The Role of Price Appreciation in Reverse Mortgage Terminations

Transcription

1 The Role of Price Appreciation in Reverse Mortgage Terminations Thomas Davidoff and Gerd M. Welke August 31, 2013 Abstract Reverse mortgages are non-recourse loans, and are thus profitable to lenders only if borrowers exit their homes quickly or home prices increase sufficiently rapidly. Over the lifetime of the dominant US reverse mortgage product, HECM, terminations have been more rapid than would be predicted based on the behavior of non-mortgagee older Americans. A key question is whether terminations can be expected to slow in the absence of strong advances in home prices. We show that there is an economically large and positive correlation between reverse mortgage terminations and home price appreciation. Moreover, we find a declining effect of price appreciation on termination with borrower age, consistent with a model of optimal move timing. This suggests that caution is warranted in structuring fees and loan to value limits. Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC V6T 1Z2; thomas.davidoff@sauder.ubc.ca Department of Finance, Real Estate and Law, California State Polytechnic University, Pomona, 3801 West Temple Ave, Pomona, CA 91768; gmwelke@csupomona.edu

2 1 Introduction Reverse mortgages provide liquid funds to older homeowners in exchange for repayment whenever the borrower sells their home or dies. 1 Because lenders have no recourse to assets other than the mortgaged home, the time to death or voluntary sale of the home and the rate of home price appreciation are critical factors in the pricing and profitability of reverse mortgages. The currently dominant reverse mortgage product is the federally sponsored Home Equity Conversion Mortgage (HECM), which has existed only since 1989 and as of mid-year 2008 had a median origination date of The actuarial performance of this program has been very strong, which can be jointly attributed to the fact that home prices have been rising and because borrowers have terminated their loans at a high rate. That is, the accumulated principal and interest on reverse mortgages loans has not had time to exceed the value of homes, and in many states, the gap between collateral value and the loan amount due has risen, rather than fallen, with time. The rapid rate of reverse mortgage terminations has been surprising. Initial models created by the US Department of Housing and Urban Development (HUD) projected that loan terminations would occur at a rate of 1.3 times the mortality rate for the age and gender of the youngest (or only) borrower in a household. To date, terminations have not only exceeded this level, they have also exceeded the termination rates for otherwise similar households, as documented by Szymanoski et al (2007), and Davidoff and Welke (2007). The relatively high mobility of HECM borrowers is puzzling because the act of taking on reverse mortgage debt would seem to induce moral hazard on the dimension of length of stay in the home, while adverse selection would likely reinforce the effect. The case for moral hazard is that older homeowners typically dislike selling their home (see, e.g., the AARP survey cited in Venti and Wise (2000)). However, reverse mortgage borrowers typically have low incomes and little non-housing wealth. Thus, in the absence of a cash infusion, and particularly in the presence of a strong positive shock to house prices, there will be a strong temptation to sell the home, and move to smaller or rental housing to finance an improved level of non-housing consumption. The reverse mortgage 1 In the reverse mortgage program we discuss here, refinancing has generally been exceedingly expensive. 2

3 reduces the gain to selling the home, because there is more wealth in the absence of a sale and less wealth after a sale. A key to understanding the rapid mobility of HECM borrowers is that, unlike non-borrowers, HECM borrowers mobility has been significantly affected by home price appreciation. This fact was documented for HECM borrowers by Davidoff and Welke (2007). The relative sensitivity is understandable when home prices are decreasing, in that a borrower anticipating their home value falling below the loan amount should have much less incentive to move than a borrower who can move and extract equity. However, conditional on positive equity, we show below that for otherwise similar borrowers, reverse mortgage termination exceeds that found in the AHEAD/HRS survey, but that this difference declines with the age of the borrower. The fact that reverse mortgage borrowers mobility has been sensitive to home prices suggests that excess mobility will be reduced in the likely event of a sustained period in which appreciation is weaker than recently seen. In this paper, we present evidence that reverse mortgage borrowers trade off utility costs associated with moving against the financial and consumption benefits of liquidating remaining home equity. Consistent with a simple model of mobility with a reverse mortgage and a one-time shock to home prices, we find that HECM borrowers sensitivity to home prices decreases with age. This explains the fact documented in Szymanoski et al (2007) that the rapid mobility of HECM borrowers is concentrated among relatively young borrowers. We conclude with what we view as important questions for future research. 2 Home Prices and Reverse Mortgagor Mobility A consumer who will live up to date A has no financial assets other than a home with value 1, on which she has taken a reverse mortgage in the amount m < 1. The consumer can save at a zero interest rate, but the mortgage debt accrues at rate r, so that at date t mortgage debt is given by m t = m exp(rt). Further borrowing is not possible and she has no income other than the proceeds from the reverse mortgage. At date 0, after taking on the mortgage, but before making any other decisions, the consumer observes the value of the home p that will prevail at each date t until her death at date A. 3

4 The consumer maximizes a utility function given by: 2 T V = max u(c t ) dt + {c},t,s 0 A T u(c t ) dt I T <A (1) subject to max(p m T, 0) + s T m c t dt s 0 (2) 0 A T s 0 (3) c t dt 0, (4) where I indicates that a move at date T before date A entails a utility cost normalized to one. Period utility u is a concave, increasing function. A critical feature of the problem (1) through (4) is that it will either be optimal to move at date 0 or never at all (i.e., at death). To see this, note that moving after date 0 subtracts lifetime resources without softening any of the constraints (2) through (4). More formally, suppose a move occurs at date 0 < T < A. Total resources available are p m exp(rt ), and utility is reduced by the fixed cost I of moving. The consumer could instead move at date 0, set s = m, and owe zero interest on the mortgage. The disutility of moving is still incurred, but she can now spend total resources p > p m exp(rt ), spread uniformly over her lifetime A. By concavity and monotonicity of u, an immediate move must result in a higher level of lifetime utility and is preferred to a later voluntary move. 3,4 Lastly, if disutility of moving is large and/or the expected lifespan A is short, the consumer may chose a forced move at death thereby avoiding the cost I altogether. We can thus restrict our attention to utility with an immediate voluntary move, which gives utility: U 0 (p, m, A) = I + A u( p m A ) (5) 2 In the appendix, we show that the results obtained here will also hold in the presence of a post-move net rent expenditure and expected future house price appreciation, provided both are sufficiently small. 3 Note that introducing inter-temporal discounting would be more realistic, but we would have to resort to numerical solutions. See Davidoff and Welke (2007). 4 Note that the possibility of default does not change the conclusion that a move takes place immediately or never. Should the loan amount m exp(rt) ever exceed the home value p at some date t, the borrower would not move past date t, but might still move at date 0. 4

5 and a forced move at death A, which provides utility: U A (p, m, A) = A u( m ). (6) A The mortgage termination rate at date t = 0 is directly related to U 0 U A. rises: Not surprisingly we find that moving is more attractive as p increases and less attractive as m ] [U 0 U A m ] [U 0 U A p =; u ( p m A ) > 0, (7) = u ( p m A ) u ( m A ) < 0. (8) Equation (8) shows that there is moral hazard to lenders on the date at which the borrower moves: borrowers will stay longer with a larger mortgage. This is a hazard to lenders because it increases the probability that the future value of the loan will exceed the value of the home. Differentiating equation (7), we find that by concavity of u moving is made relatively more attractive to consumers with longer horizons: 2 ] [U 0 U A p A = p m A 2 u ( p m ) > 0. (9) A Borrowers with a shorter horizon find it less attractive to extract a given amount to equity than younger borrowers: by concavity, utility gains are greater if the additional non-housing consumption is spread out a longer the time span, and moving voluntarily entails a fixed utility cost. 3 Home Prices, Age, and Mobility in the HECM Program We have found that in a simple model of mobility for reverse mortgage borrowers, the effect of price on demand for mobility is stronger for younger borrowers than for older borrowers. Such a simple conclusion would be complicated by moves at interior dates if discounting, stochastic mortality, 5

6 stochastic changes in prices, and a more realistic utility framework were considered. Analytical, or even numerical solutions of such a more realistic problem would be quite difficult and sensitive to assumptions (see, for example, Henderson 2005). Here we assess if the simple model predictions Eqs. (7) (price appreciation effect) and (9) (horizon effect) are a good description of the HECM mobility data. 5 In Section 3.1 we describe the HECM data in some detail, and then illustrate the appreciation and horizon effects in Section 3.2. Section 3.3 presents a more detailed empirical model, while in Section 3.4 we return to a comparison of HECM and AHS borrowers. 3.1 The Data The loan-level HECM data we use has been obtained from the U.S Department of Housing and Urban Development (HUD). The data consists of 456,957 reverse mortgage originations, with their origination date (CLSDTE), the borrowers gender or marital status (GENDER), the borrower s age at closing (CLSAGE), 6 the U.S. state in which the property is located, and the termination date (TRMDTE), if any, of the mortgage. 7 We drop 371 observations which have a termination date on or before the loan closing date, two with termination dates after the date on which the data set was obtained, 576 with missing borrower age at closing, 893 with missing 8 state information, and 1,493 observations with missing gender/martial status. Seventeen records show a borrower age at closing greater than 110, and are also dropped. The resulting data set, with 453,613 observations, 9 is summarized in Table 1. For terminated mortgages, 86,667 in total, we also summarize the length of time the mortgage was held (MRTDUR). 3.2 Illustration of Price Appreciation and Horizon Effects This section aims to illustrate graphically our model predictions, by considering reverse mortgage termination rates for four types of borrowers: younger (age at closing a 0 < 68 years) or older 5 We have no data on m, so testing Eq. (8) is not possible. 6 The younger borrower s age, if a couple. 7 Termination occurs by death (of the surviving partner, if married), a home sale, or prepayment of the mortgage. 8 We also drop any FIPS codes outside of the 50 states and DC. 9 Eight records are dropped on more than one criterion. 6

7 Variable Obs Mean Std. Dev. Min Max GENDER Male 78,615 Female 210,510 Married 164,488 CLSAGE [yrs] 453, CLSDTE [yrs] 453, TRMDTE [yrs] 86, MRTDUR [yrs] 86, Table 1: Summary of the HECM Data. (a 0 > 77 years) borrowers, living in states that experienced high or low appreciation over the period 2000Q1 through 2008Q3. We define high (low) appreciation states as states in which the nominal OFHEO house price index rose by more than 70% (less than 60%) between 2000Q1 and its peak value (around 2006) and then dropped by at least 5% (no more than 2%) through 2008Q3, which is the end of our data set. 10 Figure 1 shows the two resulting composite nominal OFHEO house price indices. These were created from the state-specific indices, weighted by the number of open reverse mortgages in each state. The high appreciation index rises 134% from 2000Q1 through its peak in 2006Q3, and then drops 12% through the end of the sample period. The low appreciation index rises 39% with only a negligible drop at the end. Figure 2 shows the termination rate T of reverse mortgages versus duration τ = t t 0, for mortgages that were originated in the period (circles) and for the period (squares). The four panels show T for the four combinations of age group and experienced appreciation. All four cases exhibit a baseline mortgage duration effect - the termination rate starts low and rises linearly, generally leveling off later on. As expected from Eq. (7), low appreciation p (panels (b) and (d)) leads to lower overall termination rates when compared to high appreciation (panels (a) and (c)). Further, the termination rate drops dramatically in the last two years of our sample in the high appreciation regions, which also experienced large house price declines in 2007 and Low appreciation states, on the other hand, where there was very little price decline in the last two years of the sample, show far weaker 10 Specifically, high appreciation states include AZ, CA, DC, FL, HI, MA, MD, NJ, NV, RI, and VA. Low appreciation states include AL, AR, CO, GA, IA, IN, IL, KS, KY, LA, MO, MS, NE, NC, OH, OK, SC, SD, TN, and TX. Missing states fit neither definition, for example, appreciation in MI was less than 60%, but the peak through 2008Q2 drop was more than 5%. 7

8 declines in termination ((a) and (c) versus (b) and (d), respectively). The overall level of termination is higher for older borrowers than for younger ones, as can be seen by comparing panel (a) with (c), and (b) with (d). This is not surprising given the former s shorter expected life span. Next consider the horizon effect, Eq. (9). It implies that the rate of change in termination rate T in house price change p = p 1 is increasing in time till death, A, or decreasing in borrower age. To illustrate this, we group the borrowers into 2 year age cohorts. 11 Proceeding cohort-bycohort, for each loan and date, we first proxy the change in price p by the state-wide OFHEO price appreciation over the prior year. The loans in the highest appreciation decile are then used to estimate a mortality-adjusted termination rate. 12 The corresponding termination rate of these loans in the prior year then allows an estimation of T. Figure 3 shows the resulting T/ p as a function of age. T/ p is a generally decreasing function of age, with a slope of and a White t-statistic of The evidence for the horizon effect is, however, not particularly compelling since other appreciation deciles generally show insignificant slope, which is occasionally even positive. The sensitivity of mobility to price shocks and borrower age needs to be investigated more closely. 3.3 An Empirical Model To more closely assess whether the simple model s prediction seems to be a good description of mobility among HECM reverse mortgage borrowers, we now consider empirical estimates of the following equation: T ags,t0 t = α + β g D g + β s D s + β τ D τ + β t D t + β p p + β A A t + β τa (τ A t ) + β pa (p A t ) + ɛ (10) 11 Ages [64, 66), [66, 68),..., [78, 80). 12 A termination rate for 2000, for example, is the adjusted number of mortgages that terminate between EoY 1999 and EoY 2000, divided by the number of active mortgages, defined as those that have not terminated by EoY Instead of summing the number of terminating mortgages, we add the survival rates of the borrowers (determined by their age and year of birth) for that year, conditional on having survived to the end of the prior year. In the approximation that HECM borrowers mortality is similar to that of the population at large, this adjustment is meant to correct for moves by death. 8

9 Figure 1: Nominal composite OFHEO house prices indices, 2000Q1-2008Q High Appreciation States Low Appreciation States Composite Nominal OFHEO HPI [1980=100] Date 9

10 Figure 2: Reverse mortgage termination rate T versus duration τ, by origination period (circles: , squares: ), state appreciation characteristics, and borrower age at origination, a 0. T t-t0 [% y -1 ] High Appreciation a 0 < 68y (a) Low Appreciation a 0 < 68y (b) T t-t0 [% y -1 ] High Appreciation a 0 > 77y (c) Low Appreciation a 0 > 77y (d) τ = t-t 0 [y] τ = t-t 0 [y] 10

11 Figure 3: The YoY change in the mortality-adjusted termination rate for the highest decile annual change home prices, as a function of borrower age Δ T/Δ p Age 11

12 where subscript a denotes the current age of the borrower in year t, g is the gender/martial status variable, s the property s state index and t 0 the year of the mortgage origination. The variable τ t t 0 is the integer mortgage duration, i.e., the number of years the mortgage has been held in year t. The dependent variable T ags,t0 t is the date t fraction of terminated to open reverse mortgages conditional on origination in year t 0 to borrowers of age a, gender g, and resident in state s. Since we have no reason for termination, terminated means by either borrower death, loan payoff while remaining at home, or by house sale. The data set leads to 1,250,104 cells in a, g, s, t 0 and t, but we shall also limit the data set to observations that originated before July 1, 2008, which removes 27,776 records. For the independent variables, D g, D s, D τ and D t are sets of dummy variables for gender, state index, integer mortgage duration and current year, respectively. These are meant to capture any effects not related to appreciation (state duration) and life expectancy (current borrower age). For instance, termination may be related to the importance of harsh weather conditions for older borrowers, implying different baseline mobility in colder versus warmer climates. p is the OFHEO house price index appreciation for the borrowers state, from origination t 0 to the current year t, while A t is the expected remaining lifespan at current date t, based on the borrowers gender, current age a = a 0 + τ and year of birth t 0 a 0 (Human Mortality Database 2008). 13 It relates directly to the expected date of death A in Eqs. (1) (4). In the following estimations, we correct standard errors by clustering on all origination variables, viz., gender, age at origination, year of origination and state index. First, consider the main effect of house price appreciation, Eq. (7), unconditional on borrower age. To determine this, we set β A = β τa = β pa = 0 in the estimation (10). The results is shown in panel A of Table 2, and termination is found to increase significantly with house price appreciation p = HP I t /HP I t0. The larger the realized appreciation, the more incentive a borrower has to improve welfare by extracting equity immediately, rather than waiting and accruing more debt. Note that this result is unconditional on the borrower s view of future appreciation, for which we 13 In what follows, we have calculated the life expectancy of a married couple by assuming both are female. The results are qualitatively unchanged if life expectancy is for a male. 12

13 have no proxy. As pointed out in the appendix, provided the consensus view is not too high, an immediate move is still called for in our simple model. Not surprisingly, the main effect of life expectancy, A, (panel B) is negative: younger borrowers (larger A) are less likely to terminate the mortgage through death than older borrowers (smaller A). Both the positive effect of appreciation and negative effect of life expectancy remain significant when p and A t are included simultaneously in the estimation (panel C). Given that the average borrower tends to extract realized equity gains sooner rather than later, we should see a declining effect of realized appreciation for older borrowers, since there is less welfare gain the older you are, by concavity of utility (Eq. (9)). The regression coefficient of interest in equation (10) is β pτ, which asks whether estimated state level home price appreciation p has a different effect on borrowers mobility depending on their expected horizon, A t. Estimating the full model, we find results shown in panel D, Table Not surprisingly, younger borrowers will terminate more rapidly than older borrowers the higher the price appreciation, β pa > 0. Thus the younger a borrower (longer horizon A), the greater the impact of appreciation ( p) on the termination rate. Alternatively put, in high appreciation states younger borrowers move relatively faster than older borrowers when compared to low appreciation states. This difference will decline as price appreciation moderates, so that the voluntary mobility of younger borrowers will decline to levels comparable to that of older borrowers. The clear implication for lenders is that for lower appreciation than has been observed in the sample period ( ), moral hazard in the dimension of mortgage duration will strengthen, adding to any direct effect from loss of collateral value. 3.4 Mobility for HECM Borrowers versus ARS/AHEAD Homeowners An implication of the model outlined in Section 2 is that non-borrowers mobility should be less sensitive to house price appreciation than borrower mobility. When values drop, there is still equity to extract for the all-equity owner, while when they rise, the incentive to move is weaker since there 14 The direct effect of house price appreciation now changes sign, though this is merely because the interaction p A t contributes if one averages over the distribution of A t. 13

14 Var Panel A Panel B Robust Robust T Coef. Std. Err. t-stat Coef. Std. Err. t-stat p A t R-sq 3.5% 4.2% Var Panel C Panel D Robust Robust T Coef. Std. Err. t-stat Coef. Std. Err. t-stat p A t A t τ A t p R-sq 4.5% 4.9% Table 2: Model Eq. (10). Dummy variables are included for gender g, state s, current duration τ, and current year t. Standard errors are corrected for clustering on state, gender, age at closing, and date of closing (43,642 clusters). The number of observations is 229,306. is no incurred cost of debt and the disutility of moving plays a relatively larger role. Given then that appreciation was elevated over most of the sample period, we would expect higher termination rates for reverse mortgage borrowers than in the population analogue. Figure 4 compares female borrower 15 in-home survival over the period 1998 through 2004 for HECM borrowers (solid line) to that of the population analogue (the HRS/AHEAD survey, open circles), as a function of age at origination. The HECM borrowers clearly terminate at a much higher rate, and, as before, the difference diminishes with age. This dependence is consistent with reverse borrowers extracting a positive shock to equity early to avoid interest costs, while all equity owners have less of an incentive to do so. By concavity, the incentive diminishes with borrower age. Note that the higher mobility in the HECM data is not an artifact of excess termination by early death in the HECM sample. One might envisage a scenario in which a financial shock, brought on by, say, a medical crisis, leads homeowners to borrow funds, but then terminate soon after by death or a move to a nursing home. Davidoff and Welke (2007), however, show that the excess mobility is significantly dependent on asset appreciation 16 - a result one would not expect if mortgage uptake and termination was conditioned on a random poor health draw. 15 Similar results obtain for male and married borrowers. 16 More specifically, the state of residence. 14

15 Figure 4: in-home survival rates for female homeowners. borrowers; open circles: HRS/AHEAD panel data. Solid line: HECM hterm$astill[hterm$margen == 2] hterm$age[hterm$margen == 2] 15

16 4 Conclusion In the dominant US reverse mortgage product, the federally sponsored HECM program, loans are non-recourse and lenders profit only if the loan balance does not exceed the home value by the termination date. For this reason, lenders have charged high fees and insurance premiums 17 to compensate them for moral hazard in length of stay and size of loan, as well as possible adverse selection in the dimension of length of stay. Surprisingly, however, the actuarial performance of loans to date has been very good. The reason has been the generally high, positive house price appreciation along with a much higher rate of termination than expected. The key question for future performance is whether the rate of termination will slow when home prices decline and revert to historical norms. In this paper, we have presented a simple model of mobility for reverse mortgage borrowers. In addition to financial considerations in a risk averse setting, the model captures the elderly homeowners well-documented dislike of moving. Conditional on positive home price appreciation, optimal mortgage termination is either immediate, improving welfare by avoiding further interest costs and spreading the extracted equity over as large a horizon as possible, or never, which saves the disutility cost of moving before death. Under these conditions, mobility increases with initial house price appreciation since utility is increasing in wealth. Marginal termination in appreciation, however, decreases with age since utility is concave. Intuitively, older borrowers will avoid the cost of moving since it outweighs utility gains from a (large) increase in non-housing consumption spread over a small horizon. These qualitative model features are borne out by an empirical analysis of data from HUD s HECM program. This data set comprises 456,957 HECM mortgages originated from 1989 through mid-2008, each characterized by (1) their date of origination, (2) the US federal state in which the collateral is located, (3) the gender/martial status and (4) age of the borrower, along with the date the mortgage was terminated, if at all. For each year in the sample observation period we can therefore calculate the mortgage termination rate in the four variables, along with the duration since origination. A proxy for collateral appreciation for each year is achieved by merging the state- 17 The insurance program is actually administered and guaranteed by the Federal Government. 16

17 time period data with OFEHO s corresponding house price index, while mortality follows from the borrower s age, gender and year of birth. We find strong evidence that mobility increases in realized price appreciation: borrowers release equity for non-housing consumption faster, the higher the gain. Note that we would not reasonably expect this result if a borrowers original motivation for borrowing was related to a financial emergency. Consistent with model predictions, we also find that the effect in appreciation diminishes for older borrowers. In a comparison of HRS/AHEAD mobility with HECM borrower mobility, we find that the latter terminate more rapidly than the former and that the difference declines with borrower age. This result is consistent with the observation that the non-borrower faces less incentive to move immediately as there is no interest cost to save. The difference in termination rate is not due to exogenous factors such as poor health, as appreciation is a strong covariate for HECM mobility. Our results suggest that rapid HECM termination will abate in a declining price environment, such as has been experienced in the US from 2006 through the present day. This spells potential trouble for the reverse mortgage industry because mobility declines precisely when collateral values drop. An important area of future work will therefore be to repeat our estimation of mobility in a few years, once enough termination data in a depreciating market has been gathered. Clearly, this will required extending the model to handle the owner s put option more carefully a difficult task, given risk aversion and little guidance on the relative sizes of bequest motives, disutility of moving, etc. Secondly, the present paper ignores the issue of adverse (or positive) selection. The simple model here has the advantage of being clear on the effect of appreciation and borrower age, but addressing the selection will require a more detailed comparison of HECM termination with the population analogue and a model that differentiates owners, for example along the dimension of discounting. A preliminary study along these lines has been done by Davidoff and Welke (2007). We thank the Department of Housing and Urban Development for the use of the HECM data. 17

18 Appendix A: Model Solution In this Appendix, we consider the solution to the consumer maximization problem Eqs. (1)-(4) for voluntary moves 0 T < A, under the additional complications of a post-move housing expenditure h and an expected future house price appreciation g s. Consider a consumer maximization problem V = max {c},{h},t,s T 0 u(c t ) dt + T H 1 + A T (u(c t ) + H(h t )) dt I T <A (11) m T 0 p T m T + s subject to (c t + h 1 ) dt s 0 (12) A T s 0 (13) (c t + h t ) dt 0, (14) where p T = exp(g s T ) and m T = exp(rt ) are the house price and mortgage debt at the date of sale, respectively. The constraints are modeling the consumer as cash-poor with no income stream, and unable to borrow beyond the reverse mortgage. Pre-move, she spends a fixed cost of ownership h 1 and derives fixed additive utility at a rate H 1. Post-move, the consumer chooses a level of rent h t. Since there is no intertemporal discounting, c t and h t are piecewise constant, c t = c 1, 0 < t T, and c t = c 2, h t = h, T < t < A. The Lagrangian simplifies to ] L = T [u(c 1 ) + H 1 [ ] + (A T ) u(c 2 ) + H(h) + λ 2 [ max(p T m T, 0) + s (A T )(c 2 + h) ] I + λ 1 [m s T (c 1 + h 1 ) ], (15) Embedded in constraint Eq. (14) is an option to default strategically, which will affect the date of move. Default is feasible only if the debt owed exceeds the value of the home before death, i.e., if T ln m/(r g) < A. 18 In our data sample, however, strong appreciation relative to 18 In which case it would not be optimal to move after T : There wold be no equity to transfer to the post-move period, and moving is costly since h reduces other consumption and the owner incurs disutility I by moving before death. 18

19 average mortgage rates implies T > A, 19 so we can dispense with the complication of strategic default from here on. Note, however, that from 2006 on, strong appreciation ended and default may likely become an issue. The first order conditions are c 1 : T u (c 1 ) λ 1 T = 0 (16) c 2 : (A T ) u (c 2 ) λ 2 (A T ) = 0 (17) h : (A T ) H (h) λ 2 (A T ) = 0 (18) s : λ 1 + λ 2 = 0 (19) T : u(c 1 ) u(c 2 ) + H 1 H(h) λ 1 [c 1 + h 1 ] + λ 2 [g s p T r m T + (c 2 + h)] = 0 (20) λ 1 : m s T (c 1 + h 1 ) = 0 (21) λ 2 : max(p T m T, 0) + s (A T )[c 2 + h] = 0 (22) Eq. (20) explicitly shows that part of the motivation to move may come from suboptimal housing consumption pre-move. To abstract from this issue, which does not concern us here, we simplify the problem by assuming that the optimal level of h sets H(h) = H 1. Since T is interior, and u and u are 1-to-1 mappings, Eqs. (16), (17) and (19) imply that c 1 = c 2 c, which simplifies Eq. (20) 0 = p T (m T m) c A h (A T ) h 1 T (23) Further, adding Eqs. (21) and (22) leaves us with r m T = g s p T + (h h 1 ). (24) 19 The mean and standard deviation of the expected borrower lifespan in our sample is 7.1 and 3.0 years, respectively, with a maximum of 16.2 years. Over the period in our data sample, the mean initial mortgage rate was 6.1% with a standard deviation of 1.0%. Assuming a generous m = 0.5 and a very high average mortgage rate of two standard deviations above the mean, we find Tmin = 15.4 years for the lowest average OFHEO house price appreciation in our data set, viz., 3.6%, for Connecticut. Note also that in our data set the lowest value of realized house price p at any time is

20 The optimal (voluntary) move date is determined by Eq. (24), which weighs the expected marginal future appreciation g s exp(g s T ) plus marginal rent savings net of ownership cost, h h 1, against marginal cost of carry rm exp(rt ) This condition implies that for sufficiently small appreciation and/or post-move net housing expense, the borrower will either move immediately, or die in the home. To see this, first note that the conditions Eqs. (23)-(24) are valid for interior moves only. Therefore, the choice of (voluntarily) moving now or sometime before death both imply a disutility I, and (expected) financial gains Eq. (24) are the sole criterion in making that decision. Secondly, an immediate move (T = 0) happens when g s + h h 1 rm, 20 and it is optimal to wait only if g s + h > rm. 21 Therefore, if appreciation and/or rent savings are small compared to mortgage costs, any voluntary move is immediate. Alternatively, however, a borrower might wait till death if the disutility I from a voluntary move is large enough to force the corner solution. 20 It is simple to show that the solution T to Eq. (24) is strictly negative if g s + h h 1 rm and g s > 0. For g s sufficiently negative (depending on h), the structure of the solution is more complicated, but we can ignore this case for our purposes. 21 In which case T may be positive. In particular, as g s approaches r from below, a voluntary move is postponed indefinitely (T ); for sufficiently large appreciation, the borrower s stay will be terminated by death. 20

21 References Davidoff, Thomas and Gerd Welke, 2007, Selection and Moral Hazard in the U.S. Reverse Mortgage Market, Haas School of Business Working paper, U.C. Berkeley. Henderson, Vicky, 2005, Explicit Solutions to an Optimal Portfolio Choice Problem with Stochastic Income, Journal of Economic Dynamics and Control 29(7), p Human Mortality Database, 2008, University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at or (data downloaded on June 2, 2008). Edward J. Szymanoski, Edward J., James C. Enriquez and Theresa R. DiVenti, 2007, Home Equity Conversion Mortgage Terminations: Information To Enhance the Developing Secondary Market, Cityscape: A Journal of Policy Development and Research, 9(1), Venti, Steven and David Wise, 2000, Aging and Housing Equity, NBER Working Paper