Risk Management and Payout Design of Reverse Mortgages

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1 Risk Managemen and Payou Design of Reverse Morgages Daniel Cho, Kaja Hanewald and Michael Sherris School of Risk and Acuarial Sudies and Ausralian Research Council Cener of Excellence in Populaion Ageing Research (CEPAR), Universiy of New Souh Wales, Sydney, Ausralia. 13h May 2013 Paper prepared for he 48h Acuarial Research Conference (ARC), Temple Universiy, Philadelphia Absrac We analyze he risk and profiabiliy of reverse morgages wih lump-sum or income sream paymens from he lender s perspecive. Reverse morgage cash flows and loan balances are modeled in a muli-period sochasic framework ha allows for house price risk, ineres rae risk and risk of delayed loan erminaion. A VAR model is used o simulae economic scenarios and o derive sochasic discoun facors for pricing he no negaive equiy guaranee embedded in reverse morgage conracs. Our resuls show ha lump-sum reverse morgages are more profiable and require less risk-based capial han income sream reverse morgages, which explains why his produc design dominaes in mos markes. The loan-o-value raio, he borrower s age, moraliy improvemens and he lender s financing srucure are shown o be imporan drivers of he profiabiliy and riskiness of reverse morgages, bu changes in hese parameers do no change he main conclusions. Keywords: Reverse morgage; Income sream; Equiy release; Vecor auoregressive model; Sochasic discoun facor; Risk-based capial JEL Classificaions: G12; G21; G32 [Corresponding auhor], Posal address: Ausralian Research Council Cenre of Excellence in Populaion Ageing Research, Ausralian School of Business, Universiy of New Souh Wales, Sydney NSW 2052, Ausralia; Phone: ; Fax:

2 1 Inroducion Populaion ageing is a global phenomenon and he quesion of how o finance he reiremen and healh care coss of a rapidly growing older populaion is becoming a major challenge. Households are expeced o rely more and more on privae savings. A subsanial par of household wealh is held as real esae. Homeownership raes are high among he elderly in mos developed counries (Chiuri and Jappelli, 2010). Reverse morgages allow reirees o ransform heir housing wealh ino liquid asses while saying in heir home. Reverse morgages are increasingly used by reirees. The produc is available in numerous counries including Ausralia, Canada, he US, he UK, India and Singapore. The financial crisis has slowed down marke growh, especially in he US, bu several markes including Ausralia and he UK have recovered and show srong growh raes (Deloie and SEQUAL, 2012; Key Reiremen Soluions, 2013). From a lender s perspecive reverse morgages differ from (forward) morgages largely because of he dependence of cash flows on longeviy risks, an area aracing increased ineres in he banking and finance lieraure (Horneff e al., 2010, 2009). Reverse morgages were iniially designed o provide a regular reiremen income and/or a line of credi for major expenses such as healh care coss or home repairs (Chinloy and Megbolugbe, 1994; Consumer Financial Proecion Bureau, 2012; Veni and Wise, 1991). Mos markes oday are dominaed by reverse morgages wih lump-sum paymens (Clerc-Renaud e al., 2010; Consumer Financial Proecion Bureau, 2012; Deloie and SEQUAL, 2012). The Consumer Financial Proecion Bureau (2012) repors ha U.S. borrowers increasingly use heir reverse morgage loans o refinance radiional morgages. Psychological aspecs also play a role: he life-ime reverse morgage income may only moderaely increase household income, whereas he equivalen lump-sum paymen would increase liquid wealh by a large fracion (Veni and Wise, 1991). Also, over-housed reirees may use he lump sum o reduce heir house price risk exposure and o diversify across asse classes (Pelizzon and Weber, 2009). Anoher imporan reason is complexiy. Reverse morgages have been criicized as oo complex for consumers (see, e.g., Consumer 2

3 Financial Proecion Bureau, 2012) and his applies paricularly o income sream reverse morgages. We ake a closer look a his growing marke and analyze he risk and profiabiliy of reverse morgage loans wih differen payou opions from he lender s perspecive. In paricular, we invesigae how lump-sum reverse morgages and income sream reverse morgages wih fixed or inflaion-indexed paymens are impaced differenly by house price risk, ineres rae risk and erminaion risk. Our sudy exends he growing lieraure on he pricing and risk managemen of reverse morgages (see, e.g., Alai e al., 2013; Chen e al., 2010; Hosy e al., 2008; Shao e al., 2012). Two previous sudies have developed pricing frameworks for reverse morgage loans ha provide regular income paymens, focusing on valuing he cross-over opion (Chinloy and Megbolugbe, 1994) and he fair value of he regular paymens (Lee e al., 2012). We assess he lender s ne financial posiion and required risk-based capial for hree payou ypes of reverse morgages. Our sudy adds a new conribuion o he limied lieraure on risk-based capial requiremens for residenial morgage porfolios (see, e.g., Calem and LaCour-Lile, 2004; Qi and Yang, 2009). We employ a muli-period sochasic framework for modeling and pricing reverse morgage cash flows ha exends he models used in Alai e al. (2013) and Shao e al. (2012). Loan erminaion probabiliies are derived from a muli-sae Markov model. A vecor auoregressive model is used o generae economic scenarios and o derive sochasic discoun facors ha reflec he key risk facors of reverse morgage cash flows and heir dependencies. The sochasic discoun facors are used o price he no negaive equiy guaranee ypically embedded in reverse morgage conracs and o deermine he risk premium lenders should charge for his guaranee. We compue risk measures such as he Value-a-Risk (VaR) and he Condiional Value-a-Risk (CVaR) o deermine he amoun of risk-based capial he lender should se aside for each ype of reverse morgage. The resuls of our sudy show ha lump-sum reverse morgages have lower risks and are more profiable han income sream producs. Lump-sum reverse morgages sar wih 3

4 higher loan balances and accumulae o higher levels early on. However, hey are also less exposed o longeviy risk han income sream producs. The effecive loan amoun for an income sream is deermined by he number of paymens made, which is subjec o longeviy risk. When an individual lives longer, he accumulaed loan amoun of he income seam exceeds he equivalen lump-sum loan a he older ages. This main resul is robus o changes in he conrac characerisics and o key model assumpions esed in he sensiiviy analysis. The resul provides an addiional explanaion as o why lump-sum producs are he mos popular ype of reverse morgage inernaionally. The sensiiviy analysis confirms he loan-o-value raio and he borrower s age as imporan pricing facors for all hree reverse morgages ypes. The remainder of his aricle is organized as follows. Secion 2 inroduces he differen reverse morgage conracs and describes heir cash flows. Secion 3 ses ou he muliperiod sochasic reverse morgage pricing and risk managemen framework. Secion 4 repors he resuls and Secion 5 concludes. 2 Reverse Morgage Produc Design Deailed descripions of exising reverse morgages producs can be found, for example, in Chen e al. (2010) for he US marke, in Hosy e al. (2008) for he UK and in Alai e al. (2013) for reverse morgages in Ausralia. In he following, sylized producs wih ypical produc characerisics are described. Under a reverse morgage, he lender advances he borrower cash and akes a morgage charge over he borrower s propery. Borrowers reain he righ o say in heir home unil hey die or sell he propery. In eiher case, he conrac is erminaed, he propery is sold and he loan and he accumulaed ineres are repaid. Conracs also ofen allow for refinancing or early repaymen. Reverse morgage loans are ypically non-recourse: borrowers are proeced from providing asses oher han he house by he no negaive equiy guaranee (NNEG). The maximum loan amoun is deermined by he age and gender of he borrower and he appraisal value of he propery. Reverse morgages can 4

5 be issued o couples or single borrowers and can carry fixed or variable ineres raes. Reflecing ypical loan characerisics, we model reverse morgages wih variable ineres raes issued o a single female borrower. Reverse morgages expose lenders o house price risk, ineres rae risk and he risk of delayed erminaion (longeviy risk). These inerrelaed risks impac lump-sum and income sream reverse morgages o differen exens. For example, a longer loan duraion increases he probabiliy ha he loan balance exceeds ( crosses over ) he propery value a mauriy for boh reverse morgage ypes and resuls in a more paymens o he borrower for income sream producs. The following subsecions describe he producs cash flows in more deail. The cash flows are modeled on a quarerly basis. 2.1 Lump-Sum Reverse Morgage The mos common ype of reverse morgage pays ou he loan amoun as a lump sum a he beginning of he conrac. We denoe he ousanding loan balance a ime as L LS and he value of he propery a as H. The lump-sum paymen o he borrower is P LS = L LS 0. Each quarer, he loan balance increases by a variable morgage rae, r κ and by he riskadjused premium rae for he NNEG, π (see Secion 2.3). The loan balance a ime is given by: L LS { } = L LS 0 exp (ri κ + π), (1) i=0 The reverse morgage lender finances he lump-sum payou wih equiy and wih borrowed capial. The borrowing raio is denoed by ϕ. Borrowed capial is assumed o accumulae wih he shor rae, r (1). The financing cos for a single lump-sum reverse morgage loan 5

6 a ime is given by: C LS = ϕl LS 0 exp { i=0 r (1) i } + (1 ϕ)l LS 0. (2) The payoff he lender receives a he dae of loan erminaion, T, is capped by he sale proceeds of he propery by he NNEG. We assume ha here is a proporional ransacion cos, γ, of selling he house. The dae of loan erminaion is random, deermined by he borrower s healh sae and prepaymen or refinancing decisions. We model he randomness using he probabiliy of loan erminaion for a borrower iniially aged x, q c x. The expeced presen value of he lender s ne payoff from he lump-sum reverse morgage is given by: EP V LS = ω x 1 =0 q c xexp { i=0 r (1) i } [min(l LS where ω is he maximum aainable age of he borrower. ], (1 γ)h ) C LS, (3) 2.2 Income Sream Reverse Morgage Figure?? compares he developmen of he loan balance over ime for lump-sum and income sream reverse morgages wih fixed paymen amouns. Income sream reverse morgages pay regular paymens o he borrower unil he conrac is erminaed. We model wo ypes of paymens: fixed and inflaion-indexed. In boh cases, he loan balance sars very low wih he firs paymen and increases quarerly wih he accrued variable morgage rae, each new paymen o he borrower, and he risk-adjused premium rae for he NNEG, π. Income sream reverse morgages accumulae ineres raes slower, which preserves he equiy of he collaeralized propery and helps miigae longeviy risk as described laer. To make he income sream produc comparable wih he lump-sum reverse morgage, we calibrae he quarerly income paymens such ha he expeced presen value of 6

7 Illusraion: Developmen of he loan balance over ime. (Loan principal and accumulaed ineres and NNEG premiums) T T Quarers (a) Lump-sum reverse morgage. Quarers (b) Income sream reverse morgage. all paymens equals he lump-sum paymen, P LS. The quarerly fixed paymens are calculaed as: ω x 1 P LS = P IS =0 { } p c xexp r () 0, (4) where r () 0 are quarerly zero-coupon yields for mauriy a ime zero and p c x he probabiliy ha he reverse morgage loan is in-force in year. The quarerly inflaion-indexed paymens are derived similarly by: P LS = P IIS ω x 1 =0 p c xexp { r () 0 + } d ln CP I i. (5) where d ln CP I is he quarerly inflaion rae and CP I is he consumer price index. Inflaion-indexed paymens are iniially lower han fixed paymens and increase quarerly. 1 i=0 The ousanding loan balances of he fixed income sream reverse morgage a ime, 1 The paymens in Equaions (4) and (5) are discouned using zero-coupon yields for mauriy a ime zero. Zero-coupon yield daa for June 2011 published by he Reserve Bank of Ausralia ( Zero-coupon Ineres Raes - Analyical Series, accessed Augus 2012) was used. The original daa is provided for mauriies of up o 10 years. The Nelson-Siegel funcion (Nelson and Siegel, 1987) was fied o exrapolae yields for higher mauriies. The Nelson-Siegel funcion is a parsimonious model for yield curves, which was found o provide a very good fi. 7

8 L IS, and of he inflaion-indexed income sream reverse morgage, L IIS, are given by: L IS L IIS = P IS = P IIS { i } exp rk κ + π i=0 k=0 (6) { i } exp rk κ + π + d ln CP I k. (7) i=0 k=0 As in he case of he lump-sum reverse morgage, we assume ha he lender finances he paymens o he borrower wih capial and borrowings/deposis. The lender mainains he required amoun of capial and borrowing o mee each paymen as i is made. Borrowed capial accumulaes wih he shor rae. The oal cos of financing for income sream reverse morgages is given by: C IS C IIS = ϕp IS = ϕp IIS { i exp i=0 k=0 { i exp i=0 k=0 r (1) k r (1) k } } + (1 ϕ)p IS (8) + (1 ϕ)p IIS. (9) Similar o he lump-sum reverse morgage, he ousanding loan balance is fully paid only if he ousanding loan amoun balance is lower han proceeds from he propery less ransacion coss. The expeced presen value of he lender s ne payoff from he wo income sream reverse morgages is: EP V IS = EP V IIS = ω x 1 =0 ω x 1 =0 q c xexp q c xexp { { i=0 i=0 r (1) i r (1) i } } [min(l LI, (1 γ)h ) C IS ] (10) [min(l LI, (1 γ)h ) C IIS ]. (11) 2.3 Pricing he No Negaive Equiy Guaranee The reverse morgage conracs considered in his sudy are non-recourse. The no negaive equiy guaranee (NNEG) limis he loan repaymen o he sale proceeds of he 8

9 propery. The lender s payoff from he NNEG a he ime of loan erminaion, T, is: NNEG T = max(l T (1 γ)h T, 0), (12) where L T is he ousanding loan balance a erminaion, H T is he value of he propery and γ are he proporional sale ransacion coss. NNEG T is he same for lump-sum or income sream reverse morgages. The srucure of he NNEG is similar o ha of a series of European pu opions wih uncerain mauriy T (Chen e al., 2010; Chinloy and Megbolugbe, 1994). Previous research has priced he NNEG using he Black-Scholes opion pricing framework (Ji, 2011). Li e al. (2010) and Chen e al. (2010) sugges ha he Black-Scholes assumpions are no appropriae for he dynamics of he underlying house price. We adop he pricing approach used in wo recen sudies, where risk-adjused sochasic discoun facors are used o discoun he cash flows arising from he NNEG (Alai e al., 2013; Shao e al., 2012). Using he same noaion, he expeced presen value of he NNEG is given by: NN = ω x 1 =0 [ ] E (m s ) qx c max (L (1 γ)h, 0), (13) s=0 where m is he quarerly sochasic discoun facor a ime. The esimaion of he discoun facors, which reflec house price risk, ineres rae risk, renal yield risk and inflaion risk, is described in Secion 3.3. We assume ha coss of providing he NNEG are charged o he borrower in he form of a quarerly premium a a fixed rae, π, applied o he loan amoun. The premiums are accumulaed and paid a he erminaion of he conrac. The expeced presen value of all premiums payable hroughou he loan duraion is given by: ω x 1 MIP = π E =0 [ ] (m s ) p c xl. (14) The fair premium rae, π, is calculaed by seing he expeced presen value of he NNEG s=0 9

10 equal o he expeced presen value of he oal insurance premium: NN = MIP. From he equaions above, he value of NNEG depends on how he loan balance accumulaes over ime. The morgage insurance premiums for lump-sum and income sream reverse morgages will differ as a resul. 3 The Reverse Morgage Pricing Framework This secion describes he framework used o simulae reverse morgage cash flows and o analyze he lender s ne financial posiion. We adop and exend he pricing mehod used in wo recen sudies (Alai e al., 2013; Shao e al., 2012). Ausralian marke and moraliy daa is used o calibrae he model. The Ausralian reverse morgage marke has nearly ripled in erms of he oal loan book size over he las decade and is expeced o coninue growing (Deloie and SEQUAL, 2012). 3.1 The Muli-Sae Markov Terminaion Model The probabiliy of reverse morgage loan erminaion for a single female borrower iniially aged x is derived from he Markov erminaion model developed in Alai e al. (2013) based on work by Ji (2011). We exend he model by Alai e al. (2013) by including prepaymen and refinancing as causes of loan erminaion in addiion o deah and long-erm care move-ou. All four causes were also considered by Ji (2011). There is no publicly available daa on reverse morgage erminaions in Ausralia, so we adop several assumpions made by Ji (2011) for he US and he UK. A-home moraliy raes are derived by scaling down he underlying age-specific moraliy raes wih a facor θ x o represen he beer healh of reirees who have no moved ino a long-erm care faciliy. The probabiliy of a move ino long-erm care is derived by muliplying he moraliy rae wih an age-varying adjusmen facor, ρ x, based on he UK experience repored in Insiue of Acuaries UK (2005). Boh he probabiliy of prepaymen and he probabiliy of refinancing are assumed o depend on he in-force duraion (in years) of he reverse morgage loan (Hosy e al., 2008; Insiue of Acuaries UK, 2005). The loan 10

11 Table 1: Assumpions on reverse morgage loan erminaion based on Ji (2011). A-home moraliy LTC incidence Prepaymen Refinancing Age facor θ x facor ρ x Duraion Probabiliy Duraion Probabiliy % % % % % % % % % % % % % erminaion assumpions are summarized in Table 1. Parameers for ages no repored in he able are obained by linear inerpolaion. The underlying age-specific moraliy raes are modeled by he Gomperz law of moraliy. The model assumes ha he force of moraliy µ x of an x-year-old is given by: µ x = α exp {γx}, (15) where α and γ are wo parameers. We esimae hese parameers based on moraliy daa for Ausralian females of ages for he period from he Human Moraliy Daabase (2012). We approximae he insananeous force of moraliy, µ x, by he deah rae, which is calculaed as he number of deahs, D x,, in a given year divided by an esimae of he populaion exposed o he risk of deah, E x,. A Poisson regression was fied o he naural logarihm of deah couns, D x,,: ln D x, = ln E x, + β 0 + β 1 x + ɛ x,. (16) The esimaed Gomperz parameers are ˆα = and ˆγ = Using he esimaed force of moraliy, ˆµ x, from he Gomperz model and he annual loan erminaion assumpions, he probabiliy p c x ha he reverse morgage loan is in in-force in policy year is given by: { p c x = exp 1 0 } (θ x+s + ρ x+s )ˆµ x+s ds (1 P(P repaymen))(1 P(Ref inancing)), (17) 11

12 where P(P repaymen) and P(Ref inancing) are he age-specific probabiliies ha he loan is erminaed because of prepaymen or refinancing, respecively. The annual probabiliy of loan in-force was convered ino a quarerly frequency by cubic spline inerpolaion. Finally, he quarerly probabiliy of loan erminaion is calculaed as qx c = +1p c x p c x. The resuling average conrac in-force duraion is 16.1, 9.3 and 4.4 years for borrowers iniially aged 65, 75 and 85. These duraions are slighly shorer han hose repored in Alai e al. (2013, Table 3), because we include prepaymen and refinancing as addiional reasons for reverse morgage erminaion. 3.2 VAR-Based Economic Scenario Generaion A vecor auoregressive (VAR) model is used o joinly model house prices, ineres raes and oher relevan economic variables, o projec economic scenarios and o derive sochasic discoun facors using he same daa and mehodology as described in Alai e al. (2013). In addiion, we include he consumer price index in he model, as a driver of ineres rae and house price dynamics. The esimaion resuls can also be compared o Sherris and Sun (2010), who esimae a VAR model using similar daa over a differen ime period (Mar Dec-2008). Table 2 summarizes he raw daa, variable names and daa sources. The daa was accessed in Augus The sample period is Jun Jun-2011, he longes period for which all variables are available. Daily and monhly series are convered o quarerly series. The en-year erm spread is calculaed as he difference beween en-year and hree-monh zero-coupon yields: r (40) r (1). Growh raes of he house price index, of he renal index, of GDP and of CPI are deermined by differencing he log series and are denoed as d ln. Figure 2 plos he hree-monh zero-coupon yields r (1) and he variable morgage rae M R. The wo variables are highly correlaed, wih correlaion coefficien of To avoid mulicollineariy in he VAR model, Alai e al. (2013) only include he shor rae in he VAR model and derive variable morgages raes, r κ, by adding a fixed lending 12

13 Table 2: Definiions, daa sources and frequency. Variable Definiion Source Frequency r (1) Three-monh zero-coupon yield Reserve Bank of Ausralia Daily r (40) Ten-year zero-coupon yield Reserve Bank of Ausralia Daily M R Nominal variable morgage rae Reserve Bank of Ausralia Monhly H Nominal Sydney house price index Residex Py Ld. Monhly R Nominal Sydney renal yield index Residex Py Ld. Monhly GDP Nominal Ausralian gross domesic produc Ausralian Bureau of Saisics Quarerly CP I New Souh Wales consumer price index Ausralian Bureau of Saisics Quarerly The sample period is Jun Jun Figure 2: Three-monh zero-coupon yields, r (1), and variable morgage raes, Jun Jun margin, κ, o he shor rae: r κ = r (1) + κ. (18) We follow his approach and esimae he lending margin as he average difference MR r (1) = 1.65% over he sample period. Assuming coninuous compounding, he quarerly lending margin is calculaed as κ = 0.41%. The ime series were esed for saionariy using he augmened Dickey-Fuller es and he Phillips-Perron es. Boh ess correc for possible serial correlaion in he error erms of he es equaion. The Phillips-Perron es is also robus o unspecified heeroscedasiciy in he error erms. The resuls of hese ess, given in Table 3, indicae ha all variables excep he renal yield growh raes d ln R are saionary a a 10% significance level. We include d ln R in he VAR model o avoid over-differencing. To deermine he opimal lag lengh, we esimae he VAR model for differen lag lenghs 13

14 Table 3: Resuls of he saionary ess. ADF es PP es Variable -saisic p-value -saisic p-value r (1) r (40) r (1) d ln H d ln R d ln GDP d ln CP I PP de- ADF denoes he augmened Dickey-Fuller es. noes he Phillips-Perron es. Table 4: Model selecion crieria and residual analysis for VAR models wih differen lag lenghs. Model Selecion Crieria Auocorrelaion Heeroscedasiciy Normaliy Lag lengh AIC BIC HQC p-value p-value p-value The model selecion crieria are denoes as: AIC-Akaike Informaion Crierion, BIC-Schwarz s Bayesian Informaion Crierion and HQC-Hannan-Quinn Crierion. The model residuals are esed for serial correlaion using he mulivariae Lagrange Muliplier es, for heeroscedasiciy using he Whie es and for normaliy using he mulivariae Jarque-Bera es. and compare hree commonly used model selecion crieria (Akaike Informaion Crierion, Schwarz s Bayesian Informaion Crierion and Hannan-Quinn Crierion). To suppor he model choice we also analyze he esimaed residuals of each VAR model. We es for serial correlaion using he mulivariae Lagrange Muliplier es, for heeroscedasiciy using he Whie es and for normaliy using he mulivariae Jarque-Bera es. Based on he es resuls, which are repored in Table 4, and in accordance wih previous lieraure using similar daa (Alai e al., 2013; Shao e al., 2012; Sherris and Sun, 2010), we choose a VAR(2) model. The model is given by: z = c + φ 1 z 1 + φ 2 z 2 + ɛ, (19) where z denoes he vecor of he economic variables lised in Table 3, c, φ 1 and φ 2 are parameer vecors and marices and ɛ is a vecor of mulivariae normally disribued error erms wih ɛ N (0, Σ). 14

15 Table 5: Esimaed parameers of he VAR(2) model. z c (6 1) φ 1 (6 6) r (1) r (40) r (1) d ln H d ln R d ln GDP d ln CP I φ 2 (6 6) Σ (6 6) The model equaion is given by: z = c + φ 1 z 1 + φ 2 z 2 + ɛ wih ɛ N (0, Σ). The VAR(2) model was esimaed using SAS s varmax procedure. The esimaed parameers are given in Table 5. The model exhibis a good fi wih an average R 2 of 72.5% across he six equaions in he VAR sysem. There are several significan dependencies beween he economic variables. The growh rae of he house price index is very volaile, wih an esimaed variance of The resuls are comparable wih hose in Alai e al. (2013), where a VAR(2) is esimaed for he same variables excluding CPI. Based on he VAR(2) model, 10,000 simulaion pahs of he economic variables over 40 years were generaed wih he MATLAB procedure vgxsim. The disribuion of he simulaed variables closely maches he empirical disribuion of hisoric daa, as shown in Figure 3. The disribuion funcions were smoohed wih he MATLAB package ksdensiy, a kernel smoohing procedure. Figure 4 plos he hisorical pahs of he economic variables and he simulaed mean values ogeher wih 90% confidence inervals. The graphs are similar o hose in Alai e al. (2013) based on a VAR(2) model wihou inflaion. House price growh is he mos volaile of he economic variables. 3.3 Deriving Sochasic Discoun Facors Building on previous work by Ang and Piazzesi (2003) and Ang e al. (2006), Alai e al. (2013) develop a VAR-based mehod o derive sochasic discoun facors for pricing reverse morgages. The key idea of he mehod is ha he discoun facors should reflec 15

16 Figure 3: Probabiliy densiy funcion of he economic variables: hisorical daa (lines wih dos) and simulaed daa (solid lines). Figure 4: Hisorical pahs of he economic variables and simulaed mean values wih 90% confidence inervals (dashed). 16

17 he main drivers of reverse morgage cash flows and should accoun for he risk facors inerdependencies. This is realized by deriving sochasic risk facors from he VAR model used o projec he economic variables. There is no allowance for longeviy risk or oher componens of erminaion risk in he pricing framework. Idiosyncraic longeviy risk is assumed o be fully diversifiable and sysemaic longeviy risk is assumed o be hedgeable hrough reinsurance or securiizaion. A calibraion procedure for he sochasic discoun facor model was developed in Shao e al. (2012). We denoe wih ζ +1 he Radon-Nikodym derivaive ha convers beween he real-world probabiliy measure P and he risk-neural measure Q. Tha is, for any variable X +1 a ime + 1: E Q [X +1 ] = E [ζ +1 X +1 ] /ζ. (20) ζ is assumed o follow a log-normal process: ζ +1 = ζ exp { 12 λ λ λ ɛ } +1, (21) where λ are ime-varying marke prices of risk associaed wih he random shocks, ɛ +1, o he economic variables in he VAR model. The vecor of he marke prices of risk, λ, is modeled as a linear funcion of he economic sae variables in he VAR model: λ = λ 0 + λ 1 z, (22) where λ 0 is a 6-dimensional vecor and λ 1 is a 6 6 marix. The pricing kernel (or sochasic discoun facor), m, is given by: m +1 = exp { r } ζ +1 /ζ = exp { e 1z 12 λ λ λ ɛ } +1, (23) 17

18 whih e 1 = (1, 0, 0, 0, 0, 0). Using he sochasic discoun facors, he price P of an asse wih a payoff X +1 a ime + 1 is given by: P = E [m +1 X +1 ]. (24) The ime price of an n-period nominal bond can be derived using he following recursive formula: P (n) = E [ m +1 P (n 1) +1 ], (25) wih he iniial condiion P (0) = 1. The bond price can be wrien as an exponenial linear funcion of he sae variables in he VAR model: P (n) = exp {A n + B nz + C nz } 1, (26) where A n, B n and C n are given by he difference equaions: A n+1 = A n + B n(c Σ 1 2 λ0 ) B nσb n, (27) B n+1 = δ 1 + (φ 1 Σ 1 2 ) B n + C n, C n+1 = φ 2B n, wih iniial esimaes of A 1 = 0, B 1 = δ 1 and C 1 = 0 (for he proof see Shao e al., 2012). The coninuously compounded yield r (n) on an n-period zero-coupon bond is given by: r (n) = log P (n) n = A n n B n n z C n n z 1, (28) In order o derive he sochasic discoun facors, he marke prices of risk, λ, need o be esimaed. λ follows he recursive formula given in Equaion (22). The saring values λ 0 and λ 1 are esimaed by minimizing he squared deviaions of he fied bond yields 18

19 Table 6: Fied values of he marke prices of risk λ 0 and λ 1. λ 0 (6 1) λ 1 (6 6) r (1) r (40) r (1) d ln H d ln R d ln GDP d ln CP I Table 7: Correlaion beween sochasic discoun facors and economic variables. Variable r (1) r (40) r (1) d ln H d ln R d ln GDP d ln CP I m from he observed yields: min λ 0,λ 1 T N =1 n=1 ( ˆr (n) ) 2 r (n). (29) The model is calibraed using zero-coupon yield for four mauriies: hree monhs, one year, five years and en years, i.e. N = 4. The calibraed values of λ 0 and λ 1 are given in Table 6. Figure 5 plos he fied sochasic discoun facors ogeher wih hisorical house price growh raes and hree-monh zero-coupon yields. The sochasic discoun facor is negaively correlaed wih he shor rae, GDP growh and wih inflaion. I is posiively correlaed wih house price growh and he erm spread (see Table 7). 4 Reverse Morgage Risk and Profiabiliy Analysis To assess how risk and profiabiliy differ for reverse morgages wih differen payou designs, quarerly cash flows of reverse morgage loans are compued based on 10,000 pahs of he economic variables simulaed from he VAR(2) model over a 40-year period along wih he projeced probabiliies of loan erminaion from he Markov model. Risk and profiabiliy are assessed on a represenaive loan basis. The key drivers of he reverse morgage cash flows, such as he ousanding loan balance and house prices, are 19

20 Figure 5: Fied sochasic discoun facors, m, house price growh raes, d ln H, and hreemonh zero-coupon yields, r (1). analyzed separaely o show how hese facors impac he lender s financial posiion. Two commonly used risk measures, he Value-a-Risk (VaR) and he Condiional Valuea-Risk (CVaR), are compued a he 99.5% level o deermine he amoun of risk-based capial he lender should se aside for he differen ypes of reverse morgage. Deailed sensiiviy analysis is conduced o es he impac of conrac seings including he loan-o-value raio (LTV), he borrower s age and key model assumpions on he resuls. 4.1 Base Case Seings In he base case, we consider a single female borrower aged 75, who is subjec o he Ausralian moraliy experience. The borrower s maximum aainable age is ω = 105. The iniial house price is se o H 0 = $600, The LTV is se o 40%, resuling in an iniial loan amoun of L 0 = $240, 000. In he sensiiviy analysis, we consider oher borrower ages, oher LTVs and also allow for moraliy improvemens. Sale ransacion coss are se o γ = 6% as in Alai e al. (2013). The reverse morgage loan lender is assumed o borrow ϕ = 92% of he loan principal(s) and o use capial o finance he remainder. Alernaive risk-baaed capial raios are considered in he sensiiviy analysis. 2 The median price of esablished house ransfers in Sydney was $595,000 in he second quarer of 2011 (see Ausralian Bureau of Saisics, House Price Indexes: Eigh Capial Ciies). 20

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