie from Revere Mortgage Product for Elderly Homeowner. Journal of te American Real Etate and Urban Economic Aociation 22: Morgan, B. A. and I. F. Megbolugbe, and D. W. Ramuen Revere Mortgage and te Economic Statu of Elderly Women. Te Gerontologit 36: Piggott, J., and O. Mitcell Unlocking Houing Equity in Japan. Journal of te Japanee and International Economie 18: Pu, M., G. Z. Fan and Y. H. Deng Breakeven Determination of Loan Limit for Revere Mortgage under Information Aymmetry, fortcoming in Te Journal of Real Etate Finance and Economic 48(3): Ramuen, D. W., I. F. Megbolugbe and B. A. Morgan Uing te 1990 Public Ue Micro Data Sample to Etimate Potential Demand for Revere Mortgage Product. Journal of Houing Reearc 6: Ramuen, D. W., I. F. Megbolugbe and B. A. Morgan Te Revere Mortgage a an Aet Management Tool. Houing Policy Debate 8: Redfoot, D., K. Scolen and S. K. Brown Revere Mortgage: Nice Product or Maintream Solution? Waington, DC: AARP Public Policy Intitute Reearc Report. Soda, S Houing-Rik, Income-Poor: Te Potential of Houing Wealt in Old Age. A paper for Houing Acro te Lifecycle. World Wide Web: ttp://ippr.nviage.uk.com / e c o mm / file/ouing-ric.pdf. Speare A. Jr Te Demograpy of Revere Mortgage. Conference Paper for te Annual Meeting of te Population Aociation of America, Denver, Colorado. Syzmanoki, E. J Rik and te Home Equity Converion Mortgage. Journal of te American Real Etate and Urban Economic Aociation 22: Szymanoki, E. J., J. C. Enriquez and T. R. DiVenti Home Equity Converion Mortgage Termination: Information to Enance te Developing Secondary Market. Journal of 36

39 Policy Development and Reearc 9: Te, Y. K Modelling Revere Mortgage. Aia Pacific Journal of Management 12: Vaicek, O An Equilibrium Caracterization of te Term Structure, Journal of Financial Economic 5: Venti, S. F. and D. A. Wie Aging and Houing Equity. National Bureau of Economic Reearc. Working Paper #7882. Von Weizacker, R. K Ditributive Implication of an Aging Society. European Economic Review 40: Weinrobe, M An Analyi of Home Equity Converion in te RAM Program. Journal of te American Real Etate and Urban Economic Aociation 15: Zai, D.H Revere Mortgage Securitization: Undertanding and Gauging te Rik, Structure Finance, Moody' Invetor Service Special Report. Appendix A: According to Equation (17), te expected lo from te croover rik i a put option wit underlying aet () and trike price (). Baed on te option pricing teory, te expected value of a put option, denoted a B( 0, ), can be expreed a follow: were B( 0, ) E[exp( r( u) du)( ( ) ( )) ] 0 F P 0, ) ( ) N( d (0, )) (0) e N( d (0, )), (A1) ( 2 1 P ( 0, ) E[exp( r( u) du) ], i te price of a zero coupon bond; 0 (0) ln( ) ( ln( P(0, )) ( ) d1(0, ) v(0, ) d (0, ) d (0, ) v(0, ). 2 1 F v (0, ) ; and Te debt amount can ten be rewritten a follow: 1 () V ( 1 r RM ) a, r ( (0) S(0) D(0)) RM a T, r RM ( 1 r RM ) a, r RM a ( ) (0) a, rrm T, rrm ( 1 r RM ). (A2) In a continuouly time framework, we ave r ( 1 rrm ) e RM. In addition, te preent value of 37

40 te annuity i () a, rrm Terefore, we ave: rrm ( e 1). Tu, we ave rrm r e RM ( ) (0) A (, T, ). (A3) ln( ) ln( A(, T d (0, ) 1 Let r RM 1, rrm )) ln( P(0, )) v 2 v(0, ) 2 (0, ). (A4) B(0, ) (0, ), te croover rik per $1 of ouing value. For implicity, (0) we let te volatility of oue return be contant. We ten ave: ( 0, ) ( ) A(, T, ) P 0, ) N( d (0, )) e N( d (0, )). Ti i te Equation (18) in te paper. r RM (

41 Appendix B We firtly ow ow to etimate te parameter of te interet rate and ouing procee, employing te maximum likeliood metod. Te jointed probability denity function of two Wiener procee i expreed a follow: 1 f ( x( t) ) 2 V 1 exp( ( x( t) x 2 ( t)) V 1 ( x( t) x ( t))), were x( t) [ r( t) r ( t) ], repreent a vector for te real data for te interet rate and te cange rate of ouing price, repectively. x (t) and V repreent teir mean and variance-covariance matrix repectively. According to Equation (5) and (10), we ave 2 1 ( ) [ ( 1) ( ( 1)), ( ( ) 2 r r x t r t a r r t dt r t ) dt], and V dt. 2 2 r Finally, let be te vector of parameter needed to be etimated; we ave [ r a ]. r We ten define te log-likeliood function a: L ( ) log( f ( x( t) )). t L( ) Te parameter i etimated if 0. In te etimation for te urvival function, we minimize te mean quare error between te actual data and teoretical model to obtain te location and diperion parameter for te Gompertz urvival probability. Tat i Min m, 1 T t T 0 ut fˆ( u) f ( u) 2 ( ), were f ˆ( u ) i te actual data of deat probability. fˆ( u) 0 39

42 Figure 1: Relationip between te average lo from te croover rik and a determinitic age of deat Note: Te y-axi repreent te expected average loe from te croover rik, calculated from Equation (19). Te x-axi repreent te elder deat age. According to our pecification, te baic parameter are: t 0 70 year, T 100 year, 1, r 10%, r 0.5%, , , r ( 0) 0. 04, ( 0) $100, 000, C I r = , a = , r = , , = and =0.2105, m = and =

43 Figure 2: Analyi of te optimal principal limit factor (PLF) for an uninured RM Note: Te y-axi repreent te expected average revenue or te expected average loe. Te x-axi repreent te PLF value ( ). Te dotted line repreent te expected average earning calculated from Equation (27), te olid line repreent te expected average loe calculated from Equation (24), and te daed line repreent te total profit calculated a te difference between te expected average earning and te expected average loe. Te optimal PLF i under te aumption tat te lender make maximum profit and if te lender break even. Oter definition are preented in te note to Figure 1. 41

44 Figure 3: Analyi of te inurer expected inurance premium and expected average loe from croover rik Note: Te y-axi repreent te lender expected inurance premium or expected average loe. Te x-axi repreent te value for te principal limit factor (PLF), repreented a. Te olid line repreent te expected average loe calculated from Equation (22), and te daed line repreent te expected inurance premium per monetary unit calculated from Equation (23). Te croover point of tee two line i te fair PLF, obtained by te metod in Szymanoki (1994). Baed on our etimated parameter for interet rate and ouing price, and te baic contract parameter in te current ytem, te fair PLF i calculated a Oter definition are preented in te note to Figure 1. 42

45 Figure 4: Analyi of te optimal principal limit factor (PLF) for an inured RM Note: Te y-axi repreent te expected average earning or te expected average loe. Te x-axi repreent te PLF value ( ). Te dotted line repreent te expected average earning calculated from Equation (27). Te olid line repreent te expected average loe calculated from Equation (32). Te daed line repreent te expected profit calculated a te difference between te expected average earning and te expected average loe. Te PLF i uing te metod in Szymanoki (1994). Te PLF i if te lender make a maximum profit and if te lender break even. Oter definition are preented in te note to Figure 1. 43