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1 This documen is downloaded from DR-NTU, Nanyang Technological Universiy Library, Singapore. Tile A Bayesian mulivariae risk-neural mehod for pricing reverse morgages Auhor(s) Kogure, Asuyuki; Li, Jackie; Kamiya, Shinichi Ciaion Kogure, A., Li, J., & Kamiya, S. (214). A Bayesian Mulivariae Risk-Neural Mehod for Pricing Reverse Morgages. Norh American Acuarial Journal, 18(1), Dae 214 URL hp://hdl.handle.ne/122/2239 Righs 214 Sociey of Acuaries. This is he auhor creaed version of a work ha has been peer reviewed and acceped for publicaion by Norh American Acuarial Journal, Sociey of Acuaries. I incorporaes referee s commens bu changes resuling from he publishing process, such as copyediing, srucural formaing, may no be refleced in his documen. The published version is available a: [hp://dx.doi.org/1.18/ ].

2 A Bayesian Mulivariae Risk-Neural Mehod for Pricing Reverse Morgages Asuyuki Kogure, Jackie Li, and Shinichi Kamiya Absrac In his paper, we propose a Bayesian mulivariae framework o price reverse morgages which involve several risks in boh insurance and financial secors (e.g. moraliy raes, ineres raes, and house prices). Our mehod is a mulivariae exension of he Bayesian riskneural mehod developed by Kogure and Kurachi (21). We apply he proposed mehod o Japanese daa o examine he possibiliy for a successful inroducion of reverse morgages ino Japan. The resuls sugges some promising fuure for his new marke. 1 Inroducion A reverse morgage has long been considered as an effecive way for an individual o manage longeviy risk, especially in rapidly aging counries like Japan. This marke has become paricularly imporan under he coninuing rend of moving away from radiional defined benefi (DB) reiremen plans o defined conribuion (DC) plans. Simply speaking, a reverse morgage allows a reiree o borrow agains he value of his or her home propery. The loan is hen repaid when he borrower dies and he propery is sold. Usually, he loan is non-recourse, which means ha he lender canno claim he borrower s asses oher han he morgaged propery. In Japan, he maor source of pos-reiremen income is public pensions (Fuisawa and Li 212). As he pension sysem is pay-as-you-go and he old-age dependency raio is expeced o rise significanly from 38% in 21 o 8% in 25 (OECD 29), wheher he sysem can susain in he long erm has become a serious concern. Since housing wealh consiues abou half of privae oal asses (Creighon e al. 25), a naural soluion o he ageing problem is o unlock he housing equiy. I hus appears ha here is a very large poenial marke for reverse morgages. However, a reverse morgage involves several risks, mos noably longeviy, house prices, and ineres raes. Recenly, hese risks have been increasingly uncerain, which may parly explain he presen saggering sae of developmen of he reverse morgage marke. In

3 paricular, i is no a sraighforward exercise o price he non-recourse provision under hese uncerainies. In order o price reverse morgage plans appropriaely, a mulivariae framework which can allow for muliple risks simulaneously is required. In his paper, we propose a Bayesian mulivariae mehod by generalizing he univariae Bayesian mehod suggesed in Kogure and Kurachi (21) and apply his new mehod o examine he possibiliy for a successful inroducion of reverse morgages ino Japan. For his paper we assume ha ineres raes are fixed and consider he oin disribuion of house prices and moraliy raes. Some earlier sudies on pricing reverse morgages include Hosy e al. (28), Li e al. (21), Ji e al. (212), and Lee e al. (212), which adoped a univariae risk-neural mehod for one paricular risk. In conras, we underake a mulivariae risk-neural approach for boh house price risk and longeviy risk under a Bayesian seing. There have been several pricing mehods proposed in he lieraure, e.g. Wang e al. (28) and Chen e al. (21). Our mehod differs from he exising ones in ha we ake a fully Bayesian approach. As discussed in Li (213), here are a number of disinc advanages of using he Bayesian approach. Firs, cerain prior or reference informaion can be incorporaed ino he modeling process in a formal manner. Second, differen model srucures can be aken ino accoun coherenly wihin he same framework. Moreover, his approach allows for process error, parameer error, and possibly model error in pricing. Specifically, we obain he disribuions of he risk facors and hen ranslae hem ino a riskneural form by applying he maximum enropy principle. Noe ha he expeced reurn in a risk-neural world is equal o he risk-free ineres rae. See Cairns e al. (26) for some discussion on parameer uncerainy and risk-neural valuaion in evaluaing longeviy risk. The res of he paper is organized as follows. In Secion 2, we presen a mehod o conver he oin disribuion o is risk-neural form by applying he minimum cross-enropy principle. The risk-neural disribuion hus obained is deduced o facor ino he produc of wo marginal disribuions. Afer a shor explanaion of reverse morgage plans in Secion 3, we fi a Bayesian Lee-Carer model in Secion 4 and a ime series model in Secion 5 o Japanese daa and deduce he risk-neural disribuion of each risk facor. By combining hese resuls, we examine he feasibiliy for an inroducion of reverse morgages ino Japan in Secion 6. The figures sugges some promising fuure for he inroducion. Secion 7 concludes he paper. 2

4 2 Bayesian risk-neural mehod As ways o hedge risks such as sock price movemens or accidenal deahs, financial opions and insurance producs are widely used. Such hedging insrumens are coningen conracs which pay off a cerain amoun of money dependen on observed risk facors such as sock prices or deahs, and are hus called derivaives. In his secion, we give pricing principles for derivaives based on Bayesian predicive pricing. 2.1 Risk-neural disribuions For ease of explanaion, we consider a single period framework wih curren ime and a ime poin T > in he fuure. Le X (a coninuous random variable) denoe he value of a risk facor a ime T and wrie F(x) for he disribuion funcion of X. We wish o price a derivaive which pays off C(X) a ime T dependen on X. Assuming ha he risk-free ineres rae r is consan over he whole invesmen period, he expeced presen value of he derivaive a ime can be given by he formula: where as: F x T r CxdF x V 1, (1) is so-called he risk-neural disribuion funcion of X, which can be expressed df x pxdfx, via a sae price densiy p(x). Noe ha all he inegrals in his paper cover he enire range of he variable of ineres. The sae price densiy p(x) represens a ransformaion from F(x) o F x. In he sandard opion pricing heory, he concep of compleeness is evoked under he no-arbirary principle o deermine he sae price densiy p(x) and hus he risk-neural disribuion F x. The marke is complee if here are many securiies raded in he marke and any cash flow can be replicaed via dynamic hedging sraegies. However, i would be unrealisic o assume such compleeness for non-financial risks such as moraliy risk or naural hazard risk, as he marke of securiizaion of hese risks is currenly far from having sufficien liquidiy. In he insurance risk heory, he sae price densiy is hen chosen based on he paricular problem being invesigaed. Two principles have been widely applied. One is he Esscher ransform, in which he sae price densiy akes he form: p x x e. X e 3

5 Here E( ) is he expecaion under F(x) and γ is a consan o be deermined consisenly wih some marke condiions. The oher is he Wang ransform, where he sae price densiy is of he form: p x 1 x, x Fx x, where Φ( ) denoes he sandard normal disribuion funcion and ϕ( ) denoes is densiy. See Gerber and Shiu (1994) for discussions on he Esscher ransform and Wang (2) on he Wang ransform. On he oher hand, Li (21) argued ha for he Wang ransform, subecive decisions are ofen required in deermining he marke prices of risk, and i is no sraighforward o include parameer error. Moreover, Kogure and Kurachi (21) commened ha he Esscher ransform can usually incorporae only one consrain for a marke price. Comparaively, our Bayesian maximum enropy approach se forh in Secion 2.3 is more flexible and does no have hese limiaions. 2.2 Bayesian risk-neural disribuions In pracice, he disribuion funcion F(x) is unknown and we se up a cerain saisical model d f x, o which f x Fx urns ino: wih x px f x f. is assumed o belong. Then he formula (1) dx T 1 r Cx f x V dx, (2) In he sandard frequenis approach, he maximum likelihood esimaor ˆ of θ is obained based on daa D and plugged ino (2) o yield: However, when he specificaion x ˆ T 1 r Cx f x ˆ V dx. (3) f, involves many parameers, parameer uncerainy in (3) becomes a real concern. Then i would be more desirable o ake he Bayesian approach. Wih a poserior densiy g D of θ given daa D, he Bayesian pricing formula becomes: ~ V V g D d r T Cx f x g D 1 r T Cx f x D 1 dx. d dx 4

6 Here, f x D f x g D d px f x D wih f x D f x g D d. Thus x D, (4) f is he risk-neural version of he Bayesian predicive densiy f x D. MCMC simulaion can hen be used o draw samples from he predicive densiy. For more informaion abou he MCMC algorihms, see Kogure and Kurachi (21) and Li (213). 2.3 Risk-neuralizaion of predicive disribuions To obain he sae price densiy p(x) for (4), we apply a non-parameric echnique proposed by Suzer (1996) insead of using parameric mehods such as he Esscher or Wang ransform. Suppose ha here are m securiies. For each 1 i m, le h i (X) and v i denoe he payoff funcion a ime T and he marke value a ime of he ih securiy. Under he risk- neural predicive densiy f x D, we have he momen condiions: T hi X D hi x f x D dx r vi 1 for i = 1, 2,, m. Suzer (1996) suggesed using x D Leibler informaion or he cross-enropy: f x D ln f f, (5) f ha minimizes he Kullback- Assume ha here are m securiies which in general depend on boh X and Y. For each 1 i m, le h i (X,Y) and v i denoe he payoff funcion a ime T and he marke value a ime of he ih securiy respecively. Then, following he idea of Suzer (1996), we wish o minimize he 5 x D x D dx, subec o he momen condiions (5) and also he probabiliy densiy consrain: x D 1 dx f. (6) The resulan minimum cross-enropy densiy is shown o be of he form: f x D f x D exp h x... h x 1 1, (7) where he consans γ, γ 1,, γ m are deermined by (5) and (6). This approach was aken in Kogure and Kurachi (21) o price longeviy risk in Japan. For some applicaions, several risk facors are involved and we need o consider he risk-neuralizaion of a mulivariae predicive densiy. Suppose ha, in addiion o X, we have anoher risk facor Y. Le x, y D f denoe he wo-dimensional predicive densiy. m m

7 wo-dimensional cross-enropy: wih respec o f x, y D f x, y D ln f f x, y D x, y D, subec o he momen condiions: dx dy, T hi X, Y D hi x, y f x, y D dx dy r vi 1, (8) for i = 1, 2,, m and he consrain: x y D dx dy 1 f,. (9) The resuling wo-dimensional minimum cross-enropy densiy can be deduced as: f x y D f x, y D exp h x, y... h x, y, 1 1, (1) where he consans γ, γ 1,, γ m are deermined by (8) and (9). If X and Y are independen under f x, y D, hey are also independen under he risk-neural disribuion (1) under some echnical condiions (e.g. Kapur 1993). Since he reverse morgage marke in Japan is sill in is infancy sage, for compuaion convenience, we follow some earlier sudies (e.g. Lee, Wang, and Huang 212) in assuming ha he wo risks are independen in he riskneural world. m m 3 Reverse morgages 3.1 The non-recourse provision A reverse morgage loan is non-recourse in he sense ha he propery owner s obligaion o repay he loan is limied o he proceeds from he sale of he propery when he loan is erminaed. Suppose ha an individual aged x eners a reverse morgage conrac a ime. Denoe by L he ousanding balance of he loan and HP he value of he morgaged propery a ime. If he borrower dies a ime, he loan is erminaed a ha ime and he lender will receive: min(l, HP ) = L max(l HP, ). (11) We assume here ha L is given deerminisically as L = L (1 + r L ) wih r L being he ineres rae charged on he reverse morgage loan. The second erm on he righ hand side of (11) represens a cash flow from he non-recourse provision. I may be regarded as a pu opion on he house price HP wih a srike price L. Thus he non-recourse provision is he borrower s righ o exercise a pu opion on his deah. For oher ypes of reverse morgages, see he Equiy Release Repor (25). 6

8 3.2 Evaluaion of reverse morgages We wish o evaluae he reverse morgage loan for a cohor group of age x a ime. Le τ denoe he highes aained age and pu T = τ x. Furhermore, le I denoe an index which represens he proporion of he cohor group who will die from ime 1 o. The per capia cash flow from he reverse morgage is given as: (L max(l HP, )) I for = 1, 2,, T. We assume ha I and HP are uncerain, bu discoun raes are fixed. Le d() = (1 + r) be he -year discoun facor. We also assume ha moraliy experience and house prices are homogeneous wihin he group. A each ime, a cerain porion of he group will die, which will erminae some loans and lead o some cash flows. The expeced presen value a ime of each reverse morgage loan of he group (on average) is herefore: T 1 d T T max I. (12) 1 1 L L HP, I d L I d max L HP, The erm E ( ) is he expecaion under he risk-neural measure. This expeced presen value mus be more han L for he reverse morgage o be provided in a susainable manner. We furher assume ha I and HP are independen under he risk-neural predicive disribuion. Then he second erm on he righ hand side of (12) can be wrien as: T 1 d I L HP, max, (13) and so all he valuaion comes down o he calculaion of E [I ] and E [max(l HP, )]. 4 Bayesian pricing of I In he following, we se: I = (1 q x ()) (1 q x+1 (1)) (1 q x+ 2 ( 2)) q x+ 1 ( 1), where q x () is he probabiliy ha an individual aged x a ime will die wihin one year. Accordingly, we consider modeling of q x (). 4.1 The Lee-Carer model Following Lee and Carer (1992), he force of moraliy μ x () is modeled as: μ x () = exp{α x + β x κ }, where α x, β x, and κ are parameers o be esimaed. The ime-specific parameer κ represens a sysemaic effec influencing all ages; he age-specific parameer β x represens he 7

9 percenage change of moraliy in response o an infiniesimal change in κ for each age x; he oher age-specific parameer α x represens an inrinsic effec of age x. We assume ha: μ x+s ( + u) = μ x (), (14) for inegers x and, and for all s, u < 1. Then, under he Lee-Carer framework, he deah probabiliy q x () is given as: q x () = 1 exp{ exp{α x + β x κ }}. 4.2 Bayesian modeling of moraliy To obain he predicive disribuion of I, we consider a saisical modeling of moraliy raes. Le m x be he crude cenral deah rae: m x = Deah x / Exposure x, (15) where Deah x is he number of deahs recorded a age x in year, and Exposure x is he maching exposure o he risk of deah. As a saisical model, he sandard Lee-Carer mehodology assumes ha y x = ln m x, he naural logarihm of he observed moraliy rae, follows a regression model: y x = α x + β x κ + ε x for x = x min, x min + 1,, x max and for = min, min + 1,, max, (16) where he error erms ε x s are disribued independenly and idenically as he normal disribuion wih zero mean and a consan variance. Since he regression equaion (16) is bilinear in β x and κ, his model is no idenified. To make i idenifiable, he parameers {β x } and {κ } are resriced such ha: xmax x x min max 1 and. (17) x For each, we define y o be a vecor: y = (y xmin,, y xmax ), consising of he log moraliy raes for all ages observed in year. Here denoes he ranspose operaion. The saisical framework of he sandard Lee-Carer mehodology can be se up as a sae-space model: observaion equaion: y = α + β κ + ɛ, ɛ ~ N M ( M, σ 2 ɛ I M ), sae equaion: κ = λ + κ 1 + ω, ω ~ N(, σ 2 ω ), (18) where α = (α xmin,, α xmax ), β = (β xmin,, β xmax ), M = x max x min + 1, I M is he ideniy marix of dimension M M, λ is he drif erm of he random walk, and ɛ and ω are independen. From (18), he likelihood funcion is given as: min 8

10 L max xmax y,,,,, f y x x, x,, min xx min 1 exp JM max xmax y min xx x min 2 2 x x 2, (19) where J = max min + 1 and κ = (κ min,, κ max ). The prior disribuions for he parameers α and β in he observaion equaion are chosen as: α ~ N M ( M, σ 2 α I M ), β ~ N M ((1/M)1 M, σ 2 β I M ), where M and 1 M are M-dimensional vecors of all zeros and all ones respecively. Here, he underlined characers sand for hyperparameers, which are parameers of he prior disribuions. The prior for σ 2 ɛ is se up as: σ 2 ɛ ~ IG(a ɛ, b ɛ ), where IG(a, b) denoes he inverse Gamma disribuion wih shape parameer a and scale parameer b. The priors for λ and σ 2 ω in he sae equaion are aken as: λ ~ N(λ, σ 2 λ ), σ 2 ω ~ IG(a ω, b ω ). 4.3 Risk-neural predicive disribuion Suppose ha we generae N pahs in he MCMC sampling, each of which is disribued as he process {m x, = 1, 2,, T}, and denoe hem by: {(m () x1, m () x2,, m () xt ), = 1, 2,, N}. Now consider a simple life annuiy in which an individual receives a consan paymen, which is se o one moneary uni for simpliciy, a he end of each year over T years as long as he or she is alive. The presen value of his annuiy for an individual aged x a ime is defined o be: a x q... 1 q T d 1 x x (2) Le a marke x be he marke value of he life annuiy a ime. For a probabiliy π over {m () x } o be risk-neural, i mus saisfy: wih N marke ax ax, (21) 1 a calculaed by {m x () }. Denoe by π he empirical disribuion of he N pahs from he x MCMC sampling, which ypically pus an equal mass of 1/N on each pah. The maximum 9

11 enropy principle sipulaes ha he risk-neural disribuion π should minimize he Kullback-Leibler informaion divergence: subec o (21) and: N 1 N 1 ln, (22) 1 and for = 1, 2,, N. (23) I is well known ha he soluion o his consrained minimizaion problem is given as: N 1 exp a x ˆ for = 1, 2,, N, (24) exp a x where γ is a consan obained by solving he equaion: a marke x N 1 N a 1 x exp a exp a x x. 4.4 Evaluaion of E [I ] for Japanese daa The daa se used are he deah raes of Japan for ages 65 o 99 over he period from 1973 o 28 for boh males and females, aken from he Human Moraliy Daabase (HMD 213). In he MCMC simulaion, we discarded he firs 5, ieraions o eliminae iniial effecs and used he 1, ieraions aferwards for sampling. We calculaed E [I ] under he seup below: age of cohor: x = 65, 7, 75; discoun facor: d() = (1 + r), r =.5. As of 1 April 213, he 1-year Japan governmen bond yield was merely.57% p.a. The low ineres rae environmen is expeced o coninue under Japan s recen decision o use a very loose moneary policy o lif he economy ou of deflaion. Hence we ook.5% p.a. as he risk-free ineres rae r in our illusraion. In Japan, valuaion assumpions are generally specified by he regulaor. To deermine he marke price, a marke x, we used he deah raes for annuiy producs in he life insurance sandard able 27, which was consruced by he Insiue of Acuaries of Japan. Mos Japanese life insurance companies price heir insurance producs using his sandard able. We also used he specified valuaion discoun rae of 1.5% p.a. in compuing he marke price. Figure 1 depics he compued values of E [I ] as well as E[I ]. 1

12 Figure 1: Compued E [I ] (solid lines) and E[I ] (dashed lines) for Japanese males (lef) and females (righ) 5 Bayesian pricing of he non-recourse provision In his secion, we evaluae E [max(l HP, )] for Japanese daa. We firs find a suiable ime series model and deduce is predicive disribuion. We hen ransform i ino he riskneural version o calculae E [max(l HP, )]. 5.1 Modeling house price index We realize here are srong auocorrelaions and changing volailiy in he log reurns of our 11

13 house price index daa (see Secion 5.3). Accordingly, as in Chen e al. (21) and Li e al. (21), we consider he ARIMA(p, d, q)-garch(u, v) models. Le r be he log reurn a ime and he general model is specified as: d r p q d i r i i1 1, 2 where, ~ u i1 2 i v 2, (25) i 1. Following Rosenberg and Young (1999), we assume ha he prior disribuion for he unknown parameers is mulivariae normal. (Noe ha p, d, and q here are orders of he ARIMA model, and are no relaed o he moraliy daa in Secion 4 or he discoun facor.) Risk-neural predicive disribuion Once we have he predicive disribuion of r, we can change i o he risk-neural form exacly he same way as in Secion 4.3. Suppose ha we generae N pahs in he MCMC sampling, each of which is disribued as he process {r, = 1, 2,, T}. Denoe hem by: {(r () 1, r () 2,, r () T ), = 1, 2,, N}, and se: HP () () = HP 1 exp(r () ) for = 1, 2,, T. For a probabiliy π over {HP () } o be risk-neural, i mus saisfy: d N HP HP. (26) 1 According o he minimum cross-enropy principle discussed earlier, he risk-neural disribuion π should minimize (22), subec o (23) and (26). The soluion is given as: N 1 d exp d HP ˆ for = 1, 2,, N, exp HP where γ is a consan obained by solving he equaion: HP d N 1 N HP 1 exp d exp d HP HP. 12

14 5.3 House prices in Japan We used he Japanese version of he S&P / Case-Shiller Home Price Indices as a represenaive for he value of he morgaged propery. The daa consiss of monhly prices from January 1995 o December 28. Figure 2 shows he values of he S&P / Case-Shiller Home Price Indices and he log reurns for Japan as well as he US. Figure 2: S&P / Case-Shiller Home Price Indices (lef) and log reurns (righ) for Japan (solid lines) and US (dashed lines) Figure 3 shows ha here are significan auocorrelaions in he log reurns, which sugges ha ARIMA models are poenial candidaes for modeling he house price reurns. The Pormaneau saisic of he log reurns gives a p-value of.. As noed in Rosenberg and Young (1999), AR models are more convenien o work wih in a Bayesian seing han MA models, and if a sufficienly large number of AR erms are included, he effecs are similar o a given ARIMA model. As such, afer inspecing he sample parial auocorrelaion funcions, we fied an ARIMA(6,, ) model (i.e. AR(6) model) o he log reurns. The lef panel of Figure 4 reveals ha he fied model capures he auocorrelaions reasonably well. The sample auocorrelaion funcions of he residuals are largely wihin he 95% confidence inervals (doed lines), and he Pormaneau saisic gives a p-value of The righ panel of Figure 4, however, indicaes ha here seems o be some exen of condiional heeroskedasiciy in he daa. The Pormaneau saisic of he squared residuals shows a p-value of.787, which is us a lile higher han he 5% significance level. This feaure may poin o he use of a GARCH-ype model. Afer examining he sample parial auocorrelaion funcions of he squared residuals and esing a number of models, we seleced an ARIMA(6,, )-GARCH(2, ) model for he log reurns. This model is relaively parsimonious, and as refleced in Figure 5, i provides a reasonable allowance for boh he 13

15 auocorrelaions and he heeroskedasiciy propery generally. The Pormaneau saisic of he squared sandardized innovaions shows a p-value of.579, which reassures ha he model fi is saisfacory. Figure 3: Sample auocorrelaion funcions of log reurns Figure 4: Sample auocorrelaion funcions of residuals (lef) and squared residuals (righ) of fied ARIMA(6,, ) model Figure 5: Sample auocorrelaion funcions of sandardized innovaions (lef) and squared sandardized innovaions (righ) of fied ARIMA(6,, )- GARCH(2, ) model 14

16 We ran 1, seps of MCMC simulaion. We discarded he firs 5, seps o remove he effecs of he saring values and used he nex 5, seps for sampling. Figures 6 and 7 display he auocorrelaions of he successive simulaed samples and he sampling hisory respecively for he model parameers. (In he figures, p[1] o p[9] sand for ϕ o ϕ 6, γ, and γ 2 respecively.) I can clearly be seen ha he auocorrelaions are minimal and he level of convergence is saisfacory (i.e. here are no paricular rends in he sampling hisory), which means ha he simulaion process has been carried ou appropriaely. Figure 6: Auocorrelaion plos of MCMC samples p[1] 2 4 lag p[2] 2 4 lag p[3] 2 4 lag p[4] 2 4 lag p[5] 2 4 lag p[6] 2 4 lag p[7] 2 4 lag p[8] 2 4 lag p[9] 2 4 lag 5.4 Evaluaion of E [max(l HP, )] We conver he predicive disribuions of fuure house prices ino he risk-neural form as explained in Secion 5.2 and hen evaluae E [max(l HP, )] in (13). We se up he inpus as follows: HP = 4 (million yen), L = 2, 24, 28, 32, 36, 4 (million yen), L = L (1 + r L ), d() = (1 + r), r =.5. The resuls for r L =.2,.3,.4,.5,.6 (wih L = 2) are shown in Figure 8. 15

17 Figure 7A: Hisory plos of MCMC samples p[1] ieraion p[2] ieraion p[3] ieraion p[4] ieraion p[5] ieraion 16

18 Figure 7B: Hisory plos of MCMC samples p[6] ieraion p[7] e ieraion p[8] 4.E-5 3.5E-5 3.E-5 2.5E-5 2.E-5 1.5E ieraion p[9] ieraion 17

19 Figure 8: Compued E [max(l HP, )] (solid lines) and E[max(L HP, )] (dashed lines) 6 Pricing reverse morgages in Japan We finally proceed o pricing reverse morgages in Japan by combining he resuls in Secions 4 and 5. We add τ = 1 o he inpus se up in Secion 3. Table 1 gives he values of (12) for L = 2, 24, 28, 32, 36, 4, r L =.2,.3,.4,.5,.6, and x = 65, 7, 75. We noice he following: (a) The expeced presen value of he reverse morgage for he male cohor is larger han ha for he corresponding female cohor. 18

20 (b) For boh sexes, he expeced presen value of he reverse morgage ends o decrease wih age. (c) For boh sexes, he expeced presen value of he reverse morgage increases wih he loan ineres rae. (d) A paricular aenion should be paid on wheher each expeced presen value is more han L, as i indicaes ha he reverse morgage is provided in a susainable manner. Those cases wih he value lower han L (i.e. no susainable) are highlighed in Table 1. I can be seen ha a higher loan ineres rae allows a higher loan o value (LTV) raio. As of November 24, UK loan ineres raes were around 7% p.a. (Equiy Release Repor 25), 1-year bond yield was abou 4.5% p.a., and so he premium embedded was 2.5% p.a. Applying his figure o our case leads o a loan ineres rae of 3% p.a., which implies a maximum LTV raio of abou 8% for males and 6% for females, based on Table 1. Moreover, as he Japan marke is relaively new and is no as sauraed as he UK marke, he Japanese insurers can poenially charge higher ineres raes iniially, which hen allows a higher LTV raio, even up o 8%-9% as in Table 1. These observaions appear o suppor a successful inroducion of reverse morgages ino Japan. 7 Concluding remarks In his paper, we have proposed a Bayesian mulivariae framework o price reverse morgages and applied he proposed mehod o Japanese daa in order o examine he feasibiliy for an inroducion of reverse morgages ino Japan. We have obained desirable empirical resuls, indicaing some promising fuure for reverse morgages in Japan. Neverheless, due o he scope of he paper, here are cerain areas ha are no covered, including sochasic modeling of ineres raes, oin lives, having more han one decremen, model error, and oher produc designs. Furher research is required o address hese issues. 8 Acknowledgemens We are graeful o Mr. Yoshimisu Takamasu and Mr. Takafumi Fushimi for assisance wih he compuaions. We grealy acknowledge financial suppor from SCOR on he Asia-Pacific longeviy proec, from which his paper is exraced. The longeviy proec has been implemened under he Insurance Risk and Finance Research Cenre a Nanyang Business School. 19

21 Table 1: The expeced presen values of he reverse morgages L = 2 (LTV = 5%) Males Females x \ r L L = 24 (LTV = 6%) Males Females x \ r L L = 28 (LTV = 7%) Males Females x \ r L L = 32 (LTV = 8%) Males Females x \ r L L = 36 (LTV = 9%) Males Females x \ r L L = 4 (LTV = 1%) Males Females x \ r L

22 References Cairns, A. J. G., D. Blake, and K. Dowd. 26. A Two-Facor Model for Sochasic Moraliy wih Parameer Uncerainy: Theory and Calibraion. Journal of Risk and Insurance 73: Chen, H., S. H. Cox, and S. S. Wang. 21. Is he Home Equiy Conversion Morgage in he Unied Saes Susainable? Evidence from Pricing Morgage Insurance Premiums and Non-Recourse Provisions Using he Condiional Esscher Transform. Insurance: Mahemaics and Economics 46: Creighon, A., H. H. Jin, J. Piggo, and E. A. Valdez. 25. Longeviy Insurance: A Missing Marke. Singapore Economic Review 5: Equiy Release Working Pary. 25. Equiy Release Repor 25. The Acuarial Profession. Fuisawa, Y., and J. S. H. Li The Impac of he Auomaic Balancing Mechanism for he Public Pension in Japan on he Exreme Elderly. Norh American Acuarial Journal 16(2): Gerber, H. U., and E. S. W. Shiu Opion Pricing by Esscher Transforms. Transacions of he Sociey of Acuaries 46: Hosy, G. M., S. J. Groves, C. A. Murray, and M. Shah. 28. Pricing and Risk Capial in he Equiy Release Marke. Briish Acuarial Journal 14(1): Human Moraliy Daabase (HMD) Universiy of California, Berkeley (USA) and Max Planck Insiue for Demographic Research (Germany). Ji, M., M. Hardy, and J. S. H. Li A Semi-Markov Muliple Sae Model for Reverse Morgage Terminaions. Annals of Acuarial Science 6(2): Kapur, J. N Maximum-Enropy Models in Science and Engineering. Wiley Easern Limied. Revised Ediion. Kogure, A., and Y. Kurachi. 21. A Bayesian Approach o Pricing Longeviy Risk Based on Risk-Neural Predicive Disribuions. Insurance: Mahemaics and Economics 46: Lee, R. D., and L. R. Carer Modeling and Forecasing U.S. Moraliy. Journal of he American Saisical Associaion 87: Lee, Y. T., C. W. Wang, and H. C. Huang On he Valuaion of Reverse Morgages wih Regular Tenure Paymens. Insurance: Mahemaics and Economics 51: Li, J An Applicaion of MCMC Simulaion in Moraliy Proecion for Populaions wih Limied Daa. Demographic Research (forhcoming). 21

23 Li, J. S. H., M. R. Hardy, and K. S. Tan. 21. On Pricing and Hedging he No-negaiveequiy Guaranee in Equiy Release Mechanisms. Journal of Risk and Insurance 77(2): Li, J. S. H. 21. Pricing Longeviy Risk wih he Parameric Boosrap: A Maximum Enropy Approach. Insurance: Mahemaics and Economics 47: OECD. 29. Sociey a a Glance Asia/Pacific Ediion. OECD Korea Policy Cenre. Rosenberg, M. A., and V. R. Young A Bayesian Approach o Undersanding Time Series Daa. Norh American Acuarial Journal 3(2): Suzer, M A Simple Nonparameric Approach o Derivaive Securiy Valuaion. Journal of Finance 51: Wang, L., E. A. Valdez, and J. Piggo. 28. Securiizaion of Longeviy Risk in Reverse Morgages. Norh American Acuarial Journal 12: Wang, S. 2. A Class of Disorion Operaors for Pricing Financial and Insurance Risks. Journal of Risk and Insurance 67:

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