The pricing analysis of reverse mortgage with redemption option

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1 Available online Jounal of Chemical and Phamaceuical Reseach, 04, 6(6):83-89 Reseach Aicle ISSN : CODEN(USA) : JCPRC5 The picing analysis of evese mogage wih edempion opion Yanxia Zhu*, and Li Gong School of Economics and Managemen of China Univesiy of Geosciences, Wuhan, China College of Public Adminisaion of Zhejiang Univesiy, Hangzhou, China ABSTRACT The pape builds he acuaial picing models of evese mogages. The CIR model and jump diffusion model ae used o descibe he sochasic pocesses of inees ae and he gowh ae of housing value, especively, and hough Mone Calo simulaion, he annuiy paymens of evese mogages wih edempion opion and no edempion opion ae obained. An impoan conclusion is ha he embedmen of he edempion opion will educe he level of paymen of evese mogage o a lage exen; howeve, he paymen sill can supplemen he social secuiy benefi. Keywods: evese mogage; edempion opion; CIR model; jump diffusion model INTRODUCTION In he conex of aged populaion, i is a ealisic opion fo he govenmen o implemen he housing evese mogage (RM) so as o elieve he fiscal pessue and supplemen he dop o insufficiency of social secuiy income. On one hand, wih he coninuous incease in he dependency aio fo he aged populaion, he dop in he social secuiy income becomes an ineviable end o mee he goal of maco acuaial balance, so in ode o peven he povey of he aged populaion and guaanee he sufficiency of he eiemen income, he policy make shall exploe ohe eiemen income souces fo make-up. On he ohe hand, he esidens in China own elaively high housing self-owneship ae. Accoding o he invesigaion of Chinese Academy of Social Science, he housing self-owneship ae of he families in China in 03 eached up o 90%, 30% highe han he wold aveage level. Unde he pecondiion of no affecing he igh of dwelling of he eiees, RM can no only help he eiees ansfom he housing ino income flows o suppo hei eiemen lives and solve he liquidiy pedicamen which has been descibed as wih house bu wihou cash, bu also povide he longeviy isk guaanee o he eiees wih he paymens as a life annuiy. Theefoe, i will be of gea significance o solve he social pension poblems wih RM in China fom he acual need of esablishing he muli-pilla pension sysem. Howeve, RM is being confoned wih some pacical challenges in China. Diffeen fom he adiional home mogage, he paymen of RM depends on he housing value a mauiy (usually he house owne dies). Theefoe, no mae he lende (such as he insue) o he boowe (housing owne) is confoned wih he isks ha housing value a em does no confom o he accumulaive loan amoun. Specifically speaking, hese isks include housing pice flucuaion isk, inees ae isk and longeviy isk. In addiion, he beques moive in China is also a key faco confining RM []. So he ial RM in some egions in China is embedded wih he clauses egading edempion opions, namely when he conac expies, if he housing value exceeds he accumulaive loan amoun, he housing owne o is childen can selec edeem he house, in his way, i can help he boowe eliminae he woy abou he ise in housing pice and he housing no funcioning as he beques. So i is impoan o compae he annuiized paymen of RA wih edempion opion o wihou edempion opion, and fuhe analyze whehe he RA can supplemen he social secuiy benefis o no. 83

2 Yanxia Zhu and Li Gong J. Chem. Pham. Res., 04, 6(6):83-89 Based on his, his pape calculaes he picing of RM wih edeemable opion and non-edempion opion unde he condiion of acuaial neualiy. The emainde of his pape is oganized as follows. The nex secion is he lieaue eview. In secion 3, we descibe he andom pocess of changes in which he inees ae and he housing pice gowh ae wih CIR Model and he jump-diffusion model coespondingly, and calculae he paymen of RM wih edempion opion and non-edempion opion unde he acuaial condiion and make he sensiiviy analysis. The final pa is he conclusion.. Lieaue eview In ems of RM picing, wha one concens mos is he change in housing pice. I is held by so many sudies adoping he coninuous ime model ha he housing pice obeys he geomeic Bownian moion [-3], howeve, i is held by he laes sudy ha he housing picing akes on elaively gea jump, which is sudied Chen e al. [4]. Based on his, Lee e al. descibed he dynamic changes in housing pice wih jump-diffusion model [5]. As o he andom behavio of he inees ae, Vasicek model and CIR model ae mos widely adoped; howeve, Vasicek model may mee wih he negaive inees ae, so CIR model is adoped in mos sudies [5-6]. Since he exising RM picing mainly adops he peiodic life ables, esuling in lile descipion of he longeviy isk, Lee e al. adops he model esablished by Lee and Cae [7] o descibe he isk [5]. Geneally, he sudies above menioned mainly make RM pice fom he pospecive of acuaial neualiy and focus on he influences of he andom pocess of hee isks above menioned on RM picing, which lay a heoeical foundaion fo he sudy on RM picing. Howeve, he edempion opion of RM is seldom aken ino accoun and he individuals wills of paicipaion in RM is no analyzed. Some sudies in China jus simply descibe he inees ae and he housing pice gowh ae, and he jump-diffusion pocess of he changes in housing pice has no been involved in RM picing. Alhough a few of eseaches have consideed he edempion opion [8], howeve, he acuaial picing of RA wih edempion opion consideing sochasic pocesses of he changes of inees ae and house pice is ae. Based on he foegoing sudies, his pape descibes he sochasic pocesses of he inees ae and he housing pice gowh ae in China wih CIR Model and Jump-Diffusion Model coespondingly and hen esimaes he paamees of he above-menioned models wih Genealized Mehod of Momens (GMM) and Maximum Likelihood Mehod (ML) especively. Based on i, his pape simulaes he annuiy paymen of RM wih edempion opion and non-edempion opion unde he acuaial condiion wih Mone-Calo mehod. 3. RM picing unde he condiion of acuaial neualiy 3. Theoeical model (Ⅰ) Acuaial model fo RM In RM conac, he old housing owne, wih he house popey as he mogage, obains he annuiy paymen flow in lump sum o he fixed annuiy paymen flow, and i is no necessay fo he boowe o epay he annuiy in he peiod of esidence ill deah, house selling o pemanen moving ou of he house, a his momen, he conac expies, he boowe epays he pincipal, inees and ohe expenses of he loan wih he income fom house selling. This pape mainly evaluaes he evese mogage (RM) loan epaid wih he life annuiy, which can poec he longeviy isk of he eiees and is he beneficial supplemenaion of he paymen fo he social secuiy. In RM, he insuance agens povide he life fixed annuiy paymen flow o he old housing owne unil he boowe dies. Unde he acuaial condiions, accoding o he adiional mehod, RM picing can be indicaed ha he pesen value of he expeced income of he boowe is equal o he one of he expeced expendiue; in case of no edempion opion, i can be indicaed as: ω x0 ω x0 x0 x0 = = H ( ) q e = A( ) p Whee H ( ) is he housing value in ime, =,..., ω x0, ω is he max. age of suvival, x 0 is he age of he eiee paicipaing in RM, qx = px - + p x is he pobabiliy of deah of he eiee aged x0 in ime and ω x0 = q x0 = 0 0 0, is he loan inees ae of coninuous compound inees, A( ) is he paymen of fixed annuiy in each ime. Accoding o : ω x0 ω x0 + + A( ) = H ( ) qx px () = = Howeve, he picing mehod above-menioned ignoes one impoan clause in RM conac, namely no-ecouse clause and some acuaial analysis does no ake his poblem ino accoun [6] [8]. I implies ha when RM conac 84

3 Yanxia Zhu and Li Gong J. Chem. Pham. Res., 04, 6(6):83-89 expies, even if he housing value is lowe han he accumulaive amoun of loan, as o he insufficiency, he lende has no igh of ecouse o he asses of he boowe in ohe fom. Theefoe, in RM conac, he lende is confoned wih he isks ha he housing value is no so sufficien o epay he accumulaive amoun of loan in expiaion of he conac, as is an impoan eason why he financial insiuions feel an objecion o cay ou RM [5]. In ode o encouage he financial insiuions especially he insues o povide RM poducs, he govenmen ofen allows hem o chage an addiional pemium o compensae he poenial condiional losses. Theefoe, consideing he no-ecouse clause in RM conac, he pesen value of he insuance pemium colleced by he lende due o he igh of non-ecouse shall be equal o he pesen value of he expeced losses in he fuue. Given he lende allocaes he pemium o each insallmen equally, he exa pemium occued in each insallmen due o no-ecouse clause shall be: ω x0 ω x0 + P( ) = [ BAL( ) H ( )] qx px (3) = = BAL( ) is he accumulaive amoun of loan in insallmen, indicaed as BAL( ) = A( i) i= i, when he boowe dies in insallmen and BAL( ) > H ( ), BAL( ) H ( ) is he loss of he lende in insallmen. Theefoe, in case of no edempion opion, accoding o Fomula () and (3), he fixed annuiy paid o he boowe in each insallmen is A( ) P( ). In case of edempion opion, given he cos of edempion opion is also allocaed o each insallmen evenly, o he lende, he cos o be colleced fo he edempion opion in each insallmen shall be: ω x0 ω x0 + π ( ) = [ H ( ) BAL( )] qx px (4) = = When he boowe dies in insallmen and H ( ) > BAL( ), he boowe s childen can selec o edeem he housing, and H ( ) BAL( ) is he beques he boowe leaves o is childen. As a esul, in case of edempion opion, he fixed annuiy paid o he boowe in each insallmen shall be A( ) P( ) π ( ). () Inees ae model CIR Model is adoped hee o descibe he flucuaion of inees ae in coninuous and dispese fom, indicaed as: d( ) = k ( µ ( )) d + σ ( ) dw ( ) (5) ( + ) ( ) = k ( µ ( )) + σ ( ) W ( ) (6) Whee, W ( ) is a sandad Wiene pocess, dw can be deemed as a nomal vaiable wih he mean of 0 and he vaiance of d, k,µ and σ ae he consans, among which µ is he long-em mean value of he inees ae, σ is he flucuaion aio of he inees ae and k is he egulaing speed. Theefoe, E( d) = k ( µ ( )) d, va( d) = σ ( ). Fuhemoe, Fomula (6) is wien as: ( ) ( ) = α + β ( ) + ε ( ) (7) Based on Fomula (7), k µ andσ need o be esimaed; and E( ε ( )) = 0,, ε = σ. va( ( )) ( ) (3) Housing pice model Jump-diffusion model is adoped o descibe he flucuaion of he housing pice in China, which can be indicaed as: dh = µ d + σ dw + JdN (8) H In jump-diffusion model, he insananeous eaning ae of he asses is composed of a simple diffusion pocess and andom jump iem, among which µ, σ and dw ae same o he ones in CIR Model, N is he Poisson pocess wih he inensiy λ, namely P( dn = ) = λd, P( dn = 0) = λd, J is he ampliude of andom jump, indicaed wih he pecenage of aleaion in pice. Fomula (8) is subjec o logaihm, hen: 85

4 Yanxia Zhu and Li Gong J. Chem. Pham. Res., 04, 6(6):83-89 Whee µ = µ σ aleaion in pice of he asses in jump plus, Fomula (9) is disceized, hen: d(ln H ) = µ d + σ dz + JdN (9), dz ~ N (0,), J is he jump ampliude, indicaed wih he logaihm of he pecenage of he J ~ N( θ, δ ) and N ~ P( λ ). Whee is he ime ineval indicaed wih he yea, η ~ N(0,), σ = σ, θ = θ and δ = δ. N ln( H ) ln( H ) = µ + σ η + J µ + σ η + J (0) N i i i= i= Ji ~ N(, ) N θ δ, ~ P( λ ), µ = µ, Then, in ime, he change of housing pice in coninuous compounding wih he ime ineval of is: ( ) = ln( H ) ln( H ) = µ + σ η + J () H i i= 3. Paamee esimaion and Mone Calo simulaion Paamee esimaion of inees ae model The daa abou he inees ae ae fom he monhly weighed mean ine-bank offeed ae fom Jan. 998 o Dec. 03 in CSMAR, oally 9 daa. The inees ae is ansfome ino coninuous compound inees. The above-menioned daa is egessed wih GMM; he insumenal vaiable is se as/ ( ), wih he egession esuls shown in Table. Table Esimaed esuls of Fomula (7) Paamee Esimaed value Sandad deviaion Appox P > α β σ.5560e I can be seen ha all of he paamees in Fomula (7) pass he significance es a he significance level of %; fuhemoe, he esimaed values of hee paamees in Fomula (5) can be obained, namely k = , µ = 0.04 and σ = especively. (Ⅱ) Paamee esimaion of housing pice model The daa abou he housing pice ae fom he monhly daa of he housing pice indexes in China fom Jan. 994 o Dec. 0 issued by Naional Bueau of Saisics, oally 6 daa. Based on Fomula H ()= ln( H / H -), he eaning ae of coninuous compound inees of he housing pice can be obained in each monh. In jump-diffusion model, 5 paamees shall be esimaed. Compaed wih he paamee esimaion of ohe andom pocesses, he paamee esimaion of jump-diffusion model is moe complicaed. In he esimaion mehods, alhough he maximum likelihood mehod has geae difficuly in calculaion, i is moe effecive han he GMM because he lae may mee wih he poblem of non-convegence [9]. Some sudies adop non-paamee esimaion mehod, such as Johannes [0] and Mancini and Renò []. This pape ies o adop Conjugae Gadien in he maximum likelihood mehod fo esimaion jus because i is one of he mos effecive mehods o sele he lage-sized non-linea opimizaion poblems. The paamee esimaion esuls unde he condiion of convegence ae finally obained hough 0 ieaions (See Table ). Table Paamee esimaion esuls of Fomula () Paamee Esimaed value value Appox P > λ µ H µ J σ H σ J N 86

5 Yanxia Zhu and Li Gong J. Chem. Pham. Res., 04, 6(6):83-89 I is shown by he esuls ha ohe paamees pass he significance es a he significance level of % excep he mean µ of he jump ampliude. J (Ⅲ) Mone-Calo simulaion fo RA picing Given he value of he housing of he boowe a he beginning of peiod is RMB,000,000 Yuan, and he Expeience Life Table of China Life ( ) is adaped o calculae moaliy aes. Since he paamee esimaion of CIR model of inees ae adops he isk-fee inees ae of inebank lending, wih which an inees ae pemium needs o be added o obain he loan inees ae, usually highe han he isk-fee inees ae. Hee i is fisly supposed he inees ae pemium is 300BP. Accoding o he housing pice flucuaion model, inees ae flucuaion model and RM picing model unde he acuaial condiion fo he above-menioned esimaion, we calculae A( ), P( ) and π ( ) wih Mone-Calo mehod. Afe he simulaion fo 000 imes, he fixed paymen o he boowes a diffeen ages paicipaing in RM in each yea is shown in Table 3. Table 3 Picing fo RM of he boowes a diffeen ages (RMB Yuan, %) Sex Age A( ) P( ) π ( ) Male Female I can be seen fom Table 3: A( ) P( ) () A( ) P( ) π ( ) ()/ No mae he men o women, he annuiized paymen of RM wih edempion opion is fa lowe han he one of RM wihou edempion opion. Fo example, as o he male housing owne aged 60, he annual paymen fo RM wih edempion opion is RMB 6,484 Yuan, only equivalen o 45.6% of he paymen fo RM wihou edempion opion. Howeve, consideing he benefi he exising social secuiy including he social pooling and he individual accouns povide, he sum of hem can ealize ove 70% of he income eplacemen ae, which can guaanee he eiees mainain he living sandad befoe eiemen. I can be seen ha even hough he edempion opion esuls in he educion in paymen of RM o a lage exen, i can also povide he sufficien supplemenaion o he social secuiy benefi. Wih he incease of age when he boowes paicipaing in RM, he aio of he paymen of RM wih edempion opion o he paymen of RM wihou edempion opion incease as well, when he boowe is a he age of 65, 70 and 75, his aio ises by 3%, 5.7% and 7.7% especively. The paymen o he female boowe paicipaing in RM is commonly lowe han he one o he male boowe; as o he female boowe aged 55, he paymen of RM wih edempion opion is less han RMB 8,000 Yuan evey yea and he aio of he paymen of RM wih edempion opion o he one wihou edempion opion is only 39.3%. Since he paymen will no be adjused wih he inflaion ae o he gowh ae of he social aveage wage, i is pediced ha he insufficiency of eiemen income may ake place in he lae sage afe eiemen. Jus like he male boowe, he aio of he paymen of RM wih edempion opion o he one wihou edempion opion inceases wih he age in same manne; howeve, as o he female boowe a he same age, his aio is commonly less han he one of he male boowe. Fo insance, as o he female boowe a he age of 60, 65, 70 and 75, his aio is especively 3.3%,.9%,.% and.3% lowe han he one of he male boowe. Howeve, wih he incease in age, he gap beween he aio of he male boowe and he female boowe is being educed because he lowe moaliy ae of he female boowe han he one of he male boowe. Theefoe, he younge he boowe paicipaing in RM is, he lowe he fixed paymen. 3.3 Sensiiviy analysis Sensiiviy analysis of aleaion in he mean of inees ae The inees ae isk is an impoan issue in he implemenaion of RM and even is mino change will have gea influences on RM picing. Hee he male and female nomal boowes aged 60 ae egaded as he compaison efeence o analyze he influences of he change in inees ae on RM wih edempion opion and he one wihou edempion opion hough expanding he pemium beween he loan inees ae and he inees ae of boowing 87

6 Yanxia Zhu and Li Gong J. Chem. Pham. Res., 04, 6(6):83-89 money among banks. I can be seen fom Table 4 ha he incease in loan inees ae will significanly educe he paymen fo RM wih edempion opion and he one wihou edempion opion; in addiion, wih he expansion of he inees ae pemium, he ampliude of dop in paymen fo RM wih edempion opion exceeds he paymen fo RM wihou edempion opion. When he diffeence beween he loan inees ae and he inebank offeed ae exceeds 6%, he paymen of RM wih edempion opion o he male boowe is only RMB 6,000 Yuan while he one o he female boowe is less han RMB 4,000 Yuan, especively educing by 39% and 38%. Due o no inflaion adjusmen, i can be foecas ha i is difficul fo he paymen of RM wih edempion opion o povide he sufficien eiemen income supplemenaion o he eiees wih he incease in age of he paicipans in case of high loan inees ae. Table 4 RM picing in expanding inees ae pemium (RMB Yuan, %) Sex Magin A( ) P( ) π ( ) Male Female A( ) P( ) () A( ) P( ) π ( ) ()/ 00BP BP BP BP BP BP BP BP BP BP () Sensiiviy analysis of change in he mean of housing pice gowh ae The change in housing pice gowh ae also has impoan influences on he paymen fo RM. Theoeically, he highe he gowh of housing picing, he highe he paymen of RM wih edempion opion and he one wihou edempion opion, which is jus veified by he changes in paymens in Table 5. Wihou change in flucuaion and jump, Table 5 shows he influences on he paymen of RM when he mean of he housing pice gowh ae flucuaes abou ~%. When he mean of he housing picing gowh ae inceases by %, he paymen o he male and female boowes fo RM inceases o RMB 37,7 Yuan and RMB 3,554 Yuan, wih he gowh ae of 0% and 4% especively compaed o he efeence, he aio of he paymen of RM wih edempion opion and he one wihou edempion opion especively dops by 4.3% and 5%; while he mean of housing pice gowh ae dops by %, he paymen of RM wih edempion opion o he male and female boowes dops by 37-38% while he aio of paymen of RM wih edempion opion and he one wihou edempion opion does no change gealy, among 0.5~%. Table 5 RM picing in change in he mean of housing pice gowh ae (RMB Yuan, %) Sex change of µ H A( ) P( ) π ( ) Male Female A( ) P( ) () A( ) P( ) π ( ) ()/ CONCLUSION The pape builds he acuaial picing models of evese mogages wih and wihou edempion opion. The CIR model and jump diffusion model ae used o descibe he sochasic pocesses of inees ae and he gowh ae of housing value, especively, and hough Mone Calo simulaion, he annuiy paymens of evese mogages wih edempion opion and no edempion opion ae obained. The embedmen of he edempion opion will educe he level of paymen of RM o a lage exen; howeve, he ampliude of educion will fall down by degees wih he incease in age of he eiee. In addiion, he paymen of RM o he female boowe is lowe han he one o he male boowe; wih he ise of loan inees ae and he dop in housing pice gowh ae, paymen by RM dops down, he gap in paymen of RM wih edempion opion and he one wihou edempion opion expands as well. 88

7 Yanxia Zhu and Li Gong J. Chem. Pham. Res., 04, 6(6):83-89 Acknowledgemens The pape is funded by China Posdocoal Science Fund pojec (03M54975) and he Fundamenal Reseach Funds fo he Cenal Univesiies(CUGW005), he views expessed ae he auhos alone. REFERENCES []Jian Chen, Shan-an Huang, Economic Reseach Jounal, vol. 9, pp , 03. []EJ Szymanoski, Jounal of Ameican Real Esae and Uban Economics Associaion, vol., pp , 994. [3]A Badhan, R Kaapandza, B Uosevic, Jounal of Real Esae Finance and Economics, vol. 3, pp. 9-0, 006. [4]MC Chen, CC. Chang, SK Lin, SD Shyu, Jounal of Risk and Insuance, VOL.77, PP , 00. [5]Yung-Tusung Lee, CW Wang, HC Huang, Insuance: Mahemaics and Economics, vol.5, pp , 0. [6]Jing Lu, P Gao, and JC Dong, Managemen Review, vol., pp. 3-8, 00. [7]RD Lee, LR Cae, Jounal of he Ameican Saisical Associaion, vol. 87, pp , 99. [8]Bing-zhen Chen, YL Deng, P Qin, 03 China Inenaional Confeence on Insuance and Risk Managemen, Kunming, China, 03. [9]Nik Tuzov, GMM paamee esimaion fo he Double Exponenial Jump-Diffusion Pocess, Pudue Univesiy, 006. [0]M Johannes, Jounal of Finance, vol.59, pp. 7-60, 004. []C Mancini, R Renò, Jounal of Economeics, vol.60, pp.77-9, 0. 89