Reverse Mortgage Market and Moral Hazard

Transcription

1 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.

2 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

3 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, The french word for the arrangement is viager. 2 See Blacker (1998), for example. 2

4 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

5 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

6 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 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

7 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 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

8 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

9 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

10 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

11 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

12 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 > 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

13 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

14 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

15 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

16 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

17 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

18 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 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 < This assumes that the reverse mortgage is available at no less favorable terms at all times. 17

19 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 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

20 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 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 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