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

Transcription

1 Developing Equiy Release Markes: Risk Analysis for Reverse Morgages and Home Reversions Prepared by Daniel Alai, Hua Chen, Daniel Cho, Kaja Hanewald, and Michael Sherris Presened o he Acuaries Insiue Acuaries Summi 2-2 May 23 Sydney This paper has been prepared for Acuaries Insiue 23 Acuaries Summi. The Insiue Council wishes i o be undersood ha opinions pu forward herein are no necessarily hose of he Insiue and he Council is no responsible for hose opinions. Daniel Alai, Hua Chen, Daniel Cho, Kaja Hanewald, and Michael Sherris The Insiue will ensure ha all reproducions of he paper acknowledge he Auhor/s as he auhor/s, and include he above copyrigh saemen. Insiue of Acuaries of Ausralia ABN Level 7, 4 Marin Place, Sydney NSW Ausralia 2 +6 () f +6 () e [email protected] w

2 Developing Equiy Release Markes: Risk Analysis for Reverse Morgages and Home Reversions Daniel Alai 2, Hua Chen, Daniel Cho 3, Kaja Hanewald 2, and Michael Sherris Paper prepared for he Acuaries Summi An earlier version of his paper has been submied for publicaion. --- 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. [email protected]. 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: [email protected] (Daniel Alai), [email protected] (Kaja Hanewald), [email protected] (Michael Sherris). 3 Mercer Ausralia, address: [email protected] (Daniel Cho).

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 (Brownfield, 22). 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 3

5 regression-based erminaion models used in boh sudies have several drawbacks. Firs, hey rely heavily on availabiliy of daa. Second, hey assume ha 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 4

6 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 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), 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 he iniial house price, moraliy improvemens, and he leverage raio on he payoffs and risk profiles of reverse morgage and home reversion conracs. We find ha he maximum LVRs offered o cusomers in he Ausralian marke are 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 he 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 pricing formulas 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 6

8 Secion 5, numerical examples are used o compare hese wo equiy release producs in erms of payoffs and risks. Secion 6 concludes he paper. 2. Produc Review in Ausralia 2.. The Reverse Morgage Marke 2... Produc Review The reverse morgage marke in Ausralia grew seadily in recen years, despie he impac of he global financial crisis. 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 Ausralia-wide, 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, he Consumer Credi Legislaion Amendmen (Enhancemens) Bill 22 ensures ha all reverse morgages providers 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. 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 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 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). 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 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 balance han anicipaed, which increases he possibiliy of parial non-repaymen when he loan evenually erminaes. 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 US 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. Prior o 25, a sale-and-lease produc was available in Ausralia, however, he provider, Money for Living, was sen 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).

12 Homesafe Soluions Py Ld, a join venure of Bendigo and Adelaide Bank Ld and Ahy Py Ld, has launched Homesafe Deb Free Equiy Release since 25. We review is feaures in Secion 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 is impaired when realized renal yields are greaer han hose assumed a conrac incepion. House Price Depreciaion Risk: Lenders of home reversion conracs are eniled o sell he propery and secure a par of he sale proceeds when homeowners die or volunarily move ou. Therefore, lenders face he risk of house price depreciaion Advanages of Home Reversions From he consumer s poin of view, home reversion producs have unbeaable advanages over reverse morgages. Oliver Wyman Financial Services (28) 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,

13 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, 28). 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 Review of Home Reversion Produc offered in Ausralia This secion reviews he feaures of he home reversion scheme currenly offered by Homesafe Soluions Py Ld. 6 Condiions: The homeowner mus be aged 6 and over. 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 6 More deails can be found on he websie of Homesafe Soluions Py Ld: hp:// see also Brownfield (22). 2

14 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 lump sum paymen in exchange for a fixed proporion of he fuure value of heir home. The paymen is generally in he range of 35% o 7% of he curren marke value of he home. The percenage of marke value is differen for single life conracs and join life conracs, and varies by age and gender. The discoun from marke value implicily allows for he value of he lease for life agreemen ha allows homeowners o live in he house for life, or unil hey sell he home. Homeowners may be eligible for an early sale rebae if hey die or sell heir home in he iniial years of he conrac. Terminaion: The conrac erminaes when homeowners die or sell he propery. Homesafe is eniled o he agreed percenage of he sale proceeds of he house, less any rebae, and homeowners reain he share of he sale proceeds ha hey have no sold o Homesafe. 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. 3

15 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 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 expx. () 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 denoes he conrac erminaion rae. x c x x x, (2) 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 guaranee prices, he difference is no significan (see Figure 3 in Ji e al. 22). 4

16 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 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 mx x x,, (3) PR: ln Dx ln Ex x x,, (4) where ln E is he offse for he Poisson regression based on he survival couns, x 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 : Gomperz 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 Moraliy Raes Age 8 hp:// 5

17 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 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 q x p x s p x x sds, (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. 6

18 Table 3: Average in-force Duraion Age Average in-force duraion 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 (GBM) (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 7

19 (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 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 adjused 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 3-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 8

20 model. Insead, morgage raes in our simulaion sudy are compued as he 3-monh zerocoupon rae plus a fixed margin.648%. 9 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 3-monh zero-coupon raes, r (), o proxy for he level of he yield curve, and he -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., 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. 9

21 h log HPI log HPI, y log RYI log RYI, and g log GDP log GDP. The vecor of sae variables can be expressed as () (4) () z r, r r, h, y, g. The plos of he raw daa and plos of he quarerly housing price growh, renal yield growh, and GDP growh are given in Figures 3 and 4. Figure 3: Plos of Raw Daa r () 2 r (4) 6 r (4) -r () 8 4 (% ) 6 (% ) 8 (% ) Quarers HPI Quarers RYI Quarers 4 x 5 GDP Index Value 5 Index Value 3 2 Index Value Quarers Quarers Quarers Figure 4: Plos of Transformed Daa for House Price Index, Renal Yield Index and GDP h y g (%).5 (%).5 (%) Quarers Quarers Quarers Before esimaing he VAR model, we es saionariy of all variables using he augmened Dickey-Fuller (ADF) es and he Phillips-Perron (PP) es, he resuls of which 2

22 are repored in Table 5. Boh he ADF and PP es resuls indicae ha all variables are saionary a he % significance level, excep for he quarerly renal yield growh rae, y. However, no profound rend is found in he ime series plo of his variable. Sims (99) argues ha he ordinary leas square (OLS) esimaors of VAR parameers are asympoically normally disribued, even if some variables are found o be non-saionary and/or coinegraed. Therefore, we proceed o fi he VAR model wihou any modificaion on he variable y in order o keep consisency and o avoid loss of informaion. Table 5: Saionary Tes Saisics Variables ADF saisic probabiliy saisic probabiliy PP We hen proceed o choosing he opimal lag lengh of he VAR model. This sep is imporan as underfied lag may disregard imporan dynamics of he mulivariae process, whereas overfied lag may violae parsimony (Kilian 2). We compare he Akaike Informaion Crierion (AIC), Schwarz Informaion Crierion (SIC) and Hannan-Quinn Crierion (HQC) o deermine he appropriae lag. Lags of one o six are esed for he above crieria. From Table 6, AIC suggess an opimal lag order of six, whereas boh SIC and HQC indicae an opimal lag order of wo. Lükepohl (25) argues ha SIC and HQC are preferred over AIC as hey are consisen even if he daa series are non-saionary. Ivanov and Kilian (25) illusrae ha he frequency of daa series should be aken ino accoun 2

23 when choosing a lag selecion crierion. They sugges ha HQC is beer when examining monhly or quarerly daa. We choose o fi a lag order of wo based on he HQC. Table 6: Lag Selecion Crierion Lag Order AIC SIC HQC * * * * indicaes lag order seleced by he crierion The VAR (2) model is given by z c z z, (6) /2 2 where z is a ( n ) vecor of sae variables, /2 he Cholesky decomposiion of he covariance marix ha capures he dependence srucure of he sae variables, and ~ (, ) N I. The parameer esimaes are summarized in Table 7. 22

24 Table 7: Esimaed Parameers of VAR (2) VAR(2): (5x) (5x5) (5x5) (5x5) (5x5) The esimaed VAR (2) model is used o simulae he sae variables. We simulae, pseudo random sample pahs of he sae variables for a period of 4 years. As shown in Figure 5, he cumulaive disribuion funcion (CDF) of each of he simulaed sae variables is found o be comparable o is empirical disribuion. We plo he hisorical daa of each variable for he period of June June 2 and he mean simulaed pahs for he period of Sepember 2 - Sepember 25 (as log differences) wih he 9% confidence inerval in Figure 6. The mean simulaed pahs look remarkably sable due o he averaging effec of simulaed pahs. From he visualized confidence inerval, we can see ha he simulaed values of variables span reasonable range of values. We also ransform he quarerly growh raes of house price indices, renal yield indices, and GDP back o he index values in Figure 7. The plos clearly show ha he mean simulaed fuure pahs of he index variables follow he hisorical dynamics. 23

25 Figure 5: CDF of Hisorical and Simulaed Sae Variables..8 Hisoric Simulaed r ().8 r (4) -r ().8 h (%) (%) (%) y g (%) (%) Figure 6: Hisorical and Mean Simulaed Pahs of Sae Variables wih 9% CI 3 r () 2 r (4) -r () h 2 5 (% ) Quarers Quarers Quarers y g (% ) Quarers Quarers 24

26 Figure 7: Hisorical and Mean Simulaed Pahs of Indices (HPI, RYI and GDP) wih 9% CI 3 HPI 4 RYI 5 x 6 GDP 2 3 Index Quarers Quarers Quarers 3.3. Sochasic Discoun Facors In his paper, we follow Ang and Piazzesi (23) and Ang e al. (26) o develop a pricing kernel ha can be used o price all nominal asses in he economy. Denoe he Radon-Nikodym derivaive ha convers he risk-neural measure o he daa-generaing measure. Thus, for any + variable X we have ha E X E X, (7) Q / where he expecaion is aken under he risk-neural measure Q. Assume ha follows he log-normal process exp 2, (8) where are he ime-varying marke prices of risk associaed wih he sources of uncerainy. We parameerize as an affine process of he sae variables z, (9) where is an n-dimensional vecor and is a n n marix accouning for ime-variaion in he risk premia. The pricing kernel or sochasic discoun facor, m, is defined as 25

27 where e,,,,. m exp r exp e z 2, () For an asse having a payoff X a ime +, he price of he asse, P, is given by P E m X. () Paricularly, he price of an n-period nominal bond a ime can be solved recursively by he following formula ( n) ( n ) P E m P, (2) () wih he iniial condiion P. The resuling bond prices are exponenial linear funcion of he sae variables in he VAR, ha is, where A n, B n and ( n) n n n P exp A B z C z (3) C n follow he difference equaions: A A B c BB 2 B B C C /2 n n n n n /2 n n n B n 2 n (4) wih he saring values A and B e and C. ( n ) Given he nominal bond price P, he coninuously compounded yield r on an n- ( n ) period zero-coupon bond is given by ( n) ( n) log P An B n C n r z z. (5) n n n n From he above equaion, i is clear ha he parameer only impacs average erm spreads and average expeced bond reurns, while conrols he ime variaion in erm Please refer o Shao e al. (22) for a deailed proof. 26

28 spreads and expeced reurns. The risk parameers (i.e., and ) can be esimaed condiional on he VAR parameers. This is done by minimizing he sum of he squared differences beween he fied yields of he erm srucure model and hisorical zero-coupon yields, i.e., T N 2 ( n) ( n) min rˆ r. (6) {, } n Besides he 3-monh and he -year zero-coupon yield raes, we calibrae he model o - year, 2-year, and 5-year zero-coupon yields. The esimaed parameers in he marke price of risk are repored in Table 8. Table 8: Esimaed Parameers in he Marke Price of Risk Variables (5x) (5x5) Based on he fied marke price of risk, we calculae he sochasic discoun facors and show is plo in Figure 8. We also show a sample pah of simulaed sochasic discoun facors in he same figure. The correlaions beween he fied sochasic discoun facor and sae variables are repored in Table 9. I can be seen ha he sochasic discoun facor has a high negaive correlaion wih he shor rae, which is inuiive. In addiion, he house price growh posiively conribues o he sochasic discoun facor. 27

29 Figure 8: Sochasic Discoun Facors and Bond Risk Premiums 99 Hisorical SDFs 99 A Sample Pah of Simulaed SDFs Quarers Quarers Table 9: Correlaions beween Sochasic Discoun Facors and Sae Variables Correlaion SDF Risk Analysis In he previous secion, we have described a erminaion model and a VAR model for economic variables. We use hese models o simulae he inpu variables and calculae he provider s capial a some fuure daes. We esimae an empirical disribuion of he capial amoun by running he simulaion procedure a large number of imes. The capial disribuion is hen used o calculae he arge solvency capial level. This simulaion-based approach was also used in Daykin e al. (994), Lee (2) and Tsai e al. (2). Various measures can be used o decide risk-based capial level for solvency requiremen and here is no general consensus as o which one is he mos appropriae. We consider wo commonly used risk measures, VaR and CVaR, o calculae he solvency capial in his paper. 28

30 4.. Payoff Srucure of Reverse Morgages 4... Pricing he No Negaive Equiy Guaranee In a reverse morgage conrac, borrowers are ypically proeced by he provision of he no negaive equiy guaranee. When he loan erminaes, if he ne proceeds from he sale of he propery are sufficien o pay he ousanding loan balance, he remaining cash usually is paid ou o he borrower or his/her beneficiaries. If he proceeds are insufficien o cover he loan balance, he no negaive equiy guaranee prevens he lender from pursuing oher asses belonging o he borrower. Denoe L and H he loan ousanding balance and he value of he propery a ime, respecively. Suppose here is a ransacion cos of selling he house,, given by a percenage of he house value. The payoff of he no negaive equiy guaranee a loan erminaion ime is NN max L H,. (7) In our analysis, we consider a lump sum payou opion, which is mos popular payou in Ausralia. The maximum iniial loan amoun is deermined by he LVR ha is se as a proporion of he value of he propery. LVRs increase wih he age a which he loan is aken ou. Suppose he borrower always akes ou % of he allowable limi, i.e., L H LVR. The loan accrues quarerly wih ineress and morgage insurance premiums. As aforemenioned, he variable morgage rae is compued by adding a fixed margin on op of he shor rae (3-monh zero coupon rae). Thus, L is given by s L L exp ri, (8) i s where r denoes he 3-monh zero-coupon rae, is he lending margin and is he morgage insurance premium rae. 29

31 As he erminaion ime is random, we use he probabiliy of conrac erminaion, q, o model he randomness of loan erminaion. We hen use sochasic discoun facors, c x m, o discoun he value of he no negaive equiy guaranee a an arbirary erminaion ime o he ime of loan originaion, aking ino accoun he uncerainy in he fuure developmen of house prices, renal yields, and ineres raes. Hence, he value of he no negaive equiy guaranee, NN, is given by x NN E m q L H s c s x max,. (9) The no negaive equiy guaranee is usually financed by morgage insurance premiums paid by he borrowers. Previous sudies usually assume a zero up-fron premium and a fixed premium rae each period. This assumpion is in line wih marke pracice in Ausralia, where he morgage insurance is implicily charged via a higher ineres rae on reverse morgage loan. However we noe ha reverse morgage providers in Ausralia se he ineres rae equal for all loans regardless of heir iniial loan-o-value raio, alhough i is clear from equaion (9) ha he value of no negaive equiy guaranee depends on he size of he loan. The acuarial presen value of morgage insurance premiums, MIP, is hen given by x MIP E m p L s c s. (2) x The acuarially fair quarerly premium rae can be calculaed by equaing he value of morgage insurance premiums wih he value of he no negaive equiy guaranee Cash Flows of he Reverse Morgage Conrac We assume ha he provider of a reverse morgage conrac finances he payou hrough is exising capial and leveraging. The proporion of borrowed capial, or he leverage raio (LR), is denoed by. The borrowed capial accrues wih he shor rae. Therefore, he oal financing cos a ime can be wrien as 3

32 The provider receives min L, RM C L exp r L s i. (2) i H from he sale proceeds of he propery when he loan erminaes. Is ne payoff discouned back o ime zero can be calculaed as x c exp s RM x i min,. (22) RM q r L H C i 4.2. Payoff Srucure of Home Reversions Pricing he Lease for Life Agreemen Under a home reversion conac, he provider buys a share of he propery a a discouned price and offers he cusomers a lease for life agreemen. The agreemen can be valued using annuiy pricing echniques, where he annuiy is indexed o he propery s renal yield rae. For he purpose of comparison, we assume ha he acquisiion raio is he same as he LVR in he reverse morgage. For a cerain lifespan, he value of he lease for life agreemen a ime can be expressed as a funcion of he erminaion ime T, where T LL E m H R LVR s R denoes he renal yield rae in year. s, (23) Again, he erminaion ime T is random. Therefore, he acuarial presen value of he lease for life agreemen can be wrien as x LL E m p H R LVR s Cash Flows of he Home Reversion Conrac H c s x. (24) In a home reversion conrac, he provider purchases a share of he equiy ha is worh LVR and discouns i by he value of he lease for life, LL. The resuling lump-sum paymen a conrac originaion is H LVR LL. Again, he provider is assumed o finance 3

33 he payou by borrowing % of he required capial. A he ime of loan erminaion, he propery is sold and he provider receives a share of he sale proceeds, which is Thus he provider ne presen value of payoffs a ime zero is given by HR q r H LVR C i H LVR. x c s HR x exp i, (25) HR s where he oal cos C H LVR LLexp r H LVR LL 5. Numerical Illusraion. i i In his secion, we compue he value of he no negaive equiy guaranee in he reverse morgage conrac and he value of he lease for life in he home reversion conrac. We hen compare hese wo equiy release producs wih respec o profiabiliy and risk under various scenarios. We conduc sensiiviy analyses o idenify he impacs of key facors, such as age a conrac originaion, he iniial house value, moraliy improvemen and he leverage raio, on cash flows and risk profiles of boh equiy release producs. 5.. The Base Case Scenario In he base case scenario, we assume a single female aged 65 residing in Sydney, Ausralia, wih an iniial house value of $6,. To finance her reiremen consumpion and/or aged care, she can eiher ener a reverse morgage conrac or sell a share of he equiy by enering a home reversion conrac. If she decides o paricipae in he home reversion scheme, he acquisiion raio is se o be he same as he LVR for he purpose of comparison. We assume ha he equiy release provider finances he lump-sum payou o he homeowner compleely hrough borrowed capial, i.e., he leverage raio is %. Noe ha he prevalen maximum LVRs in Ausralia are much lower han hose used in he US. Figure 9 compares ypical maximum LVRs for differen borrower ages in Ausralia and in he US HECM program. The maximum LVR increases wih age because he Median Price and Number of Esablished House Transfer, Ausralian Bureau of Saisics. 32

34 ime horizon for he loan accumulaion is shorer. The US marke is overwhelmingly led by HECM producs, which offer significanly more generous LVRs han comparable producs in foreign markes. For example, he ypical US LVR is more han quadruple ha of Ausralia for borrowers aged 65 and more han double for age 75 and 85. Many lenders have recenly reduced heir HECM ineres rae margins o arac addiional sales, which has produced even higher LVRs. We will show laer ha his disincion makes he Ausralian equiy release producs carry a quie differen payoff and risk srucure compared o he US producs. Figure 9: LVRs in Ausralia vs. LVRs in he US 8 LVR in Ausralia LVR in he US (% ) Age 65 Age 75 Age 85 We projec he probabiliy of loan erminaion based on he erminaion model presened above and simulae, pahs of he economic variables based on he VAR(2) model for 4 years. We assume he provider of he reverse morgage charges a zero up-fron premium and annual premiums wih an acuarially fair rae. We hen calculae he value of no negaive equiy guaranee. For he home reversion, we calculae he value of he lease for life agreemen. We obain he disribuion of he acuarial presen value of payoffs of he provider for boh producs. Given he payoff disribuions, we assess riskiness of each program by compuing VaR and CVaR a he 99.5% level. Table summarizes he resuls in he base case scenario. 33

35 Table : Payoffs and Risks in he Base Case Scenario Assumpion: Age=65, H=$6,, LR =%, No moraliy improvemen LVR Reverse Morgage Home Reversion NN E[RM] VaR CVaR LL E[HR] VaR CVaR 5% 29,623 35,764 25,96-3,873-6,564 64% 39,28 82,55-78,849-93,94 52,593,533-6,524-28,5 Noe: NN is he value of he no negaive equiy guaranee and LL is he value of he lease for life agreemen. E[RM] (or E[HR]) denoes he average acuarial presen value of he reverse morgage (or home reversion) conrac. VaR and CVaR are calculaed a he 99.5% level. When we use he maximum LVR ypically found in Ausralia (5% for age 65), he no negaive equiy guaranee has no values, which shows he reverse morgage loans has virually no likelihood of losses. As a resul, he acuarially fair premium for he guaranee is zero. However, he fac is ha reverse morgage providers in Ausralia charge more han % insurance premiums o proec hemselves from crossover risk (Bridge e al. 2). Our resuls show ha here is a possibiliy of reducing ineres raes for reverse morgage loans o be closer o hose for sandard home loans. The VaR and CVaR a he 99.5% level are boh zero, implying ha reverse morgage providers do no need o se aside risk-based capial. This finding is consisen wih he commens from many brokers ha LVRs in Ausralia are se oo conservaive and ha he premium or fees could be lowered given he very low risk of defaul or even of negaive equiy being reached (Bridge e al. 2). On he conrary, our resuls show ha home reversion providers do bear some risks and need o reserve some solvency capial. The risk mainly comes from he housing price depreciaion. 2 2 The resuls are similar when we change he age o 75 and 85 and use he corresponding maximum LVRs in Ausralia (i.e., 3% and 35%). 34

36 Figure : Loan Ousanding Balance L and he Sale Proceeds of he Propery H (LVR=64%) 6 x 6 5 (-r)h L Reverse Morgage: (-)h versus L 4 ($) Quarers Figure : Disribuions of he Acuarial Presen Value of Ne Payoffs (LVR=64%) 5 x 5 Quaniles of EPV: Reverse Morgage and Home Reversion 4 3 Final Payoff ($) 2 - RM HR Quaniles (%) We also produce resuls assuming a high LVR ha can be found in he US HECM program (64% for age 65). The LVRs are subsanially higher in he US and his has a significan impac on he risk profiles of equiy release producs. The simulaion resuls show ha negaive equiy resuls in several scenarios, which suggess ha he reverse morgage 35

37 providers offering a high LVR would face crossover risks. In order o beer undersand he developmen of negaive equiy in a high LVR case, we plo he loan ousanding balance, L, versus he sale proceeds of he propery, H, over ime in Figure. Compared wih he variabiliy of house price oucomes, he loan balance (driven by ineres rae flucuaions) is much less volaile. Negaive equiy arises when he accumulaed loan balance crosses over he sale proceeds of he propery. Crossover risk occurs afer 2 years of he loan duraion. If we consider a severe housing marke downurn (represened by he lower 5% quanile of he house price disribuion), negaive equiy occurs afer circa five years. Figure gives he quanile disribuion of he acuarial presen value of ne payoffs for boh equiy release producs. The graph shows ha he home reversion conrac is more profiable and less risky han he reverse morgage when a LVR of 64% is assumed as found in he US marke. The comparison beween reverse morgages and home reversions yields conradicing resuls when using he LVR found in Ausralia versus ha ypical of he US. The appropriae seing of LVRs is a key issue. In order o furher invesigae how he LVR affecs he payoff and risk srucure of hese wo producs, we fix he iniial age o be 65 and he iniial house value o be $6, and vary he LVR from 5% o 64%. The resuls are shown in Table. 36

38 Table : The Impac of he LVR Assumpion: Age=65, H=$6,, LR =%, No moraliy improvemen LVR Reverse Morgage Home Reversion NN E[RM] VaR CVaR LL E[HR] VaR CVaR 5% 29,623 35,764 25,96-3,873-6,564 25% 49,2 59,67 43,77-6,454 -,939 35% 64 68,23 83,449 6,447-9,37-5,36 4%,66 76,262 95,37 69,82 -,328-7,54 45% 3,636 83,52-7,293 7,292 77,79 -,68-9,69 5% 7,456 88,3-2,84-27,45 9,23 86,354-2,99-2,879 55% 4,78 9,87-34,94-49,778 3,34 94,989-4,2-24,67 64% 39,28; 82,55-78,849-93,94 52,593,533-6,524-28,5 Noe: NN is he value of he no negaive equiy guaranee and LL is he value of he lease for life agreemen. E[RM] (or E[HR]) denoes he average acuarial presen value of he reverse morgage (or home reversion) conrac. VaR and CVaR are calculaed a he 99.5% level. The change in payoff and risk for home reversion schemes has a clear rend, i.e., he average payoff increases wih he LVR and so does he risk. This is inuiive since wih a higher LVR, boh he payoff and risk are magnified. We need o ake a closer look a reverse morgages since LVRs play a more imporan role in reverse morgages and cause some rend changes. The value of he no negaive equiy guaranee increases wih he LVR since a larger LVR reduces he gap beween he house price and he loan balance, resuling in a higher crossover risk. When he LVR is low, he guaranee has a zero or a small value, indicaing no or low crossover risk. In his case, he provider would receive he ousanding loan balance a loan erminaion. So he provider s payoff is mainly he accumulaion of he lender s margin based on he iniial loan amoun. As a resul, a larger LVR leads o a higher payoff for he provider. However, when he LVR increases above a criical level, negaive equiy can occur and reduce he payoff. For he same reason, he risk measure sars a zero bu increases when he LVR is higher han 5%. We conclude ha reverse morgage providers receive higher 37

39 average payoffs han home reversion providers and bear nearly no risk for LVR levels lower han 5%. For higher LVR levels, expeced payoffs from reverse morgages become less and he risk urns ou o be higher han home reversions Sensiiviy Analysis In he following analysis, we use LVRs se by he US HECM program in order o avoid zero risk in reverse morgages and observe clear rends on comparaive resuls Sensiiviy o he Iniial Age The borrower s age has wo compeing effecs on he risk/payoff srucure: an increase in age reduces he average ime of in-force duraion and hus lowers he crossover risk; a he same ime he resuling increase in LVR raises he iniial loan amoun and leads o higher crossover risk. We find ha he value of he no negaive equiy guaranee is lower for reverse morgage loans wih a higher borrower age, showing ha he age s effec on loan erminaion dominaes he age s effec on LVRs. For he same reason, he risk (measured by VaR and CVaR) decreases wih age. As o he expeced payoff, he provider has less ime o accumulae profis when he loan is issued o an older borrower, whereas he increase in he LVR, or a larger iniial loan amoun, resuls in a higher margin accumulaion unil loan erminaion. The dominan effec of loan duraion resuls in he payoff decreasing wih age. The same logic applies equally o home reversion schemes, bu we should keep in mind ha he age effec on loan erminaion akes over. The value of he lease for life decreases wih age because an older age means a shorer ime period ha rens are payable. Home reversion providers gain from he fuure house price appreciaion. Neverheless, a higher age a conrac originaion allows less ime for he propery value o appreciae. So he payoff decreases wih age. The risk increases wih age for a similar reason. Compared wih he reverse morgage provider, he home reversion provider receives a higher payoff on average and bears a lower risk. 38

40 Table 2: Sensiiviy o he Iniial Age Assumpions: H=$6,, LR=%, No moraliy improvemen Age LVR Reverse Morgage Home Reversion NN E[RM] VaR CVaR LL E[HR] VaR CVaR 65 64% 39,28 82,55-78,849-93,94 52,593,533-6,524-28,5 75 7% 29,523 59,254-56, -7,39 6,934 85,663-25,35-36, % 8,3 33,583-42,686-5,783 72,86 46,588-4,972-48,7 Noe: NN is he value of he no negaive equiy guaranee and LL is he value of he lease for life agreemen. E[RM] (or E[HR]) denoes he average acuarial presen value of he reverse morgage (or home reversion) conrac. VaR and CVaR are calculaed a he 99.5% level Sensiiviy o he Iniial House Value Changing he iniial house price has a monoonic effec on he payoff and risk srucure. I is eviden ha he value of he no negaive equiy guaranee and ha of he lease for life decrease proporionally wih he iniial propery value. The average payoff and he ail risk decrease wih he house price for boh producs, bu payoffs from he home conrac are higher for he provider and his conrac bears less risk han he reverse morgage. Table 3: Sensiiviy o he Iniial House Value Assumpions: Age=65, LVR=64, LR=%, No Moraliy Improvemen H Reverse Morgage Home Reversion NN E[RM] VaR CVaR LL E[HR] VaR CVaR 6, 39,28 82,55-78,849-93,94 52,593,533-6,524-28,5 54, 35,352 73,94-7,964-84,547 37,333 99,479-4,872-25,25 48, 3,424 65,724-63,79-75,53 22,74 88,426-3,29-22,44 Noe: NN is he value of he no negaive equiy guaranee and LL is he value of he lease for life agreemen. E[RM] (or E[HR]) denoes he average acuarial presen value of he reverse morgage (or home reversion) conrac. VaR and CVaR are calculaed a he 99.5% level Sensiiviy o Moraliy Improvemens Table 4 illusraes he effec of moraliy improvemen on payoff and risk. The erminaion model used o deermine conrac erminaion probabiliies is based on populaion moraliy raes. Moraliy improvemens can lenghen he conrac duraion and herefore 39

41 increase he value of he no negaive equiy guaranee and ha of he lease for life agreemen. Moraliy improvemen has a relaively small impac on he average payoff and he risk embedded in he equiy lease producs. Table 4:Sensiiviy o Moraliy Improvemen Assumpions: Age=65, LVR=64%, H=$6,, LR=% Moraliy Improvemen Reverse Morgage Home Reversion NN E[RM] VaR CVaR LL E[HR] VaR CVaR % 39,28 82,55-78,849-93,94 52,593,533-6,524-28,5 % 43,367 82,376-84,832 -,979 58,69 3,69-5,765-27,6 2% 46,523 82,594-9,338-6,769 62,558 6,28-5,479-27,582 Noe: NN is he value of he no negaive equiy guaranee and LL is he value of he lease for life agreemen. E[RM] (or E[HR]) denoes he average acuarial presen value of he reverse morgage (or home reversion) conrac. VaR and CVaR are calculaed a he 99.5% level Sensiiviy o he Leverage Raio Lasly, we change he leverage raio given by he percenage of he payou ha he equiy release provider finances hrough exernal sources. The decrease in he leverage raio has no impac on he value of he no negaive equiy guaranee and ha of he lease for life (which one would expec and we do no repor in Table 5), bu resuls in an increase in average payoffs and a decrease in risk for boh producs. Table 5:Sensiiviy o he Leverage Raio Assumpions: Age=65, LVR=64%, H=6,, No moraliy improvemen Leverage Raio Reverse Morgage Home Reversion E[RM] VaR CVaR E[HR] VaR CVaR % 82,55-78,849-93,94,533-6,524-28,5 9% 3,72-57,83-72,434 23,257-3,79-5,3 8% 24,286-35,427-5,92 35,98-2,7 Noe: NN is he value of he no negaive equiy guaranee and LL is he value of he lease for life agreemen. E[RM] (or E[HR]) denoes he average acuarial presen value of he reverse morgage (or home reversion) conrac. VaR and CVaR are calculaed a he 99.5% level. 4

42 6. Conclusions and Discussions The acuarial lieraure on pricing of equiy release producs is sill raher limied. In his paper, we analyse cash flows and risk profiles for equiy release producs from he provider s perspecive. We assume a single female policyholder who inends o make use of eiher he reverse morgage or he home reversion scheme o liquidae her equiy and finance her reiremen consumpion and care coss. We find ha wih a low LVR, reverse morgages provide a higher payoff and deliver less risk o he provider han home reversions. This finding jusifies he dominan marke share of reverse morgage schemes in Ausralia and many oher counries, such as he UK. When we use a high LVR, as found in he US HECM program, we find ha home reversions are beer in erms of he payoff and risk srucure for he provider han reverse morgages. The appropriae seing of LVRs plays an imporan role in he produc risks. Our resuls indicae ha reverse morgage providers in Ausralia could consider increasing maximum LVRs and decreasing insurance premium raes or on-going fees, in order o expand he reverse morgage marke. Usually, he LVR depends on he age of he borrower a loan originaion. Our sensiiviy analysis indicaes ha among all he facors ha we consider, he iniial age of homeowners has a profound and significan impac on payoffs and risks of equiy release produc providers. I affecs boh he conrac erminaion ime and he LVR (hus he iniial payou o consumers) and resuls in wo compeing effecs on he risk and payoff profile. Cauion has o be used when deermining he LVR based on age. Our resuls have imporan implicaions o policymakers and regulaors in many oher counries ha face he issue of aging populaion and underfunded pensions. For example, he UK has a similar, conservaive paern of LVRs as in Ausralia. UK providers have he poenial o increase LVRs o simulae he reverse morgage marke. Though our resuls indicaes a high LVR as found in he US makes reverse morgage producs less profiable and 4

43 riskier han home reversion schemes, his has been based on economic scenarios from Ausralia experience. The US housing marke and economic condiions have been quie differen in recen years and his has o be considered when assessing he US markes. In addiion, in he US, he HECM providers are insured by he federal governmen and can ransfer he risk o FHA. As a newly developed equiy release produc, he home reversion scheme has advanages o boh homeowners and invesors. I usually ses a limi on he share of equiy ha can be sold o a home reversion company, leaving a remainder o consumers which can be used o fund aged care afer he propery is sold. As an asse class, much of he risk aached o radiional propery invesmen is eiher irrelevan in home reversion conracs such as enancy or defaul risk, or can be diversified in a pooled residenial propery pool, for example, duraion risk and locaion risk (Deloie 2). However, he privae marke for home reversions has been developing slowly. Lack of awareness and low financial lieracy among consumers are he main reasons on he demand side. In paricular, he implici lease for life agreemen in he home reversion conrac may be poorly undersood. On he supply side, liquidiy is he major concern of invesors. In addiion o providing an appropriae framework of regulaion and educaion, governmens should consider policies o suppor he developmen of he equiy release marke such as providing liquidiy for providers. Acknowledgemen The auhors acknowledge he suppor of ARC Linkage Gran Projec LP Managing Risk wih Insurance and Superannuaion as Individuals Age wih indusry parners PwC and APRA and he Ausralian Research Council Cenre of Excellence in Populaion Ageing Research (projec number CE29). Hua Chen also acknowledges he financial suppor 42

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