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1 SUBJECT SA0 OF THE INSTITUTE AND FACULTY OF ACTUARIES Man On Wong Essay on Welfare Effecs of Developing Reverse Morgage Marke in China Subjec SA0 Advisors Bing Zheng Chen James Orr Prepared a School of Economics and Managemen, Tsinghua Universiy

2 Absrac Absrac Populaion ageing and inadequae social securiy have inroduced serious problems for China. As a resul, seeking alernaive soluions is becoming increasingly imporan. Some developed counries, such as U.S., Canada, and Ausralia have developed reverse morgage markes o relieve relaed financial pressures. A naural quesion is wheher or no he reverse morgage sysem should be implemened in China o deal wih ageing and social securiy problems. From he perspecive of welfare economics, i amouns o asking if reverse morgage markes can help improve boh residenial and social welfare in China. Based on parameers from a daase which reflecs he curren saus of China s populaion and economy, his disseraion draws on scenarios ha help explore he above quesion. Building on he life-cycle model, we invesigae he role of reverse morgages in improving he welfare of reired elderly residens. Simulaion resuls show ha, reverse morgage can improve he welfare of residens who are house-rich and cash-poor. However, due o high ransacion coss, i is no always he case. Timing he enry of a reverse morgage conrac can help residens aain a higher level of welfare improvemen. On average, i is beer for residens o wai unil afer he age of 65 o paricipae in reverse morgages. Sensiiviy analysis indicaes ha reverse morgages can miigae he adverse effecs on residenial welfare caused by longeviy risk; i could exacerbae he negaive effecs caused by moving risk. This implies ha mainaining ransacion coss a a reasonable level is necessary for reverse morgages o reain heir welfare-enhancing funcion in differen economic condiions. Building on he overlapping generaion model, we invesigae how reverse morgages influence he equilibrium allocaion of social capial. Simulaions reveal ha reverse morgages provide subsanial income for elderly residens. This in urn, reduces he burden on younger pension axpayers, which promoes and smoohes he life-cycle consumpion of residens. In equilibrium, his reduces he opimal pension ax and improves social welfare. We furher demonsrae ha he inensiy of he moive o beques deermines wheher residenial welfare improves under equilibrium. Sensiiviy analysis indicaes ha he welfare-enhancing role of reverse morgages remain I

3 Absrac significan as he severiy of he ageing problem increases. Finally, drawing on heoreical analysis and our inernaional experience, we provide suggesions on policies ha would assis in developing an effecive reverse morgage marke in China as well as open opics for fuure research. Key words: reverse morgage; social securiy; welfare analysis; life-cycle; overlapping generaion II

4 Conens Conens Absrac... I Conens... 1 Chaper 1 Inroducion Background Moivaion Relaed Lieraure Indusry Pracices Relaion o Life-cycle model Relaion o OLG model... 8 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Model Seup Variables and Parameers Simulaions for he case wihou iming decisions Simulaions for he case wih freedom in iming decisions Summary Chaper 3 The Welfare Effecs of Reverse Morgage on Individual and Sociey Model Seup Variables and parameers Simulaion Resuls Summary Chaper 4 Concluding Remarks Main resuls Poenial Topics for Fuure Research References Appendix A RM produc in major markes Appendix B Moraliy Raes Appendix C Housing Price Growh and Reurn on Risky Asses Appendix D Moving Probabiliy Appendix E Minimum Non-housing Consumpion

5 Chaper 1 Inroducion Chaper 1 Inroducion 1.1 Background Since 1999 China has enered an era where sociey is ageing. According o he repor Forecasing China s Ageing Problem from he China Ageing Commiee, he ageing elderly 1 will soon oal 248 million by 2020, accouning for 17.17% of he whole populaion where million (or 12.37% of ageing elderly) will be over 80 years-old; by 2050, he elderly populaion will exceed 400 million, over 30% of he whole populaion, wih million (or 21.78%) being over 80 years-old. China s social securiy sysem is no equipped for such rapid and massive ageing challenge, poenially causing he naional reasury o go ino heavy deb. Currenly, he defici reaches 1.2 rillion RMB in 2013, and he social securiy sysem can only provide limied suppor for he elderly people. China s ageing problem has imposed grea pressure on is social securiy sysem and oher sources of annuiy income need o be explored in order o increase he living sandards of elderly people. To ackle he ageing problem, some counries such as he U.S., Canada and Ausralia, have developed a soluion based on securiy hrough house, also known as he Reverse Morgage (RM) marke. The basic mechanism of RM is ha elderly people sign a conrac wih a financial insiuion o use heir home equiy as a morgage in exchange for a financial lump sum. Unlike regular morgage conracs, he borrower will be allowed o live in he house unil his or her deah; he financial insiue will receive he house afer hen. Since he RM conrac does no require he debor o pay for heir deb when he or her is sill living in he house, his produc can also be used o finance many oher long-erm projecs such as educaion or help relieve unemploymen burdens. RM conracs are paricularly suiable for elderly people mainly because: (1) hey usually own home equiy; (2) hey usually have a shor expeced fuure lifeime, so 1 Several developed counries caegorizes people over 65 as elderly (for example visi he China Naional Commiee on Aging sandard, which is 60 (see hp://old.cncaprc.gov.cn:8080/en/info/1897.hml) is used as he crieria of elderly person. 1

6 Chaper 1 Inroducion ha annuiy payous can be high; (3) hey usually prefer o live in heir own house as hey age; (4) many elderly people are saring o open heir minds o new securiy ideas for he ageing. All hese reasons increase he araciveness of RM 1. The Unied Saes is one of hese counries wih he mos maure and well-developed RM marke. The firs RM conrac in he U.S. appeared in In 1987, he Deparmen of Housing and Urban Developmen sared he RM marke, and he governmen has been providing a guaranee for Home Equiy Conversion Morgages (HECM). The marke scale has increased dramaically in he U.S., from 1 billion USD in 1999 o 32.4 billion USD in Annual applicaions have increased from 7,900 o 114,000 during he same ime period. Home Equiy Conversion Morgages accoun for 90% of all reverse morgages originaed in he U.S. As of May 2010, here were 493,815 acive HECM loans 2. There are hree basic caegories of RM producs in he U.S. marke: HECM, he Home Keeper Plan, and privae producs. HECM was he original RM produc, and so far he mos maure wih a marke share of over 90%. The Canadian Home Income Plan (CHIP) is he firs, as well as he only, privae RM business operaor in Canada. I sared is business in 1986 and has issued CAD conracs worh over 2 billion. The RM marke in Canada is enirely marke-oriened wihou he need of any financial suppor from he governmen. Reverse Annuiy Morgages, Line of Credi Reverse Morgages, and Fixed Term Reverse Morgages are he hree main caegories in he Canadian marke. The Ausralian RM marke began o operae a he sar of I was esablished by several RM business operaors and supervised by he Senior Ausralian Equiy Release Associaion of Lenders (SEQUAL). In Ausralia, he RM marke was chiefly led by he major financial insiuions, wihou much inervenion or financial suppor from he governmen. The Deloie repor, commissioned by SEQUAL, shows ha as of 31 December 2010 he Ausralian reverse morgage marke comprised of more han 41,000 1 The RM produc of each counry briefly described below is abulaed in more deail in Appendix A. 2 See Home Equiy Conversion Morgages Monhly Repor (May 2010), hp:// 2

7 Chaper 1 Inroducion loans wih oal ousanding fund of $3bn. The oal represens 11% growh in he marke from 31 December 2009, and he average loan size has also increased o $72,500, up from $51,148 in All of hese counries have ailored heir RM business o elderly people who own home equiy. Currenly, 90% of families in China have heir own home equiy, of which 60% were provided by governmen and corporaions as forms of subsidies and 40% were purchased from public housing markes 2. This ranslaes o huge poenial for he RM developmen in China. In 2013, China s cenral governmen plans o launch a pilo program ha will offer reverse morgages o he elderly in he year afer. 1.2 Moivaion Given ha RM has been proven o be a valid complemen of he radiional social securiy sysem in foreign developed counries, i is imporan o ask wheher RM can help solve China s ageing problem, aking ino accoun China s economic and demographic characerisics, such as geing old before becoming rich, housing price boom, and one-child policy. From he perspecive of welfare economics, i is imporan for he governmen and praciioners o know wheher developing RM is welfare enhancing. In he lieraure of mainsream economics 3, welfare refers o he economic well-being of an individual, group, or economy. Individual welfare is commonly measured by he uiliy gained hrough he achievemen of economic aciviies (e.g., labor supply, income and asse allocaion), and he social welfare is commonly measured by he oal uiliy gained hrough aggregaing he uiliies of all paricipans. Here, uiliy is measured using a uiliy funcion, which describes individual preference based on an axiomaic foundaion 4. In his hesis, parameerized uiliy funcions were used o deermine 1 See more a: hp:// 3-billion/#shash.Tzfu5KyA.dpuf 2 The daa were provided by he office of he Deparmen of Housing and Urban and Rural Consrucion on March 9h, Addiionally, according o he China Household Finance Repor issued by Souhwesern Universiy of Finance and Economics and China s People s Bank, on May 13 h, 2013, 90% of Chinese families own home equiy. 3 See Samuelson P A, Nordhaus W D. Economics, McGraw-Hill, See Von Neumann J, Morgensern O, Theory of Games and Economic Behavior, Princeon, NJ, Princeon Universiy Press, 1944, second ed. 1947, hird ed

8 Chaper 1 Inroducion individual and social welfare. A deailed formula will be presened in he following chapers. A his juncure, welfare can be broadly undersood as well being or uiliy gained. This paper aemps o explore he above quesion based on China s social parameers from boh individual welfare and social welfare perspecives, using boh heoreical and empirical mehods. The deailed conens of he sudy include: (1) The impac of he RM marke on reired residens welfare. A he very beginning of he RM marke developmen, he major paricipans of he RM program will be elderly people. So RM will only affec residens asse allocaion afer heir reiremen. We will sudy he caegory of residens who will benefi from he RM program, and o wha exen reirees welfare will improve. Using China s demographic daa, our empirical sudy will provide useful insighs ino his quesion. (2) The impac of RM on he life cycle and welfare of residens. When he RM marke maures, residens will anicipae he changes ha RM producs will have on heir asse allocaion as hey age, and will plan for such changes when hey are young as well. This allows us o sudy he opimal life-cycle asse allocaion for an individual in a maured RM marke. In fac, he impacs of RM on young residens asse allocaion are mixed. Since RM will increase he income of elderly people, i reduces he younger generaion s burden of aking care of heir parens and promoes consumpion in heir presen ime. However, RM also reduces heir parens beques, i encourages young residens o consume less and save more. Which impac is more dominan depends on specific economic condiions. The empirical sudy will give us a clear picure of how RM will influence Chinese residens life cycle consumpion profile, and wheher such changes can improve heir life cycle welfare. (3) The impac of RM on social welfare. As will be shown in he resuls addressing he quesion above, when he RM marke maures, he residens life-cycle consumpion and asse allocaion would change. We will furher sudy how such changes affec he opimal ax rae in he social securiy program. More precisely, we will explore wheher he opimal ax rae can be reduced 4

9 and wheher he social welfare improves. Chaper 1 Inroducion In summary, Quesion (1) invesigaes he welfare effecs of RM from an individualisic and shor-erm perspecive. Quesion (2) discusses he welfare effecs of RM from an individualisic and long-erm perspecive. While Quesion (3) focuses on he welfare effec of RM from a social and long-erm perspecive. 1.3 Relaed Lieraure Indusry Pracices RM as a subjec has araced a grea deal of aenion from academics. One of he mos imporan and basic quesions mos scholars are concerned wih is wheher he RM marke deserves financial suppor and encouragemen from he governmen. Mayer and Simons (1994) invesigaed people who are: (1) older han 62;(2) own home equiy;(3) have relaively low curren income; (4) have no bequeah moivaion (no children); (5) live alone; (6) have no deb; (7) have been living in heir curren home for more han 10 years (no srong moving inenion). Their sudy shows ha, hrough he RM produc, six million U.S. residens could ge a 20% increase in heir annual cash income, and 1.4 million of hem can ge ou of povery 1. Of five million residens from he caegory, he morgage income paid in a lump sum is wice as much as heir oal liquid asses. Merrill, Finkeland and Kuy (1994) saed ha 18% of he elderly people who used o live below he povery line can ge enough cash income o keep hem above povery. Kuy (1998) showed ha RM in he U.S. can shif he proporion of populaion below he povery line by 29%. Rasmussen, Megbolugbe and Morgan (1995), and Michell and Moore (1998) presened furher evidence for he posiive effecs of he RM marke on he social securiy sysem. Wih he naure of RM conracs, and only he RM recipien s house equiy as collaeral, he insiue issuing RM faces muliple risk ypes causing he conracs o be expensive. The risks associaed o RM mainly come from ineres rae risk, housing price risk, adverse selecion, longeviy risks, and moral hazards. 1 According o he official repor from he Unied Saes Census Bureau (hp:// he povery hreshold, or povery line, is he minimum level of income deemed adequae in a paricular counry. 5

10 Chaper 1 Inroducion Boehm and Ehrhard (1994) believe ha RM is a long erm conrac, where small flucuaions in he ineres raes affec he issuing insiue drasically. Chinloy and Megbolugbe (1994) found ha pricing of a RM conrac is mainly on he value of house equiy, which may cause he issuing insiue o suffer losses when he housing marke flucuaes, his risk is hard o diversify hrough derivaives. Chen, Chang, Lin and Shyu (2010) had used marke daa from 1986 ill 2008 proved ha he volailiy of house price affecs he pricing of RM. Davidoff and Welke (2004) says ha RM consumers ge more money from he conrac if hey live longer, his will naurally arac people wih beer healh condiions, and who end o have a longer expeced lifeime. Longeviy risk is eviden when considering ha RM conracs are long erm making i ime sensiive, and would usually encouner an adverse selecion where more people in beer healh are araced, while he conrac enables an improvemen in living sandard would increases heir lifespan, pricing a RM conrac for unforeseen increasing paymens becomes very complex. Miceli and Sirmans (1994), Shiller and Weiss (2000) menions ha moral hazard exiss when a paricipan signing a RM conrac foregoes he incenive o upkeep he residence afer he propery righ has been ransferred, causing acceleraion in he depreciaion of he house value. However, insiues may reasonably increase he price of RM o offse he risk inheren o a RM conrac. Many oher researchers also noed he high expense of RM conracs. Caplin (2000) saes ha ransacion coss is abou 15% of he oal RM value. Levion (2002) saes ha he ypical cos for a median HECM loan of $105,000 USD is esimaed a $5,400 USD. High expense is also affeced by model risk, such as China being a developing counry where no much daa is available, wih he effecs of urbanizaion and indusrializaion baring unknown consequence. All of hese poins ac as a driving facor for he high RM expense Relaion o Life-cycle model In welfare economics, he life-cycle model is ofen used for sudying consumers decisions hroughou heir lives or in some specific periods. The life-cycle model was firs presened by Modigliani and Brumberg (1954) for sudying consumers savings behavior in differen periods. Since consumers decisions among differen periods are 6

11 Chaper 1 Inroducion no independen, he life-cycle model is quie suiable for sudying income disribuion, consumpion and savings, invesmen, housing, securiy, ec. The basic idea behind he life-cycle model is ha he asse allocaion of invesors will affec heir coninuing life. So, he life-cycle model is no only used o analyze consumers lifeime decisions, bu also for sudying asse allocaion wihin specific periods, such as used by Michelangeli (2007). The life-cycle model is he sandard analyical framework o sudy household finance. Research on household finance mainly focuses on finding ways for people o opimize heir consumpion choices wih various financial producs over he whole lifeime. Cocco (2005) confirms he crowding ou effec of house price risk on sock invesmen, and furher found ha his effec is more significan or younger and poorer invesors. Yao and Zhang (2005) found ha residens wih home equiy will have a higher proporion of risky-asses o liquid asses. Li and Yao (2007) documened ha he flucuaion in home equiy values would have significan influence on residens life-cycle consumpion and welfare. Many oher sudies are currenly being conduced on household asse opimizaion in China. For example, Yang and Chen (2006) sudied he opimal savings decision problem using a dynamic simulaion model. Zhao and Zeng (2006) sudied he urban residens opimal invesmen porfolio. Huang and Liu (2007) modelled iner-emporal opimal decision-making on house leasing and purchasing. These researches mainly sudied opimal decisions from he perspecive of consumpion-invesmen allocaion, while leaving ou RM producs and he resuling welfare effecs. Michelangeli (2007) firs brough RM ino he sudy of residens consumpion-invesmen decisions based on he life-cycle framework. He made careful invesigaions ino he welfare change of reired people who paricipaed in he RM marke. He used daa from he U.S. Healh and Reiremen Sudy and Consumpion and Aciviies Mail Survey Daabase ( ). The argeed group for he RM producs was single elderly people (older han 62) who have reired, owned home equiy, and had a complee consumpion and invesmen record. The model was buil on he life-cycle of 64 o 95 years old and has he following characerisics: (1) residens preferences regarding housing consumpion and non-housing consumpion is described by he 7

12 Chaper 1 Inroducion Cobb-Douglas uiliy funcion; (2) i akes he mainenance cos, ransacion cos, and moving cos ino accoun when deermining he residens budge consrain; and (3) residens welfare gain is defined as he equivalen increase of iniial non-housing asse brough abou by he RM conrac. Michelangeli (2007) saed ha RM can possibly reduce he welfare of some marke paricipans. However, his research has wo shorcomings. The firs one is he auhor assumes elderly people only paricipae in he RM program a he beginning of heir reiremen and does no consider heir abiliy o choose he iming of paricipaion. Second, he auhor only focuses on he consumpion and asse allocaion of elderly people, and does no address he influences of RM on young people. As saed above, when he RM marke becomes maure, i will also influence residens asse allocaion during heir younger years if hey anicipae ha hey will paricipae in RM when older. This research overcomes hese shorcomings by allowing people o choose he bes ime o paricipae in he RM program and exending he life cycle o include he younger periods Relaion o OLG model The OLG model is used for sudying allocaion efficiency and welfare resuls for consumers in differen generaions. This model was presened by Samuelson (1958) for sudying consumpion and economic growh. Then, Diamond (1965) inroduced governmen deb and capial accumulaion ino his model o discuss he reasonable size of governmen liabiliy. Similar o he life-cycle model, he OLG model also became one of he imporan welfare models for sudying social resource allocaion. The OLG model is also used in social securiy ax rae sudies. Longeviy risk and populaion srucure were inroduced ino he model o make a more realisic sudy of he ax rae design. Fehr (2013) provides a deailed exploraion of his opic. Feldsein (1985) firs used he OLG model in a sudy of he opimal social securiy ax rae and welfare changes. As he RM marke will change people s consumpion profile during heir life-cycle, i will naurally resul in changes in resource allocaion and welfare gain of each generaion. From he social perspecive, Coleman (2010) buil an overlapping generaion (OLG) model o sudy he welfare influence of he RM marke wih longeviy risk. A randomly seleced residen was assumed o live for four ime periods, 8

13 Chaper 1 Inroducion 0-25 years old, years old, years old, and years old. Residens have o decide on housing invesmen beween 25 years old and 65 years old. They will possess home equiy before he age of 65 if hey chose o purchase he house. Housing qualiy can be separaed ino wo caegories, high and low, which are boh relaed o housing price. Young residens are divided ino wo groups, depending on wheher hey will receive beques from heir parens. Elderly residens are also divided ino wo caegories, according o heir beques moive. Housing prices are deermined using he balance in supply and demand of producion and housing. Applying marke daa from New Zealand, Coleman found ha he main influence of he RM marke is ha i makes he residens in every generaion inves more in high-qualiy housing insead of leasing or purchasing low-qualiy housing. There is an 11% increase in invesmen in high-qualiy housing for he residens who have an expeced remaining lifeime of 20 years. The influence of he RM marke on he working ages depends on heir ype. For hose who will no receive a beques from heir parens, paricipaing in he RM marke encourages heir consumpion in he earlier period of life because RM reduces precauionary savings. However, for hose who will receive beques from heir parens, he RM marke may decrease heir consumpion in heir earlier par of life because heir parens may use RM o increase heir living sandards and as a resul reduce he beques. The empirical resuls based on he marke daa from New Zealand shows ha he young beween 20 years old and 25 years old will reduce heir invesmen in high-qualiy housing o increase heir consumpion, whereas he older people beween 40 years old and 65 years old will increase heir invesmen in high-qualiy housing o prepare for he RM program. Coleman (2010) made a good aemp a invesigaing how he RM marke will influence he residens consumpion profile in each generaion. His paper focused on analyzing he influence of he RM marke on residens housing choices, bu no on welfare changes. The sudy was based on New Zealand, and he inerpreaion of resuls canno be sraighforwardly ranslaed for China s usage. This paper will sudy he impacs of RM on individual welfare and social welfare based on he demographic and economic characerisics of China. In essence, he basic idea of hese models is o simulae he raional behavior in 9

14 Chaper 1 Inroducion financial marke, and find he opimal soluion based on budge consrain. The model seup in his research follows his direcion. The remainder of his paper is organized as follows. In Chaper 2, we sudy he effecs of RM on he welfare of reired residens, where we allow reired residens o choose he bes ime o paricipae in he RM program. Chaper 3 exends he life-cycle model o include he younger periods, which sudies how RM affecs residens consumpion a differen ages, he resuling life cycle welfare changes, and formulae an OLG model o illusrae how RM affecs he opimal ax rae and social welfare. We conclude his paper in Chaper 4. 10

15 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens In his chaper, we exend he framework of Michelangeli (2007) and Cocco (2005) by formulaing a life-cycle asse allocaion model o sudy how reverse morgage influences he welfare of residens and he ideal ime of enering a conrac. We will incorporae Chinese demographic and economic daa ino he model, and perform an in-deph empirical sudy. Throughou his chaper, we measure he welfare of reired residens via he sum of heir discouned iner-emporal consumpion uiliy and he beques uiliy, as illusraed in Equaion (2.13) below. 2.1 Model Seup The consumpion-asse allocaion process of reired residens in our life-cycle framework is illusraed in Figure deah 死 亡 时 刻 养 Pension 老 金 收 入 reire 退 休 Housing 住 房 消 费 income 收 入 储 Savings 蓄 收 入 支 出 Expendiure Non-housing 非 住 房 消 费 投 Invesmen 资 收 入 Savings 储 蓄 Invesmen 投 资 Bequeah 死 亡 遗 赠 Figure 2.1 The consumpion-asse allocaion process of reired residens in a life-cycle framework 11

16 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens The uiliy of a reired residen is deermined by housing consumpion, non-housing consumpion, savings, invesmen, and beques. A reiree may receive income from he pension, financial asses and housing asses. Reired residens have o make consumpion decisions and invesmen decisions. They srive o maximize heir expeced uiliy by opimizing heir asse allocaions. Residens will paricipae in he reverse morgage marke only if i maximizes heir individual welfare. Key variables in our model of his chaper are shown in Table 2.1. Table 2.1 Key variables in he life-cycle model Variables Y H o I Meaning Income of reired residens in ime Average housing size of reired residens in ime Indicaor of reired residens ownership of housing in ime m I Indicaor of reired residens forced o move in ime C S B r I r D W G Non-housing consumpion of reired residens in ime Risky asses of reired residens in ime Non-risky asses of reired residens in ime Indicaor of reired residens paricipaion in reverse morgage in ime Lump-sum paymen of a reverse morgage o reired residens in ime Iniial wealh of reired residens in ime Cumulaive value of reverse morgage loans of reired residens in ime Throughou his chaper, we make he following assumpions. Assumpion 1: Reiremen sars a 60, and he life-cycle of a reiree is 60 years old o 94 years old. Each cycle conains 7 ime periods, and each period lass for 5 years. represen he period: 1 denoes he ime period when he reiree is 60 o 64 years old, 2 denoes he ime period when he reiree is 65 o 69 years old, and so on. 12

17 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Assumpion 2: Reirees live no more han 7 periods, and he probabiliy of heir remaining life is defined by he survival funcion: () N n (2.1) j1 j where n is he condiional probabiliy of he reiree who is alive in period j 1 will j be sill alive in period j ( j1,, T ). Assumpion 3: The income of he reired residens in each period is a fixed proporion o heir income of he las period Y. Y Y, YYY 0,1 (2.2) Assumpion 4: There are wo kinds of housing consumpion. One is living in a rened house, and he oher is buying a house. The housing consumpion level is measured by he living space for each residen, H (square meers), and here is a minimum living space requiremen H H. min Assumpion 5: For he same living space, he uiliy from owning home equiy will be higher han rening he house. This assumpion will be in force by fixing he preference parameer on owning a house o be higher han rening, i.e. own. Residens ren housing consumpion in period will be denoed by an indicaor funcion o o I. I 1 o if residens own home equiy, while I 0 if residens ren he house. The mainenance cos of he purchased house will be own per square meer, while he renal cos is ren per square meer. Assumpion 6: Residens face an exogenous moving shock in period, denoed by a random dummy m m I ( I 1 when residens have o move in period, and I 0 m when hey don ) 1. The probabiliy of moving is relaed o he residens age, and is denoed by m q. Also, residens can sell heir home equiy o increase heir expeced uiliy, which will be denoed by s I ( I 1 when residens sell he home equiy in s 1 Moives of moving include he need for help, lack of cash, willingness o change environmen, allergy o he climae change, serious illness and willingness o live wih heir children. 13

18 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens s period, and I 0 when residens say). When a residen has o move in period, he house will be sold, and a ransacion cos of P H will be incurred. H own Assumpion 7: The rae of reurn of house prices R H H / H P P 1 is denoed by: ln H R (2.3) H H where H is he expeced rae of reurn on house prices, H is he residual erm ha follows a normal disribuion of N 2 (0, H ) Assumpion 8: Residens have a minimum non-housing consumpion consrain such ha he raio of non-housing consumpion over he income has o be larger han a lower bound C / Y c. min Assumpion 9: There are only wo asses available in he financial marke. One is risk free and he oher is risky. The rae of reurn of he risk free asse is a consan, denoed by B, while he rae of reurn of he risky asse is random, described by ln s R (2.4) s s where he expeced rae of reurn is s, and s is he residual randomness, which 2 follows a normal disribuion N 0, s risky asse and ha of he growh rae of house price is HS.. The correlaion beween he rae of reurn of Assumpion 10: Only residens wih fully owned home equiy can paricipae in he reverse morgage program. We used a dummy variable o characerize residens RM r decision. I 1 if he residen paricipaes in a reverse morgage program in period, r and I 0 oherwise. Assumpion 11: The reverse morgage conrac will pay he residen a lump sum amoun 1 r r of D, and he ransacion cos will be revd. The ineres rae of he RM r j produc is RL 1, and he value of he conrac in period j is G j D RL. Assumpion 12: In each period, residens have a non-housing consumpion C, and 1 Annuiy paymens of reverse morgage loan is anoher common ype. However, since we assume ha he reired residens are fully raional and hey can annuiize he paymen in a uiliy-maximizing way, our qualiaive insighs do no depend on wheher he paymen is annuiized. 14

19 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens a housing consumpion Cobb-Douglas funcion: H. The residens uiliy in period u C, H 1 C 1 H 1 is characerized by a (2.5) where is he relaive risk aversion parameer, and is residens relaive preference on housing consumpion. The larger he is, he more he residen places imporance on housing consumpion. Assumpion 13: If he residen passes away a he end of period, he beques is B TW. Following Li and Yao (2007), he beques uiliy funcion is described by B TW TW L H / P 1 ren In his equaion, L measures he relaive imporance of he beques. The larger he L is, he more a beques is preferred. (2.6) Under Assumpions 1 o 13, he wealh a he beginning of ime period is W I I max 0, P H 1 G 1 I P H 1 o r H r H own 1 1 own W R B R S s f 1 1 (2.7) I is seen from his equaion ha, if residens paricipaing in he RM program move ou, hey have o eiher repay he deb of he RM or give he house o he relaed financial insiuion. The ne cash value he reiree can receive if hey move ou is max 0, P H H 1 1 own G. The oal disposable wealh in period is: Q W Y (2.8) where Y is he income, and we assume ha he real income of reired residens will be he same in each period. The budge consrain in period is: Q Q o H o o Q C B S I P H I I own ren o m m s H o o I 1 I I I P H I own I ren I I I I m O P H o m s r H 1 own 1 rev own 1 H where m 1 G / P H 1, Orev rev 1 if residens paricipae in he RM a he curren period and O 1 if residens paricipae in he RM in he pas period rev (2.9) 15

20 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Wihin he budge consrain, residens maximize heir uiliy funcion 1 V max E N u C, H N 1 N B Q A T o s r V A C, H, B, S, I, I, I YYY 1, T In his equaion, E[.] is he expecaion operaor, V[.] is he expeced uiliy value, he se of decision variables, is he uiliy discoun facor, and probabiliy ha a residen will be able o survive ill period. Le sae parameers in period, where X o 1, r 1, m, H I I I P, H 1, G 1, Q Bellman equaion for he opimizaion problem (2.10) becomes max, A (2.10) A is N is he X be he se of. Then he V X n u C H n B Q n E V X (2.11) In order o reduce he compuaional burden associaed wih simulaion, we follow Yao and Zhang (2005) o simplify he Bellman equaion. We normalize he decision variables and sae variables by using he oal wealh in each period Q. Then we ge he non-housing consumpion raio c C / Q, he housing consumpion H raio h P H / Q, he risk free asse raio b B / Q, and he risky invesmen raio s S / Q. Wih such normalizaion, he decision and sae variables reduce o a o s r c, h, b, s,i,i,i, x o 1, m,i r I I 1, w, h 1, m 1, where w W / Y is he raio H of asses o income, h / 1 P H 1 W is he raio of residens housing value o asses, H and 1 1 L 1 m G R / P H is he raio of iniial reverse morgage o housing value. We define he raio of income o oal wealh as y Y / Q, and he raio of curren reverse H morgage o house value as m G / P H. Then he residens oal wealh growh rae will be q / 1 Q 1 Q. Wih he above noaion, he budge consrains (2.9) can be rewrien as o o o 1 o m m s o o o m s r c b s h 1 I I 1 1 I own ren 1 h I I I I I I own ren 1 h y I I I I m O 1 1 own 1 rev own 1 (2.12) H We rewrie he objecive funcion as v x V X / Q / P. Then he residens opimizaion program is ransformed o be 1 1 H v x max n u c, h E v 1 x 1 q 1 R 1 1 n B1 a (2.13) 16

21 where Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens min s.. c y c,, h y h,, b 0,, s 0 (2.14) min min H c is he minimum non-housing consumpion, and h / min P Hmin Y minimum housing consumpion. is he All he opimizaions hroughou his paper will have o be solved by simulaions. To proceed, we discreize he variable w W / Y by dividing is value inerval [0,10] ino H en equal-lengh grids, discreize h / 1 P H 1 W by dividing is value inerval [0,1] H ino 5 equal-lengh grids, and discreize m 1 G 1RL / P H 1 by dividing is value inerval [0.2,0.8] ino four equal-lengh grids. A he erminal ime T, he residen s value funcion equals o he beques uiliy B 1. A cubic spline inerpolaion will be used o generae he value of he uiliy funcion when he decision variables and sae variables fall wihin he corresponding inervals. To calculae he expeced value of he uiliy funcion, we follow Rus (1997) o generae wo coninuous random variables according o a wo-dimensional normal disribuion, wih one dimension represening he randomness of house prices, and he oher dimension represening he randomness of risky reurns. We calculae he expeced value of he uiliy funcion by averaging he uiliy levels of differen random pahs. For each random pah, we use backward inducion o derive he opimal decisions in each ime period. 2.2 Variables and Parameers The parameers based on China s demographic and economic characerisics o be used in he simulaions will be defined in his chaper. The baseline model parameers and seings are summarized in Table 2.2, alernaive specificaions will be considered laer in he robusness checks. 17

22 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Table 2.2 Baseline parameers Parameers Symbol Value Moraliy rae n j Appendix B Relaive risk aversion 2 Housing preference for purchaser Housing preference for rener own 0.55 ren 0.5 Uiliy discoun facor Beques facor L 1 Mean growh rae of house price H 3.89%* % Variance of house price growh 2 H 2.87% 2 * % Correlaion of house price o risky asses HS 0 Minimum living space H 2 min 35m Mainenance cos for house owners own 1.5%*5 7.5% Renal cos for house reners ren 4%*5 20% Transacion cos of house purchase own 5% Reverse morgage loan commimen m 50% Transacion cos of reverse morgage loan rev 15% Reverse morgage ineres rae Moving risk R 5 L m q 0 Pension replacemen rae 60% Minimum non-housing consumpion Reurn on Risky asses Variance of he reurn on risky asses Risk-free rae c 20% min 4.58%*5 22.9% s 2 s 25.12% 2 * % R 5 f (1) Time periods in Life-cycle We assume ha he reired residens life-cycle consiss of seven periods, wih each period covering 5 years. The maximum remaining lifespan is 35 years. According o he 18

23 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens labor law in China, he official reiremen age is 60 for males and 50 for females. However, his policy has no been sricly enforced. For example, saring in Oc 2010, many companies in Shanghai have employed a flexible reiremen program. In hese companies, he reiremen age for males can be exended o 65, and he reiremen age for females can be exended o 60. In his paper, we ignore such flexibiliy. (2) Moraliy Risk n j To esimae he moraliy rae of residens, we employ he daa se provided by China s Populaion and Employmen Yearbook 2010 (see Appendix B). Our esimaes show ha he expeced remaining lifespan for a 60 year-old is years. (3) Relaive risk aversion Sudies on invesmen consumpion in he life-cycle heory usually define o be he relaive risk aversion parameer. Michelangeli (2007) gave an esimae of 3.87 for reired residens in he U.S. who are under 64 years old. Some local researchers in China such as Chen, Yang and Fang (2005), sudied he consumpion daa and esimaed his parameer o be Ai and Wang s (2005) in heir empirical research on China s urban residens consumpion derives an esimae of Following Ai and Wang (2005), we will have 2 for his chaper. (4) Housing preference parameer For he parameer measuring residens preference for housing, we follow Han (2008) in using own 0.55 and 0.5, indicaing ha Chinese residens usually prefer ren owning han rening a house. (5) Uiliy discoun facor The discoun facor capures he ime value preference of consumers. A higher discoun facor shows ha a consumer places much imporance on fuure consumpion. According o Chen, Fu, Ge (2006) and Han (2008), he uiliy discoun facor for Chinese residens normally is The moraliy raes in he able are he cenral moraliy, Zhu and Chen (2009) gave precise formulas for he calculaion. 2 Derived from he resuls of formula (11) and parameers in (12) of Ai and Wang (2005) 19

24 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens (6) Beques facor L Following Yao and Zhan (2005), his paper uses a beques inensiy of L 1. (7) Parameers relaed o house prices,, d, H H HS According o he housing price index of China from , he mean and sandard deviaion of house price growh raes are 4.89% and 2.87% respecively. The depreciaion rae of a house value is assumed o be d 1% per year. The correlaion beween China s housing price index and he Shanghai Sock Exchange Index is found o be -0.11, which is no highly significan. So, we se he correlaion beween house price growh and reurn on risky asses HS o be 0. See Appendix C for more deails. (8) Minimum living space H min In 2011, he China Academy of Social Sciences announced ha he minimum living space for a person shall be 35 square meers. The per capial housing area under affordable housing offered by he governmen is usually beween square meers. Hence, we will assume H 2 min 35m. (9) Parameers relaed o mainenance and renal cos own, ren, own, ren According o Zhu (2010), he mainenance cos annum, and he renal cos cos relaed o purchasing a house 0 1. ren own is 1.5% of he house value per ren is 4% of he house value per annum. The ransacion own is 5% of he house value, and none for leasing (10) Loan amoun, ransacion cos, and ineres rae of reverse morgages m,, R rev L The loan amoun of a reverse morgage is commonly 50% of he house value under American sandards, wih a ransacion cos of 14% of he loan amoun (Chaplin, 2001). We assume ha he ransacion cos rev of reverse morgage in China almos equals ha in U.S. which is 15%. Since he supplier of reverse morgage conracs would have o ake on various long-erm risks. The ineres raes of reverse morgage will be higher han he ineres of a normal morgage. We se ineres rae. R L o 4% as he reverse morgage 1 Usually one monh s renal will be he cos of ransacion for rening, which is insignifican. 20

25 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens (11) Moving risk m q Owing o ha lack of daa in China, he moving probabiliy of U.S. 1 was used as a proxy o measure moving risk in China. A survey was made by our school on RM, wih a sample of 2000 people beween 60 and 80 years old in China. A quesion wihin he survey addressed he number of shifs a person makes in heir lifeime, and he average number of shifing in a Chinese residen s lifeime is abou 3. This figure is abou 1/4 he U.S. average, and his is used o scale he average moving probabiliy in U.S. o be more appropriae for China. This annual probabiliies are hen convered ino a 5 year probabiliy. See Appendix D for more deails. (12) Pension replacemen rae We assume ha pension income afer reiremen will be 60% of he income in he period jus before reiremen. The real pension afer reiremen will say he same. (13) Minimum non-housing consumpion C min We se minimum non-housing consumpion o be 20% of income, according o he proporion of minimum living sandard o disposable income per capia in China. See Appendix E for more deails. (14) Saving and invesmen parameers for reired residens R,, f s s The average inflaion rae from 1998 o 2011 in China is 2.13%, while he average nominal ineres rae of a 5-year or longer erm ime deposi is 2.85%. So, we choose a long-erm risk-free rae The average reurn R f of 1.0%, which will be used as he rae of reurn on savings. s and sandard deviaion s of he Shanghai Sock Exchange Index from is 4.58% and 25.12%, which will be se as he average reurn and sandard deviaion of risky asses. See Appendix C for more deails. 1 Source: Unied Saes Census Bureau hp:// 21

26 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens 2.3 Simulaions for he case wihou iming decisions This subsecion assumes ha residens can only apply for reverse morgage a he beginning of heir reiremen. This assumpion capures he commonly used Now-or-Never decision rule and also helps simplify calculaions. A more refined scenario will be ha residens can apply for reverse morgage in any ime period. This scenario makes considering ime of enry a sensible issue and will be sudied in he nex subsecion. Residens who already own a house a he beginning of heir reiremen period have he following choices: 1 sell he house and ren a new one; 2 sell he house and purchase anoher one; 3 keep he house and don paricipae in reverse morgage; 4 keep he house and paricipae in reverse morgage. Amongs hese choices, 4 allows he reired o repurchase heir house by repaying he reverse morgage loan and realizing he appreciaion of house price, or receiving longeviy risk proecion provided by he reverse morgage. However, choice 4 has a disadvanage in ha he reired residen will have o pay he ransacion coss and ineres gains on he reverse morgage loan, which are usually high. Moreover, reired residens migh have o bear he consequence of moving risk. If such adverse evens occur, residens choosing 4 may be paying high cos of he reverse morgage wihou enjoying is benefis when compared o oher choices. We now assess he reired residens uiliy-maximizing choices. During he reiremen period of years old, based on differen raios of asse o income w W / Y and of residens housing value o asse h -1=P H H 1 / W, reired residens opimal housing decisions are shown in he following able. Recalling ha he raio of asse o income w W / Y is higher if he residen s oal asses are relaively larger, and he raio of residens housing value o asse h -1=P H H 1 / W is higher if he residen is 22

27 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens relaively poor in cash. The resuls are presened in Table 2.3. asse o income raio Table 2.3 reired residens opimal choice a he beginning of he reiremen (60-64) house value o asse raio Table 2.3 shows ha: (1) When reired residens have a high asse o income raio (>=5) and a high house value o asse raio (>=0.6), hey are relaively wealhy in oal asses bu poor in cash. In his case, he reverse morgage gives his caegory of reired residens he highes uiliy amongs he 4 opions, making opion 4 he opimal one. Given he low moving risk in China, reired residens wih higher house value o income raio will benefi from paricipaing in he RM program. (2) When residens have a high asse o income raio (>=5) and a low house value o asse raio (<=0.4), hey are relaively wealhy in oal asse and heir cash is no oo scarce. In his case, a small increase in cash holding would allow hem o achieve he opimal allocaion. Because of he high cos of reverse morgage, his caegory of residens would prefer o sell he original house and purchase a smaller one. Keeping in mind ha he cash hey need is no oo much, he new house hey 23

28 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens purchase needs no o be oo small. They hus avoid he cos of reverse morgage by selling a small par of living space away. (3) The caegory of residens who have a low asse o income raio (<=4) and a lower house value o asse raio (<=0.6), are poor in asses. In his case, he only feasible choice for hese residens is o ren a house. asse o income raio Table 2.4 welfare effecs of reverse morgage oward reired residens house value o asse raio The values in he able above represens he % increase from he iniial wealh, based on welfare equaions used in his chaper, i follows he principle of equivalen uiliy wih values convered o a % facor of iniial wealh The welfare gained hrough paricipaing in reverse morgage is calculaed in Table 2.4. From his able, i is seen ha reirees wih mediocre asse o income raio and a high house value o asse raio receives he mos benefi from a reverse morgage. The welfare gain decreases as he house value o asse raio decreases, or in oher words, RM is less necessary as he reiree becomes richer in cash. As a check of robusness, we now consider how he increase in lifeime expecaion (longeviy), he increase in moving risk, and he decrease in ransacion cos influences 24

29 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens he welfare funcion of he reverse morgage. In he baseline scenario, we use he moraliy raes as in Appendix B. To sudy he effec of increasing lifespans, we reduce he moraliy raes as in Appendix B by 50% in our alernaive scenario. Originally, he expeced lifespan of a 60 year old reiree is anoher years, wih he increase in lifespan, he expeced remaining lifeime of 60 year-olds is now years. The reired residens opimal choice is shown in Table 2.5, and he corresponding improvemen in welfare is shown in Table 2.6. asse o income raio Table 2.5 reired residens opimal choice when expeced lifespan increases house value o asse raio

30 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Table 2.6 welfare effecs of reverse morgage oward reired residens wih increase in lifespan asse o income raio house value o asse raio The values in he able above represens he % increase from he iniial wealh, based on welfare equaions used in his chaper, i follows he principle of equivalen uiliy wih values convered o a % facor of iniial wealh Relaive o he baseline scenario, Tables 2.5 and 2.6 shows ha as expeced lifespan increases, reirees would have higher welfare from paricipaing in reverse morgage, and hence hey are more likely o paricipae. Welfare improvemen is more significan for reirees wih a higher house value o asse raio. In he baseline scenario, we assume ha here is no moving risk (moving probabiliy=0) afer reiremen. To sudy he effecs of increased moving risk, we use he moving risk probabiliy from Appendix D for his alernaive scenario. The reired residens opimal choice is shown in Table 2.7, and he corresponding improvemen in welfare is shown in Table

31 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens asse o income raio Table 2.7 reired residens opimal choice when moving risk increases house value o asse raio Table 2.8 welfare effecs of reverse morgage oward reired residens when moving risk asse o income raio increases house value o asse raio The values in he able above represens he % increase from he iniial wealh, based on welfare equaions used in his chaper, i follows he principle of equivalen uiliy wih values convered o a % facor of iniial wealh Relaive o he baseline scenario, Table 2.7 shows ha as moving risk becomes more 27

32 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens significan, reirees have less incenive o paricipae in reverse morgage. For hose who are low in boh house value o asse raio and asse o income raio, he opimal choice is o ren a house. Welfare improvemen from reverse morgage is also less when considering moving risk, as shown in Table 2.8. Finally, we consider he effecs oward wealh when we assume ha he ransacion cos of he reverse morgage is reduced o zero (ha is, he ransacion cos is paid by he governmen). The reired residens opimal choice is shown in Table 2.9, and he corresponding improvemen in welfare is shown in Table From Table 2.9, as ransacion cos on reverse morgage decreases, he negaive impac of reverse morgage reduces, making he reverse morgage more aracive, hence increasing he paricipaion in reverse morgage. The welfare gain hrough paricipaing in reverse morgage also dominaes hose wih a non-zero ransacion cos shown in Table 2.4. asse o income raio Table 2.9 reired residens opimal choice when he ransacion cos is 0 house value o asse raio

33 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Table 2.10 welfare effecs of reverse morgage oward reired residens when ransacion cos is 0 asse o income raio house value o asse raio The values in he able above represens he % increase from he iniial wealh, based on welfare equaions used in his chaper, i follows he principle of equivalen uiliy wih values convered o a % facor of iniial wealh 29

34 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens 2.4 Simulaions for he case wih freedom in iming decisions This subsecion relaxes he assumpion ha he reiree only can paricipae in he RM program a he beginning of reiremen. This relaxaion allows us o es he benefi of varying he ime of enry. The simulaion resuls wih he mos opimal iming decisions are shown in Table Compared wih he siuaion in which residens can only apply for reverse morgage a reiremen (Table 2.3), opimal housing choice has higher significance when reirees can apply for reverse morgage a any ime. Table 2.11 reired residens opimal choice a he beginning of reiremen wih relaxed iming asse o income raio house value o asse raio For he residens whose asse o income raio is high (>=9) and has a house value o asse raio ha is high ( 0.8), he opimal choice is no o paricipae in reverse morgage a he beginning of he reiremen. They can benefi by posponing heir paricipaion of reverse morgage due o high likelihood of he house appreciaing. Table 2.12 shows he periods where differen caegories of reired residens should paricipae in reverse morgage. In his Table, he house rich cash poor residens can improve welfare by paricipaing in reverse morgage no immediaely a reiremen, bu a a laer 30

35 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens period. asse o income raio Table 2.12 opimal periods ha reired residens should paricipae in reverse morgages house value o asse raio indicaes ha reired residens should paricipae in reverse morgage beween ages of 60 o is he caegory of reirees who should no paricipae in reverse morgage in any period. Table 2.13 shows he expeced level of welfare gained by reirees who should ener in a reverse morgage a any period. Compared wih he baseline scenario (Table 2.3) where reverse morgage is only applicable a reiremen, wih opimally imed paricipaion, more reired residens will receive welfare improvemens. Especially, when he asse o income raio is low and house value o asses raio is high, he welfare gain is much higher han he former case where freedom o paricipae is resriced. 31

36 asse o income raio Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens Table 2.13 welfare effecs of reverse morgage wih relaxed iming house value o asse raio The values in he able above represens he % increase from he iniial wealh, based on welfare equaions used in his chaper, i follows he principle of equivalen uiliy wih values convered o a % facor of iniial wealh We also considered how he increase in expeced lifespan (longeviy), he increase in moving risk, and he decrease in ransacion cos influences he welfare funcion of residens wih reverse morgage in he case of relaxed iming as a robusness check. The resuls wih hese alernaive assumpions are quie similar o hose obained in he las subsecion, and are hus omied. 32

37 Chaper 2 The Welfare Effecs of Reverse Morgage on Reired Residens 2.5 Summary This chaper sudied he effec of reirees opimal choice and welfare improvemen when reverse morgage is available as a complemenary insrumen o annuiizaion. Our simulaions show ha when reverse morgage is only available a reiremen, no all reirees can benefi from i. Especially, when reirees are rich in asse bu poor in cash, he opimaliy of reverse morgage depends on he exen of cash shorage. When cash shorage is more eminen, reverse morgage is he opimal choice because i offers cash wihou reducing he reiree s living space. However, when he cash shorage is no oo significan, reverse morgage is less appealing due o is high coss. In his case, he opimal choice is o liquidae he original house and make a re-allocaion of wealh beween living space and cash. When cash shorage is no oo urgen, he reiree can sill enjoy a reasonably large living space. Through sensiiviy analysis we found ha he welfare enhancing funcion of reverse morgage becomes more subsanial when expeced lifespan increases, or when moving risk or ransacion cos decreases. We also sudied he opimal ime of RM paricipaion, and resuls show ha house rich cash poor reirees can improve welfare by enering reverse morgage laer in reiremen. On average, he bes ime of enry is abou 5 years afer reiremen, due o a high likelihood of house value appreciaion. 33

38 Chaper 3 The Welfare Effecs of Reverse Morgage on Sociey Chaper 3 The Welfare Effecs of Reverse Morgage on Individual and Sociey Building on Feldsein (1985) and Cocco (2010), his chaper inroduces housing and reverse morgage ino he overlapping generaion (OLG) model o explore he opimal asse-consumpion allocaion over he whole lifeime of a residen, he opimal social securiy ax rae in an economy ha is in a seady sae, and hen evaluae he effecs of reverse morgage on individual and social welfare. Throughou his chaper, we measure an individual s welfare in by summing up uiliy in one s young, middle-age and elderly periods of one s lifeime, as well as he uiliy from beques, as given in Equaion (3.12). Social welfare is defined as he sum of uiliies of all generaions a a specific poin in ime, as illusraed in Equaion (3.11) below. 3.1 Model Seup Residens are spli ino hree generaions: youhs, middle-aged, and elderly. Among hem, he youhs and middle-aged consiue he working populaion, and he elderly are he reirees. The life-cycle consumpion and savings of each generaion is illusraed in Figure 3.1. The youhs of period 2 are he elderly in period, wih income and expendiures described in he lef column of Figure 3.1. The youhs of period 1 are middle-aged in period, wih income and expendiure parameers in he middle column. Youhs in period are presened in he righ column. In China, considering he longeviy of people, youhs would unlikely obain heir paren s beques unil hey reach middle age, so we assume ha beques is only besowed upon during middle age. 34

39 Chaper 3 The Welfare Effecs of Reverse Morgage on Sociey Figure 3.1 asse-consumpion allocaion of each generaion under an overlapping generaion framework We now incorporae China's economic and demographic characerisics, such as geing old before becoming rich, housing price boom, and one-child policy ino he OLG model. Geing old before becoming rich has wo aspecs, one, heavy burdens on he younger generaion o care for he elderly. Two, a smaller beques will be lef behind by he elderly. China housing price boom is an effec ha would cause youhs o ren a house raher han buy one due o budge consrains. One-child policy is characerized by he high proporion of elderly in he populaion. In his chaper, W sn,, sn, C, H sn,, S sn,, TW sn, are variables ha represen wage income, non-housing consumpion, housing consumpion, savings, and beques for he generaion whose birh ime is s, and who has already live for n periods. Descripions of he variables are summarized in Table

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