Stochastic House Appreciation and Optimal Mortgage Lending

Transcription

1 Stochastic House Appreciation and Optimal Mortgage Lending Tomasz Piskorski Columbia Business School Alexei Tchistyi UC Berkeley Haas School of Business We characterize the optimal mortgage contract in a continuous-time setting with stochastic growth in house price and income, costly foreclosure, and a risky borrower who requires incentives to repay his debt. We show that many features of subprime loans can be consistent with properties of the optimal contract and that, when house prices decline, mortgage modification can create value for borrowers and lenders. Our model provides a number of empirical predictions that relate the features of mortgage contracts originated in a housing boom and the extent of their modification in a slump to location and borrowers characteristics. (JEL G2, G21, G33) The recent housing market crisis has brought a lot of attention to the subprime mortgage market, which experienced exponential growth before the crisis. 1 This unprecedented increase in mortgage lending happened during a time of significant real house price appreciation and was particularly concentrated in locations where house prices were growing the fastest. 2 Unlike traditional prime mortgages, subprime mortgages have been issued to riskier borrowers We would like to thank the Editor, Matthew Spiegel, as well as an anonymous referee, for invaluable comments that helped us improve our article. We also thank Lariece Brown, João Cocco, V. V. Chari, Michael Fishman, Douglas Diamond, Narayana Kocherlakota, Arvind Krishnamurthy, Deborah Lucas, Chris Mayer, Tano Santos, Shane Sherlund, Morten Sorensen, Neng Wang, Josef Zechner, and seminar participants at UC Berkeley Haas, Duke Fuqua, NYU Stern, Columbia Business School, Northwestern Kellogg, Federal Reserve Bank of Minneapolis, 3rd NYC Real Estate Meeting, 28 Meeting of the American Real Estate and Urban Economics Association, 15th Mitsui Life Symposium, 28 Meeting of the Society for Economic Dynamics, 28 UniCredit Conference on Banking and Finance, the Federal Reserve System Conference on Housing and Mortgage Markets, 29 AFA meeting, and 29 Utah Winter Finance Conference for helpful comments and suggestions. Send correspondence to Tomasz Piskorski, Columbia Business School, Uris Hall 81, Broadway 322, NYC, NY 127; telephone: (212) tp2252@columbia.edu. 1 The share of subprime mortgages in total originations increased from 6% in 22 to 2% in 26. As of 26, the value of U.S. subprime loans was estimated at $1.5 trillion, or 15% of the $1 trillion residential mortgage market. See, for example, Agarwal and Ho (27). 2 See, for example, Mayer, Pence, and Sherlund (29). c The Author 211. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please journals.permissions@oup.com. doi:1.193/rfs/hhq152 Advance Access publication February 4, 211

2 The Review of Financial Studies / v 24 n who bought expensive houses relative to their income and made little or no down payment. Many of these mortgages came with incentives, including lower initial teaser rates, which were to be reset higher in the future. 3 Because of high default rates among borrowers and big losses to lenders in the declining housing market, subprime lending has caused a storm of controversy. A related debate concerns how mortgage lenders should react to defaults and foreclosures when house prices decline. Government officials have repeatedly called on lenders to aid struggling homeowners by reducing their loan balances and interest rates to lessen the likelihood of foreclosures. On the other hand, mortgage lenders have emphasized that the risk of distorting borrowers incentives complicates the issue, as once borrowers expect to be helped, they might stop debt repayment. Despite the economic significance of subprime mortgages and the extent of the surrounding controversy, thus far there has been no attempt at theoretical analysis of their efficiency. In this article, we formally approach this issue by addressing the following question. Assuming rational behavior of borrowers and lenders, what is the best possible mortgage contract between a home buyer and a financial institution when house prices are expected to grow, but there is also risk of a housing downturn? Instead of considering a particular class of mortgages, we derive an optimal mortgage contract as a solution to a general dynamic contracting problem. The optimal contract also dictates whether (and if so, how) to adjust mortgage terms when house prices decline, taking into account borrowers incentives to repay their debt. We focus our attention on a simple setting that nevertheless allows us to capture main aspects that we believe are central to subprime lending. These are (i) a risky borrower who needs to be given incentives to repay his debt; (ii) costly foreclosure; and (iii) stochastic house price appreciation. The housing market is expected to go through two phases. The first phase, economic boom, is characterized by house price appreciation, possibly accompanied by growth of the borrowers income. At any time, however, the market can enter a second phase, economic slump, characterized by a housing market recession and deterioration of income growth opportunities. We characterize the optimal contract in this environment, i.e., the arrangement between the borrower and the lender that maximizes their combined surplus, and show that it can be implemented as a mortgage with a credit line commitment. The borrower can borrow more against the house up to a credit limit. Default occurs once the borrower fully exhausts his borrowing capacity. The terms of the optimal mortgage contract depend on observable changes in economic conditions. During the boom, the interest rate is scheduled to increase along with the house price. The least creditworthy borrowers are given 3 The most common examples of this are the 2/28 and 3/27 loans, which carry introductory rates for the first two or three years before resetting to a typically higher rate. See Mayer, Pence, and Sherlund (29). 148

3 Stochastic House Appreciation and Optimal Mortgage Lending an additional reduction in their interest rate during the boom. The credit limit, i.e., the maximum amount the borrower can borrow against his home, is increasing over time during the boom along with the house price. The arrival of the slump triggers a mortgage modification, which includes balance write-offs and interest rate cuts, as well as a credit limit reduction. 4 The borrower s eligibility and the extent of mortgage modification depend on his debt balance and average income. 5 The more creditworthy borrowers whose balance moderately exceeds the new credit limit will be rescued through the balance reduction and the interest rate reduction, while the most indebted borrowers default immediately due to the credit limit reduction. The default clustering will be more widespread among those highly indebted borrowers who also suffered a decline in their average income. The features of the optimal mortgage contract can be explained by the incentive-compatibility constraints and the dual optimization objective of the contracting problem. This consists of minimization of liquidation inefficiencies and maximization of the value of the option to sell the home in the future at a higher price, subject to the borrower s limited liability and the incentivecompatibility requirements, and taking into account the borrower s income growth. The access to credit provides flexibility for the borrower to cover possible low income realizations, which in turn lowers the likelihood of costly default. The credit limit, which is determined by incentive compatibility constraints, takes into account the value of the option to sell. The option value increases with the home price, as does the credit limit. During the slump, the option to sell the home loses its value, which results in the credit limit reduction. Early default by the borrower in the boom is costly to the lender, as it does not allow the borrower to use future house appreciation for loan repayment. At the same time, the lender, in order to break even, has to make money on the loan in the good times (i.e., the times with prolonged house price growth), as he is likely to lose money in the bad times (the times with the slump arriving early). Hence, the interest rate is scheduled to increase over time during the boom. On the other hand, the borrower can afford to pay higher interest rates in the future and is less likely to default in the good times, since he can borrow more as the home appreciates. The additional interest rate reduction during the boom for the risky, highly indebted borrowers is driven by the maximization of the option value to sell the home in the future at a higher price and expectations of the borrower s income growth. A lower interest rate increases the chance that a distressed 4 This result yields theoretical support in our setting for the concept of continuous-workout mortgages advocated by Shiller (28). 5 In a different setting, Dunn and Spatt (1985) show optimality of due-on-sale clauses in mortgages that can be modified depending on the observed interest rate and the time to maturity. 149

4 The Review of Financial Studies / v 24 n borrower survives long enough to improve his income, sell the home, and pay back the loan. This preferential treatment comes at a cost: During the slump, the most indebted borrowers do not qualify for mortgage modification and their mortgages are foreclosed. Intuitively, helping the most distressed borrowers in the slump phase is not incentive compatible, as it would encourage borrowers not to pay back their debt. We find that the expected house price appreciation makes it profitable to extend credit to less creditworthy borrowers, who otherwise would be shut out of the housing market. Moreover, we find that these loans could generate substantial ex-ante utility gains for such borrowers. The features of the optimal mortgage are parallel to some key aspects of subprime loans. 6 In particular, during the housing boom it is optimal to extend credit to less creditworthy borrowers, provide them with a lower initial interest rate set to increase over time, and increase the borrowers access to credit as houses appreciate. A housing slump results in the tightening of borrowers access to credit, a foreclosure wave among the least creditworthy, and increased mortgage modification efforts. Our model provides a number of empirical predictions that we discuss in more detail in Section 6. First, mortgages given to borrowers with low income, low credit scores, and low down payments (subprime loans) should be more prevalent in locations with higher expected house price growth. Second, mortgages with scheduled interest rate increases, such as 2/28 and 3/27 loans, should be more prevalent among less creditworthy borrowers (those with lower credit scores, down payments, and income) and in locations with higher expected house price growth. Third, for a given borrower, an increase in the credit limit over time on products such as home equity lines of credit should be positively correlated with growth of house prices in the borrower s location. In a downturn, the home equity line of credit limits on outstanding loans should be reduced in proportion to the regional decline in growth of house prices. Fourth, the likelihood of mortgage modification should be an inverted U-shaped function of the borrower s combined loan-to-value (CLTV) ratio; i.e., it is first increasing with CLTV ratio before it starts declining. Finally, borrowers with high CLTV ratios will be less likely to qualify for modification if their income is lower. Our article contributes to a real estate and finance literature that addresses the implications of various constraints on the design of mortgages and the behavior of borrowers and lenders. 7 To our knowledge, this is the first study of 6 See Demyanyk and Van Hemert (28) for a description of the recent subprime lending episode. 7 See, among others, Chari and Jagannathan (1989), Dunn and Spatt (1985), LeRoy (1996), Stanton and Wallace (1998), Spiegel and Strange (1992), Shiller and Weiss (2), Spiegel (21), Deng, Quigley, and Order (2), Cocco (24), Campbell and Cocco (23, 27)), Lustig and Van Nieuwerburgh (25), and Ortalo-Magné and Rady (26). 141

5 Stochastic House Appreciation and Optimal Mortgage Lending optimal mortgage design in a dynamic environment with unobservable income, costly default, stochastic house price appreciation, and the option to sell. 8 It is important to mention a few limitations of our efficiency results. First, the borrower and lender need to be able to form correct expectations regarding the evolution of key variables. Second, the features of mortgages that are optimal at the individual level may have negative consequences at the aggregate level due to potential negative externalities of risky lending. Third, in deriving the optimal contract, we assumed that the borrower does not face a self-control problem. Borrowers lacking self-control might abuse access to credit, leading to inefficiently high default rates. The article is organized as follows. Section 1 presents the continuous-time model with stochastic house price appreciation and income growth. Section 2 describes the dynamic contracting problem. Section 3 derives the optimal contract. Section 4 illustrates the features of the optimal contract in parameterized examples. Section 5 implements the optimal contract using financial arrangements that resemble the ones used in the residential mortgage market. Section 6 discusses the empirical predictions of our model. Section 7 concludes. 1. The Model In a given location, a borrower (a household) wants to buy a home at date t =. Home ownership delivers to the borrower a public and deterministic utility stream θ. We assume that this utility stream remains constant as long as the borrower stays in the same house. 9 The time-zero price P of the home is greater than the borrower s initial wealth. 1 Thus, the borrower must obtain funds from the lender to finance the house purchase. We assume that the borrower and the lender are sufficiently small that their actions have no effect on house prices. 11 The lender is risk neutral, has unlimited capital, and values a stochastic cumulative cash flow { f t } as [ ] E e rt d f t, where r is the discount rate. 8 The paper most closely related to ours is by Piskorski and Tchistyi (21), who study the optimal mortgage design in a similar setting but with constant house prices. Related papers on dynamic contracting are by Clementi and Hopenhayn (26), Tchistyi (26), DeMarzo and Fishman (27a, b), DeMarzo and Sannikov (26, 28), Biais et al. (27), Philippon and Sannikov (27), DeMarzo et al. (28), Sannikov (28), and He (29). 9 For simplicity, we do not consider the possibility that the borrower can make adjustments that either increase or decrease the quality of the house. See Spiegel and Strange (1992), Shiller and Weiss (2), and Spiegel (21), who study the implications of such possibility. 1 It is reasonable to expect that the home price P is increasing in its utility θ, and the borrower optimizes over the set of available ( ) θ, P pairs. This optimization is not considered in the article; however, this does not lead to a loss of generality, since our analysis applies to any ( ) θ, P pair. 11 In a general equilibrium framework, actions of mortgage lenders and homebuyers on the aggregate level can affect macroeconomic variables. However, as long as the economic agents on the individual level have no market power, they should regard macroeconomic variables as exogenous in equilibrium. 1411

6 The Review of Financial Studies / v 24 n The consumption of a borrower consists of two categories. The first is necessary consumption, which includes food, medicine, transportation, and other goods and services essential to the household. The cumulative minimum level of necessary consumption is given by an exogenous stochastic process {η t } that incorporates shocks such as medical bills, auto repair costs, fluctuations of food and gasoline prices, and so on. The second is discretionary consumption, which might include, among others, such items as expensive restaurant dining, vacation trips, buying a new car, etc. We assume that the borrower must use his available funds first to cover the necessary expenses η t before spending on discretionary consumption or potential debt repayment. 12 The borrower values cumulative discretionary consumption flow {C t : dc t for all t} as E e rt dc t. The zero consumption (dc t = ) means that the borrower consumes only necessities. In a given location, the market is expected to go through two phases. The initial phase, economic boom, is characterized by the house price appreciation, possibly accompanied by growth in household income. The slump is the absorbing state, characterized by a housing market recession, price stabilization, and deterioration of income growth opportunities. Let the process {N t } denote the phase of the market in the period t in a given location: N t = means the boom continues in period t, and N t = 1 means the slump phase in period t. Formally, the process N = { N t, F 1,t ; t < } is a standard compound Poisson process with an intensity δ(n t ), such that N = and { δ if Nt = δ(n t ) = if N t = 1. The price of a house in a given location grows at the rate g > per year during the boom, while it remains constant during the slump: P t = { P e gt for t < τ h P e gτ h (1 α) for t τ h, where α [, 1] measures the extent of house price depreciation and τ h = inf{t : N t = 1} denotes the arrival time of the slump phase. The assumed price process is motivated by the recent behavior of U.S. house prices. In many locations, house prices experienced a period of fast growth 12 This specification resembles the one used by Ait-Sahalia, Parker, and Yogo (24), who propose a partial resolution of the equity premium puzzle by distinguishing between the consumption of basic goods and that of luxury goods. 1412

7 Stochastic House Appreciation and Optimal Mortgage Lending from the late 199s to 26, which ended in a crash in 27. The parameters of the price process could depend on the location-specific characteristics. 13 We note that, according to models of housing returns, a higher price growth is more likely to occur in locations with income and population growth and restrictions on new development (see, eg., Spiegel 21; Van Nieuweburgh and Weil 21). 14 Thus, it is plausible to think about the locations with higher expected house price appreciation as the ones characterized by a combination of such factors. Finally, we note that the above price process does not imply that one can do arbitrage. Unlike pure financial assets, housing is a consumption good; transactions and search costs can be high, and some trades, such as shorting, are difficult. Let Y t denote the total cumulative income of the borrower of a given credit risk profile up to time t. We will focus on the borrower s excess income Y t Y t η t, which represents a better measure of the borrower s ability to pay for a house than total income. The excess income is negative when the necessary expense shock η t is greater than the total income Y t. In this case, the borrower has to cover the deficit by borrowing more or using his savings 15 ; if unable to do so, he will declare bankruptcy. From now on, we will refer to Y t and C t simply as the borrower s income and the borrower s consumption, respectively. The borrower s income up to time t evolves according to { μt dt + σ d Z dy t = t for t < τ h λμ τ h dt + σ d Z t for t τ h, (1) where Z = { Z t, F 2,t ; t < } is a standard Brownian motion (independent of N), μ t = μe γ t is the drift of the borrower s disposable income, σ measures the volatility of the income, and γ < r + δ is the expected growth rate of income in the boom phase. The drift λμ τ h = λμe γ τ h of the income remains constant in the slump phase, where λ is an independent random variable, which takes value 1 with probability (1 p) and χ < 1 with probability p. The above specification implies that household disposable income is subject to both idiosyncratic shocks represented by the process Z as well as regional systematic shocks represented by the process N. The household income grows during the boom together with house prices, while it stabilizes during the slump. Moreover, at the beginning of the slump the household can lose a fraction (1 χ) of its average income with probability p. 13 The empirical literature on housing returns documents a large variation of returns across locations (see, e.g., Goetzmann and Spiegel 1997). 14 See also Khan (29), who shows in a general equilibrium model that changes in productivity growth in the non-housing sector may explain the large low-frequency changes in housing price trends. 15 Allowing the borrower to maintain a private savings account at a rate less than or equal to r would not affect our results. See Section

8 The Review of Financial Studies / v 24 n We assume that the lender knows the borrower s characteristics, including parameters of his income process, but does not know the realizations of the borrower s idiosyncratic income shocks Z. This implies that the realizations of the borrower s income are not contractible and so the borrower has the ability to misrepresent his ability to pay back debt. These assumptions are motivated by the observation that lenders use a variety of methods 16 to determine the characteristics of the borrower and location, but do not fully condition the terms of the contract on the borrower s income or spending shocks, likely because they are too costly or impossible to monitor. Before the purchase of the house, the borrower and the lender sign a contract that will govern their relationship after the purchase is made. The contract will obligate the borrower to report his income realizations to the lender (pay back his debt) and specify when the contract is terminated. 1.1 Option to default The borrower can default at any time on his contractual obligation, in which case he loses the home and receives a reservation value A t. For simplicity, we assume that the reservation value is equal to the expected present value of the borrower s future income: A t = A t = μeγ t r+δ γ where λ = (1 p) + χ p. We will use ( 1 + δ r λ ) for t < τ h A 1 τ h (λ) = λμeγ τ h r for t τ h, (2) A 1 τ h λμeγ τ h to denote the expected reservation value of the borrower in the slump occurring at time τ h (before realization of λ is known). The lender sells the repossessed house at a foreclosure auction and receives the payoff { L t = (1 l)p t for t < τ h L t = L 1 = (1 l)p τ h τ h for t τ h, r where l (, 1] measures the liquidation costs. The borrower defaults only when his expected utility a t from continuing the relationship is less than or equal to his outside option A t. In our subsequent analysis, without loss of generality, we impose the participation constraint on the optimal contract; that is, the borrower s continuation utility a t, which is his expected discounted utility at any time t under the contract, must be greater 16 Such as credit score, employment, current job status, and so on. 1414

9 Stochastic House Appreciation and Optimal Mortgage Lending than or equal to the outside option A t as long as the house is not repossessed. Once the continuation utility falls to the borrower s reservation value, the relationship is terminated, the liquidation happens, and the borrower receives the outside value. 1.2 Full homeownership We say that the borrower is a full homeowner if he owns the home and keeps all his income, i.e., has no debt. Let v t be the continuation utility (expected utility) of the borrower with no savings under full homeownership at time t. For simplicity, we assume that v t is equal to the expected present value of the borrower s future income and housing consumption plus the the value of the option to sell the home in the future. 17 Since the housing slump is an absorbing state, the continuation utility of the borrower who has no debt in the slump phase depends only on the time when the slump arrived and the average income realization λ: We will use v t = v 1 τ h (λ) λ μeγ τ h r v 1 τ h λ μeγ τ h r + θ r, fort τ h. to denote the expected full homeownership utility in the slump phase arriving at τ h (before the realization of λ is known). On the other hand, since the borrower has the option to sell the home in the boom phase and his income is growing, his continuation utility under full homeownership, which we denote by vt, depends on t. 1.3 Option to sell The borrower can put the home on the market. For simplicity, we assume that the home can be sold at the market price P t immediately during the boom, while it is impossible to find a buyer during the slump. The selling is more efficient than liquidation, as l < 1. The sale of the home is not contractible. 18 The borrower puts the home on the market at the time when it maximizes his expected payoff. We assume that if the borrower sells the home, he has to pay to the lender (v t a t ), i.e., the difference between the full homeownership utility v t and his continuation + θ r 17 We ignore a possibility that a full homeowner may have to declare bankruptcy, because he may not have sufficient income and house value to cover his necessary consumption. For reasonable parameter values, including those in Table 1, the bankruptcy is extremely unlikely and should have insignificant effect on v t. 18 In practice, contract clauses forcing the borrower to sell the home may be difficult to implement, since the price and the speed of sale critically depend on the borrower s effort and cooperation. 1415

10 The Review of Financial Studies / v 24 n utility a t under the existing mortgage contract. As we will see it in Section 5, this assumption means paying to the lender the outstanding balance B t on the loan, which is related to the borrower s continuation utility a t under the optimal contract as follows: B t = v t a t. (3) In the boom phase of the housing market, we have v t = vt. Thus, after the house sale, the borrower s continuation payoff is given by while the lender receives A S t (a t) = A t + P t (v t a t ), L S t (a t) = v t a t. Note that the borrower will want to sell whenever A S t (a t) a t, which is equivalent to A t + P t v t. (4) Condition (4) simply states that the borrower sells the home whenever the value of his outside option A t plus the proceeds from the sale exceed his continuation utility under full homeownership. The optimal selling time determined by Condition (4) does not depend on the outstanding balance or the borrower s continuation utility. This is because the outstanding balance is linear in the borrower s continuation utility, and they cancel each other out. The optimal selling time also does not depend on the liquidation value L t of the home, as the borrower does not take into account dead-weight costs associated with liquidation. Proposition 1. The optimal time for the borrower to sell the home in the boom phase of the housing market is given by ts = 1 g log θ ( ), (5) r 1 g r+δ P and the value of full homeownership at time t ts to in the boom phase is equal v t = A t + θ r + X t, where X t = e (r+δ)(t s t) [ P t s θ r ] is the time-t value of option to sell a home in the boom. 1416

11 Stochastic House Appreciation and Optimal Mortgage Lending Proof of Proposition 1. In the Appendix. To ensure that( the homeowner ) does not want to sell the home immediately, we assume θ r > 1 g r+δ P. We note that vt is increasing with time. This is because the borrower has the option to sell the home, and this option becomes more valuable as the price of the home increases. Home sale happens at time ts only in the boom phase. The optimal timing of home sale is thus given by τ s = { t s, if N t s =, if N t s = 1. In what follows, we require that if house prices decline before the optimal selling time according to Equation (5), there are gains of trade from continuing the borrowing-lending relationship. This amounts to imposing that L 1 ts < θ r, which given Equation (5) is equivalent to the requirement that g < (r + δ) (α + l αl). 2. Dynamic Moral Hazard Problem At time, the funds needed to purchase the home, in the amount of (P Y ), are transferred from the lender to the borrower. Let Ŷ = {Ŷ t : dŷ t dy t for all t } be the borrower s report of his income, where Ŷ is (Z, N, λ)- measurable. A contract, ξ = (τ f, τ d, C), specifies a time at which the borrower becomes a full homeowner (τ f ), a default time (τ d ), and the borrower s consumption (C). Without loss of generality, we assume that the borrower is required to pay the reported income to the lender. Thus, the net amount transferred from the borrower to the lender at time t equals dŷ t dc t. The contract is a function of the history of the borrower s income reports (repayments), Ŷ, and the observable economic conditions, (N, λ), consisting of a realized economic state and the borrower s average income. The relationship between the borrower and the lender is terminated at τ = min(τ s, τ f, τ d ), that is, the time of sale, full homeownership (debt repayment), or liquidation, whichever of these events occurs first. 19 Because the available income of the borrower is not observable by the lender, the borrower can use his income to increase personal consumption rather than to repay debt. Consequently, at any time t < τ, the change in the borrower s consumption flow equals dc t + (dy t dŷ t ). } {{ } underreporting 19 Formally, τ is a (Ŷ, Z, λ)-measurable [ stopping ] time, and C is a (Ŷ, Z, λ)-measurable continuous-time process τ such that the process E e γ s dc t F t is square-integrable for t τ and Ŷ = Y, and where F is the filtration generated by processes (Z, N) and λ. 1417

12 The Review of Financial Studies / v 24 n By the direct revelation principle, we can restrict our attention to the contracts requiring the borrower to report his income truthfully. Definition 1. A contract ζ = (τ f, τ d, C) is incentive compatible if the strategy of reporting income truthfully maximizes the borrower s expected utility under this contract. The contract is optimal if, for a given time-zero expected utility a of the borrower, no other contract can increase the expected payoff for the lender. Below we provide a formal definition of the optimal contract. Definition 2. Given the initial continuation utility of the borrower, a, a contract ζ = (τ f, τ d, C ) is optimal if it is incentive compatible and if it maximizes the lender s expected payoff: [ τ ] b = E e rt (dy t dc t ) + e rτ (1 τ=τ s Lτ S + 1 τ=τ d L τ ) F subject to delivering the borrower his initial continuation utility: [ τ a = E e rt (dc t + θdt) + e rτ (1 τ=τ s Aτ S + 1 τ=τ f v τ + 1 τ=τ d A τ ) F ]. The definition of the optimal contract implies maximization of the total surplus, and can also be reformulated in terms of maximizing the borrower s expected utility for a given payoff of the lender. The initial payoffs a and b are determined by the bargaining powers of the borrower and the lender. Our subsequent analysis is valid for any a > A, including the competitive lending industry, i.e., the lender breaks even: b = P Y, and the optimal contract maximizes the borrower s expected utility Derivation of the Optimal Contract In this section, we recursively formulate the dynamic moral hazard problem and determine the optimal contract. Methodologically, our approach is based on continuous-time techniques used by DeMarzo and Sannikov (26) and adopted to a setting with observable persistent stochastic states by Piskorski and Tchistyi (21). We first present and explain the optimal contract after the economic slump occurred. Next, given the post-slump value function, we derive the optimal contract in the boom environment. To characterize the optimal contract 2 As we show in Section 3, the key properties of the optimal contract do not depend on the relative bargaining power of the borrower and the lender as long as they take house prices as given. 1418

13 Stochastic House Appreciation and Optimal Mortgage Lending recursively, we define the borrower s continuation utility at time t under a given contract if he reports his income truthfully as [ τ a t = E e r(s t) [dc s + θds] + e r(τ t) [1 τ=τ s Aτ S t ] + 1 τ=τ f v τ + 1 τ=τ d A τ ] F t. 3.1 The optimal contract in the slump phase Let b 1 (a t, τ h, λ) be the highest expected utility of the lender that can be obtained from an incentive-compatible contract at time t that provides the borrower with continuation utility a t given that the slump phase started at τ h < t. Function b 1 (a t, τ h, λ) depends on τ h and λ, because the income and the liquidation value depend on τ h and λ, but it does not depend on time t, because of the time-homogeneous contracting environment in the slump phase. There are two boundary conditions for function b 1 (a t, τ h, λ). The first boundary condition stems from the borrower s participation constraint: The relationship must be terminated when the borrower s continuation utility falls to his reservation value A 1 (λ) λ μeγ τ h τ h r, so b 1 (A 1 (λ), τ h, λ) = L τ h τ h. The second boundary condition comes from the fact that the lender should expect no transfers from the borrower once he becomes a homeowner; that is, b 1 (v 1, τ h, λ) =. Let τ h b 1 and b 1 denote, respectively, the first and second derivative of b 1 with respect to the borrower s continuation utility a t. The optimal contracting problem in the slump phase is very similar to the one considered by DeMarzo and Sannikov (26). The main differences are that the underlying asset (home) generates a nonmonetary dividend θ next to average income λμ τ h to the agent (borrower), that both the principal (lender) and the agent (borrower) have the same discount factor, and that there is a possibility to declare an agent a full owner, which delivers to him a perpetual value of v τ The proposition h below describes the optimal contract in the slump phase. Proposition 2. The optimal contract that delivers to the borrower continuation utility a t at time t > τ h takes the following form. If a t [A 1 (λ), v 1 ], τ h τ a h t evolves as da t = (ra t dt θdt) + (dŷ t λμ τ h dt). (6) The default occurs at the first time τ d when a t hits A 1 (λ), the borrower becomes a full homeowner at time τ f when a t hits v 1 for the first time, the τ h τ h borrower sells the home at time ts, provided that t s < min(τ d, τ h ), and 21 Thus, our model in the slump phase is the limiting case of DeMarzo and Sannikov (26) with γ r and the possibility to declare an agent a full owner that delivers to him a perpetual value of the first-best equal to v 1 τ h. 1419

14 The Review of Financial Studies / v 24 n dc t = for all t τ. The lender s expected payoff at any time t τ h is given by the function b 1 (a t, τ h, λ), which is strictly concave in a t over [A 1 τ h (λ), v 1 τ h ] and solves rb 1 (a, τ h, λ) = λμ τ h + (ra θ)b 1 (a, τ h, λ) σ 2 b 1 (a, τ h, λ), (7) for a [A 1 τ h, v 1 τ h ], with the boundary conditions b 1 (A 1 τ h (λ), τ h, λ) = L τ h, and b 1 (v 1 τ h, τ h, λ) =. Proof of Proposition 2. The proof directly follows from DeMarzo and Sannikov (26), as the structure of the dynamic moral hazard problem after the house slump is similar to the one studied by them. The evolution of the continuation utility in Equation (6) implied by the optimal contract serves three objectives: promise-keeping, incentives, and efficiency. The first component of Equation (6) accounts for promise-keeping. In order for a t to correctly describe the lender s promise to the borrower, it should grow at the borrower s discount rate, r, less the payment, θdt, which he receives from owning the home. The second term of Equation (6) provides the borrower with incentives to report income truthfully. Because of inefficiencies resulting from liquidation, it is optimal to make the sensitivity of the borrower s continuation utility with respect to income shocks (income reports) as small as possible, provided that it does not erode the borrower s incentives to tell the truth. The minimum volatility required for truth-telling equals 1. Intuitively, underreporting income by one unit would provide the borrower with one additional unit of current utility through increased consumption, but would also reduce the borrower s continuation utility by one unit. Since default is costly and the borrower and lender have the same discount factors, it is optimal to delay the borrower s discretionary consumption until full homeownership, in which case default is no longer a possibility. On the other hand, when a t v 1, the first-best is reached and there are no gains τ for the lender and for the borrower h from continuing their relationship. Thus, during the slump, it is optimal to let } the borrower become a full homeowner at time τ f = inf {t τ h : a t = v 1τ. h 3.2 The optimal contract in the boom phase The contracting environment in the boom phase is different from that in the slump phase in three major ways. First, the house price, as well as the average income of the borrower, is changing over time. Second, the borrower will sell the house at time t s, provided that t s < τ h. Third, at any time, the slump can arrive with Poisson intensity δ. Let b (a t, t) be the highest expected utility of the lender that can be obtained from an incentive-compatible contract at time t that provides the borrower with 142

15 Stochastic House Appreciation and Optimal Mortgage Lending continuation utility a t in the boom phase, i.e., for t < τ h. Let b and b denote, respectively, the first and the second derivative of b with respect to the borrower s continuation utility a. Since the house price as well as the earning ability of the borrower is changing over time, b (a t, t) directly depends on t. When the house is sold, the payoff to the lender is given by b (a t, t s ) = v t s a t. The arrival of the slump will affect the payoffs of both the lender and the borrower. Let ψ t denote the instantaneous change in the continuation utility of the borrower under the optimal contract if the slump arrives at time t = τ h. The first-order condition implies that the optimal choice of ψ t = ψ(a t, t, λ) is such that b (a t, t) = b 1 (a t + ψ t, t, λ), if ψ t > A 1 τ h (λ) a t (8) ψ t = A 1 τ h (λ) a t, otherwise. (9) The average expected payoff to the lender right after the arrival of the housing slump is given by b 1 (a t, t) (1 p) b 1 (a t + ψ(a t, t, 1), t, 1) + pb 1 (a t + ψ(a t, t, χ), t, χ), while the average expected jump in the borrower s continuation utility is equal to ψ(a t, t) (1 p) ψ(a t, t, 1) + pψ(a t, t, χ). (1) The following proposition characterizes the optimal contract in the boom phase. Proposition 3. The optimal contract that delivers to the borrower continuation utility a t takes the following form. If a [A t, v t ], a t evolves for t τ h as da t = (ra t dt θdt) + (dŷ t μ t dt) + (ψ(a t, t, λ)d N t δψ(a t, t)dt), (11) where ψ(a t, t, λ) and ψ(a t, t) are defined by Equations (8), (9), and (1). The default occurs at time τ d when a t first hits A t, the borrower becomes a full homeowner at time τ f when a t first hits v t, the borrower sells the home at time ts, provided that t s < min(τ d, τ h ), and dc t = for all t τ. The lender s expected payoff at any time t < τ h is given by the function b (a t, t), which is strictly concave in a t over [A t, v t ] and solves rb (a t, t) = b (a t, t) t + μ t + (ra t θ ψ(a t, t)δ)b (a t, t) σ 2 t b (a t, t) + δ ( b 1 (a t, t) b (a t, t) ) (12) 1421

16 The Review of Financial Studies / v 24 n for a t [A t, v t ], with the boundary conditions b (A t, t) = L t, b (v t, t) =, b (a t, t s ) = v t s a t. For t > τ h, the optimal contract is given in Proposition 2. Proof of Proposition 3. In the Appendix. The evolution of the continuation utility in Equation (11) implied by the optimal contract serves three objectives: promise-keeping, incentives, and efficiency. The first component of Equation (11) accounts for promise-keeping, while the second term provides the borrower with incentives to report his true income to the lender as in the slump phase. The last term of Equation (11) captures the effect of the stochastic house price appreciation and income growth on the borrower s continuation utility. The optimal adjustment ψ t in the borrower s continuation utility, which is applicable when a slump occurs, is such that the sensitivity of the lender s expected utility, b, with respect to the borrower s continuation utility, a, is equalized just before and after an adjustment is made. 22 This sensitivity represents an instantaneous marginal cost of delivering to the borrower his continuation utility in terms of the lender s utility, and so the efficiency calls for equalizing this cost across the states. We note that these adjustments imply a compensating trend in the borrower s continuation utility, δψ(a t, t)dt, which exactly offsets the expected effect that these adjustments have on the borrower s expected utility. Since default is costly and the borrower and lender have the same discount factors, it is optimal to delay the borrower s discretionary consumption until full homeownership, in which case default is no longer a possibility. As the same applies in the housing slump, it is optimal to let the borrower become a full homeowner at time τ f = inf {t : a t = v t }. Default happens when the continuation utility a t drops to the agent s reservation value A t for the first time. Since both a t and A t are affected by the income growth, it is convenient to consider the distance to default process t a t A t, whose dynamics under the optimal contract follows directly from Proposition 3. Corollary 1. according to In the boom phase for t < τ, distance to default t evolves ( d t = r t + dŷ t θdt δ + ) dt ψ(a t, t) + A t A 1 t ) (ψ(a t, t, λ) + A t A 1 t (λ) d N t. (13) 22 Provided that the solution to Equation (8) is interior. 1422

17 Stochastic House Appreciation and Optimal Mortgage Lending 3.3 The optimal contract with hidden savings The optimal contract of Proposition 3 would remain unchanged if the borrower could privately save at any interest rate less than or equal to r. In our setting, it would be weakly inefficient for the borrower to save on the private account, as any such arrangement could be improved by having the lender save and make direct transfers to the borrower without affecting incentive compatibility. This argument is formalized in DeMarzo and Fishman (27a) and DeMarzo and Sannikov (26) and carries through in our setting. 4. Optimal Contract: A Parameterized Example In this section, we present the features of the optimal contract in a parameterized example. In this example, the average income of the borrower as well as house price grow at a 1% annual rate in the boom. When the slump arrives, the borrower s income stops growing, while a house price drops by 1%. In addition, the average income of the borrower can drop by 3% at the beginning of the slump, which happens with 2% probability. Table 1 shows the parameters of the model. The top panel of Figure 1 shows the lender s expected payoffs as a function of the borrower s continuation utility at the beginning of the contract (t = ). Three value functions are plotted: the one in the boom, b (a t, t), and the two values when the boom turns into the slump at t: b 1 (a t, t, 1) if the borrower s average income remains unchanged and b 1 (a t, t, χ) if the borrower in addition loses a fraction (1 χ) of his average income. The lender s value in the boom is greater than it would be in the slump phase, because the expected house price appreciation and income growth increase the value of the relationship. The reservation value for the borrower is greater in the boom, since it includes the possibility of average income growth. The full homeownership utility for Table 1 Parameters of the model Borrower s income process: Initial mean income μ = 1 Income growth γ = 1% Income volatility σ = 1.5 Probability of income loss p =.2 Fraction of income loss 1 χ =.3 House price process: Initial price P = 9 Price growth g = 1% Price drop in the slump α = 1% Liquidation loss l = 3% Probability of slump δ =.2 Utility from homeownership θ =.8 Interest rate r = 8% 1423

18 The Review of Financial Studies / v 24 n Figure 1 The lender s value function and the optimal adjustment of the borrower s continuation utility the borrower is also greater in the boom phase, since it includes the option to sell the home in the future and the expected income growth. The lowest amount of down payment the borrower has to make in the boom to obtain credit so that the lender breaks even equals 9.75% of the initial house price. Assuming that the outside value of the prospective borrower, if he decides not to purchase a house, is equal to the sum of his initial wealth and the expected value of his income A t, the borrower who has just enough money to make the minimum down payment would obtain a net gain in expected utility of 1.39 (15.5% of initial house price) from becoming a homeowner. The bottom panel of Figure 1 shows the optimal adjustments in the borrower s continuation utility, ψ, applicable if the housing boom turns into the slump at time t =. Two adjustment functions are plotted: the one if the borrower s average income remains unchanged in the slump, ψ(a t, t, 1), and the one when the borrower loses a fraction (1 χ) of his average income at the beginning of the slump. For illustrative purposes, Figure 1 displays jumps in the borrower s continuation utility, ψ, which occur when the slump arrives at t = given that the continuation utility of the borrower just prior to the slump 1424

19 Stochastic House Appreciation and Optimal Mortgage Lending was equal to ã. Our computations for different parameters and t show that the jump function ψ has a form similar to the one shown in Figure 1. Figure 1 shows that the adjustment in the borrower s utility at default (i.e., a t = A t ) when the slump arrives at t is equal to the present discounted value of the change in income (A t A1 t (λ)). On the other hand, for the full homeowner (a household with no debt), i.e., a t = vt, the adjustment is equal to the lost value of the option to sell the home, Xt, plus the present discounted value of the change in income (A t A 1 t (λ)). For a borrower in the intermediate range, i.e., a t (A t, v t ), the optimal adjustment in the utility applicable if the housing boom turns into the slump can be decomposed as follows: proportional split of loss of option to sell change in average income { { }} { [ }} ]{ ψ (a t, t, λ) = (A t A 1 t (λ)) Xt a t A t vt A t ( [ ]) + ψ (a t, t, λ) + (A t A 1 t (λ)) + X t a t A t vt A. (14) t } {{ } additional adjustment The first component of Equation (14) captures the expected present value of income change (A t A 1 t (λ)). The second component represents the reduction in the borrower s continuation utility due to the loss of the option to sell in proportion to the borrower s home equity stake (as measured by the distance of the borrower s utility a t from the default boundary A t ). The final term represents an additional adjustment in the borrower s continuation utility. Figure 2 shows a typical shape of ψ(a t, t, λ) ( A 1 t (λ) ) A t ; that is, the optimal jump function net of the utility adjustment due to a change in average income, which according to Corollary 1 represents the change in the distance to ( default when the slump arrives. The dashed straight line connecting points A t, ) and ( vt, X t ) in Figure 2 corresponds to the second term of Equation (14). The difference between the displayed functions and the dashed line connecting points ( A t, ) and ( vt, X t ) in Figure 2 corresponds to the third term of Equation (14). According to Corollary 1, a negative jump ψ(a t, t, λ) translates into the positive trend δψ(a t, t) in the distance to default in the boom. Thus, the first two components of Equation (14) lower the chances of default in the boom. However, this decrease in the likelihood of default in the boom implies a reduction in the borrower s continuation utility in the slump, which increases the chances of default in the slump. It follows from Figure 2 that the third term of Equation (14) is negative for borrowers with low continuation and positive for borrowers with high continuation utility. This implies that the reduction in the continuation utility in the 1425

20 The Review of Financial Studies / v 24 n Figure 2 Optimal adjustment of the borrower s utility due to the slump slump is bigger for borrowers with low continuation utility (higher debt level), and lower for borrowers with high continuation utility (lower debt level), all relative to their home equity stake. Thus, borrowers with lower continuation utility (higher debt level) receive more preferential treatment (a form of credit subsidy) compared with less indebted ones during the boom phase. However, these borrowers receive a larger relative reduction of their continuation utility in the slump. On the other hand, the less indebted borrowers receive a better treatment in the slump, as their continuation utility declines less relative to their home equity stake, which can be interpreted as a form of partial insurance against the slump, for which they pay in the boom. The optimal adjustment in the borrower s utility can be explained by the incentive-compatibility constraints and the dual optimization objective of the contracting problem. This consists of minimization of liquidation inefficiencies and maximization of the value of the option to sell the home in the future at a higher price, subject to the borrower s limited liability and the incentivecompatibility constraints, and taking into account income growth. First, it is efficient to reduce the chances of costly termination in the boom, as the relationship between the borrower and the lender is more valuable due to the possibility of house price appreciation and income growth. This manifests itself as a positive trend in the borrower s continuation utility in the boom, which lowers the chances of liquidation. The additional subsidy during the boom to the borrowers with low continuation utility (more likely to default) is driven by the maximization of the option value to sell the home in the future at a higher price and by expectations of the borrower s income growth. A subsidy increases the chances that a borrower at risk of default survives long enough to improve his income, sell the home, 1426