Stochatic Houe Appreciation and Optimal Subprime Lending Tomaz Pikorki Columbia Buine School tp5@mail.gb.columbia.edu Alexei Tchityi NYU Stern atchity@tern.nyu.edu February 8 Abtract Thi paper tudie an optimal mortgage deign problem. Auming full rationality, we characterize the optimal mortgage contract in a continuou time etting with a riky borrower, cotly default, a moral hazard problem between the borrower and the lender, and a bubble-like behavior of the houing market. We how that many feature of ubprime lending oberved in practice are conitent with rationality of both borrower and lender. In particular, preferential treatment of ubprime borrower i optimal during the houing boom, while default clutering among ubprime borrower i optimal during the houing lump. We alo nd that tochatic houe appreciation make it pro table to give loan to ubprime borrower who otherwie would be hut out of the houing market and that home ownerhip generate ubtantial ex-ante utility gain for thee borrower. 1
1 Introduction The recent houing market crii ha brought attention to the o-called ubprime mortgage market, which experienced exponential growth over the pat few year. The hare of ubprime mortgage to total origination increaed from 6% in to % in 6. A of 6, 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 to above 75% of GDP in 6. It i widely believed that ubprime lending ha played a major role in the houing market meltdown in 7. Unlike traditional prime mortgage, ubprime mortgage are normally made out to higher-rik borrower who buy pricey houe relative to their income level and make little or no downpayment. Often, thee mortgage come with incentive including low initial teaer rate (which later reet to higher rate), and low "interet only" repayment term that let borrower pay only the interet portion of the debt or even le than that. In addition, ubprime borrower often maintain a loan balance above the market value of the home. A a reult, ubprime mortgage loan have a much higher rate of default than prime mortgage loan. Becaue of high default rate among ubprime borrower and big loe among ubprime invetor, ubprime mortgage have caued a torm of controvery and criticim. 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 their mortgage to outide invetor in the form of mortgage-backed ecuritie. Mot critic agree that ubprime loan do not make ene and hould have not been made in the rt place. On the other hand, other expert argue that the fat growth of the ubprime market wa caued by the fat home price appreciation oberved in the beginning of the 1t century. To better undertand the nature of the ubprime crii, it i important to examine what caued the rapid growth of the ubprime market before the crii tarted. Thi paper eek to determine whether ubprime lending can be explained by rational behavior of BOTH borrower and lender. In particular, can the bubblelike behavior of the houing market, i.e., houing boom followed by houing lump, explain the fat growth of the ubprime market during the boom and it meltdown during the lump if both lender and borrower have rational expectation about the future tate of the houing market? In addition, we partially addre the following quetion. What hould the government do before and during the crii in the houing market? Doe ubprime lending exacerbate the "houing bubble"? To addre thee quetion, we conider a dynamic continuou time model with a riky borrower, tochatic home appreciation, cotly liquidation and a moral hazard problem. We adopt a two-tep approach. Firt, 1 See, for example, Agarwal and Ho (Augut 7). 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.
auming full rationality, we derive an optimal mortgage contract, i.e., the bet poible incentive-compatible contract between the borrower and the lender, a a olution to a general dynamic contracting problem. Then we examine whether feature of exiting mortgage contract are conitent with the propertie of the optimal contract. 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 borrower conumption i divided into two categorie: "neceary" conumption, which include grocery food, medicine, tranportation and other good and ervice eential for the houehold urvival, and "luxury" conumption, which include everything ele. We aume that the borrower i in nitely rik avere with repect to neceary conumption and rik neutral with repect to luxury conumption. The minimum level of neceary conumption i given by an exogenou tochatic proce. After paying for hi neceary conumption, the borrower i free to allocate the remaining part of hi income among luxury conumption, aving and debt repayment. The ditribution of the "exce" income, which the borrower can ue to pay back hi debt, i publicly known, however it realization are privately obervable by the borrower. We aume that the houing market at time zero i in the boom phae, during which the home appreciate at a contant rate. However, at any time the boom can turn into the lump with a certain probability, in which cae the home loe it value and the houing market become illiquid. The price proce i exogenou, and the borrower and the lender have rational expectation. 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 cot. We characterize the optimal allocation 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 nd that it i optimal to ubidize ubprime borrower, i.e., the borrower with low continuation utility, during the 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 an equilibrium.
boom. Default clutering among ubprime borrower i optimal during the lump. It i optimal to inure prime borrower, i.e., the borrower with high continuation utility, againt the lump. After characterizing the optimal allocation in term of the continuation utility of the borrower and the lender, we how that the optimal allocation can be implemented uing a home equity loan (HEL) with adjutable negative amortization limit with the following feature. In the boom phae, the negative amortization limit i increaing with the price of the home and i greater than the value of the home. The mortgage interet rate i increaing with the home price. The ubprime borrower are given additional ubidy during the boom, while the prime borrower are charged an inurance premium. The lump reult in the tightening of the negative amortization limit and default clutering among ubprime borrower. The prime borrower are partially inured againt the lump through a balance 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. The credit line 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. The option value increae with the home price a doe the credit limit. During the lump, the option to ell the home loe it value, which reult in the credit limit reduction. The interet rate goe up with the home price becaue the value of the option goe up. The ubidy of the ubprime 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. The inurance of the prime borrower i explained by the fact that the prime borrower are not likely to default during the boom, but are more vulnerable during the lump. The feature of our optimal mortgage contract are parallel to many apect of the ubprime lending. Thu, we conclude that ubprime lending doe not contradict rationality of both borrower and lender. We alo nd that houe appreciation make it pro table to give loan to ubprime borrower who otherwie would be hut out of the houing market. According to the parametrized example we conider, home ownerhip generate ubtantial ex-ante utility gain for the ubprime borrower. Thu, our reult provide theoretical evidence that the bubble-like behavior of houing price can caue rapid growth of the ubprime market. In term of policy implication, our model ugget that bailing out the mot ditreed ubprime 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. Although we do not conider a general equilibrium model in thi paper, we peculate that ubprime lending can contribute to the "houing bubble". During the houing boom, the in ow of ubprime borrower into 4
the houing market may help utain houe appreciation, poibly driving home price above the equilibrium level. However, default clutering among ubprime borrower during the houing lump may exacerbate the crii in the houing market. Thu, the feature of ubprime lending that are optimal at the individual level may have negative conequence at the aggregate level. Although the borrower and the lender properly factor expectation about home price behavior into the optimal contract, they do not take into account the potential negative externalitie that their contract might impoe. 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. Sannikov (6a) developed continuou-time technique for a principal-agent problem. The mot cloely related paper to our i Pikorki and Tchityi (7) 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 form of either an option ARM or a combination of an interet only mortgage with a HELOC, and that default rate and interet rate correlate poitively with the market interet rate. In thi paper, we aume the interet rate contant 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 (7). The two other tudie cloely related to our are DeMarzo and Fihman (4) and it continuoutime formulation by DeMarzo and Sannikov (6). 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. Biai et al. (6) tudy the optimal contract in a tationary verion of DeMarzo and Fihman (4) model and how that it continuou time limit exactly matche DeMarzo and Sannikov (6) continuou-time characterization of the optimal contract. Tchityi (6) conider a etting with correlated cah ow and how that the optimal contract can be implemented uing a credit line with performance pricing. Sannikov (6b) how that an advere election problem, due to the borrower private knowledge concerning the quality of a project to be nanced, implie that, in the implementation of the optimal contract, a credit line ha a growing credit limit. He (7) tudie the optimal executive compenation in the continuou-time agency model where the manager privately control the drift of the geometric Brownian motion rm ize. Clementi and Hopenhayn (6) and DeMarzo and Fihman (6) o er theoretical analye of optimal invetment and ecurity deign in moral hazard environment. None of the above tudie conider an environment with tochatic aet price and the option to ale the aet. There i a izeable real etate nance literature that addree the deign of mortgage in the preence 5
of aymmetric information between the borrower and lender. The bulk of thi literature focue on advere election and how it a ect the menu of mortgage being o ered to borrower with limited inurance poibilitie. Chari and Jagannathan (1989) conider a model with two private type of borrower, who di er in term of the rikine of their potential gain from elling the property, and how that the optimal contract to be choen by borrower with larger potential gain involve contractual arrangement uch a point 4 and prepayment penaltie together with a "due-on-ale" claue. Brueckner (1994) develop a model in which borrower elf-elect into di erent loan, and how that the optimal menu of mortgage will induce longer term borrower to elect loan with higher point and a lower coupon. Unlike thee two paper, LeRoy (1996) conider a tochatic interet rate environment and nd that, when borrower re nance optimally, if interet rate fall, the point/coupon choice can at bet erve only to eparate the leat mobile borrower type from all other. Stanton and Wallace (1998) how that in the preence of tranaction cot payable by borrower on re nancing, it i poible to contruct a eparating equilibrium in which borrower with di ering mobility elect xed rate mortgage with di erent combination of coupon rate and point. Poey and Yava (1) tudy how borrower with di erent private level of default rik would elf-elect between xed rate mortgage and adjutable rate mortgage, and how the unique equilibrium may be a eparating equilibrium in which the high-rik borrower chooe the adjutable rate mortgage, while low-rik borrower elect the xed rate mortgage. Unlike thee paper that focu on advere election, Dunn and Spatt (1985) conider a two-period moral hazard model, where future income realization of borrower are uncertain and private, and how that the optimal mortgage would involve a due-on-ale claue. In term of thi literature, to our knowledge, our paper i the rt tudy of optimal mortgage deign in a dynamic moral hazard environment, and the rt tudy that addree the optimality of alternative mortgage product. There i alo a izeable real etate nance literature that tudie the optimal trategy of the mortgage borrower in the tochatic houe price environment (ee for example Kau, Keenan, Mueller, and Epperon (199) and Deng, Quigley, and Van Order ()). Thee literature however retrict it attention to peci c cla of contract and thu, unlike thi paper, do not addre the quetion of optimal lending. A large growing literature focue on the choice of mortgage contract a a party of houehold rik management (for example, Campbell and Cocco ()). Unlike our paper, thi literature take a pace of contract a exogenouly given, and tudie the houehold choice within thi retricted et of contract. Another branch of reearch invetigate limited participation model, where houing collateral inulate houehold from labor income hock. Lutig and Van Nieuwerburgh (6) typi e thi approach. The paper i organized a follow. Section preent the continuou-time model with tochatic houe appreciation. Section decribe the dynamic contracting problem. Section 4 derive the optimal contract 4 Point repreent the amount paid either to maintain or lower the interet rate charged. 6
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. The Model Time i continuou and in nite. There i one borrower (a homebuyer) and one lender (a large nancial intitution). 5 The lender i rik neutral, ha unlimited capital, and value a tochatic cumulative cah ow ff t g a where r i the dicount rate. E 4 1 e r df t 5 ; The borrower conumption conit of three categorie. The rt i "neceary" conumption, which include grocery food, medicine, tranportation, helter and other good and ervice eential for the houehold urvival. The econd i houing conumption, which come from owning or renting a houe. Everything ele i "luxury" conumption, which, among many other thing, may include uch item a retaurant dining, vacation trip, buying a new car, et cetera. The 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. We aume that the borrower i in nitely rik avere with repect to the neceary conumption and rik neutral with repect to the luxury and houing conumption. That i, the borrower intantaneou utility function i given by u dc t ; dc t ; dc H t = 8 < : 1; if dc t < d t dc t + dc H t ; if dc t d t ; where Ct, fc t g and Ct H denote cumulative ow of the neceary, luxury and houing conumption. 6 We aume that houing and luxury conumption are perfect ubtitute. In other word, houing i a part of luxury conumption. Thi aumption can be juti ed by the fact that mot US houehold tend to buy houe by far exceeding their minimum houing need. The primary di erence between luxury conumption and houing i that luxury conumption can be adjuted intantaneouly, while houing conumption i very 5 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. 6 Thi peci cation i imilar in avor to the one ued by Ait-Sahalia, Parker and Yogo (4), 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
rigid. We aume that houing conumption remain contant a long a the borrower tay in the ame houe. 7 The borrower mut ue hi income to rt cover the neceary expene t before pending on luxury conumption. Let Y t denote the borrower total 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. From now on, we will refer to Y t and C t imply a the borrower income and the borrower conumption. The borrower value tochatic cumulative conumption ow fc t g and Ct H a E 4 1 e rt dc t + dc H t 5 : A tandard Brownian motion = f t ; F 1;t ; t < 1g on ( 1 ; F 1 ; m 1 ) drive the borrower income proce, where ff 1;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 + d t ; (1) where i the drift of the borrower dipoable income and i the enitivity of the borrower income to it Brownian motion component. We aume that the lender know and, but doe not know the realization of the borrower exce income hock t, o 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 8 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 dch t dt Y, i.e., Y < P. 9 =. The price P of the home i greater than the borrower initial wealth Thu, the borrower mut obtain fund from the lender to nance the houe purchae. 7 For implicity, we do not conider a poibility that the borrower can make modi cation that can either increae or decreae the quality of the houe. 8 Such a credit core, demographic variable and o on. 9 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 analyi applie to any (; P ) pair. 8
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. 1 The houing market i expected to go through two phae. The initial phae, houing boom, i characterized by fat 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 = ; P [N t+ = for all [t; t + ) jn t = ] = e ; P [N t+ = 1 for all [t; t + ) jn t = ] = 1 e ; P [N t+ = for all [t; t + ) jn t = 1] = : where 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 ;t ; t < 1g i a tandard compound Poion proce with an intenity (N t ) on a probability pace ( ; F ; m ), 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 h(1 ) for all t h where [; 1] meaure the extent of houe price depreciation. 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. 1 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. 9
Default Option If the borrower violate the term of the contract or default at time t, he loe the home and receive hi reervation value equal to A, which for implicity we aume to be equal to the expected preent value of the borrower future income, r. We aume and o A r : The lender ell the repoeed houe at a forecloure auction and receive the payo L t = (1 l)p t, where l (; 1) meaure the liquidation cot. Option to Sale In each period t, the borrower can put the home on the market. If he doe o, the probability of nding a buyer at time t equal (N t ). For implicity, we aume that the home can be old immediately during the boom () = 1, while it i impoible to nd a buyer during the lump (1) =. The houe ale generate P t. The elling i more e cient than liquidation (a l < 1). De nition 1 Let v t be the value of full homeownerhip at time t (e.g., the borrower continuation utility at time t provided the borrower ha no debt). If the houing market i in the boom phae we et v t = v t if the houing market i in the lump phae v t = v 1 = + r. and We note that a the houing lump i an aborbing tate, the continuation utility of the borrower who ha no debt in the lump phae doe not depend on t. The ale of the home i not contractible. The borrower put the home on the market at the time when it maximize hi expected payo. We aume that the borrower ha to pay the outtanding balance B t of the loan to the lender after he ell the houe. A we will verify in Section 5, the outtanding balance B t i related to the borrower continuation utility a t under the optimal contract a follow: B t = v t a t ; where we remember that v t i the ( rt-bet) value of full homeownerhip at time t. In the boom phae of the houing market we have v t = vt. Thu, after the houe ale the borrower continuation payo i given by A S t (a t ) = A + P t (vt a t ); while the lender receive 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 ; 1
which i equivalent to A + P t v t : () The optimal elling time determined by equation () 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 () 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 i given by t = 1 g log @ r 1 g r+ P 1 A ; () 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 ale at t Proof In the Appendix. Full Homeownerhip If the borrower become a full homeowner at time t (the borrower repay hi debt) the contract i terminated and the borrower receive the value of full homeownerhip equal to v t. 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. An allocation, ( 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. Let (; F; m) := ( 1 ; F 1 F ; m 1 m ) be the product pace of ( 1 ; F 1 ; m 1 ) and ( ; F ; m ). Let ^Y n = ^Yt : t o be the borrower report of hi income, where ^Y i (Y; P )-meaurable (F t meaurable). The allocation tranfer the reported amount, ^Y t ; from the borrower to the lender, and I t ( ^Y ; P ) from the lender to the borrower. Below we formally de ne an allocation. 11
De nition An allocation = ( f ; d ; I) peci e a time at which the borrower become a full homeowner f, a default time d ; and tranfer from the lender to the borrower I = fi t : t g ; that are baed on ^Y and P (N). Formally, f and d are ( ^Y ; P )-meaurable topping time, and I i a ( ^Y ; P )-meaurable continuou-time proce, uch that the proce 6 E 4 min( f ; d ) e r 7 di jf t 5 i quare-integrable for t min( f ; d ) and ^Y = Y: De nition Let = min( ; f ; d ), be the expected time of termination of the relationhip due to ale (which happen at t provided that houe boom lat until ), full homeownerhip, or default, repectively, implied by the allocation ( f ; d ; I). The borrower can mireport hi income. Conequently, under the allocation ; 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 ; (4) where dc t i the borrower conumption at time t; which mut be non-negative. We recall that r: In repone to an allocation = ( f ; d ; I); the borrower chooe a feaible trategy that conit of hi conumption choice, the report of hi income, and the elling time in order to maximize hi expected utility. Below we formally de ne the feaible trategy of the borrower. De nition 4 Given an allocation = ( 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 ); (iii) the aving proce de ned by (4) tay non-negative. We haven t included explicitly the borrower elling deciion in the de nition of hi trategy. A we dicued in Section the optimal elling time i determined by equation (). Equation () imply tate 1
that it i optimal for the borrower to ell the home whenever the value of hi outide option A plu the proceed from the ale exceed hi continuation utility under full homeownerhip (which happen at t provided that houe boom lat until ). The borrower trategy i incentive compatible if it maximize hi lifetime expected utility in the cla of all feaible trategie given an allocation = ( f ; d ; I). A a reult, we have the following de nition. De nition 5 Given an allocation = ( f ; d ; I), the borrower trategy (C; ^Y ) i incentive compatible if (i) given an allocation ; the borrower trategy (C; ^Y ) i feaible, (ii) given an allocation ; the borrower trategy (C; ^Y ) provide him with the highet expected utility among all feaible trategie, that i E 4 E 4 e rt (dc t + dt) + e r (1 = A S + 1 = f v + 1 = da) jf 5 e rt (dct + dt) + e r (1 = A S + 1 = f v + 1 = da) jf 5 for all the borrower feaible trategie (C ; ^Y ); given an allocation : The above de nition doe not explicitly include the participation contraint impoing the condition that the borrower utility from the continuation of the allocation hould be at leat a large a the borrower outide option, A; which he can receive at any time by quitting. A the borrower can alway under-report and teal at rate ra until a termination time, any incentive compatible trategy would yield the borrower utility of at leat A. The above de nition of an incentive compatible trategy allow u to de ne the incentive compatible allocation a follow. De nition 6 An incentive compatible allocation i an allocation = ( f ; d ; I), together with the recommendation to the borrower, (C; ^Y ); where (C; ^Y ) i a borrower incentive compatible trategy given an allocation. The allocation i optimal if it provide the borrower with hi initial expected utility a and maximize the expected pro t of the lender in the cla of all allocation that are incentive compatible. Below we provide a formal de nition of the optimal allocation. 1
De nition 7 Given the continuation utility to the borrower, a, an allocation = ( f ; d ; I ), together with a recommendation to the borrower (C ; ^Y ) i optimal if it maximize the lender expected utility: E 4 e rt (d ^Y t di t ) + e r (1 = L S + 1 = dl ) jf 5 in the cla of all incentive-compatible allocation that atify the following promie keeping contraint: a = E 4 e rt (dc t + dt) + e r (1 = A S + 1 = f v + 1 = da) jf 5 : We note that maximizing the lender expected utility i equivalent to maximizing the lender pro t, which equal the lender expected utility le the loan amount to the borrower [P Y ], which we take a given. In the following lemma, we how that earching for optimal allocation, we can retrict our attention to allocation in which truth-telling and zero aving are incentive compatible. Lemma 1 There exit an optimal allocation 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 allocation can be improved by having the lender ave and make direct tranfer to the borrower. Therefore, we can look for an optimal allocation in which truth-telling and zero aving are incentive compatible. 4 Derivation of the Optimal Allocation In thi ubection, we formulate recurively the dynamic moral hazard problem and determine the optimal allocation. Firt, we conider a problem in which the borrower i not allowed to ave. We determine the optimal allocation 11 in thi environment, achieving thi in two tep. Firt, we preent and explain the optimal allocation after the houe price lump occurred. Next, given the pot-lump value function, we derive the optimal allocation in the boom environment. We know from Lemma 1 that it i u cient to look for optimal allocation in which the borrower report truthfully and maintain zero aving, and o the optimal allocation of the problem with no private aving, 11 Thi i the allocation atifying the propertie of De nition 7 and the additional contraint that S =. 14
for a given continuation utility to the borrower, yield to the lender at leat a much utility a the optimal allocation of the problem when the borrower i allowed to privately ave. We will conclude by howing that the optimal allocation 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 (6) and extended to a etting with Lévy procee by Pikorki and Tchityi (7). 4.1 The Optimal Allocation without Hidden Saving Conider for a moment the dynamic moral hazard problem in which the borrower i not allowed to ave. Firt, we will nd a convenient tate pace for the recurive repreentation of thi problem. For thi purpoe, we de ne the borrower total expected utility received under the allocation = ( f ; d ; I) conditional on hi information at time t, from tranfer and termination utility, if he tell the truth and follow the optimal elling rule: V t = E 4 e r [di + d] + e r (1 = A S + 1 = f v + 1 = da) jf t 5 : Lemma The proce V = fv t ; F t ; t < g i a quare-integrable F t -martingale. Proof follow directly from the de nition of proce V and the fact that thi proce i quare-integrable, which i implied by De nition. Below i a convenient repreentation of the borrower total expected utility received under the allocation = ( f ; d ; I) conditional on hi information at time t, from tranfer and termination utility, if he tell the truth. Let M = fm t = N t t(n t ); F 1;t ; t < 1g be a compenated compound Poion proce. Propoition There exit F t -predictable procee (; ) = f( t ; t ); t g uch that V t = V + V + t t t e r d + e r dm = e r dy d t + e r {z } (dn (N )d): (5) d Proof We note that the couple (; N) i a Brownian-Poion proce, and it i an independent increment proce, which i a Lévy procee, on the pace (; F; m): Then, Theorem III.4.4 in Jacod and Shiryaev () give u the above martingale repreentation for a quare-integrable martingale adapted to F t taking value in a nite dimenional pace (the proce V ). According to the martingale repreentation (5), the total expected utility of the borrower under the allocation, truth telling, and optimal option execution time conditional on hi information at time t equal 15
it unconditional expectation plu two term that repreent the accumulated e ect on the total utility of, repectively, the income uncertainty revealed up to time t (Brownian motion part), and the houe price uncertainty that ha been revealed up to time t (compenated compound Poion part). According to Propoition, when the borrower report truthfully, hi total expected utility under the allocation conditional on the termination time equal V = V + e r dy d + e r dm : A I and ( f ; d ) depend excluively on the borrower report ^Y and the public houe price proce P, when the borrower report ^Y ; by (5) he get the expected utility, a ( ^Y ), which equal a ( ^Y ) = E 6 4 V + E V + e rt dyt t e rt d ^Y! t dt t + dt + e 1 rt e rt tdm t + t dy t e rt (dy t d ^Y t ) jf 7 5 = {z } utility from tealing e rt d ^Y t + tdm t jf : (6) Note that becaue the proce (; ) = f( t ; t ); t g i F t predictable; a for any t, ; E [ t+ t jf ] = E [M t+ M t jf ] = ; and given that E [V jf ] = V ; we have that a ( ^Y ) = V + E e 1 rt t dy t Repreentation (7) lead u to the following formulation of incentive compatibility. d ^Y t jf : (7) Propoition If the borrower cannot ave, truth-telling i incentive compatible if and only if t (m a::) for all t : Proof Immediately follow from (7). It i important to tre that in providing incentive for truth-telling one can neglect an impact of reporting trategie on the magnitude of the adjutment, ; in the borrower continuation utility that occur when the houe price boom end. It follow from (6) that, though in principle the reporting trategy of the borrower doe a ect the magnitude of thee adjutment, from the perpective of the borrower uch adjutment have zero e ect on the borrower expected utility, whatever hi reporting trategy. Thi property coniderably impli e the formulation of incentive compatibility. 16
To characterize the optimal allocation recurively, we de ne the borrower continuation utility at time t if he tell the truth a a t = E e r( t) [di + d] + e r( t) [1 = A S + 1 = f v + 1 = da] jf t : t Note that for t we have that V t = t e r (di + dt) + e rt a t : But thi, together with (5), implie the following law of motion of the borrower continuation utility: da t = ra t dt dt di t + t d t + t dm t = (ra t t (N t )) dt di t + t d t + t dn t : (8) Below we dicu informally, uing the dynamic programming approach, how to nd out the mot e cient way to deliver to a borrower any continuation utility a A. Propoition 4 and Propoition 5 will formalize our dicuion below. The Optimal Allocation in the Houe Slump Phae Let b 1 (a; h ) be the highet expected utility of the lender that can be obtained from an incentive compatible allocation that provide the borrower with utility equal to a given that the houe price lump tarted at h. To implify our dicuion we aume that the function b 1 i concave and C in it rt argument. Propoition 4 will etablih thee argument formally. Let b 1 and b 1 denote, repectively, the rt and econd derivative of b 1 with repect to the borrower continuation utility a. We tart by oberving that tranferring lump-um di from the lender to the borrower with continuation utility a; move an allocation to that of the borrower continuation utility of a that di: The e ciency implie b 1 (a; h ) b 1 (a di; h ) di; (9) which how that for all a [A; 1) the marginal cot of delivering the borrower hi continuation utility can never exceed the cot of an immediate tranfer in term of the lender utility, that i b 1(a; h ) 1: De ne a 1 h a the lowet value of a uch that b 1(a; h ) = 1: Lemma For any h [; t ], we have that a 1 h = v 1 = + r. 17
Proof Since the borrower and lender have the ame dicount factor there i no lo of e ciency in delaying the tranfer to the borrower. However paying early to the borrower i cotly a long it a ect the likelihood of cotly liquidation. A long a a t < v 1, the borrower cannot be declared a full homeowner a thi would be inconitent with the borrower continuation utility. But thi implie that a long a a t < v 1, due to incentive compatibility contraint, there i a poitive chance of liquidation. Therefore no tranfer would be optimal to the borrower in thi region a they would lower the borrower continuation utility and thu increae the likelihood of liquidation. On the other hand, when a t v 1 the borrower can be declared a full homeowner (with accompanied tranfer from the lender of a t v 1 ). A there are no gain for the lender (and for the borrower) from delaying full homeownerhip, we conclude that a 1 = v 1 : h Then, conditional on a houe price boom ending at h ; we have that f = infft h : a t = v 1 g: Full ownerhip and the option to terminate keep the borrower continuation utility between A and v 1 : But thi and (8) imply that when a [A; v 1 ]; and when the borrower i telling the truth, hi continuation utility evolve according to da t = (ra t ) dt + t d t ; (1) where we ue the fact that dn t = and (N t ) = for all t h. We next characterize the optimal choice of proce ( t ); where t borrower continuation utility with repect to hi report. Uing Ito lemma, we nd that determine the enitivity of the db 1 (a t ; h ) = (ra t )b 1(a t ; h )dt + 1 t b 1(a t ; h )dt + t b 1(a t ; h )d t : Uing the above equation, we nd that the lender expected cah ow and the change in the value he aign to the allocation are given a follow: E dy t + db 1 (a t ; h ) jf t = + (a t )b 1(a t ; h ) + 1 t b 1(a t ; h ) dt: From Propoition, we know that if t for all t then the borrower bet repone trategy i to report the truth, that i, ^Y = Y: Becaue at the optimum, at any time t; the lender hould earn an intantaneou total return equal to the interet rate, r, we have the following Bellman equation for the value function of the lender after the houe price boom end at h : rb 1 (a t ; h ) = max + (ra t )b 1(a t ; h ) + 1 t t b 1(a t ; h ) : Given the concavity of the function b(; h ), etting t = 18
for all h t i optimal. The lender value function therefore ati e the following di erential equation rb 1 (a t ; h ) = + (ra t )b 1(a t ; h ) + 1 b 1(a t ; h ): (11) We need ome boundary condition to pin down a olution to thi equation. The rt boundary condition arie becaue the relationhip mut be terminated to hold the borrower value 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 ) = : Finally we have that b 1(v 1 ; h ) = 1. The propoition below formalize our nding. Propoition 4 Let b 1 (:; h ) be a function (in a) that olve: rb 1 (a; h ) = + (ra )b 1(a t ; h ) + 1 b 1(a; h ); (1) when a i in the interval [A; v 1 ]; with boundary condition b 1 (A; h ) = L h, b 1 (v 1 ; h ) =, and b 1(v 1 ; h ) = 1: Then the optimal allocation that deliver to the borrower the value a h given that the houing lump occurred at h ; take the following form. If a h [A; v 1 ]; a t evolve for t > h a da t = (ra t dt dt) + (d ^Y t dt): (1) The default occur at rt time d when a t hit A. The borrower become a homeowner at rt time f when a t hit v 1 : The lender expected utility at any time t h i given by the function b 1 (a t ; h ) de ned above, which i trictly concave in a t over [A; v 1 ]. Proof Directly follow from DeMarzo and Sannikov (6) and Bia et al. (7) a the tructure of dynamic moral hazard problem after the houe lump correpond to the one tudied in thee paper. The evolution of the continuation utility (1) implied by the optimal allocation erve three objective: promie-keeping, incentive, and e ciency. The rt component of (1) 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. The econd term of (1) 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 19
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 d t. The Optimal Allocation in the Houe Price Boom Let b (a; t) be the highet expected utility of the lender that can be obtained from an incentive compatible allocation that provide the borrower with utility equal to a given that we have the houe price boom at t (t < h ). To implify our dicuion we aume that the function b i concave and C in it rt argument. Propoition 4 will etablih thee argument formally. Let b and b denote, repectively, the rt and the econd derivative of b with repect to the borrower continuation utility a. We tart by oberving that tranferring lump-um di from the lender to the borrower with continuation utility a t move an allocation to that of the borrower continuation utility of a t that di: The e ciency implie b (a; t) b (a di; t) di; (14) which how that for all a [A; 1) the marginal cot of delivering the borrower hi continuation utility can never exceed the cot of an immediate tranfer in term of the lender utility, that i b (a; t) 1: De ne a t a the lowet value of a uch that b (a; t) = 1: Lemma 4 For any t [; min( h ; t )), we have that a t = vt. Proof Since the borrower and lender have the ame dicount factor there i no lo of e ciency in delaying the tranfer to the borrower. However paying early to the borrower i cotly a long it a ect the likelihood of cotly liquidation. A long a a t < vt, the borrower cannot be declared a full homeowner a thi would be inconitent with the borrower continuation utility. But thi implie that a long a a t < vt, due to incentive compatibility contraint, there i a poitive chance of liquidation. Therefore no tranfer would be optimal to the borrower in thi region a they would lower the borrower continuation utility and thu increae the likelihood of liquidation. On the other hand, when a t vt the borrower can be declared a full homeowner (with accompanied tranfer from the lender of a t vt ). A there are no gain for the lender (and for the borrower) from delaying full homeownerhip, we conclude that a t = vt :
Then, we have that f = infft : a t = v t g: The full homeownerhip and the option to terminate keep the borrower continuation utility between A and v t : But thi and (8) imply that when a [A; v t ]; and when the borrower i telling the truth, hi continuation utility for t < h evolve according to da t = (ra t t ) dt + t d t + t dn t ; (15) where we ue the fact that (N t ) = for t < h : We need to characterize the optimal choice of proce ( t ; t ); where t determine the enitivity of the borrower continuation utility with repect to hi report, and t determine the adjutment of the borrower continuation utility when the houe price boom end. Uing Ito lemma, we nd that db (a t ; t) = @b (a t ; t) dt + (ra t @t t )b (a t ; t)dt + 1 t b (a t ; t)dt + t b (a t ; t)d t + [b 1 (a t + t ; t) b (a t ; t)] dn t : Uing the above equation, we nd that the lender expected cah ow and the change in the value he aign to the allocation are given a follow: E [dy t + db (a t ; t) jf t ] = @b (a t ; t) + + (ra t @t t )b (a t ; t) + 1 t b (a t ; t) + (b 1 (a t + t ; t) b (a t ; t)) dt: From Propoition, we know that if t for all t then the borrower bet repone trategy i to report the truth, that i, ^Y = Y: Becaue at the optimum, at any time t; the lender hould earn an intantaneou total return equal to the interet rate, r, we have the following Bellman equation for the value function of the lender max t, t A at rb (a t ; t) = @b (a t ; t) + + (ra t @t t )b (a t ; t) + 1 t b (a t ; t) + (b 1 (a t + t ; t) b (a t ; t)) ; (16) where t [; t ]: Given the concavity of the function etting t = for all t i optimal. The concavity of the objective function in t in the RHS of the Bellman equation 1
(16) alo implie that the optimal choice of t i given a a olution to b (a t ; t) = b 1(a t + t ; t); (17) provided that t > A a t ; and otherwie t = A a t. Note that the optimal jump can be expreed a (a t ; t): The lender value function in the houe price boom therefore ati e the following di erential equation for a t [A; vt ] : rb (a t ; t) = @b (a t ; t) ++(ra t (a t ; t))b @t (a t ; t)+ 1 b (a t ; t)+ (b 1 (a t + (a t ; t); t) b (a t ; t)) (18) with peci ed above. We need ome boundary condition to pin down a olution to thi equation. The rt boundary condition arie becaue the relationhip mut be terminated to hold the borrower value to A, o b (A; t) = L t. The econd boundary condition come from the fact that the lender hould expect no tranfer from the borrower once the borrower become a homeowner, that i b (v t ; t) = : Moreover b (v t ; t) = 1. The nal boundary re ect the payment to the lender at the elling time, that i b (a t ; t ) = v t a t : The propoition below formalize our nding. Propoition 5 Let b be a C function (in a) that olve: rb (a t ; t) = @b (a t ; t) ++(ra t (a t ; t))b @t (a t ; t)+ 1 b (a t ; t)+ (b 1 (a t + (a t ; t); t) b (a t ; t)) (19) when a i in the interval [A; v t ] with boundary condition b (A; t) = L t ; b (v t ; t) = 1; b (a t ; t ) = v t a t ; where 8 i a C 1 (in a) olution to b (a; t) = b 1(a + ; t) for all (a; t) >< (a; r) = for which the olution i uch that (a; t) > A a : () >: otherwie it i equal to A a Then the optimal allocation that deliver to the borrower the value a take the following form. If a