Stochastic House Appreciation and Optimal Mortgage Lending

Transcription

1 Stochatic Houe Appreciation and Optimal Mortgage Lending Tomaz Pikorki Columbia Buine School Alexei Tchityi UC Berkeley Haa December 28 Abtract We characterize the optimal mortgage contract in a continuou time etting with a riky borrower who need to be given incentive to repay hi debt, cotly forecloure and tochatic houe appreciation. We how that many feature of ubprime lending oberved in practice are conitent with rational behavior of both borrower and lender. In particular, when houe price are expected to rie, it i optimal to provide the riky borrower with a lower initial rate, which i to increae over time, and to increae the borrower acce to credit. A houing lump reult in the tightening of borrower acce to credit and default clutering among the leat creditworthy one. We alo nd that houe appreciation make it pro table to give loan to le creditworthy borrower who otherwie would be excluded from the houing market, and thee loan could generate ubtantial ex-ante utility gain for uch borrower. Bailing out the mot ditreed borrower in the lump phae i not incentive compatible, a it encourage irreponible nancial behavior in the boom phae. On the other hand, it make ene to help borrower who were in good tanding before the crii by reducing their mortgage balance and interet rate. We thank Lariece Brown, Joao Cocco, V.V. Chari, Michael Fihman, Dougla Diamond, Narayana Kocherlakota, Arvind Krihnamurthy, Deborah Luca, Chri Mayer, Tano Santo, Shane Sherlund, Morten Sorenen, Neng Wang, Joef Zechner, and eminar participant at UC Berkeley Haa, Duke Fuqua, NYU Stern, Columbia Buine School, Northwetern Kellogg, Federal Reerve Bank of Minneapoli, 3rd NYC Real Etate Meeting, 28 Meeting of the American Real Etate and Urban Economic Aociation, 15th Mitui Life Sympoium, 28 Meeting of the Society for Economic Dynamic, 28 UniCredit Conference on Banking and Finance, and the Federal Reerve Sytem Conference on Houing and Mortgage Market for helpful comment and uggetion. 1

2 1 Introduction The recent houing market crii ha brought attention to the ubprime mortgage market, which experienced exponential growth over the pat few year. The hare of ubprime mortgage to total origination increaed from 6% in 22 to 2% in 26. A of 26, the value of U.S. ubprime mortgage wa etimated at $1.5 trillion, or 15% of the $1 trillion reidential mortgage market. 1 Subprime mortgage account for a igni cant part of the recent increae in houehold mortgage debt in the United State, from about 6% of GDP in 23 to above 75% of GDP in It i widely believed that ubprime lending ha played a major role in the houing market meltdown of 27. Unlike traditional prime mortgage, ubprime mortgage have been iued to higher-rik borrower who bought pricey houe relative to their income level and made little or no downpayment. Many of thee mortgage came with incentive including lower initial teaer rate, which were to be reet higher in the future. The mot common example of thi are the 2/28 and the 3/27 loan, which carry a lower introductory rate for the rt two or three year. After that, the interet rate reet to a higher level for the remaining 28 or 27 year of the loan. Becaue of high default rate among ubprime borrower and big loe to ubprime invetor in the declining houing market, ubprime lending ha lately caued a torm of controvery. Some critic accue ubprime lender of predatory lending to naive borrower who do not fully undertand mortgage term. Other ay that ubprime underwriter iued mortgage to people who could not a ord to pay them back, and then quickly old the mortgage to outide invetor in the form of mortgage-backed ecuritie. Mot critic agree that ubprime loan do not make economic ene and hould have not been originated in the rt place. On the other hand, other expert argue that during the houing boom many home were bought with little or no money down becaue both buyer and lender bet on additional home-price appreciation to create equity. Thee bet, while riky, gave le creditworthy borrower a chance at homeownerhip. A related controvery concern how mortgage lender hould react to default and forecloure when houe price decline. Federal Reerve Chairman Ben Bernanke repeatedly called on lender to aid truggling homeowner by reducing their principal to leen the likelihood of forecloure. 3 Similarly, the Houe of Repreentative paed legilation in May 28 encouraging mortgage companie to reduce the principal on troubled loan. 4 On the other hand, many mortgage lender and invetor are grappling with which loan to modify and how. They emphaize that rik of ditorting borrower incentive complicate the iue, a once borrower anticipate to be bailed out they might top debt repayment. 1 See, for example, Agarwal and Ho (Augut 27). 2 The mortgage debt data are from Flow of Fund Account of the United State, Federal Reerve Board, and the GDP data are from Bureau of Economic Analyi. 3 See for example Bernanke (28). 4 See Houe of Repreentative (28). See alo Hubbard and Mayer (28) for a recent outline of the mortgage modi cation program featuring mortgage balance and interet rate reduction. 2

3 Depite the economic igni cance of ubprime lending and the extent of the urrounding controvery, there ha been thu far no attempt at formal analyi of e cient lending in a tochatic houe appreciation environment. Such an analyi would addre a number of quetion. Could a houing boom followed by a houing lump explain certain feature of ubprime lending and it ubequent meltdown if both lender and borrower have rational expectation about the future tate of the houing market? Can ubprime mortgage be more bene cial to both borrower and lender than traditional mortgage? How hould a mortgage contract be adjuted when houe price tart to decline, taking into account borrower incentive to repay their debt? In thi paper, we formally approach thi iue by addreing a more general normative quetion. Auming rational behavior of borrower and lender, what i the bet poible mortgage contract between a home buyer and a nancial intitution when houe price are expected to grow but alo there i rik of a houing crii? Intead of conidering a particular cla of mortgage, we derive an optimal mortgage contract a a olution to a general dynamic contracting problem without impoing retrictive aumption on the payment between the borrower and the lender or the circumtance under which the home i repoeed. We then compare feature of exiting mortgage contract with the derived bet poible contract. We focu our attention on a imple etting that neverthele allow u to capture three apect that we believe are central to ubprime lending. Thee are: (i) a riky borrower who need to be given incentive to repay hi debt, (ii) cotly forecloure and (iii) tochatic houe price appreciation. Thi etting allow u to focu on the fundamental feature of the borrowing-lending relationhip with collateral in a tochatic houe price environment, which i how to e ciently provide a borrower with incentive to repay hi debt uing a the threat of a cotly liquidation taking into account the poibility of houe appreciation and the rik of price decline. We adopt a two-tep approach. Firt, we derive an optimal mortgage contract in the tochatic houe appreciation environment, i.e., the bet poible incentive-compatible contract that maximize a total urplu of the relationhip between the borrower and the lender, a a olution to a general dynamic contracting problem. Then we examine whether main feature of exiting mortgage contract are conitent with the propertie of the optimal contract. A the optimal contract contain proviion of what hould be done in the houing lump, we can addre thi quetion a well. Speci cally, we conider a continuou-time etting in which a borrower with limited liability need outide nancial upport from a rik-neutral lender in order to purchae a houe. Home ownerhip generate for the borrower a public and determinitic utility tream. The ditribution of the "exce" income, which the borrower can ue to pay back hi debt, i publicly known, although it realization are not obervable by the lender. The moral hazard problem i that the borrower can ue hi income for peronal conumption rather than to repay debt. We aume that the houing market at time zero i in the boom phae, during which time a home 3

4 appreciate at a contant rate. At any time, however, a boom can turn into a lump, in which cae a home loe it value and the houing market become illiquid. The price proce i exogenou, and the borrower and the lender have rational expectation. 5 We aume that the borrower and the lender are u ciently mall o that their action have no e ect on macroeconomic variable uch a the market price of the home. Before the purchae of the houe, the borrower and the lender ign a contract that will govern their relationhip in the future. The contract peci e tranfer between the borrower and the lender, conditional on the hitory of the borrower income report and the tate of the houing market, a well a the circumtance under which the lender would forecloe the loan and eize the home. The borrower ha limited liability and can default on the mortgage contract at any time. The borrower ha alo the option to ell the home. We aume that elling the home in the boom phae i more e cient than liquidating it through the repoeion proce due to aociated dead-weight loe. We characterize the optimal contract, i.e. the arrangement between the borrower and the lender that maximize their combined total urplu, uing three tate variable: the tate of the houing market (i.e., the boom or lump), the market home price, and the borrower continuation utility (i.e., the expected payo to the borrower provided he act optimally given the term of hi contract with the lender). We then how that the optimal contract can be implemented uing a mortgage in the form of a home equity loan (HEL) with embedded acce to credit (or negative amortization limit). The feature of the optimal contract do not depend on the competitive tructure of the lending indutry. During the boom, the interet rate i increaing with the value of the home. The leat creditworthy borrower are given an additional reduction in the interet rate during the boom, while the more creditworthy borrower are charged an additional fee, which can be interpreted a an inurance premium. The credit limit, i.e., the maximum amount the borrower can borrow againt hi home, i greater than the value of the home and i increaing over time during the boom. The arrival of the houing lump trigger a mortgage modi cation, which include balance write-o, interet rate cut and a credit limit reduction. The more creditworthy borrower (thoe with lower debt level) get a bigger balance write-o, while mot indebted borrower default immediately due to the credit limit reduction. The feature of the optimal mortgage contract can be explained by the incentive-compatibility contraint and the dual-optimization objective of the contracting problem: minimization of liquidation ine ciencie and maximization of the value of the option to ell the home in the future at a higher price, ubject to borrower limited liability and the incentive-compatibility requirement. The interet rate i increaing with time during the houing boom becaue the lender, in order to break 5 Such a proce can be derived in a general equilibrium framework. For example, Kahn (28) introduce a Markov regimewitching peci cation for productivity growth in the non-houing ector, and how that uch regime witche are a plauible candidate for explaining both qualitatively and quantitatively the large low-frequency change in houing price trend. 4

5 even, ha to make money on the loan in the good tate (i.e., the tate with high home price), a he i likely to loe money in the bad tate (tate with the houing lump arriving early). Hence, the lender increae the interet rate a the home appreciate. On the other hand, the borrower can a ord to pay higher interet rate and i le likely to default in the good tate, ince he can borrow more a the home appreciate. The acce to credit provide exibility for the borrower to cover poible low income realization, which in turn lower chance of default ine ciencie. The credit limit, which i determined by incentive compatibility contraint, take into account the value of the option to ell. The option value increae with the home price and o doe the credit limit. During the lump, the option to ell the home loe it value, which reult in the credit limit reduction. The additional interet rate reduction of the leat creditworthy borrower during the boom i driven by the maximization of the option value, while the default clutering during the lump happen becaue of the borrower limited liability and the incentive-compatibility contraint. Bailing out the mot indebted borrower in the lump phae i not incentive compatible, a it would encourage irreponible nancial behavior in the boom phae. The inurance of more creditworthy borrower i driven by the minimization of liquidation ine ciencie, ince thee borrower are not likely to default during the boom, but are more vulnerable during the lump. The feature of the optimal mortgage are parallel to ome key apect of ubprime lending. In particular, during the boom it i optimal to increae interet rate over time. Thi nding i in line with the fact that many ubprime mortgage originated in the US had teaer rate, which were to be reet to higher rate. It i alo optimal to provide the borrower with increaing acce to credit over time a long a houe price continue to grow. Thi i conitent with the increaing indebtne of houehold during the houing boom. We alo nd that houe appreciation make it pro table to give loan to le creditworthy borrower who otherwie would be hut out of the houing market. Thi i conitent with an oberved extenion of mortgage credit to le creditworthy borrower during the houing boom. 6 According to the parametrized example we conider, the extenion of credit to le creditworthy borrower could generate ubtantial exante utility gain for thee borrower. The e cient lending alo implie default clutering among the leat creditworthy borrower during the lump, which i conitent with an oberved rapid increae in default among the borrower with the highet loan-to-value ratio once houe price tarted to decline. We conclude that many feature of ubprime lending are conitent with rational behavior of both borrower and lender in a tochatic houe appreciation environment. Thu, our model may help to explain the emergence of thi market in recent year. It i important, however, to tre one limitation of our e - ciency reult. The feature of mortgage lending that are optimal at the individual level may have negative conequence at the aggregate level, ince the borrower and the lender do not take into account the potential 6 See Mayer and Pence (28). 5

6 negative externalitie that their contract might impoe on other. In term of policy implication, our model ugget that bailing out the mot ditreed borrower in the lump phae i not incentive compatible, a it encourage irreponible nancial behavior in the boom phae. On the other hand, it make ene to help borrower who were in good tanding before the crii by reducing their debt balance and interet rate. Related Literature Thi paper belong to the growing literature on dynamic optimal ecurity deign, which i a part of the literature on dynamic optimal contracting model uing recurive technique that began with Green (1987), Spear and Srivatava (1987), Abreu, Pearce and Stacchetti (199) and Phelan and Townend (1991), among many other. 7 To the bet of our knowledge our paper i a rt tudy in the optimal nancing literature that conider a dynamic environment with tochatic aet appreciation and an option to ell the aet. The paper mot cloely related to our i Pikorki and Tchityi (27) who tudy the optimal mortgage deign in a tationary continuou time etting with volatile and privately obervable income of the borrower and a tochatic market interet rate. They how that the optimal mortgage take the form of an option ARM, and that default rate and interet rate correlate poitively with the market interet rate. In thi paper, we aume a contant interet rate and focu on a tochatic houe price environment where the borrower ha the option to ell the home. The nding of thi tudy are complementary to thoe of Pikorki and Tchityi (27). The two other tudie cloely related to our are DeMarzo and Fihman (27a) and it continuoutime formulation by DeMarzo and Sannikov (26). Thee paper tudy long-term nancial contracting in a etting with privately oberved cah ow, and how that the implementation of the optimal contract involve a credit line with a contant interet rate and credit limit, long-term debt, and equity. 8 There i a izeable real etate and nance literature on the deign of mortgage. The bulk of thi literature, which include Chari and Jagannathan (1989), LeRoy (1996), Stanton and Wallace (1998), focue on advere election and how it a ect the menu of mortgage being o ered to borrower with limited inurance poibilitie. In a moral hazard etting with unobervable income, Dunn and Spatt (1985) how optimality of due-on-ale claue that can be modi ed depending on the oberved interet rate and the time to maturity. To our knowledge, our paper i the rt tudy of optimal mortgage deign in a dynamic moral hazard environment with cotly default, tochatic houe appreciation and the option to ell. There are alo a number of tudie that addre the optimal trategy of the mortgage borrower in environment with tochatic houe price (ee for example Kau, Keenan, Mueller, and Epperon (1992) and 7 Sannikov (26) provide a recent treatment of continuou-time technique for a principal-agent problem. 8 See alo related paper by Biai et al. (26), Clementi and Hopenhayn (26), DeMarzo and Fihman (27b), DeMarzo and Sannikov (28), Tchityi (26), He (27), DeMarzo, Fihman, He and Wang (28), Philippon and Sannikov (27). 6

7 Deng, Quigley, and Van Order (2), Campbell and Cocco (27)). Thi literature retrict it attention to a peci c cla of contract, however, and thu, unlike thi paper, doe not addre the quetion of optimal lending. Lutig and Van Nieuwerburgh (25) tudy conumption inurance and rik premia in a model with houing collateral and limited commitment of the houehold. Unlike thi paper they do not conider a moral hazard problem and their model doe not feature default in equilibrium. The paper i organized a follow. Section 2 preent the continuou-time model with tochatic houe appreciation. Section 3 decribe the dynamic contracting problem. Section 4 derive the optimal contract uing the three tate variable: the tate of the houing market, the market home price and the borrower continuation utility. Section 5 preent the implementation of the optimal contract uing nancial arrangement that reemble the one ued in the reidential mortgage market. Section 6 conclude. 2 The Model Time i continuou and in nite. intitution). 9 ff t g a There i one borrower (a homebuyer) and one lender (a large nancial The lender i rik neutral, ha unlimited capital, and value a tochatic cumulative cah ow where r i the dicount rate. Z 1 E e r df t ; The borrower conumption conit of two categorie. The rt i "neceary" conumption, which include grocery food, medicine, tranportation, and other good and ervice eential to urvival. cumulative minimum level of neceary conumption i given by an exogenou tochatic proce f t g that incorporate hock uch a medical bill, auto repair cot, uctuation of food and gaoline price, and o on. The econd i dicretionary conumption, which include, among other, uch item a retaurant dining, vacation trip, buying a new car, et cetera. We aume that the borrower mut ue hi available fund rt to cover the neceary expene t before pending on "dicretionary" conumption or potential debt repayment. 1 The 9 Without lo of generality, we can think about the lender a a group of invetor who maximize their combined payo from the relationhip with the borrower. How the invetor divide proceed among themelve i not relevant for the purpoe of deigning an optimal contract between the borrower and the invetor. 1 Thi peci cation reemble the one ued by Ait-Sahalia, Parker and Yogo (24), who propoe a partial reolution of the equity premium puzzle by ditinguihing between the conumption of baic good and that of luxury good. In their model, houehold are much more rik avere with repect to the conumption of baic good, of which a certain amount i required in every period, which i conitent with the ubitence apect of baic good and the dicretionary apect of luxurie. 7

8 The borrower value cumulative dicretionary conumption ow fc t g a 2 E 4 Z 1 3 e rt dc t 5 ; where dc t. The zero conumption (dc t = ) mean that the borrower conume only neceitie. Let Y t denote the borrower total cumulative income up to time t. We will focu on the borrower "exce" income Y t Y t t, which repreent a better meaure of the borrower ability to pay for a houe than the total income. A tandard Brownian motion Z = fz t ; F 2;t ; t < 1g on ( 2 ; F 2 ; m 2 ) drive the borrower income proce, where ff 2;t ; t < 1g i an augmented ltration generated by the Brownian motion. The borrower income up to time t, denoted by Y t, evolve according to dy t = dt + dz t ; (1) where i the drift of the borrower dipoable income and i the enitivity of the borrower income to it Brownian motion component. The exce income i negative when the neceary expene hock t i greater than the total income Y t. In thi cae, the borrower ha to cover the de cit by drawing on hi aving account or by borrowing more; if unable to do o he will declare bankruptcy. From now on, we will refer to Y t and C t imply a the borrower income and borrower conumption. We aume that the lender know and, but doe not know realization the borrower exce income hock Z t. The borrower ha the ability to mirepreent hi income. Thu, realization of the borrower income are not contractible. Thee aumption are motivated by the obervation that lender ue a variety of method 11 to determine a type of the borrower (repreented here by (; ) pair) before the loan i approved, but henceforth do not condition the term of the contract on the realization of the borrower income, likely becaue the borrower neceary pending hock and poibly hi total income a well are too cotly or impoible to monitor. The borrower i allowed to maintain a private aving account. The private aving account balance S grow at the interet rate, where r. The borrower mut maintain a non-negative balance in hi account. The borrower want to buy a home at date t =. Home ownerhip would generate him the public and determinitic utility tream. We aume that thi utility tream remain contant a long a the borrower tay in the ame houe. 12 The time-zero price P of the home i greater than the borrower initial wealth Y, i.e., Y < P. 13 Thu, the borrower mut obtain fund from the lender to nance the houe purchae. 11 Credit core, current job tatu and o on. 12 For implicity, we do not conider the poibility that the borrower can make modi cation that either increae or decreae the quality of the houe. 13 The price P i conidered a a macroeconomic variable, which i not a ected by action of the borrower and the lender. It i reaonable to expect that the home price P i increaing in it utility, and the borrower optimize over the et of available (; P ) pair. Thi optimization i not conidered in the paper. Thi clearly doe not lead to a lo of generality, ince our 8

9 We aume that the borrower and the lender are u ciently mall o that their action have no e ect on macroeconomic variable uch a the market interet rate. 14 The houing market i expected to go through two phae. The initial phae, houing boom, i characterized by rapid houing appreciation. Houing lump i the aborbing tate, characterized by a houing market receion and price tabilization. Let the proce fn t g denote the phae of the houing market in the period t: N t = mean the houing boom continue in period t, and N t = 1 mean the houing lump phae in period t. Formally, the proce N = fn t ; F 2;t ; t < 1g i a tandard compound Poion proce with an intenity (N t ) on a probability pace ( 2 ; F 2 ; m 2 ), uch that N = and 8 < if N t i even (N t ) = : if N t i odd : The topping time h = infft : N t = 1g denote the arrival time of the houing lump phae. The market price of the home grow at the rate g > per year during the boom, while it remain contant during the lump: 8 < P e gt for all t < h P t = ; : P e g h (1 ) for all t h where 2 [; 1] meaure the extent of houe price depreciation. 15 The aumed price proce i motivated by the behavior home price in the pat two decade, ee Appendix A.2. Home price tayed approximately contant from 1987 to 1996, then experienced a period of fat and teady growth from 1997 to 26, which ended in a crah in 27. Before purchae of the houe, the borrower and the lender ign a contract that will govern their relationhip after the purchae i made. The contract obligate the borrow to report hi income realization to the lender. Conditional on the hitory of houe price and the borrower income report, the contract peci e tranfer between the borrower and the lender and the circumtance under which the lender repoee the home and the circumtance under which the borrower become a full homeowner. Option to Default The borrower ha the option to default at any time, in which cae he loe the home and receive a reervation value equal to A. For implicity, we aume that the reervation value i equal to the expected preent value of the borrower future income, r plu any private aving he might have. We aume and o A r : analyi applie to any (; P ) pair. 14 In a general equilibrium framework, action of mortgage lender and homebuyer on the aggregate level can a ect macroeconomic variable. However, a long a the economic agent on the individual level have no market power, they hould regard macroeconomic variable a exogenou in equilibrium. 15 Note that thi aumption doe not imply that one can do arbitrage. Unlike pure nancial aet, houing i a conumption good; tranaction and earch cot can be high, and ome trade, uch a horting, are di cult in the houing market. 9

10 The lender ell the repoeed houe at a forecloure auction and receive the payo L t = (1 l)p t, where l 2 (; 1] meaure the liquidation cot. The borrower default only when hi continuation utility under the contract a t i le than or equal to hi outide option A. In our ubequent analyi, we do not explicitly conider default a a part of the borrower trategy. Intead, without lo of generality, we impoe the participation contraint on the optimal contract. That i the borrower continuation utility a t, which i hi expected dicounted utility at any time t under the contract, mut be greater than or equal to the outide option A a long a the houe i not repoeed. Full Homeownerhip We ay that the borrower i full homeowner if he own the home and keep all hi income, i.e., ha no debt. Let v t be the continuation utility of the borrower with no aving under full homeownerhip at time t. For implicity, we ignore the poibility of bankruptcy of a full homeowner, which i poible due to negative income hock but highly unlikely, and aume that v t i equal to the expected preent value of the borrower future income and houing conumption plu the the value of the option to ell the home in the future. Since the houing lump i an aborbing tate, the continuation utility of the borrower who ha no debt and no aving in the lump phae doe not depend on t and i given by v t = v 1 + r. On the other hand, ince the borrower ha the option to ell the home in the boom phae, hi continuation utility under full homeownerhip, which we denote by v t = v t, doe depend on t. Option to Sell The borrower can put the home on the market. For implicity, we aume that the home can be old at the market price P t immediately during the boom, while it i impoible to nd a buyer during the lump. The elling i more e cient than liquidation, a l < 1. The ale of the home i not contractible. 16 The borrower put the home on the market at the time when it maximize hi expected payo. We aume that if he ell the home the borrower ha to pay to the lender (vt a t ), i.e., the di erence between the full homeownerhip utility vt and hi continuation utility a t under the exiting mortgage contract. A we will ee it in Section 5, thi aumption mean paying to the lender the outtanding balance B t on the loan, which i related to the borrower continuation utility a t under the optimal contract a follow: B t = v t a t. (2) In the boom phae of the houing market we have v t = vt. Thu, after the houe ale the borrower 16 In practice, contract claue forcing the borrower to ell the home may be di cult to implement, ince the price and the peed of ale critically depend on the borrower e ort and cooperation. 1

11 continuation payo i given by while the lender receive A S t (a t ) = A + P t (v t a t ); L S t (a t ) = v t a t : Note that the borrower will want to ell whenever A S t (a t ) a t ; which i equivalent to A + P t v t : (3) The optimal elling time determined by equation (3) doe not depend on the outtanding balance or the continuation utility of the borrower. Thi i becaue the outtanding balance i linear in the borrower continuation utility, and they cancel each other out. The optimal elling time alo doe not depend on the liquidation value L t of the home. Thi i due to the fact that the borrower doe not take into account dead-weight cot aociated with liquidation. Equation (3) imply tate that the borrower ell the home whenever the value of hi outide option A plu the proceed from the ale exceed hi continuation utility under full homeownerhip. Propoition 1 The optimal time for the borrower to ell the home in the boom phae of the houing market i given by t = 1 g r 1 g r+ P 1 A ; (4) and the value of full homeownerhip at time t t in the boom phae of the houing market i equal to vt = v 1 + e (r+)(t t) P t : r {z } value of option to ell at t Proof In the Appendix. Note that v t i increaing with time. Thi i becaue the borrower ha the option to ell the home, and thi option become more valuable a the price of the home increae. Home ale happen at time t only in the boom phae. The optimal timing of home ale i given by 8 < t, if N t = = : 1, if N t = 1 : 11

12 In what follow we require that if houe price decline before the optimal elling time (4), there are gain of trade to continue the borrowing-lending relationhip, i.e., for h t t the combined payo from liquidation A + L t i le than the full homeownerhip value v 1. Thi amount to impoing that L h =t = (1 l)(1 )P e gt < r ; which given (4) i equivalent to the requirement that g < (r + ) ( + l l) : 3 Dynamic Moral Hazard Problem At time, the fund needed to purchae the home in the amount of P Y are tranferred from the lender to the borrower. A contract, = ( f ; d ; I); peci e a time at which the borrower become a full homeowner, f, a default time, d ; and tranfer between the lender and the borrower, all of which are baed on the borrower report of hi income and the realized houe price proce. The relationhip between the borrower and the lender i terminated at = min( ; f ; d ). Let (; F; m) := ( 1 2 ; F 1 F 2 ; m 1 m 2 ) be the product pace of ( 1 ; F 1 ; m 1 ) and ( 2 ; F 2 ; m 2 ). Let n ^Y = ^Yt : t o be the borrower report of hi income, where ^Y i (Y; P )-meaurable (F t meaurable). Without lo of generality, we aume that the borrower i required to pay the reported income to the lender. 17 Thu, the contract tranfer the reported amount, ^Y t ; from the borrower to the lender, and I t ( ^Y ; P ) h R i min( from the lender to the borrower. We further require E f ; d ) e r di jf t to be quare-integrable for t min( f ; d ) and ^Y = Y: The borrower can mireport hi income. Conequently, up to time t, the borrower receive a total ow of income equal to (dy t d ^Y t ) + di t ; {z } mireporting and hi private aving account balance, S, grow according to ds t = S t dt + (dy t d ^Y t ) + di t dc t ; (5) where dc t i the borrower conumption at time t; which mut be non-negative. De nition 1 Given a contract = ( f ; d ; I); a feaible trategy for the borrower i a pair (C; ^Y ) uch that (i) ^Y i a continuou-time proce adapted to (Y; P ), (ii) C i a nondecreaing continuou-time proce adapted to (Y; P ), i.e., dc t ; 17 Since we do not impoe retriction on I, thi aumption doe not retrict the pace of contract. 12

13 (iii) the aving proce de ned by (5), uch that S t. De nition 2 Given a contract = ( f ; d ; I), the borrower trategy (C; ^Y ) i incentive compatible if (i) given a contract ; the borrower trategy (C; ^Y ) i feaible, (ii) given a contract ; the borrower trategy (C; ^Y ) provide him with the highet expected utility among all feaible trategie, that i E R e rt (dc t + dt) + e r (1 = A S + 1 = f v + 1 = da + S ) jf E R e rt (dc t + dt) + e r (1 = A S + 1 = f v + 1 = da + S ) jf for all the borrower feaible trategie (C ; ^Y ); given a contract : De nition 3 An incentive compatible contract i a contract = ( f ; d ; I), together with the recommendation to the borrower, (C; ^Y ); where (C; ^Y ) i a borrower incentive compatible trategy given a contract. The contract i optimal if for a given time-zero expected utility a for the borrower no other contract can increae the expected payo for the lender. Below we provide a formal de nition of the optimal contract. De nition 4 Given the continuation utility to the borrower, a, a contract = ( f ; d ; I ), together with an incentive compatible trategy (C ; ^Y ) i optimal if it maximize the lender expected payo : Z b = E e rt (d ^Y t while the expected utility for the borrower i equal to: di t ) + e r (1 = L S + 1 = dl ) jf Z a = E e rt (dc t + dt) + e r (1 = A S + 1 = f v + 1 = da + S ) jf : The de nition of the optimal contract implie maximization of the total urplu, and can be alo reformulated in term of maximizing the borrower expected utility for a given payo of the lender. The initial payo a and b are determined by the bargaining power of the borrower and the lender. Our ubequent analyi i valid for any a > A, including the competitive lending indutry, i.e., the lender break even: b = P Y, and the optimal contract maximize the borrower expected utility. In the following lemma, we how that earching for optimal contract, we can retrict our attention to contract in which truth-telling and zero aving are incentive compatible. 13

14 Lemma 1 There exit an optimal contract in which the borrower chooe to tell the truth and maintain zero aving. Proof In the Appendix. The intuition for thi reult i traightforward. The rt part of the reult i due to the direct-revelation principle. The econd part follow from the fact that it i weakly ine cient for the borrower to ave on hi private account ( r) a any uch contract can be improved by having the lender ave and make direct tranfer to the borrower. Therefore, we can look for an optimal contract in which truth-telling and zero aving are incentive compatible. 4 Derivation of the Optimal Contract In thi ection, we formulate recurively the dynamic moral hazard problem and determine the optimal contract. Firt, we conider a problem in which the borrower i not allowed to ave. We determine the optimal contract 18 in thi environment, achieving thi in two tep. Firt, we preent and explain the optimal contract after the houe price lump occurred. Next, given the pot-lump value function, we derive the optimal contract in the boom environment. We know from Lemma 1 that it i u cient to look for optimal contract in which the borrower report truthfully and maintain zero aving, and o the optimal contract of the problem with no private aving, for a given continuation utility to the borrower, yield to the lender at leat a much utility a the optimal contract of the problem when the borrower i allowed to privately ave. We will conclude by howing that the optimal contract of the problem with no private aving i fully incentive compatible, even when the borrower can maintain undicloed aving, jutifying our approach. Methodologically, our approach i baed on continuou-time technique ued by DeMarzo and Sannikov (26) and extended to a etting with a Lévy proce by Pikorki and Tchityi (27). To characterize the optimal contract recurively, we de ne the borrower continuation utility at time t under the contract if he tell the truth a Z a t = E e r( t) [di + d] + e r( t) [1 = A S + 1 = f v + 1 = da] jf t : t 4.1 The Optimal Contract in the Slump Phae Let b 1 (a t ; h ) be the highet expected utility of the lender that can be obtained from an incentive compatible contract at time t that provide the borrower with continuation utility a t given that the lump phae tarted 18 Thi i the allocation atifying the propertie of De nition 4 and the additional contraint that S =. 14

15 at h < t. Function b 1 (a t ; h ) depend on h, becaue the liquidation value of the home depend h, but it doe not depend on time t, becaue of time homogeneou contracting environment in the lump phae. There are two boundary condition for function b 1 (a t ; h ). The rt boundary condition tem from the borrower participation contraint: the relationhip mut be terminated when the borrower continuation utility fall to A, o b 1 (A; h ) = L h. The econd boundary condition come from the fact that the lender hould expect no tranfer from the borrower once he become a homeowner, that i b 1 (v 1 ; h ) = : Let b 1 and b 1 denote, repectively, the rt and econd derivative of b 1 with repect to the borrower continuation utility a t. The optimal contracting problem in the lump phae i very imilar to the one conidered by DeMarzo and Sannikov (26). The main di erence are that the underlying aet (home) generate a nonmonetary dividend next to average income to the agent (borrower), that both the principal (lender) and the agent (borrower) have the ame dicount factor, and there i a poibility to declare an agent a full owner that deliver to him a perpetual value of v 1 = + r :19 The propoition below decribe the optimal contract in the lump phae. Propoition 2 The optimal contract that deliver to the borrower continuation utility a t at time t > h ; take the following form. If a t 2 [A; v 1 ]; a t evolve a da t = (ra t dt dt) + (d ^Y t dt): (6) The default occur at rt time d when a t hit A. The borrower become a full homeowner at time f when a t hit v 1 for the rt time: The lender expected payo at any time t h i given by the function b 1 (a t ; h ), which i trictly concave in a t over [A; v 1 ] and olve: rb 1 (a; h ) = + (ra )b 1(a t ; h ) b 1(a; h ); (7) for a 2 [A; v 1 ]; with the boundary condition b 1 (A; h ) = L h, and b 1 (v 1 ; h ) =. Proof Directly follow from DeMarzo and Sannikov (26) and Bia et al. (27) a the tructure of dynamic moral hazard problem after the houe lump i imilar to the one tudied in thee paper. The evolution of the continuation utility (6) implied by the optimal contract erve three objective: promie-keeping, incentive, and e ciency. The rt component of (6) account for promie-keeping. In order for a t to correctly decribe the lender promie to the borrower, it hould grow at the borrower dicount rate, r; le the payment, dt, which he receive from owning the home, and le the ow of payment, di t ; from the lender. 19 Thu our model in the lump phae i the limiting cae of DeMarzo and Sannikov (26) with! r and poibility to declare an agent a full owner that deliver to him a perpetual value of the rt bet equal to + r : 15

16 The econd term of (6) provide the borrower with incentive to report hi true income to the lender. Becaue of ine ciencie reulting from liquidation, reducing the rik in the borrower continuation utility lower the probability that the borrower expected utility reache A, and thu lower the probability of cotly liquidation. Therefore, it i optimal to make the enitivity of the borrower continuation utility with repect to it report a mall a poible provided that it doe not erode hi incentive to tell the truth. The minimum volatility of the borrower continuation utility with repect to hi report of income required for truth-telling equal 1. To undertand thi, note that under thi choice of volatility, underreporting income by one unit would provide the borrower with one additional unit of current utility through increaed conumption, but would alo reduce the borrower continuation utility by one unit, o that thi volatility provide the borrower with jut enough incentive to report a true realization of income. Note that when the borrower report truthfully, the term d ^Y t dt i driftle and equal dz t. Since default i cotly and the borrower and lender have the ame dicount factor it i optimal to delay any tranfer to the borrower until he become full homeowner, in which cae the default i no longer an iue. On the other hand, when a t v 1, the rt bet i reached and there are no gain for the lender and for the borrower from continuing their relationhip. Thu, it i optimal to let the borrower become a full homeowner at time f = inf t h : a t = v The Optimal Contract in the Boom Phae The contracting environment in the boom phae i di erent from that of the lump phae in three major way. Firt, the houe price i changing over time. Second, the borrower will ell the houe at time t, provided t < h. Third, the houing lump arrive with Poion probability at time h. Let b (a t ; t) be the highet expected utility of the lender that can be obtained from an incentive compatible contract at time t that provide the borrower with continuation utility a t in the boom phae, i.e., t < h. Let b and b denote, repectively, the rt and the econd derivative of b with repect to the borrower continuation utility a. Since the home price i growing over time, b (a t ; t) directly depend on t. When the houe i old, the payo to the lender i given by b (a t ; t ) = v t a t : The arrival of the houing lump will a ect the payo of both the lender and the borrower. Let (a t ; t) denote the intantaneou change in the continuation utility of the borrower under the optimal contract if the houing lump arrive at time h = t. Then, the lender continuation payo will jump from b (a t ; h ) to b 1 (a t + (a t ; t); h ). The following propoition characterize the optimal contract in the boom phae. 16

17 Propoition 3 The optimal contract that deliver to the borrower continuation utility a t at time t < h ; take the following form. If a 2 [A; v t ]; a t evolve a da t = (ra t dt dt di t ) + (d ^Y t dt) + (a t ; t)(dn t (N t )dt); (8) The borrower become a full homeowner at rt time f when a t hit v t. The borrower default at rt time d when a t hit A. The borrower ell the home at time t, provided that t < min( d ; h ): The lender expected payo at any time t < h i given by the function b (a t ; t), which i trictly concave in a t over [A; v t ] and olve: rb (a t ; t) (a t ; for a t 2 [A; v t ]; with the boundary condition + + (ra t (a t ; t))b (a t ; t) b (a t ; t) + (b 1 (a t + (a t ; t); t) b (a t ; t)) (9) b (A; t) = L t ; b (v t ; t) = ; b (a t ; t ) = v t a t ; where 8 >< (a; t) = i a C 1 olution to b (a; t) = b 1(a + ; t) for all (a; t) for which the olution i uch that (a; t) > A a : (1) >: otherwie i equal to A a Proof In the Appendix. The evolution of the continuation utility (8) implied by the optimal contract erve three objective: promie-keeping, incentive, and e ciency. The rt component of (8) account for promie-keeping while the econd term provide the borrower with incentive to report hi true income to the lender. The lat term of (8) capture the e ect of the tochatic houe price appreciation on the borrower continuation utility. The optimal adjutment,, in the borrower continuation utility, which are applicable a when houe lump occur, are uch that the enitivity of the lender expected utility, b, with repect to the borrower continuation utility, a, i equalized jut before and after an adjutment i made. 2 Thi enitivity repreent an intantaneou marginal cot of delivering to the borrower hi continuation utility in term of the lender utility, and o the e ciency call for equalizing thi cot acro the tate. We note that thee 2 Provided that the olution to (1) i interior. 17

18 adjutment imply the compenating trend in the borrower continuation utility, (N t ) (a t ; t)dt, which exactly o et the expected e ect thee adjutment have on the borrower expected utility. Since default i cotly and the borrower and lender have the ame dicount factor it i optimal to delay any tranfer to the borrower until he become full homeowner, in which cae the default i no longer an iue. On the other hand, when a t vt during the boom, the rt bet i reached and there are no gain for the lender and for the borrower from continuing their relationhip. Thu, a the ame applie in the houe lump, it i optimal to let the borrower become a full homeowner at time f = inf ft : a t = v t g. 4.3 The Optimal Contract with Hidden Saving So far, we have characterized the optimal contract under the aumption that the borrower cannot ave. Now we how that, given the optimal contract of the problem with no hidden aving, the borrower ha no incentive to ave at the olution. Thu, the contract of Propoition 2-3 i alo optimal in the environment where the borrower can privately ave. Propoition 4 Suppoe that the proce a t 2 [A; v t ] olve da t = ra t dt dt di t + (d ^Y t dt) + t (dn t (N t )dt) (11) until topping time = min( f ; d ; ); where f = infft > : a t = v t g and d = infft > : a t = Ag, and where t i an F t predictable proce. Then the borrower expected utility from any feaible trategy in repone to a contract ( f ; d ; I) i at mot a : Moreover, the borrower attain the expected utility a if the borrower report truthfully and maintain zero aving. Proof In the Appendix. The above propoition how that the optimal contract of Propoition 2-3, remain incentive compatible even if the borrower i allowed to privately ave. 4.4 A Numerical Example In thi ection we illutrate the feature of the optimal contract in a parametrized example. Table 1 how the parameter of the model. Table 1. Parameter of the model Borrower type Home type Houing market Interet rate Y A P g l r

19 The left hand-ide of Figure 1 how the lender continuation payo a a function of the borrower continuation utility in the boom phae at time zero and the lender continuation payo when the boom turn into the lump at time zero. For a given continuation utility to the borrower, the lender payo in the boom phae i alway greater than it would be in the lump phae. Thi i becaue the liquidation value of the home i higher in the boom phae and keep growing. In addition, the borrower i le likely to default in the boom phae. The full homeownerhip utility for the borrower i alo greater in the boom phae, ince it include the option to ell the home in the future. The right hand-ide of Figure 1 how the optimal adjutment in the borrower continuation utility, t, applicable if the boom turn into the lump at time t for t =. For t >, function t ha a imilar pro le. For any t, the adjutment i zero at termination, i.e., a t = A, while for the full homeowner, i.e., a t = vt, the adjutment i negative and equal to the lot value of the the option to ell the home vt v 1. The jump function t i convex for maller a and concave for bigger a. Our imulation for di erent parameter and t how that the jump function t alway ha a form imilar to the one on Figure 1. According to equation (8), the negative jump (a t ; t) at the beginning of the lump phae tranlate into a poitive continuation utility trend " (N t ) (a t ; t)dt" in the boom phae. Figure 1: The lender value function and the optimal adjutment in the borrower continuation utility. The optimal jump t in the borrower continuation utility triggered by the lump can be partially allocated to the lo of the option to ell the home in the boom phae. It i therefore natural to decompoe 19

20 the jump a follow: t (a t ) = v t v 1 vt A (a t A) + t(a t ) + v t v 1 vt A (a t A) : (12) The olid line connecting point (A; ) and vt ; (vt v 1 ) on Figure 2 depict the rt term of (12), which repreent the reduction in the borrower continuation utility proportional to hi take in the homeownerhip. The actual reduction in the continuation utility i bigger for borrower with low continuation utility (le creditworthy), and lower for borrower with high continuation utility (more creditworthy), all relative to their take in homeownerhip (their continuation utility). Thu, the le creditworthy borrower receive more preferential treatment, or ubidy, compared to more creditworthy one during the boom phae, however they receive a larger reduction of their continuation utility in the lump. On the other hand, the more creditworthy borrower receive a better treatment in the lump a their continuation utility decline le relative to their take in homeownerhip, which can be interpreted a a form of partial inurance againt the lump. Figure 2: Decompoition of the optimal utility jump t jump f a t, tfi (v (t) v 1 ) ubidy inurance A Borrower expected payoff a t v (t) Thee adjutment can be explained by the incentive-compatibility contraint and the dual optimization objective of the contracting problem: minimization of liquidation ine ciencie and maximization of the value of the option to ell the home in the future at a higher price ubject to borrower limited liability and the incentive-compatibility contraint. It i e cient to reduce the chance of cotly termination when houe price grow, a the relationhip between the borrower and the lender i more valuable due to the poibility of houe appreciation (which 2

21 increae the future liquidation value and increae the value of option to ell). The preferential treatment of the borrower during the houing boom, which manifet itelf a a poitive trend in the borrower continuation utility, accomplihe thi tak by helping to puh away the borrower continuation utility further away from the liquidation boundary. However, the threat of repoeion mut be u ciently real in order for the borrower to hare hi income with the lender. A a reult, it i optimal to increae the chance of repoeion during the lump, by intantaneouly decreaing the borrower continuation utility, in order to compenate for the weakened threat of repoeion during the boom. The negative jump in the continuation utility at the beginning of the lump phae lead to the default clutering among the leat creditworthy borrower. Since le creditworthy borrower receive more preferential treatment in the boom phae, bailing out ditreed ubprime borrower in the lump phae i not incentive compatible, a it would encourage irreponible nancial behavior. Indeed, borrower in good tanding would prefer not to pay the mortgage in order to qualify for the ubidy in the boom phae and then get help in the lump phae. The partial inurance againt the lump of more creditworthy borrower i driven primarily by the minimization of liquidation ine ciencie, ince more creditworthy borrower are not likely to default during the boom, but are more vulnerable during the lump Utility Gain for the Riky Borrower Firt, we how that home appreciation lead to in ow of le creditworthy home buyer into the houing market. Then, we demontrate that the home ownerhip generate ubtantial ex-ante utility gain for thee borrower who otherwie would be hut out of the houing market. Our nding provide theoretical evidence that the houing boom can caue a growth of the ubprime market. For the parameter in Table 1, Figure 3 how the lowet expected income, for which the lender break even, a a function of the home price growth rate g. Fater houe appreciation make it pro table to give loan to borrower who otherwie would be hut out of the houing market. For example, with the zero home appreciation (g = ), only the borrower with :92 can take a mortgage and purchae the home. On the other hand, fat home appreciation with g = 12% make homeownerhip poible for the borrower with the mean income :7. For the parameter lited in Table 1 (with g = 12%), Figure 4 how the net utility gain (a A) from homeownerhip a a function of the borrower mean income. Borrower with < :7 cannot qualify to take the mortgage and purchae the home. However, for borrower with > :7 buying the home reult in ubtantial expected utility gain. Note that, even for the borrower with the lowet qualifying income, the gain are igni cantly bigger than zero. Thi i becaue of the limited liability of the borrower and the incentive compatibility requirement. When the borrower expected utility after the initialization of the contract i u ciently cloe to hi 21