Developing Equity Release Markets: Risk Analysis for Reverse Mortgages and Home Reversions

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2 Developing Equiy Release Markes: Risk Analysis for Reverse Morgages and Home Reversions Daniel Alai 2, Hua Chen, Daniel Cho 2, Kaja Hanewald 2, and Michael Sherris 2 Absrac: Equiy release producs are sorely needed in an ageing populaion wih high levels of home ownership. There has been a growing lieraure analyzing risk componens and capial adequacy of reverse morgages in recen years. However, lile research has been done on he risk analysis of oher equiy release producs, such as home reversion conracs. This is parly due o he dominance of reverse morgage producs in equiy release markes worldwide. In his paper, we compare cash flows and risk profiles from he provider s perspecive for reverse morgage and home reversion conracs. An a-home/in long-erm care spli erminaion model is employed o calculae erminaion raes, and a vecor auoregressive (VAR) model is used o depic he join dynamics of economic variables including ineres raes, house prices and renal yields. We derive sochasic discoun facors from he no arbirage condiion and price he no negaive equiy guaranee in reverse morgages and he lease for life agreemen in he home reversion plan accordingly. We compare expeced payoffs and assess riskiness of hese wo equiy release producs via commonly used risk measures, i.e., Value-a-Risk (VaR) and Condiional Value-a-Risk (CVaR). Key Words: Reverse Morgage, Home Reversion, Vecor Auoregressive Models, Sochasic Discoun Facors, Risk-Based Capial [Conac auhor] Deparmen of Risk, Insurance, and Healh Managemen, Temple Universiy 8 Liacouras Walk, 625 Aler Hall, Philadelphia, PA 922, Unied Saes. 2 School of Risk and Acuarial and ARC Cenre of Excellence in Populaion Ageing Research (CEPAR), Universiy of New Souh Wales, Sydney NSW 252, Ausralia. addresses: (Daniel Alai), (Daniel Cho), (Kaja Hanewald), (Michael Sherris).

3 . Inroducion Home equiy release producs allow reirees o conver a previously illiquid asse ino cash paymens which can be used for home improvemens, regular income, deb repaymen, aged care and medical reamens as well as a range of oher uses which improve qualiy of life for reirees. There has been a growing lieraure addressing risk facors and capial adequacy of reverse morgage producs in recen years, including bu no limied o,boehm and Ehrhard (994), Chinloy and Megbolugbe (994), Szymanoski (994), Rodda e al. (24), Ma and Deng (26), Wang e al. (28), Chen e al. (2), Sherris and Sun (2), and Li e al. (2). However, lile research has been done on risk analysis of oher equiy release producs, such as home reversion conracs. The purpose of his paper is o inroduce home reversion schemes o he readers and compare cash flows and risk profiles from he provider s perspecive beween reverse morgage and home reversion conracs. In a reverse morgage, he provider lends he cusomer cash and obains a morgage charge over he cusomer s propery (or a share of he propery). The conrac is erminaed upon he deah or permanen move-ou of he cusomer, a which ime he propery is sold and he proceeds are used o repay he ousanding loan. Typically, a no negaive equiy guaranee is included in he conac, which sipulaes ha he cusomer is no liable in case he sale proceeds of he propery are insufficien o repay he loan. In a home reversion scheme, he provider purchases he ownership righ over he cusomer s propery (or a share of he propery). The home is sold a discoun (ypically beween 35% and 6% of he marke price), and he conrac includes a lease for life agreemen allowing he cusomer o reside in he propery unil deah or permanen move-ou. The unouched research area of home reversions is parly due o he underdeveloped marke. In he US, reverse morgage producs dominae he equiy release marke. The Home Equiy Conversion Morgage (HECM) program is considered he safes and he mos popular 2

4 program of is kind in he US, since i is insured by he US federal governmen, and accouns for 95% of he marke share (Ma and Deng, 26). The dominance of a single equiy release produc in he US sands in sark conras o he dynamics of some foreign markes. In he UK, for example, reverse morgages, home reversions and oher equiy release producs have been available for o 3 years. Among hem, reverse morgages accoun for 75% of he equiy release producs available in he marke while home reversions accoun for mos of he remaining 25% (ASIC 25). The reverse morgage marke in Ausralia consised of 42,4 loans wih a oal marke size of $3.32 billion by he end of 2. The Ausralian marke saw a % growh in he value of new lending in 2 and a 22.5% growh over he las wo years (Deloie 22). Home reversion schemes exis in Ausralia bu are relaively new and available commercially hrough jus one oule, Homesafe Soluions. They are currenly available o consumers aged 6 or over living in cerain areas in Sydney or Melbourne. From he provider s perspecive, i is imporan o esimae he probabiliy of erminaion, as delayed erminaion resuls in heavier loan accumulaion and increases he chances of negaive equiy in reverse morgages, or i causes an unexpeced longer erm for lease in home reversions resuling in he provider overpaying he cusomer when he conrac originaes. The US HECM program iniially assumed loan erminaion raes being equal o.3 imes he underlying female moraliy raes as no erminaion experience were available. Laer on, Chou e al. (2) use a complimenary log-log regression model o examine how loan erminaion is affeced by key facors based on he acual HECM loan erminaion daa. They find ha age, house price appreciaion, loan duraion, moraliy, personal asses, gender and co-borrower saus all conribue o explain loan erminaion. They also repor ha he iniial assumpion of.3 imes he female moraliy is oo low for younger borrowers and slighly oo high for older borrowers. Rodda e al. (24) find similar resuls. However, he regression-based erminaion models used in boh sudies have several drawbacks. Firs, hey 3

5 rely heavily on availabiliy of daa. Second, hey assume he probabiliy of loan erminaion remains consan afer age 9, which is raher unrealisic. Third, hese models do no make explici allowance for move-ous, healh or non-healh relaed (Ji e al. 22). Szymanoski e al. (27) sugges ha erminaion of reverse morgage loans should be modeled based on is key causes: borrower s moraliy, long-erm care move-ou, prepaymen and refinancing. In ligh of his, Ji e al. (22) develop a semi-markov model for reverse morgage erminaions for join borrowers, which incorporaes he aforemenioned modes of erminaion. We adap heir model o a single female borrower and consider only wo reasons: deah and enry o long-erm care faciliy, as prepaymen and refinancing are rare for home reversion consumers. Ineres rae risk, house price risk, and renal yield risk are oher major risks in equiy release producs. The previous lieraure examining he embedded risks in reverse morgage conracs eiher focus on analysing he house price dynamics alone (see, for example, Chen e al. 2 and Li e al. 2), or modelling he dynamics of house prices and ineres raes independenly (Chinloy and Megbolugbe 994, Ma e al. 27, Wang e al. 28, ec). This approach neglecs correlaions among hese key variables. In addiion, he derived riskneural measure fails o represen all sources of uncerainy and he dependency srucure among risks. To overcome his, Huang e al. (2) implemen a wo-dimensional volailiy vecor linking he house price and ineres rae dynamics. Chang e al. (22) propose a mulidimensional linear regression model ha capures he relaionship beween house prices and key macroeconomic facors. Sherris and Sun (2) fi a vecor auoregressive (VAR) model o examine risks embedded in reverse morgage insurance policies. Despie is simpliciy, a VAR model is sophisicaed enough o capure he linear inerdependencies among muliple ime series. We adop a VAR process o joinly model he dynamics of ineres raes, house prices, renal yields and GDP. Our approach is differen from Sherris and Sun (2) in wo major ways. Firs, GDP is added o he model o acknowledge he impac 4

6 of macroeconomic facors on oher economic variables of ineres. Second, we derive sochasic discoun facors based on he VAR model ha can capure uncerainy arising from a range of sources: ineres rae, house price and renal yield. This approach has no been used in Sherris and Sun (2) or in any oher sudies in equiy release markes before. Our mehodology is closely relaed o Ang and Piazzie (23), who use sochasic discoun facors, or pricing kernels, o exend heir VAR model wih an affine erm srucure of ineres raes. In his manner hey are able o value all asses and cash flows. Cochrane and Piazzesi (25) sudy ime variaion in expeced excess bond reurns. They consruc an affine model, i.e., prices are linear funcions of sae variables of he VAR model, ha generaes he bond yield reurns. Hoevenaars (28) also combines he VAR model wih an affine erm srucure model of ineres raes in such a way ha here are no arbirae opporuniies. He uses he model o generae macroeconomic scenarios ha serve as inpu for an asse liabiliy managemen model of a pension fund. The derived sochasic discoun facors are used for pricing he no negaive equiy guaranee and he lease for life agreemen ha are fundamenal elemens in reverse morgage and home reversion schemes, respecively. We hen simulae cash flows and calculae he acuarial presen value of ne payoffs of he provider. We also quanify risk measures such as Value-a-Risk (VaR) and Condiional Value-a-Risk (CVaR) a he 99.5% level o illusrae he amoun of solvency capial o be se aside for each ype of equiy release producs. Sensiiviy analysis is conduced o invesigae he impacs of he loan-o-value raio (LVR), he iniial house price, moraliy improvemens, and he leverage raio on he payoffs and risk profiles of reverse morgage and home reversion conracs. Following our work, Cho (22) and Shao e al. (22) use he VAR model and he sochasic discoun facor approach o sudy oher aspecs of equiy release producs. Cho (22) compares cash flows for reverse morgages wih differen payou designs. Shao e al. (22) quanify he impac of individual house price risk on he pricing of equiy release producs. 5

7 We find ha he maximum LVRs offered o cusomers in he Ausralian marke is se so low ha reverse morgage providers bear almos no risk of capial loss. This suggess ha reverse morgage providers in Ausralia could increase maximum LVRs o faciliae he expansion of he reverse morgage marke. Compared o reverse morgage conracs, providers of home reversion schemes obain a lower payoff and assume a higher risk, which jusifies he marke dominance of reverse morgages in Ausralia. An efficien risk sharing and risk ransfer mechanism needs o be developed o simulae growh of he home reversion marke. By providing an appropriae framework of regulaion, financial lieracy educaion and by promoing liquidiy o invesors, governmens can encourage privae supply of home reversions a modes public expense. Ineresingly, using higher LVRs in he range of hose offered under he US HECM program, we find exacly opposie resuls: reverse morgage conracs are less profiable and riskier han home reversion conracs. This finding confirms ha he insurance of crossover risk in reverse morgages provided by he Federal Housing Agency (FHA) is an imporan facor in he US marke. The finding also indicaes ha here is a large poenial marke for home reversion schemes in he US. The remaining body of his paper is organized as follows. In Secion 2, we review he basic feaures of reverse morgage and home reversion conracs, and discuss risks involved in hese wo producs. In Secion 3, we presen a erminaion model and use a VAR model o joinly model he dynamics of ineres raes, house prices, and renal yields. Sochasic discoun facors are derived based on he VAR model. In Secion 4, we develop he pricing formula for he no negaive equiy guaranee in reverse morgages and he lease for life agreemen in home reversions. Cash flow srucures are analysed for boh conracs. In Secion 5, numerical examples are used o compare hese wo equiy release producs in erms of payoffs and risks. Secion 6 concludes he paper. 6

8 2. Produc Review in Ausralia 2.. The Reverse Morgage Marke 2... Produc Review The reverse morgage marke has gained considerable momenum in Ausralia in recen years. According o he media release by Deloie (22), he marke size of reverse morgages climbed from $.9 billion in 25 o $3.32 billion in 2. There were 42,4 loans in he marke as of he end of 2 while his number in 25 was 6,584. The average loan size was $78,249 in 2, compared o $5,48 in 25. While he marke is Ausraliawide, hree saes make up more han 7% of he naional marke: NSW 35%, QLD 2% and VIC 8%. The main feaures of a ypical reverse morgage conrac in Ausralia are reviewed as follows. Condiions: All lenders se a minimum age for he younges person on he ile of he propery ha is being morgaged. In mos cases, his is 6 years. Some reverse morgage providers se he minimum age as 63 or 65 years (Bridges e al. 2). Alhough he specific erms and condiions vary across producs, mos conracs oblige he consumer o (ASIC 25): mainain insurance for he propery, pay all ougoings, mainain he propery o he sandard required by he provider, no leave he propery vacan for more han six o 2 monhs, no allow new non-approved residens o reside in he propery, and no sell, lease or renovae he propery wihou he provider s prior approval. 7

9 Iniial Loans: The loan amoun depends primarily on wo facors: age and value of he home. 2 The borrower s age or he younger borrower s age in case of a couple deermines he maximum LVR. The LVR increases as an individual s age increases. For example an individual aged 6 may borrow 5% of he value of heir home whereas someone aged 8 or older can borrow up o 35% of he value of heir home. Payou Opions: Depending on he conrac, he borrower can wihdraw he loan as a lump sum, income sreams, a line of credi, or a combinaion of hese paymen plans. As of 2, lump sum loans ake up 95% of he Ausralian marke and income sreams accoun for 5%. The proporions of lump sums and income sreams have been relaively sable since 28 (Deloie 2a). Terminaion: Repaymens are generally no made unil an individual moves ou of he house or dies. If he home is joinly owned, he loan is only repayable once he las surviving parner dies or moves ou. Guaranee: In Ausralia, SEQUAL-accredied members mus offer a no negaive equiy guaranee which ensures ha no maer how long he loan runs for, he borrower can never owe more han he value of he securiy, in his case, heir house. 3 4 However, he no negaive equiy guaranee can be negaed hrough a number of acions or inacions on he par of he borrower, including fraud or misrepresenaion, failing o mainain he propery in a good condiion, failing o insure he propery, or no paying he council raes on he propery. Ineres Raes: Ineres raes can be variable or fixed. Variable rae loans are he mos popular produc in Ausralia. Variable raes are on average % above he sandard variable home loan 2 In he US, Federal Housing Adminisraion (FHA) imposes a morgage limi which is $625,5 for one-family house. The iniial loan amoun is deermined by he younger borrower s age and he adjused propery value. The adjused propery value is defined as he lesser of he appraised value of your home, he FHA HECM morgage limi of $625,5 or he sales price. 3 SEQUAL is he abbreviaion of he Senior Ausralians Equiy Release Associaion. In order o proec he cusomers, SEQUAL has esablished a sric Code of Conduc ha each SEQUAL-accredied member has o agree is equiy release produc(s) adhere o. 4 The no negaive equiy guaranee is also called a non-recourse provision in he US reverse morgage marke. 8

10 rae. The margin (or morgage insurance premium) is charged o manage he risk of providing he no negaive equiy guaranee. 5 Fixed ineres raes can be se for varying erms generally 5, or 2-years or lifeime. The proporion of fixed ineres reverse morgage loans is negligible % in 2 (Deloie 2a). There is now only one SEQUAL-accredied lender (RBS) providing a fixed rae opion on heir producs (Bridge e al. 2). Fees: There are ypically seup fees, ongoing fees and exi fees associaed wih reverse morgages which vary from lender o lender Major Risks in Reverse Morgages Reverse morgages differ from radiional forward morgages in he way ha he ousanding loan balance grows due o principal advances, ineres accruals, and oher loan charges over he life of he loan. The loan balance may grow o exceed he propery value a he ime of erminaion because of muliple risks. Terminaion Risk: If a borrower lives longer han expeced, he principal advances and ineres accruals will coninue, which may drive he loan balance exceeding he sale proceeds of he propery. The mobiliy rae has he same effec on reverse morgage producs. Borrowers may move ou of heir homes because of heir healh condiion, marriage, divorce, deah of he spouse, disasers, or simply he desire o live in anoher place. Ineres Rae risk: Mos of reverse morgage producs feaure adjusable ineres raes. Therefore, he variaion of ineres raes imposes addiional uncerainy on reverse morgage providers. A rise in he ineres rae can resul in a higher rae of ineres accruals on he loan balance han anicipaed, which increases he possibiliy of parial non-repaymen when he loan evenually erminaes. 5 In he US HECM program, morgage insurance premiums consis of wo pars: an up-fron charge which is eiher 2% (HECM Sandard) or.% (HECM Saver) of he adjused propery value, and an annual rae of.25% of he ousanding loan balance for he life of he loan. FHA collecs all he insurance premiums and reverse morgage lenders are allowed o assign he loan o FHA when he loan balance equals he adjused propery value. FHA akes over he loan and pays an insurance claim o lenders covering heir losses. So lenders are effecively shifing he collaeral risk o FHA. 9

11 House Price Depreciaion Risk: The uncerainy in house price depreciaion raes is anoher risk we need o consider. If he home price remains sagnan or grows a a lower rae han anicipaed, he ousanding loan balance a mauriy may exceed he sale proceeds of he propery. Lenders or heir insurers may suffer from he losses. As indicaed by he recen U.S. housing marke downurn, home price depreciaion risk is only parially diversifiable: pooling morgage producs naionally only reduces he risk of a downurn in he regional housing marke, bu canno diversify he risk of a naional economic recession The Home Reversion Marke Produc Review Home reversion schemes allow senior homeowners o sell a proporion of equiy in heir home while sill living here. Homeowners receive a lump sum paymen in exchange for a fixed proporion of he fuure value of heir home. There are wo main ypes of home reversion schemes: a sale-and-lease model and a sale-and-morgage model. In he sale-andlease model, he ile o he propery passes o he provider a he ime of purchase and he propery is leased back o he consumer a a nominal ren. The sale-and-lease produc provider in Ausralia, called Money for Living, wen ino adminisraion in 25. The Ausralian Securiies and Invesmens Commission (ASIC) issued legal proceedings in he Federal Cour of Ausralia alleging ha Money for Living adverised is produc in a misleading and decepive manner. A resoluion was passed in December 27, placing he company ino liquidaion. In he sale-and-morgage model, he ile o he propery remains in he consumer s name even afer he provider pays. To proec he provider s ineres in he propery, he consumer is required o give he provider a morgage over he propery (ASIC, 25). Homesafe Soluions Py Ld, a join venure of Bendigo and Adelaide Bank Ld and

12 Ahy Py Ld, has launched Homesafe Deb Free Equiy Release since 25. We review is feaures in he following. 6 Condiions: The homeowner mus be aged 6 and over. For a couple, he younger parner mus be a leas 6. Currenly, i is available only o cusomers residing in cerain poscodes wihin Melbourne and Sydney. As a general rule, he home needs o be free-sanding. Oher propery ypes are subjec o approval from Homesafe. The propery is he principal place of residence for a leas one homeowner a he ime of exchange of conracs. The land value of he propery is 6% or greaer of he oal value deermined by an independen panel valuer. The homeowner mus own he home ourigh, or use some of he Homesafe funds received o pay ou he exising morgage. Funds: Under Homesafe Deb Free Equiy Release, i is possible o access any amoun beween $25, and $,,. The maximum share ha homeowners can sell, so-called acquisiion rae, is 65% of he fuure sale proceeds of he home. Homeowners can ener ino addiional conracs over ime, up o a oal share of 65%. There is no resricion as o how he funds should be used. Payou Opion: Homesafe currenly offers only a lump sum payou opion. Lease: Homeowners receive a discouned lump sum paymen (usually 35% or 6%) in exchange for a fixed proporion of he fuure value of heir home. The discoun represens he value of he lease for life agreemen ha allows homeowners o live in he house for life or unil volunarily move-ou. Homeowners may be eligible for an early sale rebae if hey sell heir home earlier han expeced. Terminaion: The conrac erminaes when homeowners die or volunarily vacae he propery. Homesafe is eniled o he agreed percenage of he sale proceeds of he house and homeowners reain he share of he sale proceeds ha hey have no sold o Homesafe. 6 More deails can be found on he websie of Homesafe Soluions Py Ld: hp://www.homesafesoluions.com.au/

13 Tile: Homeowners remain on he ile, so hey have he righ o use heir home for as long as hey wish. There is no requiremen for homeowners o underake mainenance of he propery afer enering ino a Homesafe conrac. The owners can even ren ou he home and keep he renal income. Homesafe will regiser a morgage and lodge a cavea on he ile, only o secure is share of he sale proceeds. Fees: Homesafe charges a one-off ransacion fee of $, Major Risks in Home Reversions The provider of home reversion conracs faces house price risk. For he lease for life agreemen, he uncerainy originaes from he renal yield, and he duraion of he conrac. Terminaion Risk: In a home reversion conrac, he cusomer is always beer off prolonging he duraion of he conrac. This is in conras o a reverse morgage conrac, where early erminaion may be beneficial for he cusomer under cerain circumsances. Therefore, when valuing he lease for life agreemen in an annuiy seing, i is realisic o assume ha he only modes of erminaion are deah and unavoidable enry ino a long-erm care faciliy. I should be noed ha some home reversion conracs provide a ren rebae for conracs ha erminae much earlier han expeced, bu he amoun is no of he magniude o induce erminaion. Renal Yield Appreciaion Risk: In a home reversion conrac, he propery is sold o he provider a a discouned price. The level of he discoun reflecs he value of he lease for life agreemen. The provider s payoff could be impaired if a low renal yield were assumed when calculaing he value of he lease bu he acual renal yield would urn ou o be much higher. House Price Depreciaion Risk: Lenders of home reversion conracs are eniled o sell he propery and secure a par of he sale proceeds when borrowers die or volunarily move ou. Therefore, lenders face he risk of house price depreciaion Advanages of Home Reversions 2

14 From he consumer s poin of view, home reversion producs have unbeaable advanages over reverse morgages. Oliver Wyman Financial Services (25) prediced hough equiy soluions have radiionally fared poorly in he US, opions such as home reversion producs should find a marke especially among owners of higher-value homes, for whom equiy release may be inended o diversify a porfolio raher han o free up cash. In addiion, reverse morgages involve he accumulaion of deb over he life of he conrac while home reversions are deb-free. In order o proec borrowers from negaive equiy, reverse morgage programs usually provide a no negaive equiy guaranee so loan repaymen is capped by he sale proceeds of he propery. This guaranee is financed via morgage insurance premiums paid by borrowers. In oher words, senior homeowners bear various risks, including longeviy risk, ineres rae risk and propery value risk under a reverse morgage conrac. Neverheless, hese risks are parly remied o providers under home reversion conracs. Commercial providers are generally beer posiioned o bear such risks. For example, hey can ransfer risks o he capial marke more efficienly compared wih senior homeowners. More imporanly, he ineress of invesors and consumers are aligned under home reversion schemes: boh wan he value of he home o rise (Oliver Wyman Financial Services, 25). Therefore, we believe ha here remains room for significan growh of a diversified equiy release marke and we see a grea poenial for he developmen of home reversion producs. 3. Modelling Framework 3.. The Terminaion Model Though a significan proporion of reverse morgages are issued o couples (around 4% in he US and 5% in Ausralia, see Deloie 22), he sudy of join life dependency is 3

15 no he focus of his paper. 7 For simpliciy, we assume a single, female policyholder. The join-life mulisae erminaion model can be readily incorporaed in our model framework if necessary. We do no consider volunary prepaymen or refinancing as consumers of home reversion producs are always beer off by prolonging he duraion of heir conracs. In oher words, conrac erminaion is deermined by wo major facors: deah and enry ino longerm care faciliies. We assume a Gomperz srucure for he populaion force of moraliy x for females aged x given by exp x. () x Equiy release producs are designed for a policyholder living a home. Therefore, she is suscepible o a-home moraliy, which need no equal o female populaion moraliy. Le denoe he proporionaliy consan ha produces a home moraliy from populaion moraliy. Tha is, he female a-home moraliy raes are scaled down by muliplying o represen he beer healh of reirees, who do no move ou o long-erm care. The possibiliy of enry ino a long-erm care faciliy is represened by a proporionaliy consan,. These wo parameers can be replaced by one conrac-moraliy loading facor,. Hence, he conrac force of moraliy can be wrien as follows: c where x denoes he conrac erminaion rae. c x x x, (2) The parameers and are esimaed using Ausralian female moraliy daa for he period and age 5-5 from he Human Moraliy Daabase. 8 We fi boh an 7 Ji e al. (22) compare he value of he no negaive equiy guaranee for join borrowers under he independence assumpion and he semi-markov assumpion. Though he assumpion of independence generally leads o an overesimaion of NNEG prices, he difference is no significan (see Figure 3 in Ji e al. 22). 8 hp://www.moraliy.org/ 4

16 Moraliy Raes ordinary linear regression (LR) o he log-ransformed moraliy raes as well as a Poisson regression (PR) o deah couns wih an appropriae exposure offse. LR: ln m x x x, (3) PR: ln Dx ln Ex x x, (4) where ln E x is he offse for he Poisson regression based on he survival couns, E x. Table repors he esimaed parameers and Figure presens he fi graphically. I can be seen ha he wo regressions produce very similar fis. We use he Poisson regression hereafer due o is inuiive and naural inerpreaion. Table : Comperz Parameers for he Force of Moraliy ˆ ˆ Ordinary Linear Regression (LR) Poisson Regression (PR) Figure : Regression Fi of Log-Moraliy Raes.4.2 OLR PR Age Given esimaes for and, we urn o and. Since here is no publicly available conrac erminaion daa in Ausralia, we make use of he parameer esimaes 5

17 repored by Ji e al. (22). These auhors use he daa in he Equiy Release Repor of he Insiue of Acuaries (25) o esimae he proporional facors for he deviaion from an aggregae model o he a-home/in long-erm care spli model. Table 2 reproduces heir esimaed proporional facors for females a ages 7, 8, 9, and. The proporional facors for ages 7 79, 8 89 and 9 99 are obained by linear inerpolaion, while he proporional facors for ages below 7 and ages above are se o he proporional facors for age 7 and age, respecively. Table 2: A home and In Long-Term Care Proporional Facors from Ji e al. (22) Age c c Le q T and p Pr T x Pr x, for,,... x, where T is he conrac erminaion ime and is he maximum aainable age. We have c c c c qx px s px xsds, (5) which can be solved numerically o yield he desired conrac erminaion probabiliies. We also compue he average conrac in-force duraion for differen age groups (see Table 3). I decreases wih he age of he policyholder a loan originaion. For individuals aged 65, he average in-force duraion is around 8 years. I drops o abou years for consumers aged 75 and 5 years for consumers aged 85. Table 3: Average in-force Duraion Age Average in-force duraion

18 3.2. The VAR Model House price modelling iself is a large area of sudy. Tradiionally, house price dynamics are assumed o follow a geomeric Brownian moion (see, for example, Cunningham and Hendersho 984, Kau e al. 993, Huang e al. 2). The GBM process is a very popular ool in finance for modelling asse reurns, as i provides powerful, ye simple represenaion of he dynamics. However, he GBM assumpion canno accommodae many sylized facs, for example, condiional heeroskedasiciy, serial correlaions, and volailiy clusering of observed house prices, in real esae markes. Therefore, i is naural o apply ime-series analysis o model he housing price dynamics. Chen e al. (2) and Yang (2) use he ARMA-GARCH model o fi he house price index in he US and Li e al. (2) use he ARMA-EGARCH model for he house price growh in he UK. Anoher imporan risk facor in equiy release producs is ineres rae risk. A sochasic ineres rae model wih a realisic erm srucure needs o be considered. Furhermore, many empirical sudies demonsrae ha propery reurns and ineres raes are correlaed. Joinly modelling of house price indices and ineres raes is paricularly imporan for variable ineres rae reverse morgages, which dominae he US and Ausralian markes. In ligh of his, Huang e al. (2) implemen a wo-dimensional volailiy vecor, linking he house price and ineres rae dynamics. Sherris and Sun (2) use a VAR model wih wo lags o capure he dynamic relaionships beween a house price index, renal yields, ineres raes, and inflaion. We adop he same approach in his paper. A VAR-ype model capures he linear correlaions embedded in a mulivariae ime series sysem. Popularized by Sims (98), VAR has been exensively used in economerics and various applicaions in finance, as i provides flexibiliy and simpliciy over oher radiional economeric models. Macroeconomic variables are likely o affec he dynamics of boh house prices and ineres raes. Ang e al. (23) describe he join dynamics of bond yields and 7

19 macroeconomic variables in a VAR model. Previous sudies also argue ha house prices are affeced by macroeconomic facors (see, for example, Abraham and Hendersho 994; Muellbauer and Murphy 997). Recen sudies have included GDP as a facor in predicing housing prices (Valadez 2) and he yield curve (Ang and Piazzesi 23). For his reason, we include GDP in our VAR framework. The raw daa used in his sudy include zero-coupon ineres raes (3-monh and - year), sandard variable morgage raes (MR), a nominal Sydney house price index (HPI), a nominal Sydney renal yield index (RYI), and nominal Ausralian GDP (GDP). Daa is available for he period June 993 o June 2. Because he daa for GDP is only available on a quarerly basis, oher variables are filered o quarerly frequency. Table 4 describes he variable definiions, sources and frequency of he daa. Table 4: Noaions, Definiions, Sources and Frequency of Variables Variables Definiions Sources Frequency () r 3 monh Zero-coupon yield Reserve Bank of Ausralia Daily (4) r year Zero-coupon yield Reserve Bank of Ausralia Daily MR Nominal Morgage Raes Reserve Bank of Ausralia Monhly HPI Nominal Sydney house price index Residex Py Ld. Monhly RYI Normal Sydney renal yield index Residex Py Ld. Monhly GDP Ausralian Nominal GDP Ausralian Bureau of Saisics Quarerly Morgage raes are highly correlaed wih he hree-monh zero-coupon raes, as can be seen from Figure 2. A correlaion of 77% is found based on hisorical daa of hese wo ime series. To avoid he issue of collineariy, we decide no o include morgage rae in he VAR model. Insead, morgage raes in our simulaion sudy are compued as he hree-monh zero-coupon rae plus a fixed margin.648%. 9 9 The margin is calculaed based on he average difference beween he morgage raes and he 3-monh zero coupon raes for he period June 993 o June 2. 8

20 (%) Figure 2: Comparison beween Morgage Rae and 3-Monh Zero Coupon Yield MR r^() Quarers Though we would expec ha he enire yield curve, no jus he arbirary mauriy used o consruc he erm spread, would have predicive power, i is difficul o use muliple yields in he VAR regression because of collineariy problems. The high correlaion beween yields wih differen mauriy suggess ha we may be able o condense he informaion conained in many yields down o a parsimonious number of variables (Ang e al. 26). In his paper, we use wo facors from he yield curve, he hree-monh zero-coupon raes, () r, o proxy for he level of he yield curve, and he en-year erm spread, r r (4) (), o proxy for he slope of he yield curve. Also noe ha all he variables are recorded as indices, excep for zero-coupon yields and morgage raes which are given as coninuous compounding raes. In order o keep consisency, we ransform he index variables ino coninuously compounding quarerly growh raes by aking he firs difference of he logged indices, i.e., h log HPI log HPI, y log RYI log RYI, and g log GDP loggdp. The vecor of sae variables can be expressed as () (4) () z r, r r, h, y, g. The plos of raw 9

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