Selection and Moral Hazard in the US Reverse Mortgage Industry Thomas Davidoff and Gerd Welke Haas School of Business UC Berkeley Incomplete: comments encouraged June 30, 2004 Abstract A large number of older Americans can be characterized as house rich, cash poor. Unless strong bequest motives are present, reasonably priced reverse mortgages should thus be popular because they transfer wealth to such homeowners from the relatively wealthy period after their home is sold to the relatively impoverished period before. Absence of demand has been blamed in part on the very large fees associated with reverse mortgages. These fees, in turn, are justified by concerns that borrowers will remain in their homes so long that collateral value will fall below loan balances. Moral hazard arises if the presence of a reverse mortgage makes a given borrower stay longer in their home than they would have absent the reverse mortgage or if endogenous price appreciation is weakened by the presence of the reverse mortgage. Adverse selection arises if the characteristics that make reverse mortgages attractive also make remaining in the home longer more attractive. We show in a parsimonious model that moral hazard and selection effects are likely to work in opposite directions, so the sign of the correlation between length of stay and reverse mortgage demand is ambiguous in theory. Based on loan histories and the American Housing Survey, we find empirically that single women who have participated in the most popular US reverse mortgage program (HECM) depart from their homes at a rate almost 60 percent greater than observably similar non-participating homeowners. The results suggest that rising home prices have fed strongly advantageous selection in this market through a mechanism similar to the heterogeneity in risk aversion used to rationalize advantageous selection in insurance markets. Weaker price appreciation may reverse the direction of selection to adverse.
1 Introduction Home equity plays the leading role in the wealth of most older Americans. Based on the 2001 Survey of Consumer Finances, Aizcorbe, Kennickell and Moore (2003) show that 76 percent of household heads 75 or over owned a home, with a median value of $92,500. Median net wealth among these households was $151,400. Just 11% of these households owed any mortgage debt. A large fraction of home equity appears to be retained up to death. Sheiner and Weil (1992) report a mobility rate of approximately 4% among older single women based on the Panel Study of Income Dynamics, and a similar number arises in the Survey of Income and Program Participation. Combined with mortality rates, this suggests that approximately 50% of recent retirees will die in their current home. This is consistent with the AARP survey finding, cited by Venti and Wise (2000) that 89% of surveyed Americans over 55 reported that they wanted to remain in their current residence as long as possible. In the absence of very strong bequest motives, these facts suggest that a financial product allowing consumption of home equity without requiring a move would be quite valuable to older homeowners. Artle and Varaiya (1978) implicitly show that an older homeowner will be willing to pay an interest rate in excess of the rate on savings in return for the opportunity to borrow against their home equity. This is the logic behind the reverse mortgage market. Figure 1 illustrates the sequence of payments in a simplified version of a reverse mortgage. Payments need not be made until the (both) borrower(s) die or move out of the home. M 1 denotes a cash advance made at the time of loan closing. Should the homeowners die or move out of the home, they must pay principal and interest on M 1 as well as on fixed costs F, incurred at the time of closing but typically financed. Interest accrues at rate R M 1. If the borrower stays in the home, they have the option of paying down some of the balance or drawing down a further cash advance; M 2 can be positive or negative. In no event is the amount owed greater than the property value at the time of resale. H denotes the value of the home at the time of loan closing and Π 1 is the realized rate of price inflation. Perhaps the most familiar form of contract to economists specifies that payments are constant as long as the homeowner remains in the home. This would conform with Figure 1 if we assumed that the homeowner in no event can live past period 2 and M 1 and M 2 are pre-specified to be equal. In fact, most loans under the dominant US program allow greater flexibility, as discussed below. If the interest rate exceeds the expected rate of house price appreciation, then a natural concern for lenders is that borrowers will live so long with such a low mobility rate that the present value of reverse mortgage payments will exceed the collateral value when the loan becomes due. Indeed, the famed Frenchwoman Jeanne Calmet, 1
M 2 min((m 1 + F )R 2 M + M 2R M, HΠ 2 ) Stay [p] M 1 Move [q] min((m 1 + F )R M, HΠ) 0 Dead [1-p-q] min((m 1 + F )R M, HΠ) Figure 1: Reverse mortgage design. Loan balance is repaid by the borrower as late as the date of move out of the home or death. F denotes financed closing costs. who lived to be 121 sold her apartment forward in her late eighties in what must have been a disastrous arrangement for reverse mortgagee Andre-Francois Raffray. 1 The reverse mortgage market is thus not unlike an insurance market that combines coverage for the event of a combination of a long stay in the home and a low rate of price appreciation. Short stays and high appreciation rates generate profits for lenders whereas long stays and low appreciation generate the potential for transfers to borrowers, akin to insurance payments. As discussed in de Meza and Webb (2001), selection in such markets need not be adverse. Insured parties (here borrowers) may have characteristics that render them less likely, rather than more likely, to receive transfers conditional on insurance. For example, de Meza and Webb find that individuals who purchase insurance against losing credit cards are less likely than those without insurance to lose their credit cards and suggest that the insured are more risk averse, and hence more careful to avoid loss than those without insurance. Finkelstein and McGarry (2003) find that a similar mechanism may be operative in the market for long term care insurance among the elderly. In Section 3, we present a model of the timing of move out of an older homeowner with and without a reverse mortgage. Moral hazard arises if the presence of a reverse mortgage makes staying in one s home more attractive. The hope that such a hazard exists, in fact, is part of the public justification for government involvement in the reverse mortgage market. 2 Adverse selection arises if the characteristics that render a reverse mortgage attractive also make staying in the home longer, holding the size 1 As reported by the Associated Press on August 5, 1997. The french word for the arrangement is viager. 2 See Blacker (1998), for example. 2
of reverse mortgage debt fixed or if low rates of price appreciation are associated with reverse mortgage demand. Advantageous selection arises if those who find reverse mortgages worthwhile tend to leave the home sooner or if demand is associated with high rates of appreciation. To focus on the question of whether we should expect reverse mortgagors to remain alive and in their homes longer than the rest of the population, we assume that potential borrowers have perfect information concerning events in the future. A fuller model of demand would incorporate uncertainty into borrowers thinking. Given the infancy of the industry and the critical role of a government insurer, we ignore equilibrium effects of a large reverse mortgage market on rents, interest rates and prices and we do not consider strategic behavior on the part of lenders. Inspection of Figure 1 suggests that both advantageous selection and moral hazard are likely to operate in the reverse mortgage market. If price appreciation is sufficiently large that the amount payable at the time of move is less than property value and if R M is greater than the return available on savings to the borrower or fixed costs are significant, then it is clear that demand for a reverse mortgage will exist only for homeowners whose marginal utility is greater while in the home than after moving or in death. Relatively high marginal utility in the home makes sense if the bequest motive is weak and the present value of additional expenditures incurred after moving while still alive (such as medical costs and rent on housing still consumed) do not exceed the value of the home. 3 In the absence of a reverse mortgage, or if the reverse mortgage balance is small, then selling the home before death will thus be more attractive to borrowers than non-borrowers. Hence advantageous selection is likely to occur on the ratio of wealth to home value. The analogy to insurance against accidents is as follows: small wealth to home value ratios induces a greater difference in marginal utility between the states of moving early and moving late or never, just as greater risk aversion induces a greater difference in marginal utility between the states of accident or no accident in insurance markets. It is possible however, that net selection can be adverse. For example, in the presence of a bequest motive, borrowers with a greater probability of death (and hence early loan termination), will find the reverse mortgage relatively unattractive. By design, the presence of the reverse mortgage may attenuate the effect of advantageous selection related to greater marginal utility before the move than after. In insurance market terms, the presence of insurance may thereby render the risky behavior of remaining in the home more attractive, so this is a form of moral hazard. 4 3 Interesting complications, not addressed in this paper, arise due to Medicaid eligibility rules and particularly the protected status of home equity in some state Medicaid rules. 4 There is some belief on the part of HUD staff that in the early history of the program, the presence of large reverse mortgage debt actually pushed homeowners out of their homes, such that moral hazard operated in the reverse direction. 3
Moral hazard becomes a more serious problem if price appreciation is low enough that a long stay puts the loan into default, because then the cost to remaining in the home associated with the mortgage rate exceeding the rate on savings is lessened. Presently, the interest rate R RM does not vary by market, and hence is independent of expected price appreciation. In markets with sufficiently low expected appreciation that a long stay is associated with default, advantageous selection becomes less likely to the extent that taking on reverse mortgage debt may become favorable to the borrower in expected value. We will not discuss maintenance moral hazard, which has been discussed elsewhere (Miceli and Sirmans (1994) and Shiller and Weiss (2000)), except to observe that moral hazard may not be a problem with the elderly, in the sense that with or without a reverse mortgage, older homeowners can be expected to do very little home maintenance. Indeed, Davidoff (2004) suggests that combining a reverse mortgage with a maintenance contract might generate Pareto gains to dynasties in which children might otherwise oppose a reverse mortgage. 5 Given the likelihood of moral hazard in this market, if we believe that all relevant characteristics that are associated with increased reverse mortgage demand are also associated with lengthened optimal time to move, then we would expect to find empirical evidence that reverse mortgagors delay moving relative to the rest of the population. The model presented in Section 3 instead suggests that we can not rule out either the possibility that reverse mortgagors move out early relative to the rest of the population or move out late in the absence of strong assumptions. Section 4 provides empirical analysis showing that reverse mortgagors in the most popular US program leave their homes at a rate that far exceeds the rate for comparable older homeowners. This suggests the operation of advantageous, rather than adverse, selection. An appendix demonstrates that most of the results are consistent with the predictions of the model. Throughout, we will focus on the structure of the Home Equity Conversion Mortgage (HECM), the largest reverse mortgage product in the US and the only one that enjoys a guarantee from the Federal Housing Administration. Other products are available in the US, notably the Financial Freedom reverse mortgage which allows larger loan amounts. The most popular way for older Americans to withdraw home equity in recent months appears to be through home equity loans or home equity lines of credit. Interest rate spreads for conventional home equity products other than HECM have dropped considerably over the last two years. These products require amortization 5 With prices rising, as they have for most of HECM s history, concavity of utility over consumption implies that maintenance should increase, rather than decrease in the presence of a reverse mortgage. This follows as long as repairs are long-lived because repairs transfer money from the period before the move to the period after; the reverse mortgage generates the opposite transfer. With falling prices, moral hazard reappears just as advantageous selection likely disappears. 4
during the life of the loan, but a line of credit could be structured to postpone any out of pocket repayments on a fairly large balance for a number of years. Examination of the performance of home equity loans made for consumption smoothing purposes to older homeowners would be of interest if such data were easily available. Indeed, at present it appears that the only way that a HECM could be preferred to a home equity line would be if the anticipated stay in the home were close to 15 years or if income were very low such that only a very small home equity line would be available without a reverse mortgage. 6 In the theoretical discussion below we allow for the possibility of home equity borrowing, although the simulations do not allow other forms of borrowing. 2 The HECM Product The theoretical willingness of older homeowners to pay a spread above the riskless rate to borrow against future housing sale proceeds underlies the reverse mortgage industry, which dates to 1961 in the US and the early part of 20th Century in Europe. In the late 1980s, the US Department of Housing and Urban Development ( HUD ) devised a Home Equity Conversion Mortgage ( HECM ) program which is currently the dominant reverse mortgage product in the US. The program works roughly as follows, based on a program evaluation done for HUD by Abt Associates in 2000. Borrowers must be homeowners with very little or zero outstanding mortgage debt. HECMs are originated by banks, sometimes through mortgage brokers. The banks and brokers earn upfront fees and the originators typically retain servicing rights. These lenders typically sell the cash flow rights associated with the loans to Fannie Mae. The loan cash flows are insured by the Federal Housing Agency (FHA) against default. In exchange for the guarantee, FHA receives 2 percent of the property value at the time of 1 loan closing and assesses a charge of 24 of one percent of the outstanding loan balance each month. The borrowers are obliged to make property tax payments and to perform minimal maintenance but maintenance requirements are presumably enforceable only before closing. 7 Otherwise, no payments are due until all mortgagors (a borrower and a spouse if one exists) have moved out of the home, dead or alive. There is no recourse to the lender for payments outside of the value of the home in the event that the resale value is below the outstanding loan balance. Because interest rates are likely to exceed the rate of house price inflation, loan-to-value ratios are fairly small and increase with age. 6 As noted below, the HECM has very high fixed costs but an interest rate typically lower than home equity loans. 7 One would expect considerable legal difficulty in evicting an elderly mortgagor for failing to make sufficient repairs to their home. 5
Borrowers can receive payments in several forms. They may receive a single lump sum payment, a line of credit with an increasing maximum outstanding balance, monthly payments that last for a fixed period (term payments), or monthly payments that last as long as the borrower lives in the home (tenure payments). Borrowers may receive payments in a combination of any of these forms. The line of credit is by far the most popular option (and it includes lump sum payments as a subset). The amount that may be borrowed is decreasing in interest rates and increasing in borrower age. The interest rate on HECM loans may be fixed or adjustable, but almost all existing loan rates are adjustable as Fannie Mae will only purchase ARMs under HECM. The spread over the one-year treasury rate is typically near 1.5 percent. Closing costs on the 77,007 loans issued to date vary considerably, and the size of the loan has minimal explanatory power. 8 The median ratio of closing cost to property value is 6.8 percent. These closing costs, which may be financed, are large relative to conventional loans, particularly relative to home equity lines of credit which feature closing costs of zero in some cases. No payments are due until the borrower moves or dies. Absence of demand has led to the exit of many originators, 9 but the potential size of the market is huge. The 2000 US Census reports that there are approximately 17.5 million homes owned by households with heads over age 65. Hence the total originations of 77,000 to date in the decade of HECM operation represents a very small market share. Indeed, Mayer (1994) suggests that the HECM product should be attractive to at least six million older households. From the American Housing Survey, homeowners aged over 65 had homes that averaged values of $170,000 in 2001. If 10 percent of the value of these homes were subject to reverse mortgages, the outstanding balance would be approximately $300 billion. The Appendix explores the extent to which an increase in demand is warranted by life cycle considerations and concludes that there is good reason to think that reverse mortgages should be attractive to a large segment of the population. 8 The R 2 from a regression of closing cost on maximum loan amount is just.0008 and the coefficient on maximum loan amount has the wrong sign. 9 Other problems have plagued the industry. Some reverse mortgages were designed with shared appreciation features. Some reverse mortgagors under such SAM arrangements died within one or two years of origination but enjoyed large capital gains, so that the payments received relative to the debt owed were very small. This has led to legal conflicts. Perhaps for this reason, Fannie Mae does not purchase shared appreciation mortgages. 6
3 A Model of Ambiguous Correlation Between Length of Time at Home and Reverse Mortgage Takeup Consider a retired consumer (most frequently a single woman, often a married couple and very infrequently a single man). This consumer derives utility at any moment from the hedonic quality of housing owned (H) and from the consumption of other goods (c). the move. The quantity of housing consumed can only be changed at the time of Thus an individual of age a consumes H a up to the chosen move date T and thereafter consumes H T. move. We assume for convenience that there is only one We assume further that housing enters the utility function in a similar way before and after moving. 10 To focus on asymmetric information, we make the very strong assumption that the retiree knows for certain that she will die at age A and not before and also knows the (constant) rate of house price appreciation and the interest rate on savings. The adjustability of HECM interest rates allows the abstraction from rate-based prepayment motives. Moving out of one s home and into a new home has at least three effects on utility. First, there is a financial effect, discussed below. Second, there is an immediate disruption which is likely to subtract utility. I denote this cost of disruption by µ 0, and this cost may change with time. A third effect on utility is that the new home may feature medical attention not available in the old home, which provides a benefit most likely increasing with time µ 1. U = The Lifetime utility maximization problem is thus: T a u(c(t), H a )e δ(t a) dt µ 0 (T )+ A I assume that the felicity function takes the following form: 11 T + u(c(t), H) = c(t)1 γ 1 1 γ u(c(t), H T )+µ 1 (t)e δ(t a) dt+bu(c(a+), H(A+)). (1) + H1 η 1. (2) 1 η In addition to housing, there are two assets. The first is savings s, which earns interest rate r(s, y, HP (t)). I assume that r(s) is a constant r when savings are positive, 10 One can interpret the rental cost of a unit of housing, discussed below, as including a scaling of housing consumption after moving. The restriction on the utility function is that the curvature in H is the same before and after the move. 11 It is common (see for example Chetty and Szeidl (2004), Flavin and Nakagawa (2004), Lustig and Nieuwerburgh (1993) to assume that η = γ. This affords tractability, but implies a wealth elasticity of housing demand of one, a value considerably larger than what is observed empirically (my own back of the envelope estimate based on renters in the Oakland and Washington, D.C. metropolitan areas is approximately.6, in line with the older estimates of Carliner (1973). 7
Figure 2: Interest rate on savings and debt as functions of savings, income y and house value HP. y 1 < y 2, HP 1 < HP 2 r(s,y 1,HP 1 ) r(s,y 2,HP 2 ) r r M 0 s and that r s > 0 when savings are negative. The interest rate may shift down holding debt constant with an increase in y or HP (t) through home equity lines. These are different from reverse mortgages in that if prices do not fall, the borrower is unlikely to be able to extract more loan proceeds than the amount of property value. The interest rate function is diagrammed in Figure 2. The second asset is a reverse mortgage which may only be held in negative quantity and with a debt M(t) that grows by r M per period if no new debt is taken on (if m(t) = 0). The initial conditions for savings and reverse mortgage debt are given by: M(a) = F ( M); s(a) = s a. (3) The first equation in (3) states that the initial mortgage balance is either zero or the size of the fee required to open a reverse mortgage line of credit with upper limit M. F is difficult to estimate as there has been considerable heterogeneity in reverse mortgage fees to date. As discussed above, this is a large amount and may be increasing in the value of the home as well as in M; there is, however, a considerable fixed component. The laws of motion for savings and reverse mortgage debt are given as follows. These laws reflect the jump in cash savings at the time the home is sold and the assumption there is no collateral to allow for reverse mortgage borrowing after the move. We assume that savings must be weakly positive at death whether or not there 8
is a bequest motive. ṡ(t) = r(s(t), y, HP (t))s(t) + y + m(t) c(t); Ṁ(t) = m(t) + r M M(t) t < T (4) ṡ(t ) = max(0, H a P a e g(t a) M(T )) (5) ṡ(t) = r(s(t), y, H T P (t))s(t) + y + m(t) c(t) H T fp a e g(t a) t > T (6) The structure of the problem guarantees that reverse mortgage debt can only be attractive at age a if savings are sufficiently negative that r(s, y, HP a ) > r M. It seems correct to assume the interest rate on any debt in excess of the reverse mortgage amount will accrue interest at the rate r(s M, y, HP a ). In this case, because of the decreasing marginal fixed cost of borrowing with increasing mortgage debt it will be worthwhile to borrow at least up to the amount of debt owed through the reverse mortgage. We assume that this is feasible under the program details for people who do not choose to move; that is, we assume that consumer debt is less than the upper bound M on reverse mortgage size. The reverse mortgage balance is bounded above at M(HP a ) and below at zero. We write these as constraints on the difference between the control variables and the state variables. M(t) + m(t) + s 0; M(t) + m(t) M(P a e g(t a) )e r M (t a). (7) Let π s (t) represent the shadow value of savings at time t and π M (t) the (weakly negative) shadow value of increasing the reverse mortgage burden. The Hamiltonian of the control problem, given housing H a or H T is as follows: H(t) = c1 γ 1 1 γ e δ(t a) + π s (rs c + m + y) + π M (m + r M M). (8) The Lagrangean is: L(t) = H(t) + λ M ( M m) + λ M( Me r M (t a) M m) (9) This gives rise to the following optimality conditions: L c(t) = 0 c(t) γ = π s. (10) L m(t) = 0 π s = π M + λ M + λ M. (11) π s = L s(t) = π s(r + s r s ) (12) π M = L M(t) = (r Mπ M λ M λ M ). (13) A borrowed dollar that must never be repaid has the same value as a dollar of savings. It follows that if the property value at move time T is less than the reverse mortgage amount, then π M is zero in every period. In this event, λ M must bind and 9
have the same value as the shadow value of savings. If the loan can be repaid at the move date, however, then it is clear that π M (T ) = π s (T +). Equation (12) does not hold at date T as a result of the discrete jump in savings. For consumers who are never in debt, the equation holds in practice as these consumers can treat the resale value of the home as equivalent to cash in hand. 3.1 Consumption, Housing, Bequest and Marginal Utility of Savings After Moving The optimization problem can be solved backwards, starting with the optimal bequest. With the utility function given by (2), it is not possible to solve for c and H in closed form. We can, however, derive first order conditions. In particular, we find that for the recipient of the bequest, c(a+) γ 1 η H(A+) = P η a. We can write a general indirect utility function as bv(s(a+), P a e π(a a) ). We can see from the first order condition that v 1 > 0, v 11 < 0, v 2 > 0 if η > 1. If the consumer does not wish to borrow in the period after the move (this will be assumed in the remainder of this paper), then by the maximum principle, (12) the marginal utility of wealth must be given by bv 1 (S(A+), P a e π(a a) ) = π s (T +)e r(a T ). (14) This implies the following inverse functional relationship: s(a+) = v1 1 (π s(t +)e r(a T ), P a e g(a a) r(a T ) ). (15) b v1 1 is the inverse function of marginal utility. For example, if the bequest utility v did not depend on housing consumption, and preferences were of the CRRA form such that v 1 = s(a+) γ, we would have by (14), v1 1 = s(a+) = π s (T +) 1 γ e r(a T ) γ. From the fact that v 11 < 0, we know that the inverse savings function is decreasing in marginal utility, v1 1 π s < 0. We can make sense of an absence of a bequest motive (b 0) in the context of (15) if lim v1 0 v1 1 (v 1, ) = 0 The retiree thus solves the following maximization problem after moving: A T + max {c(t)},h T U = Subject to the budget constraint: A T + u(c(t), H T )e δ(t a) dt. (16) c(t)e r(t T ) dt = s(t ) v1 1 (π s(t +)e r(a T ) A, P a e g(a a) ) H T fp a e g(t a) r(a T ) dt. b T (17) The constant f (0, ) represents a different cost of renting housing from owning housing, under the assumption that rents are a constant proportion of prices (given a constant interest rate). In the absence of taxes or other frictions, f would equal the user cost r g. 10
The marginal utility of housing must equal its price, so that H η T = π s (T +)fp a e g(t a) δ r g [e(g r)(a T ) 1][e δa e δt ] (18) Equation (10) and the absence of liquidity constraints combined with equation (12) implies that the present value as of time T of future consumption is given by: A T + c(t)e r(t T ) dt = π s (T +) 1 γ γ r(1 + γ) δ r(1+γ) δ [e γ A e r(1+γ) δ γ T ]. (19) Combining equations (17), (18) and (19), we obtain an implicit formulation for the marginal utility of a dollar of savings after moving out of one s home, π s (T +): π s (T +) 1 γ γ r(1 + γ) δ r(1+γ) δ [e γ A e r(1+γ) δ γ T ]+ (20) 1 η πs (T +)(fp a e g(t a) δ g r [e(g r)(a T ) 1][e δa e δt ] 1 ) η 1 η +v 1 1 (π s(t +)e r(a T ), P a e g(a a) ) s(t +) = 0. b While this expression is rather involved, the critical conclusion arises from visual inspection: πs(t +) s(t +) < 0. That is, the marginal utility of savings is decreasing in savings. This follows from the concavity of both felicity and bequest utility in S. It can also be seen (and is intuitive) that marginal utility is increasing in the bequest motive. Marginal utility is also unambiguously increasing in the horizon A as long as the rate of housing price growth g is small or demand for housing is inelastic in the sense that η > 1. 3.2 Adverse Selection, Moral Hazard and Reverse Mortgage and Move Date Choice A natural definition of adverse selection is that people who optimally take on positive reverse mortgage debt are likely to have lower collateral value at their optimal move date than those who do not take on reverse mortgage debt. 12 Define I M as an indication of an individual obtaining greater utility with a reverse mortgage and the attendant fee than without. We will then be interested in the following concepts. 12 An alternative definition would weight individuals by their optimal mortgage balance. This would be difficult analytically and would provide little guidance in empirical work, since reverse mortgage balances are considerably more difficult to observe than reverse mortgage take up. Further, the small fraction of the population that has taken up reverse mortgage debt and the sharp constraints on allowable balances imply that a very large fraction of the variation in mortgage balances is absorbed by the extensive margin of take-up relative to the intensive margin of size conditional on take-up. 11
Definition 1 Selection is adverse (positive) if EP a e (g r)(t a) I M M = 0 < (>)EP a e (g r)(t a) M = 0. Selection can be considered conditional on observables or unconditional on observables. The HECM product uses only collateral value and age to screen borrowers, so long as non-hecm debt does not exceed the borrowing limit. However, income and other assets are also observed to the lenders. Moral hazard is defined as the effect of allowing a borrower to have a reverse mortgage at all: Definition 2 Moral Hazard is operative if EPae(g r)(t a) M < 0. We can not know whether to think of reverse mortgagors as different from observably similar non-mortgagors because of unobserved characteristics that led them to take up the debt or as different because they were more or less randomly selected into the program. It would thus be quite difficult to distinguish moral hazard from adverse selection by simply comparing the mobility rates of HECM borrowers from non-borrowers with similar observables. However, due to the very low take up rates, we can ask if non-borrowers who are observably similar to borrowers tend to die or move with more or less net collateral than non-borrowers who are observably dissimilar from borrowers. Similar will be defined below. This comparison allows us at least to empirically distinguish between (a) selection on observables and (b) either selection on unobservables or moral hazard. The first purpose of the analysis below is to show that in the absence of very strong assumptions, it is not clear whether we should see adverse selection and moral hazard or not. A second purpose is to show that adverse selection is more likely to arise in an environment of decreasing collateral value, all else equal. Critical to studying the extent of net selection and moral hazard is determining the optimal date of move. If there is an optimal date of move T that is different from the date of death A, then this date must satisfy the following condition, using a result reproduced in Léonard and Long (1992), p. 312 as well as the envelope condition on H T and equations (5) and (12): U T = µ 0(T ) µ 1 (T )+ c(t )1 γ 1 1 γ c(t +)1 γ 1 1 γ a 1 1 η + H1 η H1 η T 1 1 η (21) +π s (T +)I HaP ae g(t a) >M(T )((g r)h a P a e g(t a) (r M r)m(t ) +π s (T )ṡ(t ) π s (T +)ṡ(t +) = 0. This three term expression can be interpreted as follows. The first line represents the change in utility attributable to a delay. There is a change in the psychic cost of moving µ 0 and an additional instant of the pre-move felicity and one less instant of 12
the post-move felicity. Included in the reduction of post-move felicity is the loss of the benefit of moving to potentially geriatric-friendly surroundings µ 1 (T ). The second line of (21) represents the housing cost of waiting to move, and this consists of two parts. First, there is the capital gain but foregone interest attributable to waiting to sell the home. Second, there is the real cost of waiting to pay off the reverse mortgage. The third line of expression (21) represents the costs and benefits of waiting to change cash accumulation patterns. If there is dissaving before the move date but not after, then extending the move date stretches a scarcer resource more thinly and thus involves an indirect utility cost. There is no guarantee that the necessary condition (21) for an internal move date is ever satisfied nor it clear that any extremum would represent a maximum, let alone a global maximum. Given the obvious presence of a positive cost of moving µ 0, it is therefore not so surprising that mobility rates among the elderly are low. If the corner solution of not moving before death is optimal and there is no bequest motive, then it is clear that the reverse mortgage is utility enhancing and that the loan balance at death will be Me r M (T a). Before proceeding with analytical evaluations of adverse selection and moral hazard, it will be helpful to record a few facts about reverse mortgage demand. Unfortunately, the complexity of the optimization problem does not afford a large number of unambiguous results. Conditional on the optimal choice of move date T, optimal behavior between some age a and T is governed by the first order conditions (10), (11), (12) and (13). We have the following: Lemma 1 If savings are positive over any interval [t ɛ, t] and if the interest rate on the reverse mortgage exceeds the interest rate on savings then the reverse mortgage balance M is zero or λ M binds. Proof Suppose that both savings and the reverse mortgage balance are positive. In that case, λ M = 0 and by (12), π s = rπ s. By (11), π s = π M + λ M through the interval. But if λ M is zero, then π M = r M π M, a contradiction, since r M > r by hypothesis. If initial savings s a are sufficiently negative we cannot rule out the possibility that a positive reverse mortgage balance is paid off before the date of the move. This can occur if δ < r M and if the marginal utility of wealth after moving is sufficiently large. Lemma 2 If the reverse mortgage balance will exceed collateral value at the move date, then M(T ) = Me r M (T a), the maximum allowable balance. 13
Proof Failure to use up the credit line reduces consumption before moving and adds nothing to consumption after moving. This violates optimality by monotonicity of utility over consumption. Lemma 3 If λ M(t) binds for some t (a, T ), if M(T ) does not exceed collateral value and if δ < r M, then optimal consumption at date T is increasing in M. Proof We have assumed that M > s. Therefore, if λ M binds, it is desirable to borrow money to support consumption above the level y for some period, since excess borrowing cannot be used as savings by Lemma 1. But by the assumption on the discount rate, it cannot be desirable to use the reverse mortgage to borrow against income before the move. Hence consumption at time T must be weakly greater than consumption before T, which implies c(t ) > y. In this case, the marginal utility of consumption at T divided by the opportunity cost, c(t ) γ e r M δ (T a)π s (T ) 1 is no less than at any other time t < T. Hence an increase in pre-move resources implies an increase in consumption at date T From the proof of Lemma 3, it appears that if δ > r M then it is possible that optimal consumption at date T may not increase with the reverse mortgage if the borrowing constraint is severe enough. 3.3 Analytical Evaluation of Moral Hazard Putting aside questions of selection, we ask whether the collateral value at the optimal move date increases or decreases with an exogenous increase in the reverse mortgage loan amount. This involves evaluating the partial derivative T. In particular, assuming that g < r and that there is no moral hazard on maintenance, 13 the change in M the expected value of collateral has the opposite sign of T M. It is clear that to an unconstrained reverse mortgagor (one for whom λ M never binds), a small increase in M has no effect on the optimization problem for T (21) as long as F ( M) = 0. On the other hand, evaluating the effect on T of a small increase in M from a starting point of zero introduces an income effect because now the fixed fee must be repaid, either sometime before or simultaneous with the move. Consider first the effect of an increase in M from a positive level, such that there is no income effect related to the fee increase. The only effect is for borrowers that will take up the additional borrowing capacity. A small increase in M can have an effect on the optimal date of move only if the consumer is at an interior optimum for T between a and A. In this case, we can assume U T T < 0. Thus by implicit differentiation, the 13 Maintenance moral hazard would add ambiguity to the results, likely undoing Theorem 1. 14
sign of the effect of an increase in M on the optimal move date will have the same sign as 2 U T M. The only effects of the increase in M at date T are on consumption before the move and through a unit reduction in savings after the move (if the increase in M is proportional to e r M (T a). An exception would be if the loan is not repaid after the move. Differentiating equation (21) with respect to M, we obtain the following: 2 U T H(T ) = M s(t ) ds(t ) d H(T ) M s(t +) ds(t +) d M (22) I HaP ae g(t a) >M(T ) ((r M r)(π s (T +) + M π s(t +) s(t +) One result is clear: ds(t +) ). d M Theorem 1 If a consumer never moves in the absence of the reverse mortgage and if there is no bequest motive, then moving remains unattractive after a discrete increase in M. Proof By assumption π s (A) = 0. Hence by the weakly positive effect of reverse mortgage debt on consumption at date T, the effect on the optimal move date is strictly positive in (22), so the solution remains at the upper corner. Starting with the case in which the loan is not repaid, we find that if there are positive savings at T, the Hamiltonian effects must cancel for a net zero change in the optimal move date. If there are no savings at T then the mortgage balance increase has the effect only of weakly increasing consumption at T. This increases pre-move utility relative to post-move utility and hence induces the moral hazard condition. If the loan is to be repaid, we know that there are effects only if there is no saving at date T. In this case, the shadow value of savings at T exceeds the value at T +, but the change in savings at T is small relative to the decrease in savings at T + (a value of one when there is no savings and the loan is repaid). Both effects render a later move more attractive by narrowing the gap in the utility level. However, the increase in mortgage debt also renders moving early more attractive by increasing the debt burden of delay and by increasing the marginal utility after the move. In sum, the effect of a no-fee increase in the allowable reverse mortgage balance has an ambiguous effect on the timing of the move for those who take on the increase and a negative effect for those who do not. A small increase in M from zero has similarly ambiguous effects on the move date T. The effect in this case is likely to be negative because of the discrete jump in the fee relative to the infinitesimal increase in the allowable mortgage balance. The fee, in combination with a large interest rate induces a large cost to waiting to move, assuming an interior optimum for T. 15
The remaining effect to consider is that a discrete increase in the reverse mortgage will move the net sales price from positive to negative. This jump wipes away all of the effects related to the increasing burden of mortgage debt with time but leaves in tact the cumulative effect of increasing the marginal value of wealth after the move π s (T +). In sum, it is not obvious that there is a moral hazard issue with respect to reverse mortgages. We can guess that the moral hazard effect is most likely to operate if the collateral value is likely to be slightly smaller than the reverse mortgage debt at the move date. 3.4 Analytical Evaluation of Adverse Selection We can describe a potential mortgagor by the set of characteristics (γ, η, δ, g, s a, H a, P a, µ 0, µ 1, a, A, b), then we can define a probability distribution of characteristics across the population as f(γ,...b). In a population of N individuals, of whom N M take on reverse mortgages, we can thus write: adverse selection γ... e (g r)t (γ,...b)(1 I M )f(γ,..., b)dγ...db < 0, (23) N MM b with positive selection associated with a reversal of the inequality in (23). There are three major impediments to an analytical evaluation of (23). First, we do not have good information on the joint distribution of the observable and unobservable characteristics. Second, we do not have a closed form solution for the indicator of reverse mortgage take-up I M as a function of characteristics. Third, not all of the characteristics can be deemed exogenous predictors of reverse mortgage choice and subsequent behavior. In particular, someone who is likely to take on a reverse mortgage in the near future has less incentive to save than someone who is not going to do so. With these caveats in mind, we can consider each characteristic separately and ask whether the characteristics that have a positive effect on the optimal move date tend also to have a positive effect on the desirability of a reverse mortgage. Desirability must be defined with care. We will assume that g is distributed independently from the other characteristics, so that we can, as in the case of moral hazard, focus only on the relationship between desirability of the reverse mortgage and the optimal move date. Assuming g < r this we then say adverse selection applies with respect to a parameter if the effect of the parameter on desirability has the same as the effect on T. 16
3.4.1 Initial savings and selection A prerequisite for a reverse mortgage to be optimal is that r(s a, y, H a P a ) > r M. If this condition does not hold, cheaper forms of debt are available. 14 Hence s a in this sense decreases the desirability of a reverse mortgage. In terms of the effect on T through 2 U T s a, s a weakly increases c(t ). If savings are zero at time T then there is no effect of s a on consumption after T. This is a form of advantageous, rather than adverse selection. If savings are positive at T, then a decrease in s a, we conjecture, will lead to an increase in T. This is because the decrease in H T will be large relative to the induced decrease in ṡ(t +). This is a form of adverse selection, however, only to the extent that the loan will be under water at T, since positive savings at T otherwise indicate that the loan has been paid off, so that there is no loss to the lender in extending T. Expanding on this last point, in considering adverse selection, we can confine ourselves to consideration of the effect of parameters on loan desirability and optimal move date among mortgagors who either have no savings at T or whose loans (evaluated at M) are under water at T. It should also be noted that repayment is unlikely, since for most borrowers, a home equity line of credit would dominate a reverse mortgage if income were sufficient to retire the debt before moving. In this sense, we are exploring necessary, rather than sufficient conditions for adverse selection, since we are excluding a portion of the mortgagor population with values of e (g r)t that are large relative to loan size. A necessary condition for absence of savings or non-repayment of the mortgage is that a small transfer of consumption from the net post-move period to the pre-move period (or a free transfer of money in the under water loan scenario), not requiring an increase in the reverse mortgage fee, would be desirable. That is, call the small transfer z, such that U z = c(t ) γ π s (T +). We can consider the following modified version of adverse selection. Definition 3 Adverse (positive) selection on characteristic α occurs if sign 2 U T α = ( )sign 2 U z α. In the case of g, this condition is necessary but not sufficient for adverse selection. 3.4.2 Bequest motive, length of life and selection Consider first the bequest strength b. Increasing b will either reduce c(t ) if there are positive savings at T or leave c(t ) unaffected. As discussed in Section (3.1), b increases π s (T +). Hence 2 U z b < 0. 14 This assumes that the reverse mortgage is available at no less favorable terms at all times. 17
As for 2 U T b, an increase in b has a large effect on housing after the move relative to its effect on consumption before or after the move (if any). This tends to render a late move more attractive by equalizing pre- and post-move utility. However, b also increases the value of savings at T +; this most likely renders an early move more attractive, particularly if the loan is to be paid off, g > r and the rental rate f is not too large. There is thus an ambiguous selection effect associated with increasing bequest strength. The same is true of increasing length of life, since A affects the trade-off between consumption before and after the move in a way identical to b. 3.4.3 Initial price and size of home and selection Consider first the case where the loan is repaid at the move date and the effect on the price level P a on the desirability of a transfer of wealth from after to before the move date. An increase in the price level has no effect on consumption before the move if savings are zero at the time of the move. The effect on marginal utility after the move depends on whether the housing trade after the move is net positive or negative. If a trade down in present value (considering the size of the new home relative to the old home and the rental rate f compared to the user cost to a reverse mortgagee (r g + (r M r)m/hp T )) occurs at the move date, as is expected of the old, then the price level reduces the marginal value of savings after the move and increases the value of a reverse mortgage. The effect of the price level on the incentive to move is ambiguous: the likely decrease in the marginal value of post-move savings makes a delay more palatable, but the increase in the capital gain associated with moving increases the urgency of the move holding the value of savings constant. Assuming that housing wealth is large relative to other wealth and to y, increasing housing wealth leads to a greater improvement in the smoothing across housing and the other good after the move. In sum, a decrease in the optimal move date seems likely but is not guaranteed. Positive selection on this dimension is thus plausible. This effect is likely enhanced by an empirically positive correlation between the level and growth of housing prices. If the mortgagor defaults, then the price level reduces the level of utility and increases marginal utility as long as η > 1. This would be associated with both increased demand for the reverse mortgage and an unambiguously longer stay in the home. So adverse selection on the dimension of price becomes more severe in the event that the mortgagor s put option is exercised. Another effect must be considered which is that increasing the price level softens demand for reverse mortgages as a debt consolidation tool and enhances the demand for reverse mortgages as a home equity extraction tool, since greater collateral value reduces the cost of non-reverse mortgage debt. This suggests that prepayment is less 18
likely. This is a phenomenon different from adverse selection, but suggests that price is a more likely contributor to adverse selection than is a low level of initial savings. Increasing the size of the home renders the reverse mortgage more attractive in the same way that increasing the price does. As for the effect on the optimal move date, increasing the initial home size renders a long stay more attractive because it increases the level of utility before the move, but this effect is outweighed by increased utility after the move since setting the new home as large as the original home is feasible as long as f is not too large. Further, as long as g < r, the initial home size renders early moving attractive by increasing the financial incentive to move and regardless of g by increasing utility after the move. Again, these effects are attenuated by reduced marginal utility after the move with increasing price, such that the financial incentive to move is softened. Positive selection seems likely, but is not unambiguous. In the event that the loan is under water, we again obtain adverse selection, since increasing home size then increases the level of pre-move utility and does nothing to the post-move optimization. 3.4.4 Income and selection Decreasing income increases marginal utility both before and after the move by concavity of the utility function. Whether this increases or decreases reverse mortgage demand depends on the magnitudes of η and b and also on the level of savings at T relative to housing wealth. If savings are close to zero before the move (as they are by assumption in the absence of a reverse mortgage) and housing wealth is large, then reduced income is likely to be associated with increased reverse mortgage demand. The same rationale justifies an early move assuming there is a positive capital gain on sale. If the loan is underwater, then the free housing provided by the reverse mortgage presumably renders a late move more attractive. This effect is reinforced by the relatively high marginal utility after moving, which is reduced by shortening the length of the post-move period. 3.4.5 Price growth and selection Price appreciation increases marginal utility after the move if there is a net sale of housing and if the bequest motive is not too strong or if η < 1. Hence in the event that the loan is not underwater, appreciation renders the reverse mortgage more attractive. If the loan is underwater, then appreciation renders the reverse mortgage less attractive by increasing post-move marginal utility. As for the optimal time to move, price growth again has complicated effects. If there is a net trade down and the bequest motive is not too strong, then the level of utility increases after the move, rendering an early move more appealing. The financial consideration is mixed, as marginal utility may 19
decrease, but increased growth renders the gain from waiting greater. If the loan is not repaid, then a later move is made more attractive by increased growth. In sum, the pure selection correlation between growth and reverse mortgage take up is ambiguous. Further, it is unclear how growth relates to the optimal move date. If the loan is underwater at the time of move, it is clear that inflation renders the loan less attractive and a late move more attractive. This is positive selection, whereas other characteristics suggested adverse selection when the loan is under water. 3.4.6 Moving costs µ 0 and µ 1 Suppose that µ 0 is made more negative. This will reduce the optimal move date and therefore has the same ambiguous sign as the direction of moral hazard. 3.4.7 Discount Rate and Curvature of Felicity Heavy discounters will find borrowing against resale value attractive. Heavy discounters most likely have low values for π s (T +), evaluating expression (20). This has an ambiguous effect on the optimal move date. Heavy discounters are likely to have a large utility gap prior to the date of move, as savings will be low before T and consumption will be high at T +. This suggests a relatively early move and would be indicative of positive selection on unobservables. A large degree of risk aversion has the clear effect of rendering borrowing from sales proceeds more attractive, assuming a weak bequest motive. There is a further effect of a desire to move more quickly for consumption smoothing purposes. Again, this suggests positive selection on unobservables. 4 Empirical Results 4.1 Claims to be Evaluated From the discussion above, there are few clear conclusions. The following seem like reasonable statements, however: 1. Conditional on all relevant covariates, we do not whether moral hazard applies such that borrowers would be expected to remain in their homes longer than non-borrowers. 2. Reverse mortgagors should have less wealth conditional on housing and more housing conditional on wealth than non-mortgagors. 3. Low wealth conditional on housing and large housing conditional on wealth should be associated with more rapid movement out of homes. 20
4. Low income is likely associated with reverse mortgage demand and with early moving. 5. Large price appreciation should be associated with reverse mortgage demand. The effect on move date is ambiguous. 6. Most of the positive selection arguments are likely to be reversed where loans tend to non-performance. Given that the bulk of loans have been issued in the midst of a housing price boom, this hypothesis is unlikely to be testable. 7. Heavy discounting should be associated with early move and with reverse mortgage demand. Hence there is reason to suspect positive selection on unobservables. 8. Rapidly decreasing marginal utility should be associated with both reverse mortgage demand and more rapid move out, again suggesting positive selection on unobservables. The remainder of this section discusses the empirical evidence on these claims. 4.2 Data The analysis below relies on data from several sources. To compare the mobility rate of reverse mortgagors to non-mortgagors, we merge two datasets. HUD has assembled a database with some detail on the performance of all of the approximately 77,000 HECM mortgages issued since 1989. Included with each loan record are the date of loan funding, the assessed value of the home, the borrower s age and marital status, the state in which the property is located and the date of loan termination, if any. 15 We compare the rate of mobility among single women in this dataset, all 62 or older, with the rate of mobility among older single woman homeowners in the American Housing Survey (AHS). The AHS has the virtue of following homes over a sixteen year period, 1985-2001 comparable with the years of HECM activity. Neither dataset allows differentiation between mortality and moving out while alive. A natural concern in matching mobility rates is that the AHS data might under- or over-report the rate at which older women exit from homeownership due to imperfect data collection. 16 To address this concern, we estimate mortality and mobility rates in three different datasets to see if AHS appears consistent with other datasets populated almost entirely by non-reverse mortgagees. 17 Table 1 compares mobility and mortality 15 Loans rarely terminate while the owner still lives in the property prepayment without a move represents roughly 2% of the roughly 50% of the minority of loans which have terminated. This absence of prepayment is not surprising given the very limited financial assets which might be used to prepay. 16 Incorrect loan termination dates in the HECM dataset seems like less of a worry. 17 Given the penetration rate of substantially less than one percent, there is no concern that large population samples have a large number of mortgagees. 21
Table 1: Two-Year Mobility and Mortality Rates among Single Women Homeowners Aged 62 + in Several US Population Surveys Data Mortality Mobility Mortality + Mobility Mean Age AHS 1985-2001 14.2% 73.6 Berkeley Mortality 1993-1995 5.5% 74 SIPP mobility survey 1996 8.0% 76 AHEAD 1993-1995 7.4% 78.8 AHEAD (Venti and Wise (2000)) 3.8% Berkeley Mortality 1993-1995 7.9% 78 Notes: Berkeley data is for women aged 73 or 77 in 1993 only. Venti and Wise use the entire AHEAD panel. rates across datasets. The AHEAD dataset is an outlier for its low mobility rates among single women homeowners. The AHS features a combined mortality plus mobility rate that is slightly higher than the implied mortality plus mobility from the Berkeley Mortality Database and the SIPP mobility survey of 1996. The AHS mobility rate is overstated yet more if we consider that mortality rates are typically lower among homeowners than renters, as suggested by comparing the AHEAD mortality rates from 1993 to 1995 to those in the general population (mortality is nonlinear, so the comparison is imperfect). It appears that, if anything, the AHS overstates the speed of implicit loan termination. Some more data from the AHEAD survey is in agreement with our conclusion concerning the role of income in mobility and suggests that the µ 0 and µ 1 terms, the changes in which reflect health status, are critical factors in determining mobility. Results are presented in Table 2. This table presents results from a probit estimate of the probability that a household moves out or dies between 1993 and 1998. These estimates are based on a survey of 2,317 households headed by an individual over age 70 in 1993. We find that income is associated with a lower probability of moving but that the interviewee-assessed probability of leaving $10,000 to heirs has no effect, nor do non-housing financial assets. 4.3 Move Out Date and Reverse Mortgage Take Up Figures 3 and 4 show Kaplan - Meyer survival time and hazard rate estimates based on the AHS and HECM data. A failure date in the AHS data is listed as the earliest date at which a respondent who is not the original single woman occupant reports moving 22
Table 2: Probit estimates for moving or dying between 1993 and 1998 based on 2,317 AHEAD households Coef. Std. Err. z P > z price growth -.0003569.0004823-0.74 0.459 bequest10k -.0001967.0009021-0.22 0.827 Medical Exp..0000136 1.97e-06 6.91 0.000 healthprobs.0936794.0358571 2.61 0.009 income -.0741869.0422975-1.75 0.079 houseval -.0642961.0420285-1.53 0.126 fin.wealth -.0020503.0069724-0.29 0.769 woman?.196514.0806384 2.44 0.015 married? -.1956254.079148-2.47 0.013 Num.children -.0025585.0167843-0.15 0.879 constant 78.45095 49.78365 1.58 0.115 Notes: A fifth order polynomial in mean age among household members is included by not reported. bequest10k asks the probability that the household will leave $10,000 to their children (this estimated probability is typically strikingly low given housing and other assets). healthprobs ranks the severity of any health problems, averaged across household members. Num.children is the number of children (not necessarily living with the household. Price growth is estimated from OFHEO based on Census region. Medixal Exp. is out of pocket medical expenditures over the past year. 23
Figure 3: 0.00 0.25 0.50 0.75 1.00 Kaplan Meier survival estimates, by hecm 0 5 10 15 analysis time hecm = 0 hecm = 1 Smoothed hazard estimates, by hecm Figure 4: 0.05.1.15.2 0 5 10 15 analysis time hecm = 0 hecm = 1 into a home, if this occurs at all. 18 A failure in the HECM data is the date at which the loan terminates, if the loan has terminated as of mid-2003. As noted above, HECM loans appear to terminate only very rarely for reasons other than a move or death We see that single women participating in HECM terminate the loans at a rate far in excess of the combined mobility and mortality rates in the AHS. This is consistent with the preliminary results in Rodda, Herbert and Lam (2000), but inconsistent with fears of moral hazard or adverse selection. With respect to the theoretical claims, either claim 1 is incorrect, or there are relevant covariates that must be taken into account. 18 On average, this date is earlier than the date at which the respondent reports having purchased the home, biasing mobility rates up, not down in AHS. 24
Table 3 incorporates some covariates that, based on the discussion, might be expected to affect the date at which loans terminate or at which move out or death occurs. Here HECM is an indicator for participation in the HECM program. Age reflects age at entry into observation (a in the notation above), so there is an implicit assumption that changes in mobility rates follow the change in the baseline hazard. Table 4 provides summary statistics. Separate means are provided for each dataset. Unfortunately, measures of health status are available in neither the AHS nor the HECM dataset. The AHEAD results suggest that these are critically important factors. A plausible story would be that HECM proceeds are used to pay medical expenses which eventually lead to death or departure from the home. We find that the presence of covariates does not alter the conclusion that older women participating in HECM leave their homes much more rapidly than do similar women who are not participants. Older single women HECM borrowers move out at a rate approximately 50 percent greater than the rate for older single women in AHS. The (unreported) effect of age is predicted by the model; advanced age (and hence a shorter remaining lifetime) renders moving more attractive (and death more likely). The shorter horizon has an ambiguous effect on reverse mortgage demand, but the reverse mortgage borrowers are older than average. HECM borrowers are typically in the sample for a shorter period than the AHS comparison group, hence identification requires that age dependence α not depend on age. Columns (2) and (4) of Table 3 shows that there is no significant effect of log income on mobility in the full sample. However, we find in column (3) that among those not taking on a HECM, there is a nearly significantly negative effect. Unfortunately, many HECM borrowers are listed as having no income; this presumably reflects a legal restriction on using social security income to support loan payments. This lack of effect does not run counter to the model, which predicted an ambiguous effect of income on length of stay. We find dramatically lower incomes in the HECM sample, but this again could reflect definition. Nevertheless, the negative effect of income on mobility among non-mortgagors and the selection of evidently low income widows into HECM suggests positive selection. As predicted by the model, we find a consistently positive effect of home value on the hazard out of homeownership. Again suggesting positive selection, we find in Table 4 that the HECM borrowers have dramatically more expensive homes than nonmortgagors. This difference overstates the true difference because the HECM dataset is on average newer than AHS and the bulk of originations have been made since the price boom starting in the mid- to late- 1990s. A clearer picture of housing value relates to wealth. We see that despite much larger housing prices and a run-up in the stock market, HECM borrowers are much less likely than AHS older woman homeowners to have $20,000 in savings (the discrete 25
Table 3: Regressions of Hazard Rates on HECM Participation and Covariates. Dependent variable is the estimated time-varying hazard rate under the Weibull distribution (1) (2) (3) (4) (5) (6) HECM 1.51** 1.47** 0** 0** (.074) (.104) (0) (0) ln Value 1.19** 1.12** 1.14 1.12** 1.21** (.019) (.014) (.095) (.034) (.022) ln Income 1.04.93 1.05 (.027) (.132) (.027) INV20K.87**.95.746.947.866** (.03) (.042) (.114) (.042) (.036) ln Price 1.08.373**.381.322** (.091) (.184) (.184) (.155) RETURN.96** 1.14 1.13 1.16 (.014) (.095) (.092) (.094) LOWRETURN.698 5.36 6.93 8.57** (.189) (5.83) (7.00) (8.64) HECM*ln Price 2.93** 3.92** (1.44) (1.90) HECM*RETURN.84**.82** (.070) (.067) HECM*LOWRETURN.095**.11** (.01) (.116) Excludes HECM? No No Yes No No Age polynomial Yes Yes Yes Yes Yes Time dependence α 1.45** 1.46** 1.47** 1.38** 1.49** (.009) *(.009) (.010) (.016) *.010) Observations 41,608 7,748 374 7,748 40,932 Notes: Z-statistics reported in parentheses - subtract one from the coefficient estimates and divide by the standard error to get something akin in magnitude to a t-ratio. These estimates come from a merge of the cross section of HECM loan performance with the American Housing Survey Panel from 1985 to 2001. State fixed effects are approximated in the sense that the state of residence in AHS is identified through the location of the central city of a metropolitan area. The polynomial in age (at the time of first observation) contains five terms. INV20K denotes non housing assets are worth at least $20,000. ln Price is the log of mean price in the state in which an individual lives. HECM*ln Price interacts an indicator for HECM status with the mean price measure. RETURN measures the total percentage change in the OFHEO state price index from 1976-2003. LOWRETURN indicates that the log total return is less than 3. 26
Table 4: Summary Statistics for Hazard Regression Covariates Variable Obs. HECM Mean HECM Obs. AHS Mean AHS AGE 41,004 76.2 1,301 72.7 House Value 41,004 144,807 1,301 58,370 Income 41,004 2,841 1,299 12,047 House Value Total Assets 40,608.95 0. House Value Income 7,416 24.64 1,280 11.66 INV20K 41,004.02 786.28 PRICE (state mean) 40,793 148,165 684 142,550 RETURN (state mean) 40,924 5.55 684 2.05 LAMERETURN 41,004.005 685.0015 Figure 5: Distribution of non-housing assets (horizontal axis) and ratio of housing to nonhousing assets (vertical) in the HECM (top panel) and AHEAD (bottom panel) datasets See faculty.haas.berkeley.edu/davidoff/houserat.pdf figure is all that is available in the AHS dataset). Figure 5 and shows the distribution of non-housing assets and the ratio of housing assets to these other assets in the HECM dataset and the AHEAD dataset, respectively. We see both less wealth and much more concentration of wealth in housing among HECM borrowers. This is hardly surprising given the program details which guarantee that it is a poor idea to take up the reverse mortgage when liquid assets are available. 5 Conclusion There appears to be substantial positive selection (or the opposite of moral hazard) in the US reverse mortgage market to date, at least among the plurality of borrowers who are older women participating in HECM. There is clear selection on observables such as house value, age and price appreciation. However, even controlling for these observables, we find a significant positive correlation between HECM participation and the rate of departure from homes. The model presented in Section 3 makes it hard to believe that this relationship is causal in the sense that a HECM loan enables early move out. Rather, it appears that some borrower characteristics, such as health status, access to unreported assets, bequest motives, localized price conditions or an attachment to home equity (perhaps due to some precautionary concerns), are both unobservable and 27
important determinants of reverse mortgage demand that are associated with early move out. At least in recent years, we might reason that reverse mortgages enable longer stays at home, but that the kind of people who want to cash out their housing wealth turn first to a reverse mortgage and relatively soon thereafter to disposal of the entire asset. The result has been that there have been very few losses paid out of the comparatively large reserves collected by the FHA as insurance against insufficient collateral. Selection on unobservables makes sense in that health status appears to be a critical determinant of mobility in the AHEAD data. Interestingly, the large fixed fee and low interest rate that characterize the HECM program would seem to guarantee long stays among borrowers, since a home equity line of credit (with almost no fixed fee and a higher interest rate) seems to dominate HECM unless the planned stay is very long. In this way, it seems that reducing the large fees would be justified both by more rapid move outs than expected to date and by likely falling move outs with reduction of the fee. The analysis presented above suggests, however, that a reduced fee might invite participation by homeowners less eager to take out home equity and thus perhaps less likely to move out conditional on a less than 100% loan to value ratio. Given the tendency for the model s predictions concerning selection to flip to adversity when loans are more likely to go under water, this result may hinge on continued conditions in which there is no need for insurance in the first place. It will be interesting to observe whether the favorable selection observed to date continues in any deflationary periods. When default is likely to occur in the sense that loan balance exceeds amount due, a longer stay becomes more attractive with falling prices. Already, we find that HECM borrowers living in states with low historical appreciation are dramatically slower to move out of their homes than HECM borrowers living in other states, and that this phenomenon does not occur among non-borrowers. References Aizcorbe, Ana M., Arthur B. Kennickell, and Kevin B. Moore, Recent Changes in U.S. Family Finances: Evidence from the 1998 and 2001 Survey of Consumer Finances, Federal Reserve Bulletin, January 2003, pp. 1 32. Artle, Roland and Pravin Varaiya, Life Cycle Consumption and Homeownership, Journal of Economic Theory, 1978, 18 (1), 38 58. Blacker, Ruth, Testimony Regarding The American Homeownership Act of 1998 Before the Housing and Community Opportunity Subcommittee of the House Banking and Financial Services Committee, July 1998. 28
Carliner, Geoffrey, Income Elasticity of Housing Demand, Review of Economics and Statistics, November 1973, 55 (4), 528 532. Davidoff, Thomas, Maintenance and the Home Equity of the Elderly, Technical Report 2004. Fischer Center Working Paper 03-288. de Meza, David and David Webb, Advantageous Selection in Insurance Markets, RAND Journal of Economics, Summer 2001, 32 (3), 249 262. Finkelstein, Amy and Kathleen McGarry, Private Information and Its Effect on Market Equilibrium: New Evidence from the Long Term Care Industry, working paper 9957, NBER 2003. Heiss, Florian, Michael Hurd, and Axel Borsch-Supan, Healthy, Wealthy and Knowing Where to Live: Trajectories of Health, Wealth and Living Arrangements Among the Oldest Old, Working Paper 9897, NBER 2003. Léonard, Daniel and Ngo Van Long, Optimal Control Theory and Static Optimization in Economics, Cambridge: Cambridge University Press, 1992. Lustig, Hanno and Stijn Van Nieuwerburgh, Housing Collateral, Consumption Insurance and Risk Premia: An Empirical Perpective, working paper 9959, NBER 1993. Mayer, Christopher, Reverse Mortgages and the Liquidity of Housing Wealth, Journal Of The American Real Estate And Urban Economics Association, 1994, (2), 235 255. Miceli, Thomas and C.F. Sirmans, Reverse Mortgage and Borrower Maintenance Risk, Journal of the American Real Estate and Urban Economics Association, 1994, 22 (2), 257 299. Rodda, David, Christopher Herbert, and Hin-Kin Lam, Evaluation Report of FHA s Home Equity Conversion Mortgage Insurance Demonstration, Prepared for US Department of Housing and Urban Development, Abt Associates 2000. Sheiner, Louise and David Weil, The Housing Wealth of the Aged, NBER Working Paper 4115, NBER 1992. Shiller, Robert and Allan Weiss, Moral Hazard in Home Equity Conversion, Real Estate Economics, 2000, 28 (1), 1 31. Venti, Steven and David Wise, Aging and Housing Equity, 2000. NBER Working Paper 7882. 29
Appendices A Should There be Any Demand for Reverse Mortgages? A Stochastic Model In this section, we provide a somewhat different model of reverse mortgage demand, and ask whether such a mortgage would be attractive to a woman facing stochastic health conditions that simultaneously engender the need to move out of the home and a need for medical expenses We consider a 75 year old single woman homeowner. We endow this woman with assets and income that make a reverse mortgage appear moderately attractive on the surface. This seems natural given that we would like to know if there is a substantial fraction of the population that would benefit from a reverse mortgage, not whether all older households would gain. Thus she has a house valued at the 75th percentile among 75 year old women in the fifth Health and Retirement Survey (HRS) wave, or 140,000. She has income (which we assume is constant across time) and assets at the 25th percentile of women 75 or over with houses worth over 90,000. This yields an annual income of 15,000 and net non-housing wealth of 17,000. The woman maximizes lifetime utility which is a sum of subutilities over annual consumption and the terminal bequest subject to a budget constraint which forbids negative wealth. Savings earn a real interest rate of 3 percent and as a baseline there is no inflation. Interest accrues on the reverse mortgage balance at the real rate plus a typical spread of 1.5 percent. To facilitate computation, we assume that the entire loan balance is withdrawn at age 75. Given positive assets and the 1.5 percent spread, this assumes a greater loan balance due than would be the case under optimal behavior where liquid assets are run down to zero before a reverse mortgage line of credit is tapped. Thus welfare gains here will be understated. We assume that annuities beyond those which provides annual income are unavailable (the tenure payments option is like an annuity, but payments end when the owner moves out of the home). In the absence of a bequest motive and any possibility of moving out of the home, the value of taking on a reverse mortgage is equal to the loan amount. This is because repayment occurs only in states of the world (death) which someone with no bequest motive does not care about. Valuing the reverse mortgage is more complicated when, as in the real world, the individual may move out of the home and thus be required to repay the balance due before death. The balance due exceeds the loan amount in present value because of the interest rate spread and the upfront fee (assumed to be 6.8% of the loan value). Thus if frontloading consumption is not sufficiently important and if the probability of a move is relatively high the reverse mortgage can be unattractive 30
even in the absence of a bequest motive. A bequest motive can be expected to detract further from the value of a reverse mortgage to the borrower because even absent a move, repayment occurs in a state of the world where money is valued. For modeling purposes, at least two important assumptions must be made regarding what occurs after a move: changes to the sub-utility function and changes to the probability of survival. We might think that marginal utility (holding wealth constant) increases after a move because moves might be occasioned by increased medical and hence financial needs. Indeed, Heiss, Hurd and Borsch-Supan (2003) show that a common transition from one s own home is to a nursing home. A move generally will require new financial burden in that housing payments must be made relative to the assumption that there was no mortgage debt on the existing home. However, there may be a negative complementarity between consumption and the state of being in ill health. We thus assume a constant utility function while alive, regardless of where one lives, but we allow for the possibility of required expenditures up to $30,000 per year. 19 A second assumption concerns the probability of death conditional on being alive. We borrow mortality probabilities for ages 75 to 105 year old from Berkeley s mortality database. To speed computation we assume death occurs with 100 percent probability at the end of the year if the woman lives to be 105. However, we multiply these probabilities by 1.5 (up to a maximum of 100 percent probability) in states of nature where the woman has moved out of her home. This is consistent with the correlations among ill health, mortality and moving out of one s home found by Sheiner and Weil (1992) and Heiss et al. (2003). As is commonly done, we assume that utility is additively separable across time and that utility over consumption c in any period has the constant relative risk aversion form of c1 γ 1 1 γ. In particular, we assume that γ is equal to 2 so that utility is given by c. This seems moderately risk averse given the responses to questions in the Health and Retirement Survey concerning willingness to gamble different fractions of something that could be interpreted as either income or wealth. We assume that the bequest motive has the same curvature as the periodic utility function and is multiplied by some number κ. Discounting arises only from mortality probabilities. Hence, lifetime utility can be written as: U = 105 t=75 1 γ + m tκ w1 γ t+1 ). (24) 1 γ q t ( c1 γ t 19 In this version, medicaid is ignored. The consequences of the presence of medicaid for the attractiveness of a reverse mortgage are unclear and depend on medicaid s treatment of housing wealth. For a permanent move out of one s home and into a nursing home, most states would capture home equity before paying for the nursing home from medicaid funds. This suggests that the presence of medicaid makes HECM more, not less attractive in that the disutility of zero wealth is bounded. 31
Here q t is the probability as of age 75 that the individual will be alive. m t is the probability that the individual dies at age t conditional on surviving to date t 1. We assume that there is no discounting of future utility over consumption or bequests. other than a weighting of future utility by survival probabilities. It is clear that an increased discount rate will increase the value of a reverse mortgage because the reverse mortgage involves the transfer of funds from later to earlier periods. w t,wealth available at time t is constrained to be positive. w t s value is given by: w t = (w t 1 + RM t 1 c t + H t 1 X t )(1 + r), (25) RM t 1 represents transfers arising from the reverse mortgage program at date t 1 and H t 1 represents the sale price of the home if the widow moves out of her home at the end of period t 1. We assume that there is no real house price appreciation; relaxation of this assumption would predictably increase the value of the reverse mortgage to the borrower. X t represents required medical or housing expenditures that do not arise while the widow lives in the home she owns but do arise after she moves out. Reverse mortgage transfers RM are equal to the maximum loan amount of 100,000 (based on interest rates of November, 2003 and the results of AARP s Reverse Mortgage Calculator software) at age 75 and equal to the outstanding loan amount (the future value of RM plus fees of 6.8 percent, with interest accruing at the real interest rate plus the interest rate spread). We vary κ between 0 and 50. At a value of 50, the bequest motive is extremely strong. In the last year of life, an individual with such a large value for κ with $60,000 in wealth would consume $1500 and bequeath the remainder. Because repayment of the reverse mortgage is triggered by a move out of the home, the state space is fairly simple. At the age of 76, the individual may die, move or live in the same home. If the individual has moved, she may remain out of the home or she may die at age 77. If the individual has stayed through age 76, she may again stay, move or die. If she has died at age 77, she remains dead. These transition possibilities are repeated up to age 105. The probability of moving, based on the results of Sheiner and Weil (1992) and panel analysis of the American Housing Survey, is equal to 4.5 percent through age 85 and 8.5 percent per year thereafter. B Optimization Procedure and Results To solve the utility maximization problem subject to a positive wealth constraint, we solve the problem backwards. We allow wealth and consumption values between $1,500 and $300,000 in each period. We solve for the optimal level of consumption conditional on wealth at the maximum possible age of 105 under the assumption that the individual no longer lives in their home (this amount depends on the strength of the bequest motive). This yields a value function mapping wealth at age 105 to a terminal 32
utility and allows an estimate of optimal consumption and attained utility conditional on wealth at age 104. We solve backwards to age 76 under the assumption that the individual no longer lives in her home. The above procedure provides a mapping from wealth to indirect utility based on transiting from alive and in one s home to alive and out of one s home for every age from 76 to 105. We then solve backwards for optimal consumption conditional on being alive and in one s home for every age from 75 to 105. The woman weighs utility over consumption in each period against the probability weighted indirect utilities of (1) being alive and in one s home, (2) being alive and moving and (3) being dead (if there is a bequest motive). Exiting any period with negative wealth generates practically infinitely negative utility. We find that there are large gains to the reverse mortgage but that these gains vary considerably depending on assumptions regarding the utility function and annual required expenditures as a result of moving out of one s home. Table 5 presents the equivalent variation associated with the introduction or elimination of reverse mortgage contracts under different assumptions. With no bequest motive and no additional spending required (scenario 1), the opportunity to participate in the HECM program has a value of 67,500 to our hypothetical widow. Introduction of a bequest motive reduces the value of the program to 15,000 or 9,000, depending on the relative weight put on bequests. It should be noted that weights of 25 and 50 imply very large bequest motives: in the last year of life consumption expenditures based on a wealth of $60,000 would be $60,000 with no bequest motive but only $10,500 with κ = 25 and $7,500 with κ = 50. A large level of required expenditures after moving out of the home reduces the value to 15,000 with no bequest motive. The logic is that the required expenditure increases marginal utility in states where the individual has already lost money due to the presence of the reverse mortgage. $30,000 seems like an extremely large negative financial shock relative to resources and the presence of medicaid, but the purpose is to consider the welfare consequences of a reverse mortgage under a broad set of circumstances for a more targeted type of consumer. A slightly negative valuation of the reverse mortgage arises when there is a very strong bequest motive and large expenditures are required when the homeowner has exited the home. The negativity of this last number may not be robust to a realistically timed drawdown of a home equity line of credit because smaller loan amounts in early years of the loan s life would reduce interest spread expenses. The natural conclusion is that it is difficult to come up with preferences and postmobility conditions such that the reverse mortgage harms a house rich and cash poor older household. It is easy to come up with assumptions whereby the gain is very large relative to wealth. 33
Table 5: Utility Gains from a HECM Reverse Mortgage of 100,000 on a 140,000 home for a 75 year old woman with 15,000 in nonhousing financial wealth and an annuitized income of 15,000. Scenario Parameter Assumed Value Value to Consumer of HECM 1 κ (increasing weight on bequest) 0 67,500 X (required annual spending after selling home) 0 2 κ 25 15,000 X 0 3 κ 50 9,000 X 0 4 κ 0 25,500 X 30,000 5 κ 50-750 X 30,000 Notes: Values are based on numerical simulation of a consumption problem starting at age 75. The woman s subjective discount rate is zero and her coefficient of relative risk aversion (in a CRRA utility function) is 2. As discussed in the conclusion, the value of the reverse mortgage falls when the ratio of liquid wealth to housing wealth rises. Presumably the desire to move out of one s home decreases when this ratio rises. In combination, these facts may explain the positive selection effects discussed below. Future work will extend the model to endogenize the date at which homeowners leave their houses with stochastic house prices. This is a complicated matter because it requires strong assumptions on complementarities between housing consumption and other consumption in the period subutility function. 34
1 Appendix: Optimization Procedure and Numerical Results To solve the utility maximization problem subject to a positive wealth constraint, the problem is solved backwards (cf Bellman Equation). Allow wealth and consumption values between $1,500 and $300,000 in each period, and solve for the optimal level of consumption conditional on wealth at the maximum possible age, T D, under the assumption that the individual no longer lives in their home (this amount depends on the strength of the bequest motive), and at several possible rent levels. This yields a value function mapping wealth at T D to a terminal utility and allows an estimate of optimal consumption and attained utility conditional on wealth at age T D 1. We solve backwards to age T 0 = 75 under the assumption that the individual no longer lives in her home. The above procedure provides a mapping from wealth to indirect utility based on transiting from alive and in one s home to alive and out of one s home for every age from 76 to T D. We then solve backwards for optimal consumption conditional on being alive and in one s home for every age from 75 to T D. The woman weighs utility over consumption in each period against the probability weighted indirect utilities of (1) being alive and in her home, versus (2) being alive and moving to an optimal level of new housing. We assume here that she dies only at age T D. Dying with negative wealth generates practically infinitely negative utility. In the case of a reverse mortgage being taken on, the owner receives at time t = T D a loan which is is some fraction of her house value (see Table 1). Fees are financed, and the reverse mortgage balance increases by (1 + r rf + r spread + π) each year the owner is still in the house (see Table 1). Upon selling the house, or dying, the owner repays the balance, but only up to the current value of the house. Clearly, we need to improve this calculation, by allowing the owner to draw down her available balance as her savings are depleted, since the spread of the RM loan over the savings 1
rate is positive. This will be done in a later version of the model. The baseline case parameters are given in table 1. The disruption function, µ 0 is chosen as the fixed one-time cost to utility W 1 γ D µ 0 = F D 1 γ < 0 (1) while the benefit to moving only kicks in at a later age, T P B = 83 W 1 γ { } B µ 1 (t) = F B p die (t) p die (T P B ) > 0 fort > T PB (2) 1 γ and accrues every year the agent is out of the house. Here, p die (t) is the probability of a single woman dying at age t. µ 1 is intended to capture the benefit of living in an assisted care facility when sufficiently frail (old). For the baseline case parameters, we find that there is a large gain to taking on a reverse mortgage. Figure 1 shows the compensated variation (wealth an individual without access to a reverse mortgage product must be given to attain the same utility as one with a reverse mortgage) as a function of initial wealth W 0. Notice that the CV decreases slightly in initial wealth, which follows from concavity. Further, in Figure 2, we see that the CV is strongly increasing in initial housing wealth, which is a direct consequence of the increased loan amount available. Figures 3 and 4 shows the optimal move-out date for individuals with differing initial wealth W 0 and housing H 0. First, consider the case of no reverse mortgage. The more housing wealth an individual has, the earlier she will move: the marginal utility gain from more housing is less than the gain from more numeraire consumption. Similarly, the more non-housing wealth, the later the move: the individual can afford to wait until the benefit of moving becomes more pronounced. Cash-poor individuals move quickly, especially if they have large housing wealth, and move down in rent. Once the reverse mortgage is taken on, all individuals (except for the very debt laden) stay in their home until death at age T D. Figure 5 shows the optimal rent chosen at the move-out date. Without a reverse mortgage, cash-poor and house-rich individuals rent cheaper 2
Table 1: Baseline case parameters for numerical solution of optimization problem Description Baseline value Age at Retirement T 0 83 Age at Death T D 83 Numeraire RRA coefficient γ 2 Housing RRA coefficient η 2.3 Housing Service Flow SF H 3%/year Disruption Factor F D 3 Disruption Wealth W D 30,000 Benefit Factor F B 1 Benefit Wealth W B 30,000 Age of positive move benefit T P B 83 Annual income y 10,000 House value H 0 140,000 Bequest Strength κ 0 House Price Appreciation π H 1%/year Personal Discount Factor δ 3%/year Inflation Rate π 0 Real Interest Rate r rf 3%/year Reverse Mortgage Spread r spread 1.5%/year Reverse Mortgage Fee r fee 6.8% Loan Amount L RM 10 14 H 0 3
dwellings at move-out: for H 0 = 170, 000, the service flow is 5,100 (SF H = 3%/year), but the initial rent chosen is 4,100. As initial wealth and house values increase, the new dwelling increases in value. Reverse mortgagors only move at (very) negative wealth, and trade down house size. As the personal discount factor rises (earlier consumption preferred), the CV of having a RM available increases (see Figure 6). The owner frees up illiquid wealth for immediate use. Further (Figure 7) heavier discounting leads to earlier move-out, even with a RM for sufficiently low wealth. CV is an increasing function of income (see Figure 8). Relative to income, though, CV decreases as income rises (see Figure 9). The former result probably follows from the fact that low-income owners want to move in the absence of a RM (see Figure 10). As a function of house price appreciation, the CV (Figure 11) and optimal move dates (Figure 12) are more complicated. At low HP appreciation and low initial wealth, owners want to move (to lower quality housing with constant rent). Since the RM induces them to stay a while longer, they take a capital loss when selling. At low HP appreciation and larger wealth levels, the effect is exacerbated since the RM induces them to stay until death (and their house service flow is decreasing), but they would have liked to move to constant housing consumption (which they do w/o a RM). At high HP appreciation, everyone moves before death because the increasing service flow from owning is sub-optimal once HPs are sufficiently high. Thus the owner gets the capital gains from selling to use in the last years of life, and gains the loan for numeraire consumption early on. Longevity increases the value of a RM (see Figure 13 and increases the likelihood of staying (see Figure 14) longer in the home. Bequest motives decrease the value of a RM (see Figure 15 and increases the likelihood of staying (see Figure 16) longer in the home. 4
4.5 x 104 CV vs W 0 for various H 0 ; other parameters = BaseLineCase 4 3.5 3 CV 2.5 2 1.5 H 0 = 110000 130000 170000 1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 W x 10 4 0 5 Figure 1: Compensated variation as a function of W 0. Other parameters as in baseline case.
4.5 x CV vs H for various W ; other parameters = BaseLineCase 104 0 0 W = -3000 0 6000 36000 4 3.5 CV 3 2.5 2 1.5 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 H 0 x 10 5 6 Figure 2: Compensated variation as a function of H 0. Other parameters as in baseline case.
30 Optimal Move Date vs W 0 for various H 0 ; other parameters = BaseLineCase 25 20 Year of Optimal Move 15 10 5 w/o RM, H 0 = 110000 w/ RM w/o RM, H 0 = 170000 w/ RM 0-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 W x 10 4 0 7 Figure 3: Optimal move-out date as a function W 0 (with and without a reverse mortgage). Other parameters as in baseline case.
8
Optimal Move Date vs Init House Value H 0 ; other parameters = BaseLineCase 35 w/o RM, W 0 = -3000 w/ RM w/o RM, W 0 = 4500 w/ RM w/o RM, W 0 = 12000 w/ RM 30 25 Year of Optimal Move 20 15 10 5 0 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 H 0 x 10 5 9 Figure 4: Optimal move-out year as a function of H 0 (with and without a reverse mortgage). Other parameters as in baseline case.
10000 Optimal Rent at Move Date vs W 0 for various H 0 ; other parameters = BaseLineCase 9000 8000 7000 Optimal Annual Rent at Move 6000 5000 4000 3000 2000 1000 0 w/o RM, H 0 = 110000 w/ RM w/o RM, H 0 = 170000 w/ RM -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 W x 10 4 0 10 Figure 5: Optimal annual rent at move-out as a function of W 0 (with and without a reverse mortgage). Other parameters as in baseline case.
6 x CV vs H for various W ; other parameters = BaseLineCase 104 0 0 W = -6000 0 6000 36000 5.5 5 4.5 CV 4 3.5 3 2.5 0 2 4 6 8 10 12 14 16 18 20 PDF [%/y] 11 Figure 6: Compensated variation as a function of the personal discount rates. Other parameters as in baseline case.
Optimal Move Date vs Personal Discount Factor; other parameters = BaseLineCase 35 w/o RM, W 0 = -6000 w/ RM w/o RM, W 0 = 4500 w/ RM 30 25 Year of Optimal Move 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 PDF [%/y] 12 Figure 7: Optimal move-out date as a function of the personal discount rate. (with and without a reverse mortgage). Other parameters as in baseline case.
4.5 x 104 CV vs y for various W 0 ; other parameters = BaseLineCase 4 3.5 CV 3 2.5 2 W 0 = -3000 6000 36000 1.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Annual Income y x 10 4 13 Figure 8: Compensated variation as a function of annual income. Other parameters as in baseline case.
11 10 CV vs y for various W 0 ; other parameters = BaseLineCase W 0 = -3000 6000 36000 9 8 7 CV/y 6 5 4 3 2 1 0 0.5 1 1.5 2 2.5 Annual Income y x 10 4 14 Figure 9: Compensated variation in units of annual income as a function of annual income. Other parameters as in baseline case.
Optimal Move Date vs y; other parameters = BaseLineCase 35 w/o RM, W 0 = -3000 w/ RM w/o RM, W 0 = 4500 w/ RM 30 25 Year of Optimal Move 20 15 10 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Annual Income x 10 4 15 Figure 10: Optimal move-out date as a function of annual income. (with and without a reverse mortgage). Other parameters as in baseline case.
8 x 104 CV vs HP Appreciation; other parameters = BaseLineCase W = -6000 0 4500 7 6 5 CV 4 3 2 1 0-4 -2 0 2 4 6 8 10 π [%/y] H 16 Figure 11: CV as a function of HP Appreciation. Other parameters as in baseline case.
Optimal Move Date vs HP Appreciation; other parameters = BaseLineCase 35 w/o RM, W 0 = -6000 w/ RM w/o RM, W 0 = 4500 w/ RM 30 25 Year of Optimal Move 20 15 10 5 0-4 -2 0 2 4 6 8 10 π [%/y] H 17 Figure 12: Optimal move-out date as a function of HP Appreciation. (with and without a reverse mortgage). Other parameters as in baseline case.
4 x CV vs T for various W ; other parameters = BaseLineCase 104 D 0 W = -6000 0 4500 3.5 CV 3 2.5 85 90 95 100 105 T D 18 Figure 13: CV as a function of age at death T D. Other parameters as in baseline case.
Optimal Move Date vs Date of Death T D ; other parameters = BaseLineCase 35 w/o RM, W 0 = -6000 w/ RM w/o RM, W 0 = 4500 w/ RM 30 25 Year of Optimal Move 20 15 10 5 0 80 85 90 95 100 105 T D 19 Figure 14: Optimal move-out date as a function of age at death T D. (with and without a reverse mortgage). Other parameters as in baseline case.
3.5 x 104 CV vs κ for various W 0 ; other parameters = BaseLineCase W 0 = -6000 4500 36000 3 2.5 2 CV 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 κ 20 Figure 15: CV as a function of bequest strength κ. Other parameters as in baseline case.
Optimal Move Date vs Bequest Strength; other parameters = BaseLineCase 35 w/o RM, W 0 = -3000 w/ RM w/o RM, W 0 = 4500 w/ RM w/o RM, W 0 = 12000 w/ RM 30 25 Year of Optimal Move 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 κ 21 Figure 16: Optimal move-out date as a function of bequest strength κ. (with and without a reverse mortgage). Other parameters as in baseline case.