Financing Retirement with Stochastic Mortality and Endogenous Sale of a Home

Financing Retirement with Stochastic Mortality and Endogenous Sale of a Home Thomas Davidoff UBC Preliminary Draft April 13, 2010 Abstract This paper considers the retirement financing problem of a single retiree with no bequest motive, facing a stochastic date of death, who owns a home out of which it is difficult to move, and chooses the date (if any) at which to sell the home. Annuities and mortgage debt are strongly complementary on the demand side, but worsen each other s adverse selection on longevity. Negatively amortizing reverse mortgages are less desirable than a bundled forward mortgage and annuity on the demand side, but face less adverse selection on longevity. Reverse mortgages are no less adversely selected on the dimension of taste for remaining in the home. Utility is globally concave in neither date of move nor the (assumed constant) value of the home. Two difficult problems in financing retirement are whether and how to extract the equity from one s primary residence and how to cope with uncertainty over length of life. Social insurance throughout the developed world provides large transfers from those who die early in retirement to those who die late in retirement. Economists have expended considerable effort to understand why many retirees fail to supplement these transfers by converting liquid savings to life annuities. 1 The US government helps finance a reverse mortgage product (the Home Equity Conversion Mortgage (HECM)), but only a minority of older homeowners owe mortgage debt, and less than one percent of US homeowners over 62 are reverse mortgagors. This paper shows that when sale of the home is a matter of endogenous choice and longevity is stochastic, even frictionless, fairly priced annuities and forward mortgages in isolation offer limited welfare benefits to households with no bequest motive and a large 1 See Brown (2007) for a survey. 1

fraction of their wealth in housing. However, combining the two products generates a very large welfare gain. Notably, the design of HECM, a negatively amortizing mortgage with only implicit annuitization appears to combine elements of forward mortgages and annuities in a way that offers substantially less consumer surplus than a simple bundle. Selection on longevity may rationalize the design of HECM. Earlier work (Davidoff (2009)) has shown that the problem of home equity and financing longevity are linked. If home equity extraction is correlated with poor health, and if poor health is correlated with old age, then the payouts from unleveraged owner housing and annuities are correlated. Owner housing is then a substitute for annuities. Simulations in which the home is sold if and only if the (single) owner is in very poor health, with mortality and health state transition probabilities derived from the model of Robinson (2002), show that annuities offer much smaller benefits to retirees with large and illiquid housing wealth. While it is well known that poor health is a critical determinant of selling a home in old age (see, e.g. Venti and Wise (2000), Davidoff (forthcoming)), there is evidence that financial considerations also matter. Engelhardt (2008) and Engelhardt and Greenhalgh-Stanley (2010) show that more generous social insurance is associated with a greater likelihood of homeownership among widows. Casual inspection of US Census ownership rates by age and income among single women reveals the same effect. I find a significant effect of wealth on the probability of HRS/AHEAD widows remaining in their homes between 2004 and 2007 with or without conditioning on age. Particularly relevant to the analysis that follows, both Davidoff and Welke (2006) and Shan (2009) find that reverse mortgage borrowers are more likely to terminate loans (through death, refinancing, or quite plausibly sale of the home) when home equity has grown more since origination. Home equity is a dominant and relatively illiquid component of the portfolio of many retirees. Quoting from Davidoff (forthcoming), 79% of retirees in the 2004 HRS/AHEAD are homeowners. Median home equity is approximately $75,000 in the full sample and $110,000 among owners. The median ratio of home equity to total wealth among 2

homeowners is 55%. In the 2004 wave, only 12% of the sampled retiree homeowners owed any mortgage debt. Among mortgagors, median debt was 50,000, and the median debt to home value was 33%. Between 1998 and 2004, despite a massive home price increase, less than ten percent of the older retired homeowners increased mortgage debt. The mean ratio of equity to home value in the 2004 sample rises with age from.84 among owners in their 60s to.96 among owners in their 90s. The notion that high home equity combined with a reason to remain in the home will affect the life cycle model behavior dates at least as far back as Artle and Varaiya (1978). In their model, older homeowners decide when and if to sell their home. Selling the home is the only way to consume home equity, but renting housing costs more than the economy s common cost of capital times the value of (fixed) housing consumed. They show that there must be zero liquid savings at the moment just before home equity is sold. If this were not true, then the move date could be delayed with a gain in resources available and no liquidity cost. While Artle and Varaiya (1978) assume a deterministic date of death, there is reason to suspect that the absence of savings at the date of home sale would weaken demand for annuities. As shown by Bernheim (1987b) and Davidoff et al. (2005), the gain to purchasing annuities is increasing in demand for savings carried over to dates late in life. If home equity weakens demand for savings held late in life, it should weaken demand for annuities. As emphasized by Davidoff et al. (2005), the effect of liquidity considerations on demand for annuities depends heavily on the numerical specification of liquidity problems. For example, several authors (e.g. Turra and Mitchell (2004)) have shown that the prospect of long-term care expenditure needs in early retirement may sharply reduce annuity demand. Davidoff (2009) shows that if illiquid home equity substitutes for long term care insurance early in retirement, annuities and long-term care insurance may be substitutes. In the case of illiquid home equity, a consideration that is missing from Artle and Varaiya (1978) is that annuities purchase a fixed bundle of future consumption at a lower cost than conventional 3

assets. It is thus not clear that optimal purchases of annuities constrained to feature constant real payouts would be zero (they are not in any simulations like those presented below) or even that they would be far below optimal purchases with liquid home equity. Optimal design of products offering longevity insurance and home equity liquidity is an interesting problem in terms of both supply and demand. On the supply side, the way in which a bundled annuity and mortgage is structured will likely affect the expected longevity and duration at home of consumers choosing the product. These effects may arise through selection and moral hazard. Below, I compare the structure of HECM, a negative amortization mortgage with some implicit annuity characteristics to a simple bundle of a forward mortgage and an annuity. Most of the analysis below is numerical and considers only a simple setting that ignores important uncertainty over health status and home prices. This simplification may be justified by two related considerations. First, there is almost no hope for meaningful analytical results due to highly non-linear survival probabilities and options embedded in homeownership. In particular, the corner solutions of moving immediately and never are salient. As shown below, the second derivative of utility in move date is unlikely to have a consistent sign, particularly once a negatively amortizing mortgage is considered. Second, the problem tends to be highly sensitive to assumptions and key parameters of the problem have not been estimated convincingly or even explicitly. 1 A Model of Annuity Demand, Stochastic Mortality, and Endogenous Move Date A single retiree with no bequest motive at date 0 owns w in fully liquid financial wealth, a home worth p (constant across time), and faces probability of living to date t equal to q(t). To make a constant real annuity a natural baseline, liquid savings earn return r, which is also the retiree s rate of time preference. While in the home, she enjoys instantaneous 4

utility u(c(t)) where c is the rate of consumption at time t. After she moves, utility over expenditures is v(c(t)). If housing consumption is fixed, a natural formulation would be v(x) = u (x prk 1 (t)) k 2 (t), with k 1 and k 2 positive and decreasing in t. 2 That is, holding housing fixed, being out of the owner home requires more expenditures than staying in place (ignoring opportunity cost of equity and any debt in the home) as well as a pure utility cost. These effects plausibly diminish as health becomes worse, so that managing a home becomes costlier and less desirable. The retiree may not borrow except possibly through mortgages considered below. If annuities or mortgages are available, they are exogenously given. I consider different constellations of retirement product endowments, varying the holdings of a constant real annuity a paying r a per period, a mortgage m with regular payments of r p m and accruing interest at rate r m. If the mortgage is negatively amortizing (r p < r m ), there is no recourse to the borrower s liquid assets. This absence of recourse is part of the design of HECM and is commonly viewed as practically true even with forward mortgages to working-age owners. Denote by T the date of any move prior to the maximum possible age A. The lifetime optimization problem is to maximize utility over choice of consumption path conditional on survival and date of move T : 2 Marginal utility after moving can be made considerably greater relative to before by offering choice over housing consumption. However, such an intrapersonal but inter-regime difference in marginal utilities does not strike me as natural. Moreover, if moves are triggered by ill health (and hence short longevity) as in Davidoff (forthcoming), marginal utility of wealth may be lower after moving. Results presented below do not appear qualitatively highly sensitive to k 1, which affects relative marginal utility. 5

T max U = u(c(t))e rt q(t)dt + {c},t 0 subject to: A T v(c(t))e rt q(t)dt (1) s(0) =w a + m (2) s(t ) 0 (3) s(a) 0 (4) ṡ =sr + ar a mr p c for t < T (5) s(t +) s(t ) = max ( p me [rm rp]t, 0 ) (6) ṡ =sr + ar a c for t > T (7) I will not consider dynamic choice of annuitization or mortgage borrowing. It is natural to think that welfare would be improved if annuities were available after a move. A justification outside of the model is that it is uncommon for retirees in good health to sell their homes. Consistent with the model choices, much of the benefit of annuitization is lost if annuities are purchased later in life (Brugiavini (1993)). 3 A social planner with ownership of the consumers assets, investing the liquid assets with return rate r and renting the home for pr on the consumer s exit at date T would face a differently constrained problem. The planner would pay for planned consumption only if the consumer survived to a given period. The problem would thus be: 3 Disallowing annuities that decline over time (e.g. nominal annuities) should have only a second order impact on the quantitative results. Weak demand for savings necessarily leads to weak demand for annuities, at least with complete markets. 6

T max U = u(c(t))e rt q(t)dt + {c},t 0 subject to: A T v(c(t))e rt q(t)dt (8) s(0) =w + p (9) s(a) 0 (10) ṡ =sr q[c pr] for t < T (11) ṡ =sr qc for t > T (12) The solution to this problem yields a constant marginal utility of consumption across time. Denote by π(t) the shadow value of savings at date t and we have at an optimum: π = r (13) u (c(t))e rt q(t) = π(t)q(t) (14) v(c(t +)) u(c(t )) + [c(t +) c(t ) pr]π(t ) = 0. (15) Conjecture If (and almost only if) u (x) = v (x + rp) for all x then the optimum can be decentralized with a constant real annuity equal to w+p and a simple forward mortgage with r p = r m = r. If marginal utility for equal expenditures (including the opportunity cost of initial housing pr) are different for v and u, then the optimum can only be decentralized if the annuity differentiates between the before and after move periods. I assume that the problem of marginal utility mismatch (which worsens with time in the baseline parameterization below) has small welfare effects. In the absence of a mortgage, depending on u and v, the consumer s liquidity constraint (3) may bind, so that π(t ) > π(t +). In this case, even if v(x) = u(x rp) a constant real annuity does not achieve the first best described by (13) through (15). In particular, it would 7

be optimal to consume more savings prior to T. The Artle and Varaiya (1978) result that all savings w would be exhausted prior to T does not hold with stochastic mortality both because of the superior return on annuities and because without annuities, a precautionary buffer might be held to T to hedge against an unexpectedly long life. In the absence of an annuity, the marginal utility of consumption will be greater late in life than earlier, conditional on survival. Optimal consumption will thus be greater earlier than later in life. In this case, the timing of payments of a forward mortgage may not be desirable. If the probability of survival to T is low, then the need to dedicate p[1 e rt ] to non-annuitized interest reserves may greatly subtract from the liquidity benefit of the mortgage. A zero coupon (r p = 0) and hence negatively amortizing mortgage (e.g. HECM) appears to solve the potential problem that interest reserves may be greater than desired savings, because no payments are required. This is not entirely true, though, because the lender must hold interest reserves in the form of reduced loan proceeds. If p is constant through time, there is no risk in a forward mortgage with proceeds p. 4 By contrast, the loan balance on a HECM with accumulation rate r m = r can be no greater than p[1 e rt ]. The optimal HECM loan almost certainly features an interest rate greater than r. This is inefficient in that it almost guarantees that the first-best optimality condition for moving (15) will not be satisfied at any internal date T. However, the greater interest rate induces a form of annuitizing the home. There is a transfer from households that die before any date t to households that survive past t but either set T > A and never move or die before the optimal date T. Thus a HECM loan generally will not replicate first best with or without an annuity, but may be preferred to a simple forward mortgage in the absence of an annuity. There is a moral hazard in the move date (Davidoff and Welke (2006)) because the lender loses rp for every period the borrower remains in the home past the date T at which me rt = p. By contrast, past the weakly earlier date S at which me rms = p, the borrower 4 There would be risk if p(t) were stochastic, but this risk might be as severe or more severe in a HECMtype loan. 8

perceives no financial cost to remaining in the home. As long as living outside the home costs more financially and psychologically than remaining in the home, a borrower with no mortgage cost will not leave. Note that absent mortgage debt, liquidity considerations will push the move date suboptimally early, so a conjecture is that a small increase in the HECM rate above r is welfare increasing. I consider an upper bound on the HECM rate r m of.11, an arguably unrealistically usurious spread over the riskless rate r =.03. 5 This leaves a considerable fraction of home value unannuitized. A reverse mortgage alone does not at all annuitize interest reserves or wealth w. Thus the bundling of annuitization with a forward mortgage in a reverse mortgage is incomplete because of all of (i) moral hazard limiting loan proceeds, (ii) incomplete annuitization of proceeds from an upper bound on r m, (iii) failure to annuitize the interest reserve gap between p and m, and (iv) failure to annuitize savings. Summarizing, there is reason to suspect that there will be a strong complementarity between mortgage debt and an annuity. A loan such as HECM is likely to be preferred to a simple forward mortgage without an annuity present, but will likely not provide as much welfare benefit as a combination of a large annuity and a forward mortgage. 2 Numerical Examples This section presents a numerical parameterization of the problem described above that explores whether two quantities are plausibly large. The quantities are: first, the complementarity between mortgage debt and annuitization; second, the gap in welfare benefits between (a) the nearly first best bundle of a 100% loan to value forward mortgage and 100% annuitization of w+p and (b) a zero coupon reverse mortgage constrained by an 11% accrual rate r m. Table 1 describes the parameterization of the retirement planning problem that I simulate. 5 The reverse mortgages I consider are approximately actuarially fair, but involve large transfers away from the dynasty of those who die early. Such explicitly annuitized reverse mortgages were the subject of lawsuits and recently unfavorable coverage in The New York Times. 9

I consider a single woman of age 72 with $75,000 in cash and a home worth $100,000. This is not far from a typical ratio of total wealth to housing wealth, based on data from HRS/AHEAD described above. Utility while in the home is u(c(t)) = c1 γ 1 γ with the coefficient of relative risk aversion γ set equal to -1. As shown below, risk aversion is not simple to define in this context. Before moving, the consumer can hedge wealth with the choice of move date. I assume that housing needs are fixed. With choice over housing, both intratemporal substitution and intertemporal substitution (the latter here governed by γ) would matter for risk aversion after moving. I assume that housing demand is fixed after moving. The cost of housing after moving is given by prq(t)/q(72), so that housing expenditures are relatively less expensive after moving as health deteriorates, with health at age t measured by the probability that a 72 year old lives to date t. A similar assumption governs an additive disutility to moving. v(x) = u(x prq(t)) [u(100, 000) u(100, 000x)] q(t), with x as a q(72) default set to.6. Thus a year out of the initial home has a utility cost equal to the difference between consuming 100,000 and 60,000 in a year, with this cost declining with health (proxied by age). The purpose of the calibration is to have no-product exits around age 90, the age by which roughly half of widows are no longer homeowners in the IPUMS data. I set the economy-wide interest rate to 3%. A fair constant (real) annuity provides a payout of approximately 7% per year. Table 2 presents the welfare benefits of various products under the baseline assumptions. The first product is a constant real annuity with no mortgage. The second product is a fair interest only mortgage for $100,000 (100% LTV). The third product is a reverse mortgage with initial proceeds equal to $66,000, r m = 11% and r p = 0. 66,000 is the maximal nonnegative loan proceeds with r m constrained to be no larger than 11%. The loan crosses over with losses to the lender conditional on survival before death, but the interest rate spread provides profits to the lender for all borrowers who die before the cross over date. The findings suggest that the considerations outlined above are economically important. 10

Table 1: Parameterization of retirement financing problem for a single 72 year old woman Symbol Meaning Value p value of home 100 w start financial wealth default 75 a Annuitized savings default 0 m mortgage default 0 r a annuity rate default fair ( 7.2%) r p periodic rate on mortgage default 0 (HECM) r m accrual rate on mortgage default r r riskless rate.03 δ discount rate.03 Also consider exogenous T u period utility pre-move CRRA γ = 2 v period post-move u(c prk 1 (t)) k 2 (t) k 1 (t) housing cost post-move 1, decay = q(t)/q(72) k 2 (t) utility cost post-move default u(100, 000) u(60, 000) decay with age. Annuities and forward mortgages provide welfare gains, but not nearly as large as they would be without either stochastic mortality (in the case of the mortgage) or illiquid housing (in the case of the annuity). Thus the combined welfare gain from a bundled product of $133,000 is much greater than the separate gains of the products in isolation ($23,000 for the annuity plus $17,000 for the forward mortgage). A reverse mortgage is better than a simple forward mortgage or a fair annuity alone, but not nearly as welfare improving as a bundled fair annuity and forward mortgage. 2.1 Features of Consumer Choice 2.1.1 Curvature of Utility in Move Date There are some noteworthy elements of the consumer optimization. First is the shape of utility in the date of move. Figure 1 shows that utility is neither globally concave nor globally convex in the move date in the absence of any product, and features a corner solution with the bundled product. For this reason, the problem of optimal retirement product design problem includes a highly complicated constraint on the optimality of the chosen move date. The problem of optimal design with selection considerations so that the first best is not 11

Table 2: Welfare gains to various retirement products Product Parameters Equivalent dollar gain no products 75,000 real annuity Baseline 23,000 100,000 forward mortgage rate r Baseline 17,000 HECM constrained (60,000, r m = 11%) Baseline 36,000 Annuity $175,000, 75,000 mortgage Baseline 133,000 75,000 real annuity High move cost 31,000 100,000 forward mortgage rate r High move cost 24,000 HECM constrained (60,000, r m = 11%) High move cost 45,000 Annuity $175,000, 75,000 forward mortgage High move cost 141,000 75,000 real annuity low survival 18,000 100,000 forward mortgage rate r low survival 33,000 HECM constrained (60,000, r m = 11%) low survival 56,000 Annuity for $175,000, 75,000 forward mortgage low survival 113,000 75,000 real annuity no move costs 66,000 75,000 constant real annuity no move costs low survival 45,000 Note: The equivalent dollar gain is the dollar amount greater than the baseline of 75,000 in non-housing wealth that would be required to attain the same level of utility as with the product in question and 75,000 in wealth. High move costs means k 1 is set to 1.2 and k 2 to.2. 12

Figure 1: Maximized utility over consumption conditional on fixed move dates with no mortgage or annuity (solid line), forward mortgage and annuity (dashed line), and negatively amortizing reverse mortgage only (bubbles). Optimal move date plotted with X. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX X utility 8 6 4 2 X X 0 10 20 30 40 date of move feasible likely must be solved numerically. Figures 1 also illustrates the presence and absence of moral hazard in the move date with a hecm and a combined forward mortgage and annuity. 2.1.2 Curvature of Utility in Wealth The presence of housing, risk over longevity, and choice over the date at which to move generate different curvature of utility, so that there is no constant relative risk aversion in the joint Yaari-Artle and Varaiya problem. Figure 2 shows that utility is less concave when the date of move is flexible rather than fixed. Some economists, e.g. Caplin et al. (1997) and Shiller (1993) have questioned why markets for sharing local price risk are not more thickly traded. Figure 3 provides a partial answer. Given the options on the extensive and intensive margin for moving (an argument sometimes attributed in a static setting to Chicago wisdom), utility is not globally concave 13

Figure 2: Utility U as a function of wealth. No products optimal choice over move (solid line); no products, move date fixed at optimal level of 15 at wealth of 75 (dash-dots); no housing, annuity (dashed); and no housing, annuity for wealth plus 100 (value of home in simulations). utility 4 3 2 1 0 50 100 150 200 wealth in price, and the curvature of in the concave range is less than the curvature over wealth. 2.1.3 Selection Given the evident advantage of a bundled forward mortgage and annuity over a negatively amortizing reverse mortgage, it is worth asking how problematic adverse selection in longevity and propensity to remain in the home is likely to be for each product. I will assume that these dimensions are linked, in that low survival is linked to a weak preference for remaining in the home by the definition of post-move utility described above. The liquidity of home equity through willingness to move and availability of mortgage debt has dramatic effects on the nature of selection for annuities. When there are no moving costs, (last two rows of Table 2) longevity is strongly associated with the willingness to pay for the right to annuitize savings. Willingness to pay is roughly 33% less when the probability of death in each period is doubled (up to a death probability of 100% at age 108). When 14

Figure 3: Maximized utility as a function of home resale value in the baseline case with no product utility 2.6 2.4 2.2 2.0 0 20 40 60 80 100 price housing is illiquid however, matters are more complicated and a doubled periodic death rate is associated with a much smaller difference in demand in dollar terms and percentage terms (18,000 versus 23,000). The reason is that with shorter longevity, the move occurs much later in the annuitized case, so that there is a much smaller problem of marginal utility mismatch. The move is later with shorter longevity because the gain to increased wealth is backloaded relative to the disutility and cost of moving. Demand for conventional forward mortgage debt alone is sharply increasing in death probability when no annuity is available. This is plausible in that the probability of repaying principal declines as mortality rises. In the presence of a forward mortgage, annuity demand rises sharply with the probability of survival, just as it does when an absence of move costs renders the illiquidity of a real annuity irrelevant. In sum, a bundled annuity and forward mortgage The favorable selection in mortality for the HECM comes from the fact that for any consumer who will not leave, the value of the $60,000 proceeds is equal to the proceeds less a 15

penalty for inefficiency of the move date. This inefficiency is smaller for a consumer unlikely to live to the optimal T. I find that when willingness to move is decreased (by changing x from.6 to.4 and increasing the rental cost of housing after moving from pr to 1.2pr) that the valuation of the annuity increases from 23,000 to 31,000. The valuation of the HECM increases from 30,000 to 38,000. Thus in levels there are approximately equal selection effects, but in percentage terms the bundle is much less adversely selected. 2.1.4 Usury in Reverse Mortgages Given an environment in which the optimal reverse mortgage is never repaid (this need not arise but does here), a larger and more welfare increasing loan can be arranged by increasing the interest rate. In the limit, the home itself is annuitized with an inifintely high interest rate. I do not consider the welfare gain to annuitization of the home separate from the forward mortgage and annuity described below. The only difference is that wealth would presumably not be annuitized in a stand alone reverse mortgage. 3 Conclusion Both stochastic mortality and liquidity of housing equity affect demand for financial products in a world with no bequest motives and freely chosen date of move. Annuities are highly valued and severely adversely selected when housing is fully liquid, in the sense that consumers with greater longevity have greater demand for annuities. When housing is illiquid, annuities have much less value and interactions between longevity, move date, and optimal savings patterns may render selection on longevity smaller in magnitude. Symmetrically, annuitization of principal and interest reserves greatly increases the benefits and adverse selection on longevity of forward mortgages. Partly bundling mortgage debt and annuitization with a zero coupon, bounded interest 16

rate reverse mortgage fails to fully annuitize mortgage principal, fails to annuitize interest reserves at all, and engenders moral hazard on duration as long as there is a move before death with positive probability when the reverse mortgage is not used. Considerably larger welfare gains are available with an equally zero profit bundle of a forward mortgage and an annuity. The reverse mortgage with a limited interest rate is more favorably selected than a bundled forward mortgage and annuity on the dimension of longevity. By contrast, selection on the dimension of preference for moving later in life appears to be at least as bad for the reverse mortgage as for the more welfare increasing bundle. References Roland Artle and Pravin Varaiya. Life cycle consumption and homeownership. Journal of Economic Theory, 18(1):38 58, 1978. B. Douglas Bernheim. The economic effects of social security: Toward a reconciliation of theory and measurement. Journal of Public Economics, 33(3):273 304, 1987b. Jeffrey Robert Brown. Rational and behavioral perspectives on the role of annuities in retirement planning. NBER Working Papers 13537, National Bureau of Economic Research, Inc, 2007. Agar Brugiavini. Uncertainty resolution and the timing of annuity purchases. Journal of Public Economics, 50(1):31 62, 1993. Andrew Caplin, Sewin Chan, Charles Freeman, and Joseph Tracy. Housing Partnerships: A new approach to markets at a crossroads. MIT Press, Cambridge, 1997. Thomas Davidoff. Home equity commitment and long-term care insurance demand. Journal of Public Economics, forthcoming. 17

Thomas Davidoff. Housing, health, and annuities. Journal of Risk and Insurance, 76(1): 31 52, 2009. Thomas Davidoff and Gerd Welke. Selection and moral hazard in the reverse mortgage market. working paper, UC Berkeley, 2006. Thomas Davidoff, Jeffrey Brown, and Peter Diamond. Annuities and individual welfare. American Economic Review, 95(5):1573 1590, 2005. Gary V. Engelhardt. Social security and elderly homeownership. Journal of Urban Economics, 63(1):280 305, 2008. Gary V. Engelhardt and Nadia Greenhalgh-Stanley. Home health care and the housing and living arrangements of the elderly. Journal of Urban Economics, 67(2):226 238, 2010. James Robinson. A long-term care status transition model. working paper, University of Wisconsin, 2002. Hui Shan. Reversing the trend: The recent expansion of the reverse mortgage market. Working paper, Federal Reserve Board, 2009. Robert Shiller. Macro Markets. Clarendon Press, Oxford, 1993. Cassio M. Turra and Olivia S. Mitchell. The impact of health status and out-of-pocket medical expenditures on annuity valuation. Working paper, Pension Research Council, 2004. WP 2004-2. Steven F. Venti and David A. Wise. Aging and housing equity. NBER Working Paper 7882, 2000. 18