Selection and Moral Hazard in the Reverse Mortgage Market

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1 Selection and Moral Hazard in the Reverse Mortgage Market Thomas Davidoff and Gerd M. Welke June 30, 2007 Abstract A common explanation for the smallness of the reverse mortgage market is that lenders charge hefty fees because they fear long borrower tenure, leading to a loan balance in excess of collateral value. Empirically, however, reverse mortgagors have exited homes unusually rapidly. Modeling demand, we show that advantageous selection on mobility combined with rapid price appreciation can explain this difference. Reverse mortgages induce moral hazard by making sale less attractive, but heavy discounters find extraction of home equity attractive both through reverse mortgage loans and through sale of the home. Consistent with our model, borrowers empirical mobility is highly sensitive to home price changes, unlike non-borrowers. Presented at the 12th Annual AsRES Conference, Macau, July Walter A. Haas School of Business, UC Berkeley, Berkeley, CA 94720; Zicklin School of Business, Baruch College/CUNY, 137 East 22nd Street, New York, NY 10010; gerd

2 1 Introduction Reverse mortgages provide older homeowners with a lump sum cash payout, scheduled payments, or a line of credit in exchange for repayment of fees, principal, and interest upon loan termination. If the home is worth less than the amount due at termination, the lender has no recourse to borrower or estate assets. The loan terminates when the borrower (both borrowers if a couple) either dies or voluntarily sells the home. These features bode both well and ill for the future of the reverse mortgage market. Reverse mortgages are potentially very popular, because they offer liquidity to older homeowners, who are typically house-rich, cash-poor, and reluctant to sell their homes. 1 While originations have grown rapidly in percentage terms over the last few years, less than one percent of eligible homeowners are reverse mortgage borrowers. 2 It is natural to blame the small market size on actuarial problems related to the market structure described above. The market structure seems to invite both moral hazard and adverse selection. A long expected loan term increases the probability of losses to lenders as long as the interest rate exceeds the rate of home price appreciation. Consumers with information that they will live a long time, or who have a particularly strong distaste for moving, may be able to obtain more in loan proceeds than the expected present value of the home upon departure. The case of famed Frenchwoman Jeanne Calment, who signed something like a reverse mortgage at age 90 and then lived to be 122 years old, in a disastrous outcome for her counterparty, illustrates the potential for adverse selection. 3 In the US, where reverse mortgages also terminate with voluntary moves while alive, moral hazard arises even without private information. 4 For house rich, cash poor homeown- 1 See Mayer and Simons (1994) and Kutty (1998) for discussions of the US market and Mitchell and Piggott (2005) for a discussion of the potential market in Japan. Back of the envelope estimates based on Davis and Heathcote (forthcoming) and the American Housing Survey suggest that older homeowners owned over $5 trillion of home equity in Originations of the largest home equity product grew from 157 in fiscal 1990 to 43,131 in As reported by the Associated Press on August 5, We do not know if Calment anticipated a long life, but Finkelstein and Poterba (2004) present evidence of adverse selection on longevity among UK annuitants. 4 Moral hazard on longevity arises if the reverse mortgage proceeds fund improved health. 1

3 ers, the liquidity that reverse mortgages provide undercuts an important financial motivation for selling homes. Indeed the primary selling point of reverse mortgages is that borrowers can afford to remain at home longer. Despite the natural fear that reverse mortgage borrowers would plan to remain in their homes long enough to trigger insolvency and hence a transfer of value from lenders, reverse mortgage borrowers have moved out of their homes at a rate significantly greater than nonborrowers. Figure 1 shows this graphically, depicting unconditional plots of the probability of death or sale while alive (top panel) and probability of still being alive and not having sold (bottom panel) against time for a sample of single borrowers (borrowers in the dominant HECM program) and a sample of eligible single non-borrowers (homeowners over age 62 in the American Housing Survey). This paper offers an explanation for HECM borrowers rapid exits based on advantageous selection on mobility through unobserved characteristics. It is well known that elderly homeowners, the eligible population for reverse mortgages, have a strong distaste for moving out of their homes. 5 Another familiar fact about this group, verified in Table 3, is that housing wealth is large relative to other financial resources. Reverse mortgages are naturally attractive to heavy discounters and those with a taste for smooth consumption, who wish to consume quickly but wish to postpone the disutility of moving. The transfer of money from later in life or from bequests to the present should be less attractive to light discounters and to those more willing to tolerate uneven consumption streams. If the reverse mortgage leaves the owner with positive home equity and refinancing is difficult, then the heavy discounters who borrow may soon find it attractive to consume remaining home equity by selling the home. With identical discount rates and curvature of utility over consumption, reverse mortgage borrowers would have less home equity, and hence less incentive to move, than non-borrowers. If, however, reverse mortgage borrowers are more eager to spend than non-borrowers, less home equity may provide borrowers with 5 An AARP survey, cited by Venti and Wise (2000), found that 89% of surveyed Americans over 55 reported that they wanted to remain in their current residence as long as possible. 2

4 a stronger incentive to move. We also show that moral hazard on mobility is virtually guaranteed: reverse mortgages make staying at home more pleasant by allowing owners to spend home equity. The optimal stay thus increases with loan size, holding characteristics constant. Advantageous selection seems to have won the horse race against this moral hazard over the years since 1992 when federally guaranteed reverse mortgages emerged, but we show that weaker price appreciation has been, and thus may continue to be, associated with diminished mobility. Our results on selection and moral hazard are in line with recent theoretical and empirical work on insurance markets. Just as de Meza and Webb (2001) argue that heterogeneity in risk aversion can make the insured a lower risk than the uninsured, so we show heavy discounting or a strong taste for smooth consumption can leave reverse mortgage borrowers at lower risk of remaining in their homes a long time. In the model of de Meza and Webb (2001), incompletely insured risk averters undertake costly steps to avoid the low utility insured states. In our formulation, those who are unwilling to defer consumption of home equity and are unable to borrow the full value of their home take the unpleasant step of moving to cash out remaining home equity and thereby avoid the low utility state of low consumption. A complication in our model is that individuals not only choose whether or not to take the reverse mortgage and at what date to move (analogous to insurance choice and preventive action choice), but they also decide at what rate to consume resources. For this reason, we use analysis to provide further intuition for why advantageous selection may outweigh moral hazard, but rely on numerical examples to prove the possibility. Recent empirical findings suggest that favorable selection on unobservables operates in other insurance markets. Following the approach of Chiappori and Salanié (2000), Finkelstein and McGarry (2003) find that older individuals who purchase long term medical care insurance are less likely to wind up in long term care than non-purchasers. The proposed explanation is that more risk averse consumers are likely both to seek insurance and to behave in a way that avoids the insured event. Cohen and Einav (2004) find complicated selection 3

5 effects relating both to underlying probability of accidents and to risk aversion in the Israeli auto insurance market. An important simplifying feature of our analysis is an assumption that there is only one provider of reverse mortgages, and we further assume that pricing does not vary on borrower characteristics other than age. In the future, the latter assumption will surely be violated. However, for the Home Equity Conversion Mortgage that we consider, the Federal Housing Administration ( FHA ) does almost no price discriminate and available documentation suggests that they have adopted a simplistic actuarial approach to date. In particular, the FHA has assumed that loan terminations will occur at the age-specific (but not gender specific) mortality rate of the borrower or (youngest borrower if a couple) times a constant. Our analysis should make clear that competitive profit maximizing pricing in this industry would be extremely difficult to characterize. We ignore the presence of other forms of home equity borrowing in the analysis. Such borrowing was rare for the elderly over our sample period, but is growing rapidly. Our paper is a departure from the existing literature on the actuarial nature of reverse mortgages, which has focused on maintenance moral hazard (Miceli and Sirmans (1994) and Shiller and Weiss (2000)) and modeling zero profit prices given interest rate and home price volatility (Szymanoski (1994) and Chinloy and Megbolugbe (1994)). These earlier papers took mobility as exogenous. To our knowledge, our paper is the first to model optimal mobility for reverse mortgage borrowers, although both Szymanoski (1994) and Rodda et al. (2000) suggest that those in ill health may find reverse mortgages attractive, so that borrowers may die quickly leave rapidly for nursing homes. 6 We have only limited data with which to assess this claim, but have some evidence that mortality differences explain a significant but relatively small part of the HECM borrowers rapid mobility. While we do not consider endogenous maintenance, 6 Taking on a reverse mortgage with the intention of leaving home soon for a nursing home may be unwise for an ordinary discounter not only because of the short amortization of fees, but also because Medicaid typically treats home equity more generously than other assets. 4

6 our framework suggests that maintenance may be increasing in reverse mortgage debt. The remainder of the paper proceeds as follows. Section 2 details the HECM program, with much of the information provided taken from Rodda et al. (2000) and Szymanoski et al. (2007). Section 3 describes the joint choice of when to move and whether or not to take on a reverse mortgage in the interim, under the assumption that a single reverse mortgage product is the only way to extract home equity without moving. In that section, we develop the intuition that heavy discounters, those with unwillingness to substitute consumption intertemporally, and those with short expected lives may all be expected to take on reverse mortgages and subsequently exit the home relatively quickly. We verify the intuition with some numerical examples. Section 4 describes data and hazard analysis that are consistent with out explanation for rapid mobility with data. A final section concludes. 2 The Home Equity Conversion Mortgage and Alternatives In the late 1980s, the United States Department of Housing and Urban Development ( HUD ) devised the Home Equity Conversion Mortgage ( HECM ) program to stimulate development of what was then a trivially small US reverse mortgage market. HECM loans are originated by financial institutions or by mortgage brokers. Almost all loans to date have been purchased and held by Fannie Mae, a large mortgage company. There has been one securitization of HECM loans, and more are planned. Borrowers must be over 62 years old and own their home with less pre-existing mortgage debt on the home than the maximal allowed HECM loan amount. Before loan closing, the borrower must participate in educational seminars.footnoteanecdotally, a dearth of seminar leaders has created a bottleneck in the origination process. Borrowers can receive payments in several forms: they may receive a lump sum payment, a line of credit, fixed monthly payments that last for a predetermined number of months, or 5

7 fixed payments that last as long as the borrower is alive and has not yet moved. The most frequently chosen and most flexible option is the line of credit. With a line of credit, the maximum outstanding balance allowed grows with time, reflecting increased mortality and hence a shorter period over which the value of the lender s position is likely to depreciate (older borrowers generate less crossover risk, all else equal). Reverse mortgages have sometimes been associated with annuities. The option with fixed payments until death or exit from the home is similar to an annuity, except that the borrower receives no payment when she is still alive but has moved. A lump sum payment can be converted into an annuity by purchasing an annuity from another provider. The maximal loan balance at the time of origination is increasing in the borrower s age and decreasing in a short treasury rate. Despite differences in mortality, men and women are offered the same interest rate and maximal loan balance; couples are offered the balance that would be offered to the younger spouse if single. As of August, 2004 an 80 year-old homeowner could obtain lump sum proceeds of approximately two-thirds of property value; a 65 year-old could obtain approximately one-half of value. The maximal loan amount is a slightly increasing fraction of appraised value, up to a maximal property value that depends on average home prices in the state of residence. No other borrower characteristics, such as sex, income, or wealth, influence loan terms. HECMs are typically repaid all at once, when the borrower dies or moves out of the home for other reasons. Borrowers are not required to make any payments until the earlier of death or permanent exit from the home. If the borrowers are a couple rather than an individual, then termination occurs at the death or exit from home of the last remaining borrower. At this point, the borrower or their estate must repay the bank the minimum of the home s resale value (in an arm s length transaction) or the outstanding loan balance. The outstanding loan balance at termination date T on a loan that was originated at time 0 is: T 1 T 1 M T = F (1 + r t ) + m t T 1 t=0 t=0 s=t (1 + r s ), (1) 6

8 where F represents costs incurred at loan closing, r t is the one month interest rate on treasuries at month t plus an annualized spread of 1.5 percent, 7 and m t is the cash taken out by the borrower in period t. m t can be a negative number, but such prepayment is very rare. Fixed rate loans are possible under HECM, but Fannie Mae only purchases adjustable rate loans, so as a practical matter all loans are adjustable. Historical closing costs have been extremely large relative to conventional home equity loans, on average close to 6.8 percent of property value, and this has been cited as a major reason for the relative weakness of demand. Part of the reason for high closing costs has been a onetime guarantee fee of 2% of appraised property value, plus an ongoing 0.5% annual premium on the loan balance, charged by the Federal Housing Authority (FHA, a subsidiary of HUD). The FHA makes up mortgagee losses in the event that the resale value at termination is less than the outstanding loan balance M T. Both the up-front insurance and ongoing insurance premia paid to FHA are far larger than the insurance payments made on conventional loans. To date, rapid mobility and house price growth have left FHA with few losses and thus large reserves. Refinancing was very costly in the period for which we have data, so the only feasible way for borrowers to extract increased equity from appreciation was to move. Recently, refinancing has become easier, but remains very expensive. 8 Technically, borrowers are required to perform a minimal level of maintenance and to pay property taxes. Practically, the maintenance requirement cannot be enforced after closing, because it would be unlikely for a court to force a senior citizen out of their home for failure to perform maintenance. A HECM is not the only product that allows older homeowners to convert home equity into cash without moving. There are other reverse mortgage products on the market which have, to date, had smaller market share than HECMs. The largest of these is Financial 7 Or the one year interest treasury rate in year t plus a larger spread if the rate adjusts annually rather than monthly. 8 In late 2004, HUD reduced FHA s mortgage insurance premium somewhat on refinanced loans, to 2% of the increase in loan amount. Previously, refinancing was as costly upfront as a new loan. 7

9 Freedom, which imposes larger fees and interest rates than HECM, but allows larger loan amounts for higher value properties. For seniors with large incomes, conventional home equity loans ( HELOCs ) can work much like reverse mortgages. While these loans require payment of interest and some principal before moving, for several years a line of credit with an increasing balance can behave like a reverse mortgage. Reverse mortgages can be more attractive, despite their high fees, because home equity lines will only provide a loan amount such that repayment is feasible without selling the home and without using annuitized income. For the many seniors with low income, the allowable HELOC loan size is thus small. Larger loans without sufficient income are possible, but failure to document sufficient income leads to higher interest rates ( Alt-A or -B loans). 3 Selection and Moral Hazard: Theory and Numerical Examples We now ask whether the HECM industry structure described above lends itself in theory to adverse or advantageous selection, and whether moral hazard or its opposite seems more likely to arise. We focus on the dimension of mobility out of the home. A large portion of the high costs associated with the HECM product stems from guarantee fees charged by the FHA. For this reason, and because the FHA bears the risk of default, we consider selection and moral hazard from the guarantor s perspective. Selection is thus adverse (advantageous) if the expected profits from guarantees on loans to individuals with characteristics associated with becoming a borrower are less (greater) than the expected profits from guarantees on loans to individuals with characteristics associated with not becoming a borrower. Under assumptions detailed below, guarantor profits per dollar of home value on a loan 8

10 to a borrower with characteristics x are: π(x) H a = F + fm T (x) a e (r M r)(t a) dt I(T > t ) ( me (r M r)(t (x) a) e (g r)(t (x) a)). (2) The first term in equation (2) represents the guarantee fee charged at closing, expressed as a fraction F of the initial home value H a. The second is the present value of the continuing fee income, taken to be a fixed fraction f of the outstanding loan balance. Here, r M is the interest rate on the reverse mortgage, r is the rate (all rates are real) at which the guarantor discounts, a is the date at which the loan is funded, and T is the date at which the borrower terminates. We assume that the reverse mortgage is a lump sum fraction of home value m and that there is no partial prepayment before T. There are thus no transfers between crossover date t. For this reason, we consider selection to be adverse (advantageous) if (1) characteristics associated with long (short) stays before moving or dying are associated with reverse mortgage take-up and (2) borrowers tend to live in geographic areas with low (high) appreciation rates. 9 Moral hazard occurs when, holding characteristics constant, the act of borrowing leads to behavior associated with reduced profits to the guarantor. Because g is assumed exogenous conditional on location, we define moral hazard to occur if, conditional on characteristics, reverse mortgage borrowers are likely to move or die at a later date than non-borrowers. We focus on mobility, assuming away private information on mortality or effects of borrowing on mortality. We will consider selection and moral hazard both analytically and with the aid of numerical examples. Analysis is difficult because the choice to take on a reverse mortgage or not is discrete. As a proxy for reverse mortgage demand, we consider the gain in utility from adding reverse mortgage debt. For any given reverse mortgage size M, the larger the value of the derivative of lifetime indirect utility with respect to mortgage size, V/ M, the likelier we consider reverse mortgage take-up to be. 9 Gyourko et al. (2004) explain how expected growth rates can differ across regions. 9

11 We thus define adverse selection as a positive correlation across characteristics x between V/ M and the gain from waiting to move V/ T, evaluated at any arbitrary combination of T and M. Advantageous selection is defined as a negative correlation. Moral hazard arises if taking on additional reverse mortgage debt makes deferred mobility more attractive, i.e. if 2 V/( T M) > Plausibility of Advantageous Selection and Moral Hazard We now provide some analytical intuition for why selection and moral hazard are likely to move in opposite directions. Even in a deterministic framework, general results are not available. We provide numerical example that match our conjectures on comparative statics below. For analytical purposes, we assume that a prospective reverse mortgagor maximizes the following objective: V = max v (T a, w, y, H a, x) + µ(t, x) + v + (A T, w +, y, e g(t a), x), (3) s + s.t. w =s a + M s +, (4) w + =f(s +, y, H a, M, T ) + h(t, M, H a e g(t a) ) (5) Indirect utility V is the maximized sum of three terms: indirect utility before moving v ; a disutility associated with moving that may be a general function of the date of move, µ(t ); and indirect utility after moving, v +. v and v + map from five arguments to the real numbers. The first argument in v and v + is the length of the period in question. The period before moving has length T a and the period after has length A T. T is thus bounded above at A. We assume that length of life A is deterministic and known to both the borrower and the lender. As discussed below, the advantageous selection seen to date in this market appears unrelated to more rapid mortality among borrowers. v + could include a bequest motive. 10

12 The second and third argument of indirect utilities v and v + relate to wealth spent in the period in question. Expenditures in the period prior to moving consists of savings at period a, plus the reverse mortgage proceeds M, minus savings (or debt), s +, carried over to the post-move period. An important observation is that for anyone who puts no weight on consumption after moving (v + 2 = 0), the value of a reverse mortgage in dollars is equal to exactly the dollar amount of proceeds from the mortgage. The budget constraints (4) and (5) show that post-move wealth w + consists of an increasing function f of savings s +, plus housing wealth h. The wealth from savings function f is affected by income level and housing assets because if s + is negative, the borrowing rate will be affected by collateral. For example, a home equity loan at lower interest rates than credit card debt will only be available if home value is sufficiently large and if there is no reverse mortgage debt. h itself is a function increasing in home resale value, H a g(t a) e and decreasing in M and T. For reverse mortgages in default at date T, the derivative of h with respect to each argument is zero. Income y is assumed annuitized and constant in real terms. The fourth argument of indirect utility relates to housing consumption. For the period prior to moving, this argument is direct housing consumption, fixed at H a. For the period after moving, v + is negatively affected by the real price of housing after moving, e g(t a). The fifth argument of indirect utility is a vector of unobservable characteristics x that affect the shape of v and v +. Such characteristics include health status, gender, bequest strength, impatience, and taste for smoothing consumption. µ(t, x) relates to the disutility of moving and to any health benefits that might arise from moving closer to care givers. Immediately, we see why moral hazard on the dimension of home maintenance would not be an obvious outcome if maintenance were included as a choice. Taking on reverse mortgage debt increases the marginal utility of pre-move wealth v 2 and reduces the marginal utility of post-move wealth v + 2. To the extent that maintenance is savings rather than consumption (e.g. absent default), this makes maintenance more, not less, attractive. 11

13 Turning to the questions of selection and moral hazard on the dimension of move date T, if V is differentiable, then using the envelope condition on s +, we have: ( V f T = v 1 + µ v T + h ) T + gha g(t a) h e v + H a eg(t a) 2 ge g(t a) v 4 +, (6) V M = v 2 + ( h M + f M )v+ 2, (7) 2 V M T = v 12 + ( h M + f M ) ( ) v ge g(t a) v 24 + (8) ( ) 2 f + T M + 2 h T M + 2 h M H a e g(t a) gha g(t a) e v s + T ( w s + v 22 + ( h M + f M )v+ 22 ). Moral hazard on T arises if 2 V/( T M) > 0. The plausibility of moral hazard is reflected in the first expression in equation (8). v 12, the effect of length of stay prior to moving on the value of expenditures allocated to the pre-move period, must be positive by concavity of direct utility. Moral hazard is not guaranteed. The second and third terms in equation (8) reflect the fact that additional time spent at home makes the reverse mortgage costlier if default is not planned. The added cost comes from the spread r M r. This consideration has the effect of making the critical cross-partial 2 V/( T M) more negative (barring very strong substitution between numeraire and housing demands and strong real home price appreciation). The fourth term reflects the concavity of wealth and the fact that savings are likely to be diminished the longer the time at home. If default is planned, then extra time at home does not make reverse mortgage debt more costly, except to the extent that the borrower also has non-mortgage debt paying an interest rate that rises with M. Moral hazard is thus more likely to operate if default is feasible and we should expect lower rates of exit for reverse mortgagors in default than for non-mortgagors. 12

14 Adverse (advantageous) selection arises if the values of the derivatives V/ M and V/ T are positively (negatively) correlated across borrower characteristics, conditional on M. These derivatives are given by equations (6) and (7). The effect of different characteristics on these derivatives are complicated and will be simulated for a particular parameterization below. The possibility of advantageous selection can be seen by considering the correlation between v1 v 1 +, which increases the value of V/ T, and v2 v 2 +, which increases V/ M. The first difference reflects whether time spent before moving is more or less pleasant than time after moving. This value should be large when consumers have adequate income and savings to remain in their home for a long time and when their second period wealth is relatively small, e.g. through a low level of housing wealth and a low growth rate g. By contrast, the second difference is likely to be large under the exact opposite circumstances. v2 v 2 + reflects whether money is more valuable before or after moving. By concavity, pre-move wealth should be useful when savings and income are low and post-move wealth should be less useful when the home value is large. Hence a negative correlation between V/ T and V M is plausible. More succinctly, house-rich, cash-poor older individuals can smooth consumption across periods both by taking on reverse mortgage debt and by moving to release any remaining home equity. Observable characteristics should thus feed advantageous selection. There is an empirical problem with using observable wealth and income to explain the rapid mobility of reverse mortgagors relative to non-borrowers. While the typical borrower is, in fact, house rich and cash poor relative to the population before taking on the reverse mortgage, this is no longer the case once the loan is funded. This is because the elderly, in general, are house rich and cash poor and loan proceeds are not small. For this reason, we now turn to consideration of unobservable characteristics. Variation in the concavity of indirect utility over wealth consumed (risk aversion) has ambiguous effects on the direction of selection. The stronger the taste for consumption 13

15 smoothing (risk aversion) the more negative will be v1 v 1 + and the more positive will be v2 v 2 + for the majority of the elderly who have most of their financial wealth in home equity. This effect, which argues for advantageous selection, could be reversed if default is planned on the reverse mortgage or the reverse mortgage is large, so that the post-move period is poorer than the pre-move period. Large variation in discounting is a feasible explanation for all of the mobility patterns we observe. In the limit, an individual who considers only present utility will wish both to postpone the pain of moving and to consume home equity. This should make a reverse mortgage attractive. In terms of our model, v 2 v + 2 is large for heavy discounters, rendering V M large. Following an initial spending spree, the level of utility before moving is much lower than the level of utility after moving if there is a large disutility to extremely low spending, so V T is small. The fact that the move will lead to prolonged disutility from having moved would be ignored by a sufficiently heavy discounter. Another possible source of advantageous selection is heterogeneity in remaining length of life. With a short remaining life, a reverse mortgage can obviate the need to move before death. Hence v 1 v + 2 may go to infinity for the short-lived with no bequest motive. For those anticipating a long remaining lifetime, a reverse mortgage may not provide sufficient cash to render remaining at home through death attractive, so that post-move marginal utility may be relatively large. In this case, a reverse mortgage may be less attractive to the Jeanne Calments of the world than to those expecting to die soon. A related unobservable characteristic is the number of children. This may proxy for bequest strength and more children may make staying at home easier. Thus those with few children may prefer reverse mortgages and move out of their homes earlier. 3.2 Numerical Examples of Selection and Moral Hazard Even with simplifications, our analysis has only provided intuition as to the direction of selection. We now turn to numerical examples to show that advantageous selection can 14

16 arise. They allow us to approximate the dollar value of the right to take on a mortgage of a given size for different individuals, and to compare the minimum of the move or death date for individuals with different characteristics. Our purpose is threefold: to show that moral hazard and advantageous selection are likely to arise; to determine under what conditions which of these two forces is likely to dominate at loan termination; and to identify testable comparative statics that can be compared with data to see if the model broadly fits with reality. We use a simple constant relative risk aversion utility function that is time separable and discounts exponentially, but introduce the complication of direct utility costs of not remaining at home. Adding such features as habit formation and non-exponential discounting might add realism but are not necessary to achieve our purposes. Individuals have the following utility function: A 1 A U = u(c t )δ t + δ A bu(c A ) δ T µ 0 δ t µ 1 (9) t=0 t=t In (9), µ 0 represents an immediate disutility that arises from moving and µ 1 represents a chronic disutility that is suffered every year that the consumer no longer lives at the original address. We simplify by assuming that consumption of more or less housing before or after moving generates no utility except through the µ terms; that is, v4 = v 4 + = 0. In formulation (9) we explicitly consider a bequest motive. This could also represent a high marginal utility state of nature that is associated with being alive but necessarily out of the home, such as a nursing home stay. This rationalizes the simultaneous prolonged failure to move and small or negative gains to reverse mortgage debt for some households. The consumer maximizes (9) over the choice of consumption and bequest streams {c t } and move date T, conditional on taking on a reverse mortgage or not. The consumer draws down the entire reverse mortgage balance at origination. 15

17 Formally, the constraints faced by the consumer are: c T = s T (10) s t+1 = [1 + r] [s t c t + y + M t + I(t = T )h(t )], (11) M 0 = αh 0, (12) M T = min( ( (M 0 + F )(1 + r M ) T, H 0 (1 + g) ) T, (13) M t = 0, t {1, 2,... T 1, T + 1,... A 1} (14) h(t ) = max ( ) 0, H 0 (1 + g) T M T. (15) Equations (10) and (11) disallow borrowing except through a reverse mortgage, ignoring the growing home equity loan market for reasons discussed above. Equations (12) through (14) detail reverse mortgage proceeds and repayment under the assumption of a lump sum payout. Equation (15) assumes constant home resale growth. A first result is immediate from our setup. When δ r, b 1, and g r is sufficiently large, advantageous selection can arise on the dimension of curvature of u(c). To see this, note that for linear utility, there is no gain to consumption smoothing. Under the condition on the growth of housing prices, it is optimal to remain in the home until death to avoid the utility cost of moving, while weakly increasing lifetime wealth. There is thus no move before death. The reverse mortgage reduces future wealth and is thus purely welfare destructive. For sufficiently small µ 0 and µ 1, a consumer with u sufficiently negative (with y and s 0 sufficiently small, but retaining δ = r and b = 1, g = r) will cash out home equity before death to smooth consumption, even with a reverse mortgage. Naturally, matters are more complicated when the required inequalities are not satisfied. g r is difficult to justify, as it implies purchasing a larger home increases available non-housing lifetime wealth. To explore the role of discounting in selection, we use some numerical examples. The parametrization of the constrained maximization (9) through (15) is detailed in Table 1. The observable parameters are chosen, for the most part, to capture realistic levels. For example, 16

18 Table 1: Parameterization of Utility Maximization Problem (9) through (15) Variable Description Values s 0 Initial savings 3,000 and 22,000 H 0 Initial home value 150,000 M 0 Reverse mortgage proceeds (net of fee) 90,000 F Reverse mortgage fee 9,000 r Interest rate on savings 3% r M Interest rate on reverse mortgage 4.5% g Home price growth rate -2% and 6% y Annual income (annuitized) 5,000 and 50,000 A Remaining lifetime 15 u(c t ) Utility over consumption 20,000 c µ 0 Utility cost of selling home 2.5 µ 1 Utility cost of not living in original home 3 δ Discounting 1,.5, and.25 b(c A ) Utility over bequest 400,000 c A the interest rate and fees for the reverse mortgage are taken from HECM averages. A life expectancy of 15 years for a senior is reasonable based on mortality tables. The unobservable parameters were chosen by trial and error to illustrate the possibility that advantageous selection would lead to rapid mobility on the part of reverse mortgage borrowers. We discuss the parameters after reviewing the results of the simulation. The results of the simulated maximization are summarized in Table 2. We find that the value of a reverse mortgage is much greater for heavy discounters. The final column in Table 2 asks how many dollars in initial wealth a consumer would need to be paid to be indifferent between taking out a reverse mortgage immediately, and never getting one. The very heavy discounters (δ =.25,.5) value the $90,000 in reverse mortgage proceeds almost dollar for dollar. The light discounters (δ = 1) value the proceeds at a much lower rate because they also value the fees and interest that must be repaid upon death or exit. Concave utility over wealth at age A makes the fees more onerous to light discounters when the home depreciates in value. We also find that the heaviest discounters (δ =.25), who value the reverse mortgage most, move out earlier than lighter discounters, even with the reverse mortgage in hand. 17

19 Table 2: Results of numerical simulations: Move date and welfare gain or loss from taking on a reverse mortgage Initial Income Price Discount Move Date Move Date Approx. Value savings growth rate no RM yes RM of RM 3,000 5,000 6% ,000 20,000 5,000 6% ,000 3,000 50,000 6% < -3,000 20,000 50,000 6% < -20,000 3,000 5,000-2% ,000 20,000 5,000-2% ,000 3,000 50,000-2% < -3,000 20,000 50,000-2% < -20,000 3,000 5,000 6% ,000 20,000 5,000 6% ,000 3,000 50,000 6% ,000 20,000 50,000 6% ,000 3,000 5,000-2% ,000 20,000 5,000-2% ,000 3,000 50,000-2% ,000 20,000 50,000-2% ,000 3,000 5,000 6% ,000 20,000 5,000 6% ,000 3,000 50,000 6% ,000 20,000 50,000 6% ,000 3,000 5,000-2% ,000 20,000 5,000-2% ,000 3,000 50,000-2% ,000 20,000 50,000-2% ,000 Notes: The first column is the consumer s starting wealth in period 0. The second column is the annual annuitized income while alive. The third column is the annual price appreciation of the home. The fourth column is the consumer s discount rate δ. The fifth (and sixth) columns report the date at which the consumer optimally sell their home without (and with) a reverse mortgage. The last column reports the approximate payment an individual without a reverse mortgage would have to obtain in order to be indifferent to getting the cash or being granted a reverse mortgage. Negative valuations imply the reverse mortgage makes the consumer worse off. Parameters are as in Table 1 unless specified. 18

20 This conclusion holds only when prices are appreciating and for sufficiently low income. With sufficiently high income, the utility cost of moving always outweighs the gain to supplementing annuity income with home sale proceeds. This follows from concave utility. Hence, even extremely heavy discounters may not move before death. When the home value falls, the gain from moving becomes smaller relative to the disutility of moving. The unobservable parameters δ, b(c A ), µ 0, and µ 1 are reverse engineered (or calibrated ) to match the fact that reverse mortgage borrowers move more quickly than non-borrowers. It should be emphasized that the result is far from guaranteed. For reasonable discount rates, the moral hazard induced by reverse mortgage debt is sufficient to keep borrowers at home until the end of life as long as the costs of moving are sufficient to keep patient consumers at home. Even for a seemingly unreasonable discount rate of 50%, moral hazard is as strong as adverse selection in that it is not possible to get reverse mortgagors to move before non-borrowers. Further, even with strong bequest strength and no discounting (δ = 1), the consumption smoothing provided by reverse mortgages is sufficient to render them attractive, albeit less so than to the impatient and only when home prices are increasing over time. We also find in Table 2 that low savings and income relative to home values and greater anticipated price appreciation are associated with both weakly more rapid mobility and weakly greater reverse mortgage valuation, consistent with the theoretical intuitions developed earlier. 4 Empirical Analysis of HECM Borrower Mobility We have seen that it is theoretically possible to explain the excess speed of exit from homes by reverse mortgage borrowers on life cycle grounds, but that theory does not unambiguously predict that reverse mortgage borrowers will move more rapidly than non-borrowers. We now test whether the data support the hypothesis that excess mobility among reverse mortgage borrowers can be explained by a relatively strong desire to spend remaining home equity. 19

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