The Dynamics of Adjustable-Rate Subprime Mortgage Default: A Structural Estimation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Dynamics of Adjustable-Rate Subprime Mortgage Default: A Structural Estimation Hanming Fang, You Suk Kim, and Wenli Li 2015-114 Please cite this paper as: Fang, Hanming, You Suk Kim, and Wenli Li (2015). “The Dynamics of Adjustable-Rate Subprime Mortgage Default: A Structural Estimation,” Finance and Economics Discussion Series 2015-114. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.114. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Dynamics of Adjustable-Rate Subprime Mortgage Default: A Structural Estimation∗ Hanming Fang† You Suk Kim‡ Wenli Li§ December 9, 2015 Abstract We present a dynamic structural model of subprime adjustable-rate mortgage (ARM) borrowers making payment decisions taking into account possible consequences of different degrees of delinquency from their lenders. We empirically implement the model using unique data sets that contain information on borrowers’ mortgage payment history, their broad balance sheets, and lender responses. Our investigation of the factors that drive borrowers’ decisions reveals that subprime ARMs are not all alike. For loans originated in 2004 and 2005, the interest rate resets associated with ARMs, as well as the housing and labor market conditions were not as important in borrowers’ delinquency decisions as in their decisions to pay off their loans. For loansoriginatedin2006,interestrateresets,housingpricedeclines,andworseninglabormarket conditions all contributed importantly to their high delinquency rates. Counterfactual policy simulations reveal that even if the Libor rate could be lowered to zero by aggressive traditional monetary policies, it would have a limited effect on reducing the delinquency rates. We find that automatic modification mortgage designs under which the monthly payment or the principal balance of the loans are automatically reduced when housing prices decline can be effective inreducingbothdelinquencyandforeclosure. Importantly,wefindthatautomaticmodification mortgageswithacushion, underwhichthemonthlypaymentorprincipalbalancereductionsare triggered only when housing price declines exceed a certain percentage may result in a Pareto improvement in that borrowers and lenders are both made better off than under the baseline, with a lower delinquency and foreclosure rates. Our counterfactual analysis also suggests that limited commitment power on the part of the lenders to loan modification policies may be an importantreasonfortherelativelysmallrateofmodificationsobservedduringthehousingcrisis. Keywords: Adjustable-Rate Mortgage, Default, Loan Modification, Automatic Modification with a Cushion JEL Classification Codes: D12, D14; G2, G21, G33 ∗We thank Shane Sherlund and seminar/conference participants at the Econometric Society World Congress (2015),UniversityofNewSouthWalesandUniversityofTechnologySydneyfortheircomments. Theviewsexpressed are those of the authors and do not necessarily reflect those of the Board of Governors of the Federal Reserve, the Federal Reserve Bank of Philadelphia, or the Federal Reserve System. †DepartmentofEconomics,UniversityofPennsylvania,3718LocustWalk,Philadelphia,PA19104andtheNBER. Email: hanming.fang@econ.upenn.edu. ‡DivisionofResearchandStatistics,BoardofGovernorsoftheFederalReserveSystem. Email: You.Kim@frb.gov. §Department of Research, Federal Reserve Bank of Philadelphia. Email: wenli.li@phil.frb.org.

1 Introduction The collapse of the subprime residential mortgage market played a crucial role in the recent housing crisis that subsequently led to the Great Recession.1 At the end of 2007, subprime mortgages accounted for about 13 percent of all outstanding first-lien residential mortgages but over half of the foreclosures. The majority of the subprime mortgages, both by number and by value, were adjustable interest rates mortgages (ARMs); and these mortgages had a foreclosure rate of 17 percent, much higher than the 5 percent foreclosure rate for the fixed-rate subprime mortgages (Frame, Lehnert, and Prescott 2008, Table 1). In response to these developments, many governmentpoliciesweredesignedandimplementedtochangethedefaultincentivesofthesubprimeARM borrowers.2 Few structural models, however, exist that can guide us in these efforts, and that can help us understand why most of the programs had limited success.3 In this paper, we develop a dynamic structural model to study the incentives of the adjustablerate subprime borrowers to default, and investigate how these incentives change under various policies. In our model, at each period, a borrower decides whether to pay the amount due (and be current) or not pay (and stay in various delinquent status), taking into account the lender’s responses such as mortgage modification, liquidation, or waiting (i.e., doing nothing). Relative to the existing structural models on mortgage defaults which we review below, our model has two key distinguishing features: first, in our model default is not the terminal and absorbing state as we allow borrowers to self cure their delinquency; second, we consider loan modification as one of the lenders’ loss mitigation practices while the existing models only allow for liquidation. We empirically implement our model using unique mortgage loan level data sets that contain not only detailed information on borrowers’ mortgage payment history and lenders’ responses, but also credit bureau information about borrowers’ broader balance sheet. We are thus one of the first to utilize borrowers’ credit bureau information to understand their mortgage payment decisions.4 To track movements in the local housing and labor markets, we further merge our data with zip code level home price indices and county level unemployment rates. 1There is no standard definition of subprime mortgage loans. Typically, they refer to loans made to borrowers with poor credit history (e.g., a FICO score below 620) and/or with ahigh leverage as measuredby either the debtto-income ratio or the loan-to-value ratio. For the data used in this paper, subprime mortgages are defined as those in private-label mortgage-backed securities marketed as subprime, as in Mayer, Pence, and Sherlund (2009). 2To name a few of such programs, the FHASecure program approved by Congress in September 2007; the Hope Now Alliance program (HOPENOW) created by then-Treasury Secretary Henry Paulson in October 2007; Hope for HomeownersrefinancingprogrampassedbyCongressinthespringof2008;MakingHomeAffordable(MHA)initiative inconjunctionwiththeHomeAffordableModificationProgram(HAMP)andtheHomeAffordableRefinanceProgram (HARP)launchedbytheObamaadministrationinMarch2009(HAMP).SeeGerardiandLi(2010)formoredetails. 3Over the first two and a half years, HARP refinancing activity remained subdued relative to model-based extrapolations from historical experience. From its inception to the end of 2011, 1.1 million mortgages refinanced through HARP, compared to the initial announced goal of three to four million mortgages. In December, HARP 2.0 was introduced and HARP refinance volume picked up, reaching 3.2 million by June 2014. http://www.fhfa.gov/AboutUS/Reports/Pages/Refinance-Report-February-2014.aspx. Similarly, HAMP was designedtohelpasmanyas4millionborrowersavoidforeclosurebytheendof2012. ByFebruary2010,oneyearinto the program, only 168,708 trial plans had been converted into permanent revisions. Through January 2012, a populationof621,000loanshadreceivedHAMPmodifications. Seehttp://www.treasury.gov/resource-center/economicpolicy/Documents/HAMPPrincipalReductionResearchLong070912FINAL.pdf 4Elul,Souleles,Chomsisengphet,Glennon,andHunt(2010)alsousecreditbureauinformationtostudymortgage default decisions in their empirical analysis. 1

Three main factors drive ARM borrowers’ mortgage payment decisions: home equity, income, and monthly mortgage payment; importantly, both the current levels of these factors and the expectations of their future changes matter. Borrowers with negative home equity have little financial gains from continuing with their mortgage payments, especially when they do not expect house prices to recover and when costs associated with defaults and foreclosures are low. Changes in incomes and expenses, including changes in monthly mortgage payments due to interest rate resets for example, affect borrowers’ liquidity position. In principle, borrowers can refinance their mortgages to lower interest rates or sell their houses to improve their liquidity positions, but these options may not be available in the presence of declining house prices, increasing unemployment rates, and/or tightened lending standards. These constrained borrowers thus may find it optimal to default on their mortgages despite the possible consequences of foreclosure. To quantify the relative importance of these different drivers of default, we simulate our structurally estimated model under various counterfactual scenarios. Our simulation results suggest that the factors that drive the borrower delinquency and foreclosure differ substantially by loan origination year. For loans originated in 2004 and 2005, which preceded the severe downturn of the housingandlabormarkets,theinterestrateresetsassociatedwithARMsaswellasthehousingand labormarketconditionsdonotseemtobeasimportantfactorsforborrowers’delinquencybehavior as they are in determining whether the borrowers would pay off their loans (i.e., sell their houses or refinance). However, for loans that originated in 2006, interest rate reset, housing price declines, and worsening labor market conditions all contributed to their high delinquency rates with housing price declines being the most significant contributing factor.5 These results arise because for loans originated in 2004 and 2005, interest rates did not reset until 2006 or 2007 at which time house prices have just begun to decline. More importantly, since house prices continued to appreciate in 2004, 2005, and part of 2006, these borrowers have accumulated some home equity by the time of their interest rates reset; in fact, in many places house price did not go all the way down to their 2004 levels until 2008. Additionally, the labor market did not deteriorate significantly until 2008 or 2009. In contrast, borrowers whose loans originated in 2006 had the perfect storm in 2008 or 2009 when their interest rates reset, as house prices had depreciated substantially and unemployment rates had risen sharply. Counterfactual policy simulations reveal that even if the Libor rate could be lowered to zero by aggressive traditional monetary policies, it would have a limited effect on reducing the delinquency rates. We find that automatic modification mortgage designs under which the monthly payment or the principal balance of the loans are automatically reduced when housing prices decline can be effective in reducing both delinquency and foreclosure. Importantly, we find that automatic modification mortgages with a cushion, under which the monthly payment or principal balance reductions are triggered only when housing price declines exceed a certain percentage may result in a Pareto improvement in that borrowers and lenders are both made better off than under the 5Our finding is consistent with those in the literature including Bhutta, Dokko, and Shan (2010), Foote, Gerardi, and Willen (2012), and Fuster and Willen (2015). Bhutta, Dokko, and Shan (2010) also find that 80 percent of the defaults in their sample (2006 loans originated in the crisis states) are the results of income shocks combined with negativehouseequity. Foote,Gerardi,andWillen(2012)findthatinterestrateresetraisedthedefaultratesof2006 loans. 2

baseline, with a lower delinquency and foreclosure rates. Our counterfactual analysis also suggests that limited commitment power on the part of the lenders to loan modification policies may be an important reason for the relatively small rate of modifications observed during the housing crisis. Thereareseveralstructuralmodelsonmortgagedefaultsandforeclosures. Bajari,Chu,Nekipelov, and Park (2013) is most related to our paper both in questions addressed and in the empirical methodology. However, there are several key differences. First, we incorporate mortgage modification as a possible lender response while they do not. Second, we allow for borrowers to self cure while they treat default as a terminal event that leads to liquidation with certainty.6 Third, we focus on adjustable-rate subprime mortgages which were much more prevalent than the fixed-rate subprime mortgages that they focus on. Fourth, the two papers differ in the way we examine the effect of counterfactual policies. There differences enable us to study the effects of exogenously changinglenders’actionsonaborrowers’behaviorandtoshedlightonwhylenderswerenotwilling to modify loans. More importantly, the effects of alternative policies such as automatic modification mortgages with a cushion can be studied in our framework because this involves changing borrowers’ expectation about the co-evolution of house prices, mortgage balances and payment sizes. Campbell and Cocco (2014) study a dynamic model of households’ mortgage decisions incorporatinglaborincome, houseprice, inflation,andinterestraterisktoquantifytheeffectsofadjustable versus fixed mortgage rates, mortgage loan-to-value ratio, and mortgage affordability measures on mortgage premia and default. Corbae and Quintin (2015) solve an equilibrium model to evaluate the extent to which low down payments and interest-only mortgages were responsible for the increase in foreclosures in the late 2000s. Garriga and Schlagenhauf (2009) study the effects of leverage on default using long-term mortgage contract. Hatchondo, Martinez, and Sanchez (2011) investigate the effect of broader recourse on default rates and welfare. Mitman (2012) considers the interaction of recourse and bankruptcy on mortgage defaults. Chatterjee and Eyigungor (2015) analyzethedefaultoflong-durationcollateralizeddebt. Noneofthesepapersmakeuseofmortgage loan level data as in our paper and in Bajari et al. (2013). There are also several recent empirical papers that use regression analysis to study lenders’ loss mitigation practices and the impact of government intervention policies on these practices. For example, Haughwout, Okah, andTracy(2010)estimatea competing riskmodelusingmodifications (excluding capitalization modifications) of subprime loans that were originated between December 2004 and March 2009. They find a substantial impact of payment reduction on mortgage re-default rates. Agarwal, Amromin, Ben-David, Chomsisengphet, and Evanoff (2015) analyze lenders’ loss mitigation practices including liquidation, repayment plans, loan modification, and refinance of mortgages that originated between October 2007 and May 2009 from OCC-OTS Mortgage Metrics data and find a much more modest effect of mortgage modification on defaults. In a subsequent paper, Agarwal, Amromin, Ben-David, Chomsisengphet, Piskorski, and Seru (2012) study the impact of the 2009 Home Modification Program on lenders’ incentives to renegotiate mortgages. Finally, our paper also adds to the growing literature on the recent subprime mortgage cri- 6Adelino, Gerardi and Willen (2013) show the importance of self-cure as a hinderance for loan modifications. 3

sis, including, among many others, Foote, Gerardi, and Willen (2008), Demyanyk and van Hemert (2011), Keys, Mukherjee, Seru, andVig(2010), andGerardi, Lehnert, Sherlund, andWillen(2008). Additionally, Piskorski, Seru, and Vig (2010) find that securitization reduced mortgage renegotiation and led to more foreclosures. In contrast, Adelino, Gerardi, and Willen (2013) show that it is information asymmetries rather than securitization that hindered mortgage renegotiation. The remainder of the paper is organized as follows. In Section 2 we describe the data sets we use in our empirical analysis and present the descriptive statistics. In Section 3 we present our model of borrowers’ behavior and their interactions with the lenders in a stochastic environment with shocks to housing prices, unemployment rates and Libor interest rates. In Section 4 we briefly discuss how we solve and estimate our model. In Section 5 we present our estimation results. In Section 6 we describe the goodness-of-fit between the predictions of our model under the estimated parameters and their data analogs. In Section 7 we present results from several counterfactual experiments. In Section 8 we conclude and discuss avenues for future research. 2 Data 2.1 Data Source Ourdataonmortgagesandtheirmodificationscomefromthreedifferentsources, theCoreLogic Private Label Securities data – ABS, the CoreLogic Loan Modification data, and the TransUnion Consumer Risk Indicators for Non-Agency RMBS data (also known as “TransUnion-CoreLogic Credit Match Data”). The CoreLogic ABS data consist of loans that were originated as subprime and Alt-A loans and represents about 90 percent of the market. The data include loan level attributesgenerallyrequiredofissuersofthesesecuritieswhentheyoriginatetheloansaswellastheir historicalperformance, whichareupdatedmonthly. Theattributesincludeborrowercharacteristics (credit scores, owner occupancy, documentation type, and loan purpose); collateral characteristics (mortgage loan-to-value ratio, property type, zip code); and loan characteristics (product type, loan balance, and loan status). The CoreLogic Loan Modification data contain information on modifications on loans in the CoreLogic ABS data. The data include detailed information about modification terms including whether the new loan is of fixed interest rate, the new interest rate, whether some principal is forgiven,whetherthemortgagetermischanged,etc. Themergeofthetwodatasetsisstraightforward as each loan is uniquely identified by the same loan ID in both data sets. TheTransUnionConsumerRiskIndicatorsforNon-AgencyRMBSdataprovideconsumercredit information from TransUnion for matched mortgage loans in CoreLogic’s private label securities databases. TransUnion employs a proprietary match algorithm to link loans from the CoreLogic databases to borrowers from TransUnion credit repository databases, allowing us to access many borrower level consumer risk indicator variables, including borrowers’ credit scores, income at origination, among many others. We then merge our data with CoreLogic monthly zip code level repeat-sales house price index and county level unemployment rates from the Bureau of Labor Statistics. Thus our constructed 4

data have several advantages over most of those used in the literature. First, the match with the mortgage modification data allows us to accurately identify lenders’ actions, and separate delinquent mortgages that are self-cured from delinquent mortgages that become current after lendermodification. Second,theTransUniondataenableustocaptureborrowers’otherliabilitiesas well as the payment history of these liabilities as summarized by credit scores, which are important for borrowers’ mortgage payment decisions. 2.2 Mortgage Loans: Summary Statistics We focus on subprime adjustable-rate mortgage loans originated in four major housing crisis states, Arizona, California, Florida, and Nevada, between 2004 and 2007.7 In particular, we take a 1.75 percent random sample of adjustable-rate mortgages with an initial fixed interest rate for a period of two or three years and a mortgage maturity of 30 years, which are for borrowers’ primary residence, are first lien, and are not guaranteed by government agencies such as Fannie Mae, Freddie Mac, the Federal Housing Administration (FHA), and Veterans Administration (VA). We follow these loans until February 2009 before the first coordinated large-scale government effort to modify mortgage loans – the“Making Home Affordable”program was unveiled. In total, we have 16,347 mortgages and 337,811 monthly observations. Of the 16,347 mortgages, 11 percent were originated in Arizona, 55 percent in California, 28 percent in Florida, and 6 percent in Nevada. Not surprisingly, the largest fraction of the loans were originated in 2005 (43 percent), followed by 2004 (37 percent), 2006 (17 percent), and then 2007 (2 percent). Table 1 provides summary statistics of the mortgage loans at origination and of the whole dynamicsampleperiod. Theaverageageoftheloanis16monthsinthesampleandthemedianis14 months. At origination, 81 percent of the sample are loans with two-year fixed-rates. Through the sampleperiod,however,76percentofthesampleareloansoriginatedwithtwo-yearinitialfixed-rate periodindicatingthatmoreofthoseloanshaveterminatedviapayoff/refinanceorforeclosure. Over 90 percent of the loans have prepayment penalty. About 40 percent of the mortgages at origination are interest-only mortgages and the fraction becomes slightly higher in the whole dynamic sample. About half of the mortgages have full documentation both at origination and through the sample period. While 43 percent of the mortgages are purchase loans at the origination, the ratio increases to48percent. Consistentwithbeingsubprime,mortgageborrowersinthesampleallhaverelatively low risk scores, averaging 445 at origination, and the scores deteriorate somewhat as the loans age.8 Additionally,boththeaverageandthemedianmortgageloan-to-valueratiosatoriginationareboth around 80 percent and they do not change much as the loans age. The annual household income estimated by TransUnion averages $72,000 at origination with a median of $67,000. Loan balances average $259,000 at origination with a median of $228,000. These numbers are not very different from their dynamic counterparts. The mortgage interest rates average 7.13 percent at origination with a median of 6.99 percent. Dynamically, both the mean and median mortgage interest rates are higher by 20 and 15 basis points, respectively, as many of these adjustable-rate mortgages reset 7The subprime mortgage market dried up after 2007. 8The risk scores are estimated by TransUnion. They range between 150 and 950 with a high score indicating low risk. 5

At Origination Dynamic Sample Variable Mean Median Std. Dev. Mean Median Std. Dev. Age of the loan (months) 0 0 0 16 14 11 Share of 2-year fixed period (%) 81 1 39 76 1 41 Prepayment penalty (%) 0.90 1 0.30 0.92 1 0.27 Interest-only mortgages (%) 40 0 49 44 0 50 Full document at origination (%) 52 1 50 52 1 50 Purchase loan (%) 43 0 50 48 0 50 Risk score 445 445 155 424 432 178 LTV ratio at origination (%) 79 80 11 81 78 21 Annual income ($1000) 72 67 26 Principal balance ($1000) 259 228 141 260 228 141 Current interest rate (%) 7.13 6.99 1.15 7.35 7.13 1.39 Remaining mortgage terms (months) 360 360 0 345 347 11 Monthly payment ($1000) 1.616 1.429 0.859 1.679 1.475 0.902 Maximum lifetime interest rate (%) 13.50 13.45 1.28 13.42 13.38 1.27 Minimum lifetime interest rate (%) 6.70 6.89 1.86 6.59 6.85 1.90 Periodic interest rate cap (%) 1.20 1.00 0.33 1.20 1.00 0.32 Periodic interest rate floor (%) 0.01 0 0.13 0.01 0 0.13 First interest rate cap (%) 2.50 3.00 0.87 2.53 3 0.91 Margin for adjustable rate loans (%) 5.74 5.95 1.17 5.67 5.90 1.21 30 days delinquent (%) 0 0 0 6.86 0.0 25.37 60 days delinquent (%) 0 0 0 3.10 0.0 17.33 90 days delinquent (%) 0 0 0 1.62 0.0 12.63 120 days delinquent (%) 0 0 0 1.40 0.0 11.73 150 days delinquent (%) 0 0 0 1.25 0.0 11.11 180 days delinquent (%) 0 0 0 1.14 0.0 10.63 180 days more delinquent (%) 0 0 0 3.86 0.0 19.27 House liquidation (%) 0 0 0 0.64 0.0 8.08 Loan modification (%) 0 0 0 0.26 0.0 5.06 Deviation local unemployment rates (%) -1.51 -1.81 1.40 Local house price growth rates (%) -0.32 -0.27 2.15 Number of observations 16,347 337,811 Table 1: Summary Statistics of Selected Mortgage Loans. 6

to higher rates after the initial fixed-rate period expires. The ARMs in our data have a lifetime maximum interest rate of 13.50 percent on average at origination, similar to the dynamic average of 13.42 percent; and the lifetime minimum interest rate averages 6.7 percent at origination and 6.59 percent in the dynamic sample. The margin above Libor rate when interest rates are adjusted averages 5.74 percent at origination and 5.67 percent in the dynamic sample. Both at origination and in the dynamic sample, the period interest rate adjustment has a cap of 1.2 percent and a floor of 0.01 percent on average. The first interest rate adjustment cap, however, is higher at 2.5 percent on average at origination and 2.53 percent in the dynamic sample. Unemployment rates tend to be lower than their recent local historical averages. Local house prices, on the other hand, all depreciate in our sample period. Two observations emerge from Table 1. First, some mortgages stay in delinquency status for a long time without being liquidated. Particularly, in our loan-month dynamic sample, close to 7 percent of loans are 30-day delinquent, 3 percent are 60-day delinquent, 2 percent are 90-day delinquent, etc. Close to 4 percent of the loans are delinquent for over half a year. The liquidation rate, in contrast, is only 0.64 percent if measured at loan-month level.9 Of course, at the loan level, 2,177 out of the 16,347 loans in our random sample were liquidated (see Table 2), resulting in a 13.3% foreclosure rate, similar to what others have documented in the literature. Second, at the loan-month level, about 0.26 percent of all mortgage loans are modified by their lenders. This ratio is obviously much higher if we condition on loans that are delinquent. At the loan level, out of 857 out of the 16,347 loans in our randomly selected sample were modified, resulting in a modification rate of about 5.24%. We elaborate on the second observation regarding lenders’ decisions in more details in the next subsection. In the appendix, we provide summary statistics of the mortgage loans separately by the origination year, both at the time of origination and over time in Tables A1 to A3. As can be seen, the loans originated in later years are riskier, more likely to have two-year interest fixed period instead of three-year, more likely to be interest-only mortgages, less likely to have full documentation, and morelikelytobepurchaseloansinsteadofrefinanceloans. Theirprincipals,theinitialinterestrate, and monthly payment are also larger. Furthermore, the maximum and minimum lifetime interest rates and margins have risen over time. Given these differences at origination, not surprisingly, mortgage delinquency rates are much higher for loans originated in later years than earlier years. 2.3 Lenders’ Choices: Descriptive Statistics FromTable1,weobservethatlendersdonotalwaysrespondtoborrowers’mortgagedelinquency immediately by liquidating them. In this subsection we describe lenders’ decisions in more details. Table 2 presents the delinquency status (in months) at the beginning of the month when the loan was liquidated and modified. It shows that mortgage liquidation typically occurs when the borrower is between 6 and 9 months delinquent. While houses with loans less than 3 months delinquent are rarely liquidated, many houses are liquidated when the mortgage is over one year 9Houseforeclosurecanbealongandexpensiveprocessespeciallyinstateswithjudicialforeclosurelaws(Li2009). Ofthefourstatesthatwestudy,Floridarequiresjudicialforeclosure. Arizona,California,andNevadaallowforboth judicial and nonjudicial foreclosures, but most of the foreclosures are nonjudicial foreclosures. 7

Begnning-of-the-Month At Liquidation At Modification Loan Status (%) (%) Current 0.00 17.09 1 months 0.05 18.71 2 months 0.05 10.74 3 months 0.87 8.55 4 months 2.39 6.12 5 months 2.71 7.39 6 months 10.98 4.62 7 months 26.32 4.50 8 months 15.48 5.31 9 months 9.00 4.04 10 months 7.35 2.54 11 months 5.19 1.96 12 months 3.81 1.50 13 months 4.04 0.92 14 months 3.12 1.73 15 months 2.02 1.15 16 months 2.07 0.81 More than 17 months 4.46 2.31 Number of observations 2,177 857 Table 2: Loan Status at the Beginning of the Month when Liquidation or Modification Occurs. delinquent; indeed, about 4.46 percent of the loans liquidated is over 17 months delinquent. As a side note, the average loan age at liquidation is 27 months; about half of the liquidation occurred in 2008, 30 percent in 2007, and 8 percent in 2006, and about 6 percent in the first two months of 2009. Loan modifications are offered generally to loans already in distress. Nearly 60 percent of the loans are three months or more behind payments at the time of modification. Close to 9 percent are one year or more behind on payments. What is interesting, however, is that about 17 percent of the loans are modified when they are listed as current at the beginning of the period. The majority of these loans (55 percent) are originated in 2005 and the rest mostly in 2006 (37 percent). Furthermore, the majority of the modifications occur within three months of interest rate reset.10 Table 3 presents the modification terms. The majority of the modification results in more affordable mortgages as 83 percent of them have a reduction in monthly payments of about $542 on average. However, 8.6 percent of the modifications produce higher payments of about $287 on average; and 8 percent of the modified loans lead to less than $50 of monthly payment changes. Capitalization in modification is very common with arrears added to the principal balance. Indeed over64percentofthemodifiedloanshaveanincreaseofprincipalbalance,averaging$12,248. About 30 percent of the modified loans experience less than $500 in the change of principal balance; and only 5.4 percent of the loans have a principal reduction averaging $34,030.11 Nonetheless, more 10Haughwout, Okah and Tracy (2010) documented similar observations but their sample is different from ours as they include fixed-rate mortgages, adjustable-rate mortgages that have more than 3 years of fixed period, and mortgages with maturity not equal to 30 years (Table 3). 11See Section 3.3 for how we model the lenders’ terms of loan modification in our empirical analysis. 8

Variable Reduction No Change∗ Increase Monthly payment (percentage) 83.41 7.95 8.64 Average change in monthly payment ($) -542 1 287 (443) (19) (1,141) Balance (percentage) 5.41 30.18 64.40 Average change in balance ($) -34,030 -73 12,248 (39,603) (143) (11,993) Interest rate (percentage) 83.11 16.89 0.00 Average change in interest rate (percentage) -2.980 0.00 (1.415) (0.00) Table 3: Terms of Modification. Notes: No change refers to changes in monthly payment of less than $50 or total loan balance change of less than $500. Standard deviations are in parenthesis. than 83 percent of the modified loans have an annualized interest rate reduction averaging 2.98 percent, leading to reduced monthly payment. No modified loans experience interest rate increases. All of the loans are brought back to being current after modification. 3 The Model In this section, we present a model of a borrower’s behavior from the time his mortgage is originated until period T which we specify later. We do not endogenously model lenders’ decisions in this paper; instead we estimate them parametrically from the data. We assume that borrowers take lenders’ decisions as given. Time is discrete, denoted by t = 1,2,...,T, with each period representing one month. We use x to denote the borrowers’ state vector in period t, which includes time-invariant borrower and t mortgage characteristics (e.g., information collected at mortgage origination, and house location) as well as time-varying characteristics (e.g., a mortgage’s delinquency status, interest rates, local housing market condition, local unemployment rates, etc.). 3.1 Choice set In each period t, after information x is realized, a borrower chooses an action j. He has three t choices: makethemonthlymortgagepayment,skipthepayment,orpayoffthemortgage(whichwe denote by“PO”). We assume that the option to pay off the mortgage is available to any borrower, regardless of their delinquency status.12,13 More specifically, a borrower has different options of making mortgage payments, depending on the number of late monthly payments he has, which we denote by d where d ≥ 0. If the borrower 12In the data, about 86 percent of those who paid off loans were current in their mortgage at the time of the payoff, and 9 percent, 2 percent and 1 percent were one-, two-, and three-month delinquent, respectively. Very few of mortgage payoffs were by borrowers who were more than three months delinquent. Our conversation with the industryexpertssuggeststhatbecauseofinformationdelay,borrowerswhohavechosentoprepaymaysometimesbe recorded as one-month delay. 13In reality, a borrower can pay off the mortgage by refinancing or by selling the house. Our data, unfortunately, does not allow us to make such a distinction. 9

is current on his mortgage payment (i.e., d = 0), then he decides whether to make one monthly payment, which we denote by P and specify it below in Equation (2); to miss the payment; or t to pay off the loan.14 If the borrower is one month behind on the payment (i.e., d = 1), then he can choose to pay just P and remain one-month-delinquent; pay 2P to bring his status to current t t again;15 to miss the payment again and thus his status will be d = 2 next period; or to pay off the loan. In general, therefore, if a borrower has d ≥ 2 unpaid monthly payments at the beginning of time t, he can choose to make payments of 0, P ,2P ,··· ,(d+1)P , or paying off the whole loan. t t t However, we simplify the problem by assuming that, for d ≥ 2, if the borrower decides to pay he only has the options to pay 0, (d−1)P ,dP , or (d+1)P to become (d+1)-month delinquent, twot t t month delinquent, one-month delinquent, or current, respectively, or to pay off the entire loan.16 Formally, a borrower’s choice set with d unpaid payments is denoted by J(d), and given by:  {0,1,PO}, if d = 0;    J(d) = {0,1,2,PO}, if d = 1;    {0,d−1,d,d+1,PO}, if d ≥ 2, where the number zero refers to the action of not making any payment, and“PO”refers to paying off the loan. In the remainder of the paper, we sometimes denote the choice set by J(x ) instead of t J(d) because x includes the loan delinquency status d. We denote the borrower’s chosen number t of payments in period t as n ∈ J(d ). t t 3.2 State Transition The evolution of the state variables is captured by the transition probability F(x |x ,j), t+1 t where, as we discussed previously, x represents the state vector, and j ∈ J(x ) represents the t t borrower’s action at time t. We now discuss each of the state variables. Interest Rate, Monthly Payment, Mortgage Balance, and Liquidation. A mortgage contract with adjustable rates specifies the initial interest rate, the length of the period during which the initial rate is fixed, mortgage maturity, the rate to which the mortgage rate is indexed, the margin rate, the frequency at which the interest rate is reset, and the cap on interest rate change in each period, and the mortgage lifetime interest rate cap and floor. As stated in Section 2, we focus on loans that have two or three years fixed interest rate and 30 years maturity. Almost all of the loans have a six-month adjustment frequency after the initial fixed period. We now describe how the interest rate evolves through the life of an ARM loan contract. Let 14Giventhatwemodelthebehaviorofaborrowerwithanadjustable-ratemortgage,amonthlypaymentispotentially time-varying, which is reflected in the time subscript in P . t 15We do not observe penalty directly in the data. In the model, we allow for different payoff for each decision, which potentially captures the disutility from penalty associated with missing payments, see subsection 3.4 for more details. 16In the data, we do not observe borrowers’ payment decisions directly. Instead, we observe their loan status. In our sample, once a loan becomes d≥2 months delinquent, we do not observe that its delinquency status goes down yet still leave him 3-or-more months delinquent. 10

i denote the initial interest rate and let i denote the new mortgage interest rate at the r-th 0 r reset. For example, i denotes the interest rate at the first reset right after the fixed-rate period. 1 The term Margin represents the margin rate, which is the margin above the index rate that the new interest will be reset to. All ARMs in our selected sample data are indexed to the six-month Libor rate, we use Libor to denote the index rate at time t. An ARM contract also specifies a t lifetime interest rate floor and a lifetime interest rate cap, which we denote by LFloor and LCap, respectively. The ARM interest rate is restricted to be within the band specified by LFloor and LCap even though Margin above the Libor rate may go outside the band. ARM loan contracts also specify a cap on the permissible interest rate adjustment in each period, which we denote by PCap; moreover, for most mortgages, the cap on interest rate change for the first reset at the end of the initial fixed-rate is different from the subsequent caps, we thus denote the cap on the interest rate change at the first reset by FCap.17 Combining all the elements, the new interest rate at the r-th reset in period t evolves as follows: (r)  (cid:110) (cid:110) (cid:111)(cid:111) max i r−1 −FCap,LFloor,min Margin+Libor t −1 ,i r−1 +FCap,LCap , if r = 1; i = (r) r (cid:110) (cid:110) (cid:111)(cid:111) max i −PCap,LFloor,min Margin+Libor ,i +PCap,LCap , if r > 1, r−1 t −1 r−1 (r) (1) where the first term in Equation (1) is the lowest interest rate the mortgage can have assuming the periodic interest change takes its maximum allowed value, the second term is the lowest lifetime interest rate the mortgage can have, and the third term is the lowest of three rates: Libor rate plus margin, last period interest rate plus the maximum allowed periodic interest adjustment, lifetime mortgage interest rate cap. Note that Libor evolves stochastically. The borrower, therefore, t (r) needs to form expectations about future values for Libor in order to predict the interest rate he will havetopay. Thevaluesfortheothermortgageparameters,{Margin,LFloor,LCap,FCap,PCap} are fixed throughout the life of the mortgage. It follows from Equation (1) that i ∈ [max{i −FCap,LFloor},min{i +FCap,LCap}] r r−1 r−1 if r = 1 and that i ∈ [max{i −PCap,LFLoor},min{i +PCap,LCap}] if r > 1. In other r r−1 r−1 words, {LFloor,LCap,FCap,PCap} put bounds on the volatility of the adjustable mortgage interest rate: even when Libor is very volatile, the mortgage interest rate may not change significantly if FCap, PCap and LCap − LFloor are low. Given the rule that determines the interest rate reset, we now specify the transition of an ARM interest rate from period t to period t + 1. With a slight abuse of notation, let r(t) denote the numberofresetsthatoccurreduptoperiodt.18 Notethateitherr(t+1) = r(t)orr(t+1) = r(t)+1. The former is true when both period t and t+1 are in between two resets, hence i = i . r(t+1) r(t) The latter is true when an interest rate is just reset in period t+1, hence i = i , where r(t+1) r(t)+1 i is calculated using the formula in (1). r(t)+1 Once the new interest rate is determined, the new monthly payment can be calculated based on 17Typically,FCapislargerthanPCap;thatis,theinterestratechangeistypicallylargerattheinitialresetthan at subsequent resets. 18Forexample,iftheinitialinterestrateisfixedforatleasttperiods,thenr(t)=0. Ifaninterestrateisresetfor the second time in period t, r(t)=2. 11

theinterestrateandthebeginning-of-the-periodmortgagebalance. Consideraborrowerinperiodt with remaining mortgage balance Bal and interest rate i . The borrower’s mortgage monthly t−1 r(t) payment P is calculated so that if the borrower makes a fixed payment of P until the 360th period t t (i.e., the end of the 30-year loan term), he will pay off the entire mortgage; specifically, Bal × i (cid:14) 12 t−1 r(t) P = , (2) t (cid:0) (cid:14) (cid:1)−(360−t+1) 1− 1+ i 12 r(t) and the new balance entering period t+1 is updated to: (cid:34) (cid:14) (cid:35) i 12 Bal = Bal × 1− r(t) . (3) t t−1 (cid:0) (cid:14) (cid:1)360−t+1 1+ i 12 −1 r(t) Remark 1. Note that the lenders’ decisions affect the transition of borrowers’ state variables, i.e., F (x |x ,j) incorporates the lenders’ responses. If the lender chooses to modify the loan, it will t+1 t lead to possible changes to the borrower’s loan status, interest rate, monthly payment and mortgage balance. We describe how modification affects the mortgage balance, interest rate, monthly payment and loan status in Section 3.3 below. If the lender chooses to liquidate the house, then the borrower will be forced to the state of liquidation. Other State Variables. Other state variables include the number of late monthly payments d , t the Libor rate Libor , house price h , changes in local unemployment rate relative to its trend t t ∆Unr , borrower credit score CS , and borrower income Y . The evolution of these state variables t t t are as follows: • Number of late monthly payments (d ): d = d −n +1, where n ∈ J(d ) is the number t t+1 t t t t of monthly payments a borrower makes at time t. • Libor Rates (Libor ): We assume that the borrower’s belief regarding the evolution of Libor t rates is that it follows an AR(1) process in logs ln(Libor t+1 ) = λ 0 +λ 1 ln(Libor t )+(cid:15)Libor,t , where (cid:15)Libor,t ∼ N(0,σ2 Libor ) is assumed to be serially independent. • House price (h): We assume that the borrower’s belief regarding the evolution of housing prices in each zip code is that it follows an AR(1) process: h = λ +λ h +(cid:15) , t+1 2 3 t h,t where (cid:15) ∼ N(0,σ2) is assumed to be serially independent. h,t h • Local unemployment rate (∆Unr ): We focus on the deviation of the current unemployment t rate Unr in a county from the average of monthly unemployment rates from 2000 to 2009 t 12

in the same county Unr, which we denote by ∆Unr = Unr −Unr. We assume that the t t borrower’s belief regarding the evolution of ∆Unr is that it follows an AR(1) process: ∆Unr t+1 = λ 4 +λ 5 ∆Unr t +(cid:15)Unr,t , where (cid:15)Unr,t ∼ N(0,σ2 ∆Unr ) is assumed to be serially independent. • Credit score (CS ): We assume that the borrower’s belief regarding the evolution of the log t of his credit score is that it has the following process: ln(CS t+1 ) = λ 6 +λ 7 ln(CS t )+λ 8 1[d t = 1]+λ 9 1[d t = 2]+λ 10 1[d t = 3]+λ 11 1[d t ≥ 4]+(cid:15)CS,t , where 1(·) is the indicator function and (cid:15)CS,t ∼ N(0,σ2 CS ) is assumed to be serially independent. 3.3 Loan Modification and Foreclosure A lender makes the following decisions each period: foreclose the house, modify the loan, or wait (i.e., do nothing). As we mentioned in the introduction, in this paper we do not endogenize these decisions; rather, we assume that lenders follow decision rules that depend on borrowers’ various characteristics and are invariant to policy changes.19 Borrowers take these decision rules as given. As we describe in detail in Section 5.1, we specify that the probability that the lenders will choose one of the three options depends on the delinquency status, and a rich set of loan and housing characteristics. We estimate these lender decision rules by flexible logit or multinomial logit regressions. If the lender chooses to foreclose a house, the borrower receives the payoff associated with liquidation (see Eq. (6) below). If the lender chooses to wait, then the borrowers’ terms of the loan stay unchanged. However, if the lender chooses to modify a loan, we need to specify the new terms of the modified loan. Here we recall from Table 3 in Section 2 that the most popular modification is recapitalization coupled with interest rate reset. Ideally we would like to estimate lenders’ bidimensional choice of the new balance and new interest rate of the modified loan; however, instead of estimating such a joint process, we assume for simplicity that the new term of the modified loan is determined as follows: • After modification, borrowers’ payment status is brought to current, i.e., d = 0; t+1 • The new balance upon modification will be the sum of the pre-modification loan balance and 19Thischaracterizationoflenderbehaviorisconsistentwiththedata. Inacompanionpaper,weendogenizelenders’ decisions and investigate why they did not respond to the various policies introduced by the government to reduce foreclosures and encourage loan modifications. 13

the arrears in late payments, i.e.,20 Bal = Bal +d ·P , if the loan is modified at time t. t+1 t t t • The modified loan is a fixed rate mortgage with the maturity equal to the remainder of the initial loan, and the new modified interest rate, and thus the new monthly payment upon loan modification, is specified as a function of the initial monthly payment, initial interest rate, initial loan balance, margin rate, and states of the property. We estimate this process for the modified monthly payment directly from the data and by the year of the mortgage origination. 3.4 Payoff Function We specify a borrower’s current-period payoff from taking action j in period t as u (x )+(cid:15) , j t jt where u (x ) is a deterministic function of x and (cid:15) is a choice-specific preference shock. The j t t jt (cid:0) (cid:1) vector (cid:15) ≡ (cid:15) ,···(cid:15) is drawn from Type-I Extreme Value distribution and we assume that t 1t J(xt)t (cid:15) is independently and identically distributed over time. t When a borrower with d late payments makes n monthly payments, but does not pay off the mortgage, we assume that the deterministic part of his period-t payoff is:  β P +β (n−1)P +β CS +β P ×CS +β (n−1)P ×CS   1 t 2 t 3 t 4 t t 5 t t if n ≥ 1 u (x ) = +β Y +β ∆Unr +β X +ξ +ζ , (4) n t 6 0 7 t 8 0 d n   ξ , if n = 0. d Thefirsttermβ P representsthedisutilityfromonemonth’spayment. Thesecondtermβ (n−1)P 1 t 2 t is the disutility of n−1 months’ payment.21 The term β CS determines the borrower’s ability (or 3 t willingness) to make a payment. Specifically, CS is the borrower’s updated current credit score t provided by TransUnion, and it captures not only the borrower’s past payment history but also his ability to obtain future credit. We also allow credit scores to interact with borrowers’ payment decisions, P and (n−1)P , and the parameters β and β capture those interaction effects. The t t 4 5 term Y represents the borrower’s income at origination; and ∆Unr captures the deviations of 0 t the current local market condition relative to its long-run average. The term X is a collection of 0 theborrower’scharacteristicsatoriginationwhichcontainsoriginalmonthlypaymentamount(P ), 0 inverse loan-to-value ratio at origination (ILTV ), the year of loan origination, and whether the 0 borrower’s income is fully documented. ξ is a dummy variable for the borrower’s payment status d 20As shown in Table 3, a small fraction of modified loans (about 5 percent) received a balance reduction in our sample. We assume that these borrowers are“surprised”by the unexpected changes in their loan balance. In our futureresearchwhereweendogenizethelenders’choices,wewillendogenouslydeterminethelenders’choicesofnew mortgage and interest rate upon modification. 21We use β P +β (n−1)P , instead of a single term β nP to allow for the possibility that paying more than a 1 t 2 t 1 t single monthly payment amount could have a different utility cost than making only one payment. 14

d at the beginning of the period. In order to reduce the number parameters to be estimated, we assume that for d ≥ 4, ξ = ξ +dξ d 4,0 4,1 Finally, ζ is a constant for taking action n. We also make the assumption that for n ≥ 4, n ζ = ζ +nζ . n 4,0 4,1 We normalize ζ = 0 because only relative utility is identified in a discrete choice model. 0 When a borrower chooses to pay off the mortgage (j = PO), the deterministic part of the flow payoff is: T u (x ) = β (cid:88) δt(cid:48) +β PPN +β CS +β Y +β ILTV +β ILTV +ζ , (5) PO t 9 10 t 11 t 12 t 13 0 14 t PO,d t(cid:48)=t+1 Where δ is the discount factor (which we set to be 0.99 in our estimation), PPN is an indicator t for whether the borrower has to pay a prepayment penalty if prepaying in period t, ILTV is the t ratio of the borrower’s current house price to the remaining balance, i.e., the inverse of mortgage loan-to-value ratio, and ILTV is the inverse mortgage loan-to-value ratio at origination.22 We 0 assume that the model is terminated when the borrower pays off the mortgage.23 ζ determines PO,d the utility from paying off depending on the borrower’s payment status d at the beginning of the period. As before, in order to reduce the number parameters to be estimated, we assume that for d ≥ 3, ζ = ζ +dζ . PO,d PO,3,0 PO,3,1 If the house is liquidated, then as we mentioned earlier the borrower’s continuation value is give by: V (liquidated) = ζ . (6) t liquid,state Note that we allow ζ to depend on state of the property in order to capture state level difliquid,state ferences that are not captured by the model such as legislative differences regarding the foreclosure process. We normalize ζ to zero. liquid,NV If the borrower does not pay off the mortgage by period T, and if the borrower’s house is not liquidated by period T, the borrower reaches the final period T.24 The model is then terminated, 22WeassumethatthehousepricefollowsanAR(1)processwiththeshockdrawnfromanormaldistribution. The inverse of a normal random variable, however, does not have mean. In the analysis, we therefore use the inverse loan-to-value ratio ILTV instead of the mortgage loan-to-value ratio. 23We make this assumption because the mortgage loan exits our database once the borrower pays off or refinance the mortgage. 24To simplify the problem, we do not follow mortgages to their actual terminal period, that is, 360 months. As shown in the data section, most borrowers either pay off their mortgages or become seriously delinquent within the first six years after mortgage origination. 15

and the borrower receives the terminal payoff:  β +β CS +β ILTV , if current at T 15 16 T 17 T V (x ) = (7) T T 0, otherwise. Remark 2. In our framework, we assume that the lender can directly affect a borrower’s currentperiod flow utility only if the lender forecloses (i.e., liquidates) the house. If the lender chooses to modify the loan terms, or wait, the borrower’s flow utility is affected only to the extent that the modified loan term affects the borrower’s monthly payment. Dynamically the lender’s choices obviouslyaffecttheborrower’sabilitytostaycurrentinthemortgageandsubsequentlytheprobability of being foreclosed. 3.5 Value Function Theborrowersequentiallymaximizesthesumofexpecteddiscountedflowpayoffsineachperiod t = 1,...,T. Letσ (x ,(cid:15) )betheborrower’schoiceattimetgiventhestatevectorx andthevector t t t t ofchoice-specificshocks(cid:15) ,suchthatσ (x ,(cid:15) ) = 1ifaborrowerchoosesactionj given(x ,(cid:15) ); and t t,j t t t t 0 otherwise. Let σ ≡(σ ,...,σ ) denote the borrower’s decision profile from period 1 to T where 1 T σ , the terminal-period decision rule is included for ease of exposition, but the borrower makes no T choices (see the discussion prior to Eq. (7)). We can then express the borrower’s value functions from decision profile σ ≡ (σ ,...,σ ) recursively as follows: for t ≤ T −1, 1 T   (cid:40) (cid:41) (cid:90) (cid:88) V t (x t ;σ) = E (cid:15)t  σ t,j (x t ,(cid:15) t ) u j (x t )+(cid:15) jt +δ V t+1 (x t+1 ;σ)dF(x t+1 |x t ,j) , (8) j∈J(xt) xt+1∈Xt and V (x ;σ) is given by (7). The borrower’s optimal decision rule σ∗ is such that V (x ;σ∗) ≥ T T t t V (x ;σ) for any possible decision rule σ, and for all x , where t = 1,··· ,T. t t t 4 Estimation We define the choice-specific value function for action j in period t ≤ T − 1, v (x ), under t,j t decision profile σ∗, as (cid:90) v (x ) = u (x )+δ V (x ;σ∗)dF(x |x ,j). (9) t,j t j t t+1 t+1 t+1 t xt+1∈Xt The value function V (x ;σ∗) can then be written as: t t   (cid:88) V t (x t ;σ∗) = E (cid:15)t  σ∗ t,j (x t ,(cid:15) t ){v t,j (x t )+(cid:15) jt }. (10) j∈J(xt) In order to solve for the optimal decision profile σ∗, we use backward induction following the standard methods in dynamic discrete choice models with a finite number of periods (see, for 16

example,Rust1987,1994a,and1994b,andKeaneandWolpin1993). Westartfromthepenultimate period T −1. The choice-specific value function in period T −1 is given by: (cid:90) v (x ) = u (x )+δ V (x ;σ∗)dF(x |x ,j), T−1,j T−1 j T−1 T T T T−1 xT∈XT where V (x ;σ∗) is given by (7), and σ∗ is null. The optimal decision rule in period T −1 is then: T T T σ∗ (x ,(cid:15) ) = 1iffv (x )+(cid:15) ≥ max (cid:8) v (x )+(cid:15) (cid:9) . (11) T−1,j T−1 T−1 T−1,j T−1 j,T−1 T−1,j(cid:48) T−1 j(cid:48),T−1 j(cid:48)∈J(xT−1) Given the functional-form assumption for (cid:15) , we can show, following Rust (1987), that T−1   (cid:88) V T−1 (x T−1 ;σ∗) = ln exp(v T−1,j(cid:48) (x T−1 ))+γ, (12) j(cid:48)∈J(xT−1) where γ is the Euler constant. Now let us consider the borrower’s optimal decision rule in period T − 2. In order to calculate v (x ), we need to know (cid:82) V (x ;σ∗)dF(x |x ,j), which can be T−2,j T−2 xT−1∈XT−1 T−1 T−1 T−1 T−2 calculated using equation (12) and the state transition function F (x |x ,j). We then derive T−1 T−2 σ∗ (x ,(cid:15) ) and V (x ;σ∗) analogous to what we did in period T −1. We repeat this T−2,j T−2 T−2 T−2 T−2 process until we reach the initial period. The borrower’s optimal decision rule in period t is: σ∗ (x ,(cid:15) ) = 1ifv (x )+(cid:15) ≥ max (cid:8) v (x )+(cid:15) (cid:9) , (13) t,j t t t,j t jt t,j(cid:48) t j(cid:48)t j(cid:48)∈J(xt) and the period-t continuation value function is:   V t (x t ;σ∗) = ln (cid:88) exp (cid:0) v t,j(cid:48) (x t ) (cid:1) +γ. (14) j(cid:48)∈J(xt) Moreover, a borrower’s conditional choice probability under the optimal decision profile σ∗ for alternative j ∈ J(x ) in period t when the state vector is x is given by: t t exp(v (x )) p (x ;σ∗) = E [σ∗ (x ,(cid:15) )] = t,j t . (15) t,j t (cid:15)t t,j t t (cid:80) exp(v (x )) j(cid:48)∈J(xt) t,j(cid:48) t We estimate the model using maximum likelihood. In the data, we observe a path of states and choices for each individual i: (xi,ai) ≡ {(xi,ai)}T , where ai ≡ {ai } , and ai is defined to t t t=1 t jt j∈J(xi) jt t be a dummy variable that equals one when individual i chooses action j in period t. The likelihood of observing (xi,ai) given initial state xi and parameter vector θ for individual i is: 1 T−1 (cid:89) L(xi,ai|xi;θ) = l (ai,xi |xi;θ), (16) 1 t t t+1 t t=1 where (cid:81)T−1l(ai,xi |xi;θ) is the likelihood of observing action ai in period t and observing the t=1 t t+1 t t 17

state to transition to xi in period t+1 given state xi and parameter vector θ, as predicted by t+1 t the model, and it is given by: l (ai,xi |x ;θ) = (cid:89) (cid:2) p (xi;σ∗(θ))f(xi |xi,j) (cid:3)ai jt. (17) t t t+1 it t,j t t+1 t j∈J(xi) t where p (·;·) is given by (15) and σ∗(θ) is the model’s predicted optimal decision profile for the t,j borrower given parameter vector θ. Parameter estimate θ∗ maximizes the log-likelihood for the whole sample, i.e, I θ∗ = argmaxlnL(θ) = (cid:88) ln (cid:0) L(xi,ai|xi;θ) (cid:1) 1 i=1 I T−1 = argmax (cid:88)(cid:88) (cid:88) ai (cid:2) ln (cid:0) p (xi;σ∗(θ)) (cid:1) +lnf(xi |xi,j) (cid:3) . (18) jt t,j t t+1 t i=1 t=1 j∈J(xi) t 5 Estimation Results 5.1 Lenders’ Decisions As previously discussed, we estimate lenders’ policy functions parametrically using logit or multinomial logit regressions. The borrower enters period t with a delinquent status d , makes t the payment decision a , after which the lender makes the decisions regarding whether to modify, t liquidate, or do nothing about the loan based on the delinquent status of the loan at the end of the period t.25 However, in the data we only observe the loan status at the beginning of the period. Thus when we observe that a loan was current in period t and was also modified in period t, we assume that the loan would have been one month late at the end of period t had the modification not taken place. Specifically, we estimate the lenders’ decisions separately for four categories of loans: Category 1: (d = 0,a = 0). Borrowers are current in the beginning of the period, but do not t t make a payment in the period; Category 2: (d = 1,a = 0). Borrowersareone month delinquent inthebeginningoftheperiod, t t but do not make a payment in the period; Category 3: (d = 2,a = 0). Borrowersaretwo month delinquent inthebeginningoftheperiod, t t but do not make a payment in the period; Category 4: (d ≥ 3,a = 0). Borrowers are three-or-more-month delinquent at the beginning of t t a period, but do not make a payment in the period. It is important to note that lenders only modify or liquidate a loan if the borrower does not make any payment in the period. Therefore, if a borrower who enters the period with loan status 25We do not separately model lenders’ decision when to start foreclosure. As long as foreclosure is not complete, we consider the lender as“waiting.” 18

d ≥ 1, and if he makes a ≥ 1 payment, the lender’s only choice is waiting even though the status t t of the loan at the end of the period may still be one or more month delinquent if a < d +1. t t In our specification of the lenders’ decisions, we recall from Table 2 that lenders almost never liquidate a house whose mortgage is less than three months delinquent. Thus we assume that for loans in categories 1 to 3, the lenders choose only between modification and waiting; and the probability of modification is specified as a logit function of the state variables that includes borrower characteristics and loan status.26 For loans in category 4, we assume that lenders decides amongthreeoptions: modification,liquidation,andwaiting. Wespecifyamultinomiallogitfunction to represent the lenders’ probabilities of choosing the three alternatives. We further condition lenders’ decisions on state and year of origination. Finally, we also estimate lenders’ decision on interest rates for modified loans. Given the much smaller number of modified loans, we only conditionthisdecisiononmortgageyearoforigination. Intotal,wehave51regressions(4statesx3 origination years x 4 loan status + 3 origination years for interest rate estimation). To save space, we only report the estimation results for lenders’ modification, foreclosure, or wait decisions for loans originated in 2006 in Florida in Appendix Tables A4 and A5. Estimation results for interest rates after modification for loans originated in year 2004 are reported in Table A6.27 Category 1 Loans. For category 1 loans originated in Florida in 2006, lenders are more likely to modify if the borrower has a high credit score, high monthly payment but low initial monthly payment, and full documentation. An older loan is also more likely to be modified. By contrast, mortgage loans with high initial mortgage loan-to-value ratios and three-year fixed interest periods are less likely to be modified. Category 2 Loans. For category 2 loans originated in Florida in 2006, the factors that explain modification probability are similar to those that are current at the beginning of the period with a few exceptions. Income at origination reduces the probability of being modified while increases in local unemployment rates relative to recent trends raise the modification probability. Category 3 Loans. For category 3 loans originated in Florida in 2006, similar factors determine thelikelihoodofbeingmodifiedbylendersasthoseforCategory2loans. Theonlyexceptionisthat loan-to-value ratio at origination and credit scores no longer matter for modification probability. Category 4 Loans. For category 4 loans, we include many more explanatory variables to our multinomiallogitregressions. Aloanismorelikelymodifiedifincomeatoriginationislow, loan-tovalue ratio is low, initial loan-to-value ratio is high, the loan is older, or it has full documentation. A loan, however, is less likely to be modified if the borrower has many missed payments. Given 26In our estimation, we dropped the few (specifically, 4 case) loans of category 1 to 3 that were liquidated. That is, we assume that the four borrowers were making choices assuming that foreclosure would not have happened yet. We did not include their terminal liquidation in the likelihood function to avoid degeneracy. 27Toincreasetheprecision,weusethefullsample,insteadofthe1.75percentrandomsample,inestimatinglenders’ decisions. 19

Coefficient Estimate Standard Errors Panel A: Libor: ln(Libor )=λ +λ ln(Libor )+(cid:15) t+1 0 1 t Libor,t λ -0.013 (0.010) 0 λ 0.996*** (0.009) 1 σLibor 0.09656*** (0.00106) Panel B: House Price h =λ +λ h +(cid:15) t+1 2 3 t h,t λ 0.671*** (0.010) 2 λ 0.997*** (0.000) 3 σ 2.5419*** (0.00979) h Panel C: Local Unemp. Rates ∆Unr t+1 =λ 4 +λ 5 ∆Unr t +(cid:15)Unr,t λ 0.049*** (0.007) 4 λ 0.959*** (0.003) 5 σUnr 0.90066*** (0.00979) ln(CS )=λ +λ ln(CS )+λ 1[d=1] Panel D: Credit Score: t+1 6 7 t 8 +λ 1[d=2]+λ 1[d=3]+λ 1[d≥4]+(cid:15) 9 10 11 cs,t λ 0.149*** (0.001) 6 λ 0.897*** (0.001) 7 λ -0.072*** (0.001) 8 λ -0.164*** (0.002) 9 λ -0.130*** (0.002) 10 λ -0.007*** (0.000) 11 σCS 0.17719*** (7.93e-05) Table 4: Coefficient Estimates for Stochastic Processes. Notes: ***, ** and * denote statistical significance at 1%, 5% and 10% respectively. the number of missing payments, high loan-to-value ratio increases the probability of modification. Most modifications occur when the loan is between 5 and 9 months delinquent. Intermsofliquidation,currentcreditscore,incomeatorigination,mortgageloan-to-valueratio, andmonthsofdelinquencyallincreasetheprobabilitysignificantly. Currentmonthlypayment,loan age, and full documentation all reduce the probability of liquidation. Given the number of missing payments, higher mortgage loan-to-value ratio reduces the liquidation probability. Finally, most liquidation occurs when the loan is eleven or twelve months behind payments. 5.2 New Interest Rate and Monthly Payment Following Modification As indicated in Table A6, for loans originated in 2006, the new interest rate increases with the interestrateatorigination, themarginrate, mortgagebalanceatorigination, incomeatorigination, mortgage loan-to-value ratio, and whether the loan has full documentation, but decreases with current credit score, remaining balance, loan-to-value ratio at origination, loan age, deviation of localunemploymentratesfromrecenttrends,andthenumberofmonthsthattheborrowerisbehind payments. Of the four states, everything else equal Florida has the lowest modified loan rates. 5.3 Estimates of the Stochastic Processes In Section 3.2, we described that borrowers and lenders have beliefs about the stochastic processes that govern the evolution of Libor rates, the local housing prices, local unemployment rates, and credit scores. We assume that the borrowers have rational expectations about these processes 20

and estimate them using the ex post realizations of these processes. The estimates are reported in Table 4. Note that the processes of log credit score is endogenous for the borrower because its evolution depend on the payment status on mortgage loans, whose evolution depends on the borrower’s payment decisions. As can be seen, all the variables depend strongly on their lagged values, i.e., they exhibit strong persistence. For credit scores, missing mortgage payments also impact significantly negatively on their values. 5.4 Borrowers’ Payoff Function Parameters Table 5 presents the coefficient estimates in the three payoff functions associated with the three payment decisions. From Panel A, we observe that a borrower overall derives negative flow utility from making more payments; moreover, his flow utility from making a single payment is higher when his credit score is higher, but the flow utility from making more than one payments is lower if he has a higher credit score. His flow utility from making a payment is lower when the local unemployment rate is high relative to its recent historical average. In terms of conditions at origination, aborrower’sflowutilityfrommakingpaymentimproveswithhisinitialincomeandthe initial amount of the payment. High house value relative to mortgages (or low mortgage loan-tovalue ratio) at origination and full document increase the propensity to make payments. Turning to the constants associated with each payment status at the beginning of the period captured by ξ to ξ , the model requires relatively larger values associated with more months delinquent in 0 4+ order to explain the payment rate for such borrowers. For constants associated with payment decisions captured by ζ to ζ , the high disutility the borrower suffers from making large number 0 4+ of payments indicates his reluctance (or inability) to do so. From Panel B, we see that the borrower’s repayment decisions are negatively correlated with prepayment penalty. A borrower with higher current credit score, high initial income, high current house value relative to mortgage, but low house value relative to mortgage at origination is more likely to payoff his mortgage. The more payments that the borrower has missed, the less likely he will be able to pay off his mortgages by either refinancing or house sales. From Panel C, we see that if the house is liquidated, the payoffs to the borrower are lower in California,and Florida than in Nevada. Finally, from Panel D, we see that borrowers’ payoff function at the terminal period T is not well identified as none of the variables are significant. 6 Model Fit In order to gauge the fit of our model, we present figures that compare the model’s predictions for the distributions of endogenous variables with empirical analogs in the data. Figure 1 compares the probabilities of missing payments, and prepayment conditional on the delinquency status at the beginning of the period in the data and those predicted by our estimated model. The model does a good job at capturing the patterns in the data. The more payments a borrower misses, the more likelythathewillmisspaymentsagain;moreimportantly,oncetheborroweristhreemonthsormore 21

Coefficient Estimate Std. Err. Panel A: Coefficients in u (x ) as specified in (4) n t P : (β ) -0.1285*** (0.0117) t 1 (n−1)P : (β ) 0.2689** (0.0062) t 2 CS : (β ) 0.0866*** (0.0040) t 3 P ×CS : (β ) 0.0003 (0.0021) t t 4 (n−1)P ×CS : (β ) -0.1077*** (0.0021) t t 5 Y : (β ) 0.0209** (0.0052) 0 6 ∆UNR : (β ) -0.0130*** (0.0013) t 7 P : (β ) 0.1154*** (0.0096) 0 8,1 ILTV : (β ) 0.0153** (0.0072) 0 8,2 Full Doc: (β ) 0.0033** (0.0017) 8,3 Constant: (ξ ) -0.4961*** (0.0496) 0 Constant: (ξ ) -1.3017*** (0.0468) 1 Constant: (ξ ) -1.3880*** (0.0481) 2 Constant: (ξ ) -1.6704*** (0.3436) 3 Constant: (ξ ) 0.5403*** (0.0518) 4,0 Constant: (ξ ) -0.0143*** (0.0033) 4,1 Constant: (ζ ) 0.0761 (0.0491) 1 Constant: (ζ ) -2.1488*** (0.0922) 2 Constant: (ζ ) -3.0994*** (0.1417) 3 Constant: (ζ ) 0.6790* (0.3540) 4,0 Constant: (ζ ) -0.5983*** (0.0318) 4,1 Panel B: Coefficients in u (x ) as specified in (5) PO t (cid:80)T δt(cid:48) : (β ) 0.1012*** (0.0081) t(cid:48)=t+1 9 PPN : (β ) -2.6110*** (0.0765) t 10 CS : (β ) 0.6368*** (0.0136) t 11 Y : (β ) 0.3302** (0.1521) 0 12 ILTV : (β ) 8.5867*** (0.1708) t 13 ILTV : (β ) -6.3451*** (0.2791) 0 14 Constant: (ζ ) -7.2565*** (0.5160) PO,0 Constant: (ζ ) -8.3419*** (0.5138) PO,1 Constant: (ζ ) -8.1471*** (0.5182) PO,2 Constant: (ζ ) -5.7989*** (0.5209) PO,3,0 Constant: (ζ ) -0.3685*** (0.0310) PO,3,1 Panel C: Coefficients in V (liquidated) as specified in (6) t ζ 0.3358 0.2363 liquid,AZ ζ -1.0123*** 0.2008 liquid,CA ζ -3.5621*** 0.2688 liquid,FL Panel D: Coefficients in V (x ) as specified in (7) T T Constant (β ) -3.3825 (65.9577) 15 CS (β ) -0.0847 (4.0928) t 16 ILTV (β ) -0.5666 (43.7830) T 17 Table 5: Coefficient Estimates for Borrowers’ Payoff Functions. Notes: ***, ** and * denote statistical significance at 1%, 5% and 10% respectively. 22

1 8. 6. 4. 2. 0 Probability of Missing Payments 0 5 10 Number of Late Monthly Payments Data Model 40. 30. 20. 10. 0 Probability of Prepayment 0 5 10 Number of Late Monthly Payments Data Model Figure 1: Probabilities of Missing Payments and Prepayment, By Beginning-of-Period Delinquency Status. behindhispaymentschedule,hewillstaydelinquentwithalmostcertainty. Themodelalsocaptures the relationship between months of delinquency and the probability of prepayment; interestingly, the model predicts that the probability of prepayment is highest among those borrowers who are one month late in their payment. Figure 2 compares the probabilities of missing payments and prepayment by loan age in the data and those predicted by our model. Note that while we capture the probability of default by loan age well, the match with the probability of prepayment is less than perfect partly because the data is more volatile. Both curves are hump shaped with the probability of default or staying default peaking at age 36 months, roughly one-year after the majority of the loans exited their fixed-teaser-rate period. The peak of prepayment, by contrast, occurs at 24 months, the time when the majority of the loans’ fixed-teaser-rate period expires. Figure 3 charts the probabilities of missing payments and prepayment by the ratio of current monthlymortgagepaymenttoinitialmonthlypayment(whentheloanwasoriginated). Thefitsare good for both charts. Interestingly, there is a large jump of about 50 percentage points in default probability when the current payment exceeds the initial payment, consistent with the observations wedocumentedearlierthataborrowerhasahigherprobabilityofdefaultshortlyafterhismortgage payment resets to a higher value. After that, the probability of default declines somewhat and then hovers at around 50 percent. The prepaymentprobability, on the other hand, increases consistently with the increase in the current mortgage payment relative to the initial mortgage payment after theinitialdropfollowingtheresetininterestrates. Sincealoanleavesoursampleafteritisprepaid, the default pattern depicted in the figure cannot be interpreted as direct evidence that interest rate reset necessarily leads to higher default rate as pointed out in Fuster and Willen (2015). We will 23

5. 4. 3. 2. 1. 0 Probability of Missing Payments 0 10 20 30 40 50 Loan Age (Months) Data Model 80. 60. 40. 20. 0 Probability of Prepayment 0 10 20 30 40 50 Loan Age (Months) Data Model Figure 2: Probabilities of Missing Payments and Prepayment, By Loan Age. Note: We group the loans into age intervals in months, 1-3, 4-6, ..., 43-45, 46+, in the calculation for the probabilities. 6. 5. 4. 3. 2. 1. Probability of Missing Payments 1 1.1 1.2 1.3 1.4 1.5 Ratio of Current Payment to Initial Payment Data Model 50. 40. 30. 20. 10. Probability of Prepayment 1 1.1 1.2 1.3 1.4 1.5 Ratio of Current Payment to Initial Payment Data Model Figure3: ProbabilitiesofMissingPaymentsandPrepayment,ByRelativeMonthlyPaymentRatio. Notes: (1). Relativemonthlypaymentistheratioofcurrentmonthlypaymenttotheinitialmonthly paymentwhentheloanwasoriginated. (2). Wegrouploansintointervalsofrelativepaymentratio, 1-1.05, 1.05-1.1, 1.1-1.15, ..., 1.45-1.50, in the calculation for the probabilities. 24

6. 5. 4. 3. 2. 1. Probability of Missing Payments 40 60 80 100 120 Loan to Value Ratio Data Model 80. 60. 40. 20. 0 Probability of Prepayment 40 60 80 100 120 Loan to Value Ratio Data Model Figure 4: Probabilities of Missing Payments and Prepayment, By the Current Mortgage Loan-to- Value (LTV) Ratio. Notes: (1). Unit for LTV is in percentage. (2). We group loans into intervals of LTVs, 50-, [50, 60), [60,70), ..., [110,120), 120+, in the calculation for the probabilities. address this issue in details in the next section. Figure 4 depicts the probabilities of missing payments and prepayment by the current mortgage loan-to-value ratio. The model does a good job at capturing the patterns in both series. As expected, the higher the current mortgage loan-to-value ratio is, the more likely the borrower will default and less likely he will prepay. Finally, Figure5chartstheprobabilitiesofmissingpaymentsandprepaymentbytheborrower’s current credit scores. The model captures the default probability better than it captures the prepayment probability. Note that credit scores capture the borrower’s past payment history as well as future payment ability. Not surprisingly, the higher the credit score is, the less likely the borrower will default. In other words, a borrower with a higher credit score is more likely to make his mortgage payments on time, and is also more likely to prepay. 7 Counterfactual Simulations In this section, we report counterfactual simulation results to address two sets of questions. The first set of simulations is aimed at a quantitative understanding of the roles of different factors that contributed to the subprime borrowers’ default and prepayment behavior during the housing crisis. The second set of simulations is aimed at the policies, particularly monetary policies and alternative mortgage designs, that may help reduce defaults. It is useful to start out with some basic facts about the changes in monthly payments, housing 25

6. 4. 2. 0 Probability of Missing Payments 2 4 6 8 Updated Credit Score (from TransUnion) Data Model 40. 530. 30. 520. 20. 510. Probability of Prepayment 2 4 6 8 Updated Credit Score (from TransUnion) Data Model Figure 5: Probabilities of Missing Payments and Prepayment, By Credit Score. Notes: (1). Credit score units are in 100. (2). We group loans into intervals of credit scores, 150-, [150, 200), ..., [650,700), 700+, in the calculation for the probabilities. Figure 6: Current Monthly Payment Transition by Loan Age and ARM Type. 26

Figure 7: Housing Price and Unemployment Rate Trends, by Year of Origination of Loans. prices and unemployment rates that the ARM borrowers in our dataset face as their loans age. In Figure 6, we show the average monthly payment amounts as loans age, for 2/28 (2 years fixed rate, 28 years adjustable rate) and 3/27 (3 years fixed rate, 27 years adjustable rate) ARM mortgages. It shows that upon the end of the initial lower teaser rate period, borrowers’ monthly payment would typically increase substantially for loans that originated in 2004 and 2005, in contrast, it will decrease substantially for loans that originated in 2006. These observations are not surprising as interest rates moved down substantially after 2007. In Figure 7, we plot the percentage changes of local housing prices and local unemployment rates by loan age for loans that were originated in 2004, 2005 and 2006, respectively. It shows that for loans that were originated in 2004, the local housing prices experienced on average more than 30 percent gains before it declined at around the time these loans reached about 24 months of loan age; for loans that were originated in 2005, there was also a modest (about 10 percent) housing price gains up to loan age of 12 months before the housing market crash. In contrast, the loans that were originated in 2006 immediately experienced housing price declines as deep as 45 percent. Similarly, the experience of the loans in terms of labor market conditions as measured by local unemployment rates also differs substantially by loan origination years. Loans originated in later years faced much tougher labor market conditions marked by high unemployment rates. The differences by loan origination year on these dimensions explain why the effects of a variety of counterfactual changes differ by loan origination years, as we discuss below. 7.1 Understanding the Factors for Defaults and Prepayments 27

Loan Baseline Fixed Rate Mortgage Age Current Paid off Delinquent [Liquidated] Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.360 0.584 0.056 0.011 0.388 0.564 0.048 0.010 24 0.220 0.735 0.045 0.016 0.253 0.698 0.049 0.014 30 0.127 0.821 0.052 0.020 0.166 0.787 0.048 0.017 36 0.075 0.871 0.054 0.025 0.115 0.838 0.047 0.020 42 0.044 0.896 0.060 0.030 0.075 0.868 0.057 0.024 Panel B: Loans Originated in 2005 18 0.444 0.457 0.099 0.015 0.498 0.405 0.098 0.014 24 0.319 0.564 0.117 0.029 0.400 0.493 0.107 0.025 30 0.220 0.636 0.144 0.045 0.300 0.565 0.135 0.036 36 0.144 0.671 0.185 0.063 0.217 0.604 0.179 0.050 42 0.094 0.689 0.217 0.082 0.140 0.628 0.232 0.073 Panel C: Loans Originated in 2006 18 0.460 0.315 0.226 0.031 0.490 0.284 0.226 0.025 24 0.315 0.351 0.334 0.063 0.372 0.322 0.306 0.055 30 0.216 0.376 0.408 0.105 0.265 0.352 0.383 0.097 36 0.156 0.393 0.451 0.156 0.190 0.374 0.436 0.142 42 0.122 0.404 0.474 0.216 0.158 0.387 0.455 0.195 Table 6: Role of Interest Rate Reset. Notes: (1) The“Baseline”panel reports the model’s prediction of the loan status under the actual loans; the“Fixed RateMortgage”panelreportsthecounterfactualresultswhenallARMswereconvertedtofixedratemortgageswith interest rate fixed at the initial teaser rate of their corresponding ARMs. (2) The numbers reported in the table are the fractions of loans in different status,“Current”,“Paid Off”or“Delinquent.”The loans in“Liquidation”are also included in“Delinquent”status. The total fractions in“Current”,“Paid Off”and“Delinquent”status sum to 1. 28

Adjustable-Rate Mortgages. The amount of the monthly mortgage payment in an ARM is fixed for a few (typically, 2 to 3) years initially and then resets every six months. The initial fixed rate is in general lower than the rate after the interest rate resets. Because of an increase in mortgage payments upon the reset, many commentators believed that the massive amount of default by subprime mortgage borrowers in the recent financial crisis was attributable to the reset ofARMinterestrates. ToquantifyhowmuchtheinitialresetofARMscontributedtothesubprime borrower’s default and prepayment rates observed in the data, we simulate the model under the counterfactualscenariothattheinterestrateisfixed attheinitialteaserratefortheentireduration of the loan. In Table 6 we report the model’s predictions regarding the fraction of loans in different status (Current, Paid Off, Delinquent, or Liquidated) at different loan ages, for loans originated in 2004, 2005and2006respectively. Thepanellabeled“Baseline”isthemodel’spredictionoftheloanstatus under the actual loans,28 and the panel labeled“Fixed Rate Mortgage”is the model’s prediction of the loan status if all of the ARMs were replaced by FRMs with interest rate fixed at the initial teaser rate of the ARM. Under the baseline, the left panel in Table 6 shows that the performance of the loans differs substantially depending on the year of origination. Loans originated in 2004 are much more likely to be paid off over time than entering delinquency or liquidation. By 36 months of loan age when the initial interest rate resets occurred, 87.1% of the loans were already paid off (i.e., refinanced or prepaid by selling); 5.4% of the loans are in various stages of delinquency, including 2.5% being liquidated. Theperformanceoftheloansthatoriginatedin2005werequitedifferent. By36months of loan age, 67% of these loans were paid off, and 18.5% would be in different stages of delinquency, including 6.3% in foreclosure. The loans originated in 2006 would face even more difficulty, as 45.1% of them would be in delinquency, including an astonishingly high 15.6% in foreclosure at 36 months of loan age. These differential outcomes of loans that originated in different years are the result of many factors, including the dynamics of the interest rates, local unemployment rates and local housing prices, as depicted in Figures 6 and 7. The right panel in Table 6 presents the performance of the loans if all the ARMs were to be converted to fixed rate mortgages at the initial teaser rate. It shows that in general changing the ARMs to FRMs alone, thus taking away the interest rate resets of the ARMs, has a very limited effectofthedelinquencyandliquidationrates. Forloansoriginatedin2004, thedelinquencyrateat 36monthsofloanagewouldbe4.7%undertheFRMsinsteadofthe5.4%undertheoriginalARMs; similarly, at 36 months of age the delinquency rates would be 17.9% and 43.6% for loans originated in 2005 and 2006, in contrast to 18.5% and 45.1% respectively under the original ARMs.29 The margin that the FRMs seem to have a bigger effect is the “Current” and “Paid Off” margin, for example, the fraction of current loans at 36 months of loan age would be 11.5% (7.5%), 21.7% (14.4%) and 19.0% (15.6%) respectively for loans originated in 2004, 2005 and 2006 under the 28We will repeatedly compare our counterfactual results with the results in the“Baseline”below. 29It is important to point out that our calculation of loan status is over all loans including those that are paid off. Bydoingso,weavoidtheselectionbiasissueraisedinFusterandWillen(2015)wheretheyarguethatwhenlessrisk loans were refinanced, the delinquency rates of remaining loans would by definition higher. 29

FRMs (respectively, under the original ARMs). Declining Housing Prices. Many researchers argued that negative home equity is important in a borrower’s default decision (see, e.g., Bhutta, Dokko, and Shan, 2010; and Fuster and Willen, 2015). In Table 7, we report counterfactual simulation results to understand the role of substantial housing price declines for the loans we study. We conduct two counterfactual experiments. In the first counterfactual experiment, we ask what would have happened to the delinquency and foreclosure rates, had the housing prices stayed unchanged from the origination of the mortgage, i.e., (cid:101)h t = h 0 for all t ≥ 1. In the second counterfactual experiment, we explore the interaction of interest rate resets of the ARMs and local housing market conditions, by assuming in addition that all the ARMs are converted to FRMs with interest rate fixed at the initial teaser rate (as in the right panel of Table 6). In the left panel, we report the results from the first counterfactual experiment, (cid:101)h t = h 0 for all t ≥ 1.AsshouldbeexpectedfromFigure7, settinghousingpriceunchangedatitslevelofmortgage originationwouldhavedeprivedthesubstantialhousingpricegainsforloansthatoriginatedin2004, and to some extent for the loans that originated in 2005. Indeed, our counterfactual experiments show that our model predicted much higher (respectively, slightly higher) delinquency rates and foreclosure rates for 2004 loans (respectively, for 2005 loans) than in the baseline (see the left panel in Table 6). Had the housing price stayed constant at the loan origination, the delinquency and liquidation rates would be 22.7% and 17.6% at 36 months loan age for loans originated in 2004 and 2005, in contrast to 5.4% and 18.5% respectively under the baseline. The liquidation rates would be 10.7% instead of 2.5% for loans originated in 2004 at 36 months loan age; for loans originated in 2005, the liquidation rates would be higher if the housing prices stayed at h than the baseline, 0 but they would be slightly lower than those in the baseline after 24 months. This precisely reflect the fact that for loans originated in 2005, housing price actually started to fall below the level at the loan origination at around the 23 months of loan age (see the left panel of Figure 7). In striking contrast, from Figure 7 we know that the 2006 loans experienced housing price declines immediately in the data; thus setting the housing prices unchanged at their origination levels would lead to much lower delinquency and foreclosure rates. Indeed, our counterfactual results for the 2006 loans confirm these: had housing price not declined so precipitously, our model predicts that the delinquency rates for loans originated in 2006 would be about 7% at all loan ages, and the cumulative liquidation rates would reach 3.3% at 42 months of loan age. In the right panel, we see that adding the additional assumption that the interest rates would be fixed at the initial teaser rates of the ARMs for the total duration of the loan only generates rathersmalleffect. Comparingwiththeresultsintheleftpanel, wefindthatmakingthemortgages of fixed rates rather than of adjustable rates makes the loans more likely to be paid off for young loans originated in 2004 and 2005 and all loans originated in 2006. Delinquency rates are slightly lower for loans originated in 2005 and 2006 at all ages, and for older loans originated in 2004. For loans originated in 2004, the delinquency rates do not change much when the loans are young. 30

Loan (cid:101)h t =h 0 (cid:101)h t =h 0 & FRM Age Current Paid off Delinquent [Liquidated] Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.530 0.324 0.146 0.038 0.499 0.349 0.152 0.040 24 0.445 0.398 0.157 0.063 0.416 0.425 0.159 0.069 30 0.322 0.495 0.183 0.083 0.335 0.489 0.176 0.088 36 0.223 0.550 0.227 0.107 0.257 0.536 0.207 0.104 42 0.140 0.589 0.271 0.132 0.175 0.572 0.253 0.129 Panel B: Loans Originated in 2005 18 0.531 0.358 0.112 0.021 0.522 0.380 0.098 0.020 24 0.436 0.450 0.114 0.034 0.429 0.465 0.106 0.031 30 0.284 0.570 0.146 0.046 0.325 0.553 0.121 0.042 36 0.182 0.642 0.176 0.062 0.233 0.612 0.155 0.055 42 0.109 0.679 0.212 0.079 0.152 0.652 0.196 0.069 Panel C: Loans Originated in 2006 18 0.459 0.471 0.070 0.012 0.449 0.487 0.063 0.007 24 0.341 0.594 0.066 0.017 0.342 0.608 0.050 0.010 30 0.195 0.735 0.071 0.021 0.211 0.738 0.051 0.013 36 0.122 0.801 0.077 0.025 0.133 0.809 0.058 0.018 42 0.088 0.835 0.077 0.033 0.079 0.859 0.062 0.027 Table 7: Role of Housing Prices and the Interaction with the Interest Rate Resets. Notes: (1) In the left panel, we assume that the housing price stayed unchanged from that at the loan origination; in the right panel, we assume in addition that all the ARMs were converted to FRMs with interest rate fixed at the initial teaser rate of the corresponding ARMs. (2) The numbers reported in the table are the fractions of loans in different status,“Current”,“Paid Off”or“Delinquent.”The loans in“Liquidation”are also included in“Delinquent” status. The total fractions in“Current”,“Paid Off”and“Delinquent”status sum to 1. Loan ∆Unr =∆Unr t 0 Age Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.409 0.536 0.054 0.008 24 0.278 0.676 0.046 0.012 30 0.162 0.793 0.045 0.015 36 0.100 0.854 0.046 0.020 42 0.056 0.890 0.055 0.025 Panel B: Loans Originated in 2005 18 0.524 0.388 0.088 0.013 24 0.437 0.468 0.095 0.022 30 0.294 0.565 0.141 0.032 36 0.192 0.615 0.193 0.052 42 0.125 0.642 0.233 0.079 Panel C: Loans Originated in 2006 18 0.549 0.260 0.191 0.029 24 0.454 0.297 0.249 0.056 30 0.337 0.330 0.333 0.098 36 0.269 0.353 0.378 0.149 42 0.222 0.371 0.407 0.201 Table 8: Role of Labor Market Conditions. Note: The numbers reported in the table are the fractions of loans in different status, “Current”, “Paid Off” or “Delinquent.”The loans in“Liquidation”are also included in“Delinquent”status. The total fractions in“Current”, “Paid Off”and“Delinquent”status sum to 1. 31

Year of Origination 2004 2005 2006 Teaser Rate (%) 6.78 7.03 8.00 Margin (%) 5.63 5.71 5.92 Libor at Initial Reset (%) 5.30 4.88 2.66 Margin + Libor (%) 10.93 10.59 8.58 Lifetime Floor Rate (%) 6.39 6.58 7.49 Table 9: Average Loan Characteristics by Year of Origination. Labor Market Conditions. In Table 8, we simulate the role of local unemployment rates on the observed borrowers’ delinquency and foreclosure decisions. Gerardi, Herkenhoff and Ohanian (2013) document that individual unemployment is a strong predictor of default using data from the Panel Study of Income Dynamics. We assume that the local unemployment rate stayed the same as that at loan origination. The results show that because of their increased payment ability, borrowers are more likely to stay current with their mortgage payment, less likely to pay off, and less likely to default. These effects are stronger for loans originated in 2006. The reason is that for loans that originated in 2004 and 2005, the local unemployment rates did not increase initially. In contrast, local unemployment rates increased almost immediately after origination for loans originated in 2006 as shown in Figure 7. 7.2 Potential Policy Responses to Reduce Defaults In this subsection, we evaluate the effectiveness of several potential policy responses to reduce default and foreclosure rates. We first consider the role of monetary policy, and then consider the role of alternative mortgage contract designs. 7.2.1 Traditional Monetary Policy There are recent works that looked at how ARM borrowers responded to a decrease in their mortgage interest rates due to a low short-term interest rate (Libor). General findings in the works are that monetary policy can have positive effects on ARM borrowers because their interest rates aretiedtoashort-terminterestrate. Inparticular,ARMborrowersarelesslikelytodefault(Fuster and Willen 2015) and more likely to increase consumption due to a larger disposable income (Keys, Piskorski, Seru and Yao 2014, and Di Maggio, Kermani and Ramcharan 2014). In Table 10, we report the counterfactual results from an experiment where Libor rate is set to zero, and as a result, the ARM borrowers’ monthly payment amount will be determined by the lifetime floor interest rate once the teaser rate period of the ARM expires. This could provide the best case scenario (or upper bound) on how much monetary policy may reduce the delinquency and foreclosure rates. It is important to point out that setting the Libor rate to zero does not necessarily imply that the borrowers’ monthly payment will be that much lower than their payment in the teaser period. The reason is that, as we mentioned in Section 3.2, the majority of the ARMs have life time floor rates, which would be applied even when Libor rate is zero. In fact, most borrowers’ monthly payment would only decrease slightly when Libor rate is zero upon the reset of the interest rate. 32

Loan Libor =0 t Age Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.383 0.560 0.057 0.009 24 0.263 0.689 0.048 0.015 30 0.160 0.794 0.046 0.019 36 0.104 0.846 0.050 0.023 42 0.062 0.881 0.057 0.027 Panel B: Loans Originated in 2005 18 0.492 0.417 0.092 0.013 24 0.391 0.504 0.105 0.025 30 0.270 0.585 0.145 0.037 36 0.183 0.627 0.191 0.053 42 0.119 0.647 0.234 0.075 Panel C: Loans Originated in 2006 18 0.483 0.287 0.230 0.029 24 0.368 0.324 0.308 0.055 30 0.241 0.355 0.403 0.096 36 0.175 0.372 0.453 0.146 42 0.134 0.387 0.479 0.204 Table 10: Impacts of Traditional Monetary Policy: Setting the Libor Rate to Zero. Note: The numbers reported in the table are the fractions of loans in different status, “Current”, “Paid Off” or “Delinquent.”The loans in“Liquidation”are also included in“Delinquent”status. The total fractions in“Current”, “Paid Off”and“Delinquent”status sum to 1. Also note that as reported in Table 9, in the data margin rates and lifetime floor rates were high and Libor rates were already low for 2006 loans, thus setting the Libor rate to zero had little effect on loans originated in 2006. Therefore, the results in Table 10 suggests that setting the Libor rate at zero would reduce the mortgage pay off rates of almost all loans; however, mortgage delinquency rates and liquidation rates change little. 7.2.2 AutomaticLoanModificationContingentonHousingPriceIndex, witha“Cushion” If a housing price downturn leads to massive default rates, then one way to mitigate this problemistolinkthemortgagemonthlypaymenttothecurrenthousepriceindex. Caplinet al. (2007), Shiller (2008), Mian and Sufi (2014), and Kung (2015) have suggested that such“continuous workout mortgages”might have reduced the mortgage default and foreclosure. Piskorski and Tchistyi (2010, 2011) show that the optimal mortgage contracts in the presence of stochastic house price appreciation or uncertain income and uncertain mortgage rates all have some forms loss sharing between borrowers and lenders such as balance or interest rate reduction when house price declines or when income decreases or interest rates jump. In their model, interest rates are exogenous and the optimal plan involves a complex home equity line. We consider two different automatic loan modification schemes in this subsection.30 30Kung(2015)studiesthegeneralequilibriumeffectof“continuousworkoutmortgage”onhousingpriceandmortgage interest rates. Borrowers in his model are only allowed to make the current monthly payment or to refinance, i.e., delinquency and foreclosure are not focus of his paper. 33

Modification of Monthly Payments Only. We first consider the case in which only the monthly payment amount is automatically modified as housing prices change. Specifically, denote P˜ as the modified monthly payment at period t, and P as the monthly payment amount in t t the absent of modification according to the original loan. Let h and h denote the housing price t 0 index at period t and at origination respectively. The first counterfactual we consider assumes that the monthly payment will be automatically modified from P to P˜ as follows: t t (cid:26) (cid:27) h P˜ = P ×min 1,κ× t , where κ ≥ 1, (19) t t h 0 while the principal balance is not adjusted.31 In (19), the parameter κ ≥ 1 can be used to adjust how much cushion is afforded to the seller in terms of housing price declines before the automatic modification of monthly payments (and loan balance below) is activated. We refer to κ as the “cushion parameter”: the higher κ is, the more housing price decline is required to trigger the automatic modification. For example, as we will experiment below, when κ = 1.15, the housing price would have to decline by 13% (≈ 1−1/κ) from that at the loan origination before monthly payment is reduced. Modification of Principal Balance (and Monthly Payments Too) In the second counterfactual, we assume that (cid:26) (cid:27) (cid:93) h t BAL = BAL ×min 1,κ× , (20) t t h 0 where as in (19), κ ≥ 1 is the cushion parameter. Because monthly payment is proportional to principal balance, as we showed in (2), the automatic modification of principal balance will also automatically adjust the monthly payment.32 A key feature of the automatic modification mortgages of both forms (19) and (20) is that modifications are triggered by the housing price declines alone, not at all by the delinquency status of the borrowers, which are subject to potential strategic behavior by borrowers. This feature distinguishes from the automatic modification from the stochastic loan modification of the form as modeled in Section 3.3. In this subsection, we are also interested in evaluating the impact of alternative mortgage contracts on the revenue of the lenders. For this purpose, we make the following assumption:33 Assumption 1. Upon foreclosure, the lender receives 75 percent of the house value. It is worth making three observations. First, the baseline corresponds to the case of κ = +∞, i.e., the monthly payment actually never deviates from those in the baseline. Second, everything else equal, as long as κ ≥ 1 the borrower is always made better off under both (19) and (20) than under the baseline in expectation. The closer κ is to 1, the easier that housing price decline triggers 31Stochastic loan modification of the form as modeled in Section 3.3 stays as in the baseline in the counterfactual experiments in this subsection. 32This is akin to“partially shared appreciation mortgage”considered in Kung (2015). 33Campbell, Giglio, and Pathak (2011) find that the average discount of a house value for foreclosures is about 27 percent. 34

reduction in monthly payment under (19), or both loan balance and monthly payment under (20), and thus the better off the borrowers are. Third, changes in the cushion parameter κ in the automatic modification mortgages (19) and (20) have two effects. First, different values of κ affects the borrowers’ payment behavior, resulting in different levels of delinquency and liquidation. Intuitively, and as we will show below, the closer κ is to 1, the smaller is the fraction of mortgages that are liquidated, and thus the smaller the social surplus destruction in the 25% loss of the house value in foreclosure (see Assumption 1). Second, κ affects how the surplus from the reduction in foreclosure is shared between the borrowers and the lender. When κ is closer to 1, while it is true that the automatic modification mortgages(19)and(20)preventsmoreforeclosure, thelenderreceivesasmallershareofthesurplus becausetheforeclosurereductionisachievedsolelybythelenderreducingmonthlypaymentand/or principals. At higher values of the cushion parameter κ, the borrower will also share some of the “sacrifices”for reduction in foreclosure as smaller declines in housing prices would not trigger the automatic reduction. As a result, while the borrowers’ ex ante expected value under automatic modification mortgages is always higher than that in the baseline, and decreasing in κ, the lender’s expected revenue is non-monotonic in κ. In particular, to the extent that it is important that lenders voluntarily adopt the automatic modification mortgages of the form (19) or (20), we would like to explore whether there are values of κ that both the borrowers and lenders are better off than the baseline. Otherwise, we would expect that lenders would have to raise interest rates to compensate for their revenue loss. Automatic Modification Mortgages without a Cushion: κ = 1. InTable11wepresentthe resultsfromcounterfactualsimulationsunderautomaticmodificationmortgages,withoutacushion, i.e., when κ = 1. We should emphasize that this version of automatic modification corresponds to most of what has been studied in the literature.34 In the left panel, we present the results under the automatic modification mortgages (19) that adjust payment sizes only, with κ = 1. We find that the automatic modification mortgages only slightlyreducethedelinquencyandforeclosureratesforloansoriginatedin2004and2005; however, the delinquency and foreclosure rates are significantly reduced for loans originated in 2006. The delinquency rate for 2006 loans under the“just payment size”automatic modification mortgages is reduced from 45.1% under the baseline to 35.0% at 36 months loan age, and the liquidation rate is lowered from 15.6% to 11.1% at 36 months of loan age and from 21.6% to 15.7% at 42 months of loan age. As we mentioned earlier, it is important to know how such automatic modification mortgages might affect lender’s revenues. We also calculate lenders’ revenue as the present value of the borrower’s expected payment. If a borrower prepays, his payment in that period will just be the remaining mortgage balance. If a borrower’s house is liquidated, we assume that the lender receives75percentoftheestimatedcurrenthousevalueasstatedinAssumption1. Asstatedinthe notesforTable11, thelender’srevenuesperborrowerunderthebaselineare$221.7K,$230.5K,and $216.3K, respectively, for loans originated in 2004, 2005, and 2006. Comparing these numbers to thoseobtainedinTable11, weseethattheautomaticmodificationloansoftheform(19)lowersthe 34For example, this is the case considered in Kung (2015)’s“partially shared appreciation mortgage.” 35

Loan Just Payment Size (κ=1) Payment Size and Balance (κ=1) Age Current Paid off Delinquent [Liquidated] Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.385 0.563 0.051 0.008 0.379 0.570 0.051 0.008 24 0.235 0.720 0.045 0.011 0.236 0.723 0.041 0.011 30 0.137 0.819 0.044 0.015 0.136 0.820 0.044 0.014 36 0.082 0.869 0.049 0.020 0.080 0.872 0.049 0.019 42 0.051 0.898 0.052 0.025 0.045 0.902 0.052 0.024 Revenue 221.7K 221.7K Panel B: Loans Originated in 2005 18 0.447 0.458 0.095 0.014 0.443 0.468 0.089 0.014 24 0.328 0.562 0.110 0.025 0.316 0.586 0.098 0.024 30 0.234 0.632 0.134 0.038 0.211 0.673 0.117 0.033 36 0.164 0.667 0.169 0.053 0.143 0.721 0.136 0.045 42 0.111 0.685 0.204 0.071 0.095 0.754 0.151 0.059 Revenue 228.8K 229.7K Panel C: Loans Originated in 2006 18 0.525 0.285 0.190 0.025 0.469 0.380 0.150 0.021 24 0.426 0.317 0.257 0.047 0.352 0.477 0.171 0.038 30 0.344 0.340 0.316 0.076 0.250 0.575 0.175 0.054 36 0.296 0.353 0.350 0.111 0.181 0.646 0.172 0.070 42 0.254 0.365 0.380 0.157 0.137 0.699 0.164 0.086 Revenue 212.5K 212.8K Table 11: Automatic Modification of Payment Size and Principal Balance, without a Cushion (κ = 1). Notes: (1). The numbers reported in the table are the fractions of loans in different status,“Current”,“Paid Off”or “Delinquent.”The loans in“Liquidation”are also included in“Delinquent”status. The total fractions in“Current”, “Paid Off” and “Delinquent” status sum to 1. (2). The numbers in the row labeled “Revenue” refer to lender’s expectedrevenueperborrower,underAssumption1. Thelender’sexpectedrevenuesperborrowerunderthebaseline are $221.7K, $230.5K, and $216.3K, respectively, for loans originated in 2004, 2005 and 2006. 36

lender’s revenue per borrower unfortunately for loans originated in 2005 and 2006. In particular, theperborrowerrevenuefor2005loansis$228.8K,about$1,700(orabout0.74%)lessperborrower than under the baseline, and the per borrower revenue for 2006 loans is $212.5K, about $3,800 (or about 1.76%) less per borrower under the baseline. In the right panel, we present the results under the automatic modification mortgages (20) that adjust the loan balance and thus also monthly payment, with κ = 1. We find that automatic modification mortgages that lower loan balance still have little effect on the outcomes of loans originated in 2004 relative to the baseline; however, the impacts on loans originated in 2005 and 2006 are much bigger than those under the“Just Payment Size”automatic modification loans. For loans originated in 2005, automatic reductions in loan balance reduce the delinquency rate at the 36 months of loan age to 13.6%, in contrast to 18.5% under the baseline and 16.9% under the“Just Payment Size” auto-modification mortgages. For loans originated in 2006, the delinquency rate is now 17.2%, in contrast to 45.1% under the baseline and 35.0% under the“Just Payment Size” auto-modification mortgages. The reduction in foreclosure rate for loans originated in 2006 is also astonishing: at 36 months loan age it is 7.0% under the balance auto-modification mortgages, in contrast to 11.1% under the “Just Payment Size” auto-modification mortgages and 15.6% under the baseline. The reductions in delinquency and foreclosure are mostly achieved by increases in the fraction of paid off loans. The lender’s expected revenues per borrower are also lower under this type of automatic modification mortgages than those under the baseline. For loans originated in 2005, the per borrower revenue under the counterfactual mortgages is $229.7K, about $800 less than that under the baseline, and for 2006 loans, the per borrower revenue is $212.8K, about $3,500 lower than that under the baseline. However, it is also interesting to note that the lender’s expected revenue per borrower is actually higher under the seemingly more generous automatic balance modification mortgages (20) than under the automatic modification mortgages (19) that only adjust the monthly payment. This is due to the fact that the more generous automatic balance modification mortgages are very successful in reducing the costly foreclosure, allowing the lenders to more than recoup the cost of the generosity in lowering the mortgage balances as well as monthly payments. Automatic Modification Mortgages with a Cushion: κ = 1.15. One issue of the automatic modification mortgages without cushion studied in Table 11 is that lender’s revenue is lower than that in the baseline. In Table 12 we show that it is possible to adjust the cushion parameter κ to κ = 1.15 so that the lender’s per borrower revenue is at least as high as that in the baseline for loans generated in all years. This ensures that lenders are also better off under the proposed automatic modification mortgages with a cushion. As is obvious qualitatively and as we will show quantitatively below in Table 14, borrowers are better off than the baseline under the cushioned automatic modification mortgages as well; thus this represents a Pareto improvement over the baseline mortgages. The left panel of Table 12 shows that at κ = 1.15, the automatic modification loans that adjust the payment size only have little impact on the borrower outcomes, leaving only very slight reductions in delinquency and liquidation rates relative to the baseline (see the left panel in Table 37

Loan Just Payment Size (κ=1.15) Payment Size and Balance (κ=1.15) Age Current Paid off Delinquent [Liquidated] Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.378 0.569 0.053 0.008 0.366 0.581 0.053 0.011 24 0.229 0.725 0.046 0.013 0.226 0.728 0.045 0.014 30 0.136 0.819 0.045 0.017 0.134 0.819 0.047 0.018 36 0.082 0.868 0.051 0.021 0.081 0.867 0.052 0.022 42 0.049 0.895 0.055 0.027 0.046 0.897 0.057 0.028 Revenue 221.8K 222.2K Panel B: Loans Originated in 2005 18 0.436 0.466 0.097 0.013 0.434 0.468 0.098 0.014 24 0.312 0.572 0.116 0.024 0.317 0.571 0.112 0.026 30 0.214 0.642 0.144 0.040 0.216 0.647 0.137 0.039 36 0.148 0.675 0.177 0.057 0.150 0.684 0.165 0.054 42 0.097 0.695 0.209 0.077 0.098 0.710 0.193 0.073 Revenue 230.8K 230.7K Panel C: Loans Originated in 2006 18 0.477 0.308 0.215 0.027 0.469 0.322 0.209 0.029 24 0.364 0.339 0.297 0.059 0.360 0.378 0.262 0.060 30 0.280 0.367 0.353 0.095 0.265 0.444 0.291 0.090 36 0.228 0.381 0.391 0.134 0.203 0.495 0.302 0.122 42 0.193 0.393 0.414 0.184 0.164 0.533 0.302 0.154 Revenue 216.7K 216.3K Table 12: Automatic Modification of Payment Size and Principal Balance, with a Cushion (κ = 1.15). Notes: (1). The numbers reported in the table are the fractions of loans in different status,“Current”,“Paid Off”or “Delinquent.”The loans in“Liquidation”are also included in“Delinquent”status. The total fractions in“Current”, “Paid Off” and “Delinquent” status sum to 1. (2). The numbers in the row labeled “Revenue” refer to lender’s expectedrevenueperborrower,underAssumption1. Thelender’sexpectedrevenuesperborrowerunderthebaseline are $221.7K, $230.5K, and $216.3K, respectively, for loans originated in 2004, 2005 and 2006. 38

6) for loans originated in all years, though the lender’s revenue also increase slightly relative to the baseline. However, the right panel shows that automatic modification mortgages that reduce the mortgage balance when the housing price declines by about 13 percent (= 1 − 1/1.15) are able to achieve moderate reductions in delinquency and liquidation rates for loans originated in 2005, and more sizable reductions for loans originated in 2006. Specifically, we find that under the automatic modification mortgages that reduce balance with κ = 1.15, for loans originated in 2005, the delinquency rate at 42 months of loan age decreases from 21.7% in the baseline to 19.3%, and the liquidation rate is reduced from 8.2% in the baseline to 7.3%; for loans originated in 2006, the delinquency rate at 43 months of loan age decreases from 47.4% in the baseline to 30.2% and the liquidation rate is reduced from 21.6% in the baseline to 15.4%. At the same time, the lender’s per borrower revenue is at least as high as that in the baseline for all years. Even though the reductions in delinquency and liquidation rates under auto-modification mortgages with a cushion parameter κ = 1.15 is not as large as those without a cushion, it should be noted that the lenders would not have to increase their interest rate under the cushioned auto-modification mortgages. It is also worthnotingthatborrowersarebenefitingfromtheautomaticmodificationmortgagesregardlessof whether they are eventually delinquent or liquidated, since one important feature of the automatic modification mortgages is that modifications are triggered by the housing price declines, not at all by the delinquency status of the borrowers, which are subject to potential strategic behavior by borrowers. 7.3 What if the Lender Can Commit Not to Modify Any Loans? In this subsection, we consider a different counterfactual: what if the lender can commit not to modify any loans? This counterfactual can shed light on whether borrowers’ strategic defaults, in order to receive loan modification, played any role in the observed delinquency and liquidation. As we will argue below, it will also shed light on why the fraction of loans that received modification (only 0.26%) during the housing crisis was so low. To implement this counterfactual, we consider two scenarios depending on whether the lender would replace each loan modification observed in the data by either the alternative of“waiting and do nothing”or the alternative of“liquidation.”The results are presented in Table 13. In the left panel, we assume that the lenders replace all modifications in the data by liquidation instead. We find that, with the additional threat of foreclosure, borrowers are much less likely to default on their mortgages than the baseline and this effect is particularly strong for loans originated in 2005 and 2006; and much higher fraction of loans are paid off. Because of the now higher probability of liquidation, although liquidation rates come down for all loans the magnitude of the reductions of liquidation rates is much smaller. Surprisingly, the lender’s per borrower revenues under this counterfactual policy are respectively $224.6K, $240.4K, and $233.2K for loans originated in 2004, 2005 and 2006, which are respectively 1.3%, 4.3% and 7.8% higher than their counterparts under the baseline. The increases in lender revenue are mainly due to more borrowers paying off their loans (despite prepayment penalty for most of the loans), and less delinquency. In the right panel, we assume that the lenders replace all modifications in the data by“waiting.” 39

Loan No Modification, More Liquidation No Modifcation, More Waiting Age Current Paid off Delinquent [Liquidated] Current Paid off Delinquent [Liquidated] Panel A: Loans Originated in 2004 18 0.336 0.630 0.034 0.008 0.340 0.625 0.035 0.009 24 0.199 0.774 0.027 0.012 0.206 0.767 0.027 0.011 30 0.111 0.860 0.029 0.015 0.119 0.853 0.028 0.014 36 0.064 0.905 0.031 0.019 0.068 0.898 0.033 0.018 42 0.035 0.930 0.036 0.024 0.039 0.924 0.037 0.025 Revenue 224.6K 224.3K Panel B: Loans Originated in 2005 18 0.390 0.562 0.048 0.012 0.408 0.540 0.051 0.012 24 0.270 0.675 0.055 0.019 0.291 0.652 0.057 0.020 30 0.183 0.750 0.066 0.031 0.202 0.725 0.073 0.030 36 0.127 0.784 0.089 0.044 0.139 0.760 0.102 0.047 42 0.079 0.801 0.120 0.062 0.084 0.776 0.140 0.065 Revenue 240.4K 238.8K Panel C: Loans Originated in 2006 18 0.433 0.439 0.128 0.031 0.455 0.416 0.129 0.034 24 0.323 0.496 0.182 0.063 0.335 0.462 0.203 0.064 30 0.235 0.529 0.235 0.101 0.221 0.497 0.283 0.107 36 0.167 0.548 0.285 0.140 0.147 0.518 0.336 0.159 42 0.120 0.560 0.320 0.189 0.106 0.530 0.364 0.217 Revenue 233.2K 230.0K Table 13: Automatic Modification of Payment Size and Principal Balance, without a Cushion (κ = 1). Notes: (1). The numbers reported in the table are the fractions of loans in different status,“Current”,“Paid Off”or “Delinquent.”The loans in“Liquidation”are also included in“Delinquent”status. The total fractions in“Current”, “Paid Off” and “Delinquent” status sum to 1. (2). The numbers in the row labeled “Revenue” refer to lender’s expectedrevenueperborrower,underAssumption1. Thelender’sexpectedrevenuesperborrowerunderthebaseline are $221.7K, $230.5K, and $216.3K, respectively, for loans originated in 2004, 2005 and 2006. 40

We find that mortgage delinquency rates are also lower than the baseline, though the reduction is less pronounced than in the previous case where lenders replace modification by liquidation. As in the previous case, borrowers pay off more often than in the baseline when there is no prospect of loan modification. Note that the liquidation rate under this counterfactual policy is actually higher than that in the baseline for loans originated in 2006; after all, waiting and doing nothing by the lender in place of modification will still raise borrowers’ months in delinquency, eventually increasing the probability of foreclosure. It may seem somewhat surprising that borrowers will be less likely to default when the lender commits not to modify any loans. The reason is actually quite simple. The ex ante value of being delinquent is a lot lower without any possibilities to get their loans modified; and as the months in delinquency rise, the house will eventually be foreclosed. So the value of being delinquent decreases substantially without modification. In other words, the presence of modification possibility led some borrowers to delinquency in the hopes of getting their interest rates reduced. Again, it is interesting to note that the lenders’ expected per borrower revenues under this counterfactual policy are $224.4K, $238.8K and $230.0K respectively for loans originated in 2004, 2005, and 2006, which are respectively 1.2%, 3.6% and 6.3% higher than their counterparts under the baseline. Discussion. The counterfactual results reported in Table 13 suggest that the lenders would have been able to raise their expected revenue if they were able to commit not to offer any modification at all. Similarly, results reported in Table 12 show that the lenders could also have raised their expected revenue if they could commit to automatically modify the loan balances whenever the housing prices decline by more than 13%. In reality, however, as we mentioned in the discussions followingTable1, only5.24%oftheloansweremodified. Thekeypuzzleiswhatexplainsthissmall presence of the loan modification? Table 13 suggests that the lenders’ revenue would have been higher if they did not offer any loan modification; yet Table 12 and many commentators suggest that they should have offered more modification and more automatic modification. We would like to point out that the results in Tables 13 and 12 are both predicated on the lenders having the ability to commit: in the case of Table 13 the lenders need to commit never to offer modification to any borrower regardless of his circumstances; and in the case of Table 12 the lender needs to be able to commit to automatically reduce the loan balance whenever the housing prices decline by more than 13% regardless of whether the borrower has shown any difficulty in making the payments. In reality the lenders do not have the commitment power. We believe that the lack of commitment power is an important contributor to understand the lender behavior and will further investigate this issue in future research. 7.4 Quantitative Assessment of the Impact on Borrowers’ Ex Ante Expected Welfare In Table 14 we summarize the borrowers’ average expected welfare evaluated at the loan age of month 1 under the baseline and different counterfactual scenarios. It shows that borrowers are 41

Loan Year of Origination 2004 2005 2006 Baseline 34.859 34.819 33.293 Automatic Modification, Just Payment (κ=1) 35.007 35.796 35.768 Automatic Modification, Both Payment and Balance (κ=1) 35.014 35.836 37.703 Automatic Modification, Just Payment (κ=1.15) 34.887 35.111 33.881 Automatic Modification, Both Payment and Balance (κ=1.15) 34.898 35.363 35.513 No Modification, More Liquidation 32.576 32.076 30.304 No Modification, More Waiting 32.867 32.472 30.556 Table 14: Borrower’s Ex Ante Expected Welfare Under the Baseline and Different Counterfactual Scenarios. Notes: (1)Unitsarein$1,000. (2)Theborrowers’expectedwelfareevaluatedatloanage1. Theexpectedutilityof theborrowerunderdifferentscenariosisconvertedintodollarunitsviadividingtheutilitybytheestimatedcoefficient for monthly payment in the utility function. alwaysbetteroffundertheautomaticmodificationmortgagesofbothforms(19)and(20), andwith or without cushion. As expected, borrowers are better off under automatic modification mortgages that adjust both payment and balance than under automatic modification mortgages that adjust onlythe payment, and also the borrowers are better off when the automatic adjustment is triggered without a cushion (when κ = 1) than with a cushion (when κ = 1.15). We also find that borrowers are significantly worse off if the lenders commit not to offer any loan modifications. 8 Conclusion One important characteristic of the recent mortgage crisis is the prevalence of subprime mortgages with adjustable interest rates and their high default rates. In this paper, we present a dynamic structural model of subprime adjustable-rate mortgage (ARM) borrowers making payment decisions taking into account possible consequences of different degrees of delinquency from their lenders. We empirically implement the model using unique data sets that contain information on borrowers’ mortgage payment history, their broad balance sheets, and lender responses. Our investigation of the factors that drive borrowers’ decisions reveals that subprime ARMs are not all alike. For loans originated in 2004 and 2005, which preceded the peak of the housing prices, the interest rate resets associated with ARMs, as well as the housing and labor market conditions were not as important in borrowers’ delinquency decisions as in their decisions to pay off their loans. For loans originated in 2006, interest rate resets, housing price declines, and worsening labormarketconditionsallcontributedimportantlytotheirhighdelinquencyrates. Counterfactual policy simulations further suggest that even if the Libor rate could be lowered to zero by aggressive monetary policies, it would have a limited effect on reducing the delinquency rates. We also examine the effectiveness of automatic modification mortgages under which the monthly payment or the principal balance of the loans are automatically reduced when housing prices decline. We show that such alternative mortgage designs can be effective in reducing both delinquency and foreclosure; and importantly, we find that automatic modification mortgages with a cushion, which will trigger the monthly payment or principal balance reductions only when housing price declines exceed a certain percentage may result in a Pareto improvement in that borrowers and lenders are 42

both made better off than under the baseline, with a much lower delinquency and foreclosure rates. Our counterfactual analysis also suggests that limited commitment on the part of lenders to loan modification policies may be an important reason for the relatively small rate of modifications observed during the housing crisis. In future research, we plan to model lender behavior explicitly, so that we can have a better understanding of what is the nature of the lender’s lack of commitment issue, and how policies may be designed to alleviate the lender’s lack of commitment power problem. It is also important to consider the general equilibrium effects of alternative mortgages on the housing market, both on the mortgage interest rates the lenders may charge and on the housing market prices, particularly taking into account the spillover effects on property prices due to foreclosed properties. References [1] Adelino, Manuel, Kristopher S. Gerardi and Paul S. Willen (2013).“Why Don’t Lenders Renegotiate More Home Mortgages? Redefaults, Self-Cures and Securitization.”Journal of Monetary Economics, 60(7): 835–853. [2] Agarwal, Sumit, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet, and Douglas Evanoff (2011). “Market-Based Loss Mitigation Practices for Troubled Mortgages Following the Financial Crisis,”Federal Reserve Bank of Chicago Working Paper 2011-03. [3] Agarwal,Sumit,GeneAmromin,ItzhakBen-David,SouphalaChomsisengphet,TomaszPiskorski, and Amit Seru (2015). “Policy Intervention in Debt Renegotiation: Evidence from the Home Affordable Modification Program,’; forthcoming, Journal of Political Economy. [4] Bajari, Patrick, Sean Chu, Denis Nekipelov, and Minjung Park (2013). “A Dynamic Model of Subprime Mortgage Default: Estimation and Policy Implications,”NBER Working Paper 18850. [5] Bhutta, Neil, Jane Dokko, and Hui Shan (2010).“The Depth of Negative Equity and Mortgage Default Decisions.”Finance and Economics Discussion Series 2010-35, Board of Governors of the Federal Reserve System. [6] Campbell, John Y., and Joao F. Cocco (2015). “A Model of Mortgage Default,” Journal of Finance, Volume 70, Issue 4, 1495-1554. [7] Campbell, John, Stefano Giglio, and Parag Pathak (2011).“Forced Sales and House Prices,” American Economic Review, 101(5) (2011), 2108-2131. [8] Caplin, Andrew, Sewin Chan, Charles Freeman and Joseph Tracy. Housing Partnerships: A New Approach to a Market at a Crossroads. Cambridge, MA: MIT Press. [9] Chatterjee, Satyajit, and Burcu Eyigungor (2015).“A Quantitative Analysis of the U.S. Housing and Mortgage Markets and the Foreclosure Crisis,”Federal Reserve Bank of Philadelphia Working Paper 15-13, Philadelphia, PA. 43

[10] Corbae, Dean, and Erwan Quintin (2015).“Leverage and the Foreclosure Crisis,”Journal of Political Economy, Volume 123, No. 1, 1-65. [11] Demyanyk, Yuliya, and Otto van Hemert (2011). “Understanding the Subprime Mortgage Crisis,”Review of Financial Studies, 24(6), 1848-1880. [12] Deng, Youngheng, John M. Quigley, and Robert van Order (2000).“Mortgage Terminations, Heterogeneity and the Exercise of Mortgage Options,”Econometrica, Volume 68, No. 2, 275- 307. [13] Elul, Ronel, Nicholas S. Souleles, Souphala Chomsisengphet, Dennis Glennon, and Robert Hunt (2010).“What ’Triggers’ Mortgage Default?”American Economic Review, 100(2), 490- 494. [14] Foote, Christopher, Kristopher Gerardi, and Paul Willen (2008).“Negative Equity and Foreclosure: Theory and Evidence,”Journal of Urban Economics, 64(2), 234-245. [15] Foote, Christopher L., Kristopher S. Gerardi, and Paul S. Willen (2012).“Why Did So Many People Make So Many Ex Post Bad Decisions? The Causes of the Foreclosure Crisis,”Public Policy Discussion Paper 12-2, Federal Reserve Bank of Boston. [16] Foster, Chester, andRobertvanOrder(1984).“AnOption-BasedModelofMortgageDefault,” Housing Finance Review, 3(4), 351-372. [17] Frame,Scott,AndreasLehnert,andNedPrescott(2008).“ASnapshotofMortgageConditions: Emphasis on Subprime Mortgage Performance,”manuscript, Federal Reserve Bank of Atlanta. [18] Fuster, Andrea and Paul S. Willen (2015). “Payment Size, Negative Equity, and Mortgage Default,”Federal Reserve Bank of New York Staff Reports, No. 582. [19] Garriga, Carlos, and Don Schlagenhauf (2009). “Home Equity, Foreclosures, and Bailouts,” manuscript, Federal Reserve Bank of St. Louis. [20] Gerardi,Kristopher,AndreasLehnert,ShaneSherlund,andPaulWillen(2008).“MakingSense of the Subprime Crisis,”Brookings Papers on Economic Activity, 69-145. [21] Gerardi, Kristopher, and Wenli Li (2010).“Mortgage Foreclosure Prevention Efforts,”Federal Reserve Bank of Atlanta Economic Review, 2, 1-13. [22] Gerardi, Kristopher, Kyle F. Herkenhoff, Lee E. Ohanian, and Paul S. Willen (2013). “Unemployment, Negative Equity, and Strategic Default,”Working Paper 2013-4. Federal Reserve Bank of Atlanta. [23] Hatchondo, Juan Carlos, Leonardo Martinez, and Juan M. Sanchez (2011). “Mortgage Defaults,”Federal Reserve Bank of Saint-Louis Fed Working Paper no. 2011-019A. 44

[24] Haughwout, Andrew, Ebiere Okah, and Joseph Tracy (2010). “Second Chances: Subprime Mortgage Modification and Re-Default,”Federal Reserve Bank of New York Staff Report No. 417. [25] Keane, Michael, and Kenneth Wolpin (1997). “Career Choices of Young Men,” Journal of Political Economy, 105 (3), 473-522. [26] Keys, Benjamin, Tanmoy Mukherjee, Amit Seru, and Vikrant Vig (2010).“Did Securitization Lead to Lax Screening? Evidence From Subprime Loans,”Quarterly Journal of Economics, 125(1), 307-362. [27] Kung, Edward (2015).“Mortgage Market Institutions and Housing Market Outcomes.”Working Paper, UCLA. [28] Li, Wenli (2009).“Residential Housing and Personal Bankruptcy,”Philadelphia Reserve Bank of Philadelphia Business Review, 1-11. [29] Mayer, Christopher, Karen Pence, and Shane M. Sherlund (2009). “The Rise in Mortgage Defaults,”Journal of Economic Perspectives, 23(1), 27-50. [30] Mian, Atif and Amir Sufi (2014). House of Debt. University of Chicago Press. [31] Mitman, Kurt (2012). “Macroeconomic Effects of Bankruptcy and Foreclosure Policies,” manuscript, Department of Economics, University of Pennsylvania. [32] Piskorski, Thomaz, Amit Seru, and Vikrant Vig (2010).“Securitization and Distressed Loan Renegotiation: EvidencefromtheSubprimeMortgageCrisis.”JournalofFinancialEconomics, 97(3), 369-397. [33] Piskorski, Tomaz and Alexei Tchistyi (2010).“Optimal mortgage design.”Review of Financial Studies, 23 (8), 3098-3140. [34] Piskorski, Tomasz, and Alexei Tchistyi (2011).“Stochastic House Appreciation and Optimal Mortgage Lending.”Review of Financial Studies, 24 (5), 1407-1446. [35] Posner, Eric A. and Luigi Zingales (2009). “A Loan Modification Approach to the Housing Crisis,”American Law and Economics Review, Vol. 11, No. 2, 575–607. [36] Rust,John(1987).“OptimalReplacementofGMCBusEngines.AnEmpiricalModelofHarold Zurcher,”Econometrica, 55(5), 999-1033. [37] Rust, John (1994a). “Estimation of Dynamic Structural Models, Problems and Prospects: Discrete Decision Processes,”in Christopher Sims and J.J. Laffont, eds., Proceedings of the 6th World Congress of the Econometric Society. Cambridge University Press. [38] Rust, John (1994b).“Structural Estimation of Markov Decision Processes,”in Robert Engle and Daniel McFadden eds., Handbook of Econometrics, Vol. IV. Amsterdam: North-Holland. 45

[39] Tracy, Joseph, and Joshua Wright (2015).“Payment Changes and Default Risk: The Impact of Refinancing on Expected Credit Losses,”manuscript, Federal Reserve Bank of New York. 46

At Origination Dynamic Sample Variable Mean Median Std. Dev. Mean Median Std. Dev. Age of the loan (months) 0 0 0 15 12 11 Share of 2-year fixed period (%) 76 1 43 71 1 45 Prepayment penalty (%) 0.99 1 0.09 0.88 1 0.32 Interest-only mortgages (%) 30 0 46 34 0 47 Full document at origination (%) 53 1 50 53 1 50 Purchase loan (%) 42 0 49 46 0 50 Risk score 446 445 159 448 457 173 Inverse-LTV ratio at origination (%) 79 80 11 69 70 14 Annual income ($1000) 70 66 26 Principal balance ($1000) 239 213 127 239 212 131 Current interest rate (%) 6.78 6.70 1.09 6.95 6.75 1.40 Remaining mortgage terms (months) 360 360 0 346 349 11 Monthly payment ($1000) 1.442 1.293 0.720 1.417 1.267 0.719 Maximum lifetime interest rate (%) 13.19 13.15 1.27 13.06 13.00 1.28 Minimum lifetime interest rate (%) 6.40 6.63 1.76 6.23 6.50 1.83 Periodic interest rate cap (%) 1.23 1.00 0.36 1.22 1.00 0.37 Periodic interest rate floor (%) 0.00 0.00 0.00 0.00 0.00 0.00 First rate cap (%) 2.50 3.00 0.94 2.54 3 1.02 Margin for adjustable rate loans (%) 5.63 5.80 1.26 5.54 5.75 1.34 30 days delinquent(%) 0 0 0 5.46 0.0 22.72 60 days delinquent(%) 0 0 0 1.83 0.0 13.42 90 days delinquent(%) 0 0 0 0.73 0.0 8.50 120 days delinquent(%) 0 0 0 0.57 0.0 7.51 150 days delinquent(%) 0 0 0 0.48 0.0 6.88 180 days delinquent(%) 0 0 0 0.40 0.0 10.63 180 days more delinquent(%) 0 0 0 1.51 0.0 12.18 House liquidation (%) 0 0 0 0.24 0.0 4.94 Loan modification (%) 0 0 0 0.54 0.0 7.30 Deviation local unemployment rates (%) -1.68 -1.80 1.03 Local house price growth rates (%) 0.01 0.01 1.80 Number of observations 6,013 108,178 Table A1: Summary Statistics for Loans Originated in 2004. 47

At Origination Dynamic Sample Variable Mean Median Std. Dev. Mean Median Std. Dev. Age of the loan (months) 0 0 0 17 12 11 Share of 2-year fixed period (%) 83 1 38 80 1 40 Prepayment penalty (%) 99 1 3 80 1 40 Interest-only mortgages (%) 45 0 50 49 0 50 Full document at origination (%) 52 1 50 51 1 50 Purchase loan (%) 44 0 50 49 0 50 Risk score 450 450 154 426 450 180 Inverse-LTV ratio at origination (%) 78 80 11 82 79 20 Annual income ($1000) 72 67 26 Principal balance ($1000) 265 236 142 268 239 141 Current interest rate (%) 7.04 6.90 1.02 7.31 7.00 1.35 Remaining mortgage terms (months) 360 360 0 344 346 11 Monthly payment ($1000) 1.624 1.444 0.831 1.614 1.445 0.803 Maximum lifetime interest rate (%) 13.38 13.30 1.15 13.28 13.20 1.18 Minimum lifetime interest rate (%) 6.59 6.80 1.83 6.45 6.75 1.85 Periodic interest rate cap (%) 1.19 1.00 0.30 1.19 1.00 0.30 Periodic interest rate floor (%) 0.03 0.00 0.18 0.03 0.00 0.19 First rate cap (%) 2.48 3.00 0.86 2.51 3 0.90 Margin for adjustable rate loans (%) 5.71 5.95 1.13 5.64 5.84 1.16 30 days delinquent (%) 0 0 0 8.96 0.0 28.57 60 days delinquent (%) 0 0 0 4.85 0.0 21.49 90 days delinquent (%) 0 0 0 2.85 0.0 16.64 120 days delinquent (%) 0 0 0 2.29 0.0 11.30 150 days delinquent (%) 0 0 0 1.20 0.0 10.89 180 days delinquent (%) 0 0 0 1.06 0.0 10.23 180 days more delinquent (%) 0 0 0 4.28 0.0 20.25 House liquidation (%) 0 0 0 0.72 0.0 8.43 Loan modification (%) 0 0 0 0.28 0.0 5.24 Deviation local unemployment rates (%) -1.65 -1.93 1.39 Local house price growth rates (%) - 0.005 - 0.004 0.020 Number of observations 7,105 157,544 Table A2: Summary Statistics for Loans Originated in 2005. 48

At Origination Dynamic Sample Variable Mean Median Std. Dev. Mean Median Std. Dev. Age of the loan (months) 0 0 0 15 15 8.84 Share of 2-yr fixed period (%) 87 1 34 87 1 34 Prepayment penalty (%) 99 1 2 83 1 77 Interest-only mortgages (%) 46 0 50 47 0 50 Full document at orig. (%) 49 0 50 50 1 50 Purchase loan (%) 47 0 50 50 1 50 Risk score 436 435 148 386 383 176 Inverse-LTV ratio at origination (%) 78 80 12 94 90 24 Annual income ($1000) 73 67 27 77 75 28 Principal balance ($1000) 281 241 159 274 235 153 Current interest rate (%) 7.99 7.88 1.06 8.02 7.94 1.14 Remaining mortgage terms (months) 360 360 0 346 346 8.84 Monthly payment ($1000) 1.922 1.661 1.061 1.869 1.634 0.995 Maximum lifetime interest rate (%) 14.30 14.24 1.20 14.23 14.12 1.18 Minimum lifetime interest rate (%) 7.49 7.75 1.87 7.42 7.65 1.84 Periodic interest rate cap (%) 1.17 1.00 0.29 1.17 1.00 0.29 Periodic interest rate floor (%) 0.01 0.00 0.08 0.01 0.00 0.08 First rate cap (%) 2.54 3.00 0.75 2.55 3 0.75 Margin for adjustable rate loans (%) 5.92 6.00 1.07 5.90 6.00 1.07 30 days delinquent(%) 0 0 0 8.97 0.0 28.56 60 days delinquent(%) 0 0 0 4.85 0.0 21.49 90 days delinquent(%) 0 0 0 2.85 0.0 16.64 120 days delinquent(%) 0 0 0 2.55 0.0 15.76 150 days delinquent(%) 0 0 0 2.32 0.0 15.07 180 days delinquent(%) 0 0 0 2.11 0.0 14.37 180 days more delinquent(%) 0 0 0 6.85 0.0 25.28 House liquidation (%) 0 0 0 1.13 0.0 10.59 Loan modification (%) 0 0 0 0.49 0.0 6.97 Deviation local unemployment rates (%) -1.02 -1.52 1.70 Local house price growth rates (%) - 1.58 - 1.42 1.68 Number of observations 2,840 64,308 Table A3: Summary Statistics for Loans Originated in 2006. 49

Category 1 Loans Category 2 Loans Category 3 Loans ( d =0,a =0) ( d =1,a =0) ( d =2,a =0) t t t t t t Variable coeff. s.d. coeff. s.d. coeff. s.d. Current Credit Score 0.0030*** 0.0017 0.0010*** 0.0002 -0.0004 0.0003 Income at origination ($1000) -0.0025 0.0017 -0.0061*** 0.0018 -0.0091*** 0.0019 Loan-to-value 0.0004 0.214 0.0004 0.0210 -0.3292 0.3997 Loan-to-value at origination -1.3357*** 0.3241 -0.4142* 0.2230 0.3478 0.3514 Initial monthly payment ($1000) -0.5874*** 0.0979 -0.5164*** 0.1106 -0.3439*** 0.1268 Monthly payment ($1000) 0.5790*** 0.0960 0.5884*** 0.195 0.5765*** 0.1254 Dummy for 3-yr fixed period -0.9434*** 0.360 -0.4273*** 0.1175 -0.4320*** 0.1275 Loan age (months) 1.0312*** 0.0580 0.6360*** 0.0490 0.3970*** 0.0419 Loan age squared - 0.0191*** 0.0012 -0.01130*** 0.0010 -0.0071*** 0.0009 Dummy for full documentation 0.1878*** 0.0640 0.1600*** 0.0672 0.1752*** 0.0718 Deviation local unemp. rate (%) -0.0443 0.0248 0.1797*** 0.0248 0.2060*** 0.0325 Constant -16.1495*** 0.7985 -11.738*** 0.6887 -8.8700*** 0.6275 Number of observations 78,568 52,154 41,221 Pseudo-R2 0.1955 0.1389 0.0991 Table A4: Lenders’ Decisions for Loans in Categories 1-3 (Florida, origination year: 2006). Notes: (1). Results are from logit Regressions where the dependent variable is a dummy for loan modification. (2).***, ** and * denote statistical significance at 1%, 5% and 10% respectively. 50

Modification Liquidation Variable coeff. s.d. coeff. s.d. Current Credit Score 0.0000 0.0003 0.0006*** 0.0001 Income at origination ($1000) -0.1178*** 0.0009 0.0024*** 0.0005 Loan-to-value -1.8278*** 0.5408 4.6127*** 0.5897 Loan-to-value at origination 0.5969*** 0.1636 -1.1419 0.1248 Deviation in local unemp. rates (%) 0.3224 0.0528 - 0.0987 0.0757 Current monthly payment ($1000) 0.0141 0.0816 -0.1620*** 0.0512 Initial monthly payment ($1000) 0.1288 0.0830 0.0703 0.0514 Loan age (months) 0.2019*** 0.0174 -0.1291*** 0.0109 Loan age squared -0.0038*** 0.0004 0.0026*** 0.0002 Months of delinquency -0.3988*** 0.1261 0.6474*** 0.0921 Months of delinquency squared 0.0132** 0.0048 -0.0170*** 0.0320 Loan to value ratio x Months of delinquency 0.4045*** 0.1061 -0.4214*** 0.0866 Loan to value ratio x Months of delinquency squared -0.0138** 0.0046 0.0101*** 0.0032 Dummy for full documentation 0.2027*** 0.0323 -0.0715*** 0.0216 Change in unemp rates x number of late payments -0.0138 0.0107 -0.0156 0.0107 Change in unemp rates x number of late payments2 -0.0000 0.0005 - 0.0008** 0.0004 Dummy for 4-month delinquency 0.6252 0.4113 -3.8769*** 0.2652 Dummy for 5-month delinquency 0.73538** 0.3514 -2.8718*** 0.2029 Dummy for 6-month delinquency 0.4998 0.2973 -2.5894*** 0.1679 Dummy for 7-month delinquency 0.6295** 0.2472 -1.8006*** 0.1295 Dummy for 8-month delinquency 0.4306** 0.2039 -1.1820*** 0.1003 Dummy for 9-month delinquency 0.3777** 0.1661 -0.5726*** 0.0765 Dummy for 10-month delinquency 0.2491 0.1366 -0.2683*** 0.0597 Dummy for 11-month delinquency 0.1405 0.1162 -0.0699 0.0478 Constant -5.6849*** 0.8695 -6.0277*** 0.6546 Number of observations 304,984 Pseudo-R2 0.0933 Table A5: Lenders’ Decisions on Category 4 Loans (Florida, origination year: 2006). Notes: (1). Results are from multinomial logit Regressions where the alternatives are modification, liquidation and waiting (omitted). (2). ***, ** and * denote statistical significance at 1%, 5% and 10% respectively. 51

Modification Variable coeff. s.d. Log (initial interest rate) 0.4131*** 0.0270 Log (margin rate) 0.2042*** 0.0141 Initial balance ($1000) 0.0018*** 0.0006 Remaining balance ($1000) -0.0022** 0.0005 Current Credit Score -0.00004** 0.00002 Income at origination 0.0004*** 0.0001 Loan-to-value ratio 0.2583 *** 0.0203 Loan-to-value ratio at origination -0.1266*** 0.0200 Local unemployment rate deviation -0.0285*** 0.0016 Loan age (months) -0.0103*** 0.0025 Loan age squared 0.0003*** 0.0001 Full documentation 0.0463*** 0.0042 Months of delinquency -0.0209*** 0.0016 Months of delinquency squared 0.0001 0.0001 Loans originated in Arizona -0.0176 0.0106 Loans originated in California -0.0136 0.0098 Loans originated in Florida -0.0197** 0.0098 Constant 0.7171*** 0.0719 Number of observations 18,646 Pseudo-R2 0.2738 Table A6: Lenders’ Decisions on Modified Interest rates (origination year: 2006). Note: ***, ** and * denote statistical significance at 1%, 5% and 10% respectively. 52