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

Transcription

1 The Dynamics of Adjustable-Rate Subprime Mortgage Default: A Structural Estimation Hanming Fang You Suk Kim Wenli Li May 27, 2015 Abstract 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 build and estimate a dynamic structural model of adjustable-rate mortgage defaults using unique mortgage loan level data. The data contain detailed information not only on borrowers mortgage payment history and lender responses but also on their broad balance sheet. Our structural estimation suggests that the factors that drive the borrower delinquency and foreclosure differ substantially by the year of loans origination. For loans that originated in 2004 and 2005, which precedes the severe downturn of the housing and labor market conditions, the interest rate resets associated with ARMs, as well as the housing and labor market conditions do not seem to be important factors for borrowers delinquency behavior, though they are important factors that determine 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 importantly to their high delinquency rates. Countefactual policy simulations also suggest that monetary policies in the most optimistic scenario might have limited effectiveness in reducing the delinquency rates of 2004 and 2005 loans, but could be much more effective for 2006 loans. Interestingly, we found that automatic modification loans in which the monthly payment and principal balance of the loans are automatically reduced when housing prices decline can reduce delinquency and foreclosure rates, and significantly so for 2006 loans, without having much a negative impact on lenders expected income. Preliminary and Incomplete. All comments are welcome. The views expressed 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. Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA and the NBER. hanming.fang@econ.upenn.edu Division of Research and Statistics, Board of Governors of the Federal Reserve System. You.Kim@frb.gov. Department of Research, Federal Reserve Bank of Philadelphia. wenli.li@phil.frb.org.

2 Keywords:Adjustable-Rate Mortgage, Default JEL Classification Codes: C1, C4, G2 1 Introduction The collapse of the subprime residential mortgage market played a crucial role in the recent housing crisis and the subsequent Great Recession. 1 At the end of 2007, subprime mortgages accounted for about 13 percent of total first-lien residential mortgages outstanding but over half of total house foreclosures. The majority of the subprime mortgages, by number as well as by value, had adjustable rates and the fraction of the adjustable-rate subprime mortgages in foreclosure at 17 percent was much higher than the fraction of the fixed-rate subprime mortgages at 5 percent (Frame, Lehnert, and Prescott 2008, Table 1). In response to these developments, many government policies have been designed and carried out that aimed at changing the incentives of these borrowers to default. 2 Few structural models, however, exist that can guide us in these efforts especially since most of them have had limited success. 3 In this paper, we first develop a dynamic structural model to study the various incentives adjustable-rate subprime borrowers have to default and how these incentives change under different policies. Our study focuses on the period between the time when either a mortgage is granted and the time when the mortgage is repaid (including refinance), or the house is foreclosed, or the end of the sample period. More specifically, at each period, a borrower decides whether to repay the loan (and be current) or not repay the loan (and stay in various delinquent status), taking as given lender s possible responses which include various loss-mitigation practices such 1 There 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 a high leverage as measured by either the debt-to-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). 2 To 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 Homeowners refinancing program passed by Congress in the spring 2008; Making Home Affordable (MHA) initiative in conjunction with the Home Affordable Modification Program (HAMP) and the Home Affordable Refinance Program (HARP) launched by the Obama administration in March 2009 (HAMP). See Gerardi and Li (2010) for more details. 3 Over 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 Similarly, HAMP was designed to help as many as 4 million borrowers avoid foreclosure by the end of By February 2010, one year into the program, only 168,708 trial plans had been converted into permanent revisions. Through January 2012, a population of 621,000 loans had received HAMP modifications. See 1

3 as mortgage modification, liquidation, and waiting (i.e., doing nothing). Relative to the existing structural models on mortgage defaults which we review below, our theoretical framework has the two key distinguishing features: first, in our model default is not the terminal event, and second, besides liquidation we also consider lenders various loss mitigation practices such as loan modification. We then empirically implement our model using unique mortgage loan level data. Our data not only contains detailed information on borrowers mortgage payment history and lenders responses, but also detailed credit bureau information (from TransUnion) about borrowers broader balance sheet and income. We are thus one of the first to utilize borrowers credit bureau information to understand their mortgage payment decisions. 4 To track movements in home prices and local employment situation, we further merge our data with zip code level home price indices and county level unemployment rates. Three main forces drive adjustable-rate mortgage (ARM) borrowers mortgage payment decisions: changes in home equity, changes in income, and changes in monthly mortgage payment. 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 monthly mortgage payments affect borrowers liquidity position. In principal, 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, rising interest rates, and/or tightened lending standards. As a result, these constrained borrowers have no choice but to default on their mortgages. To arrive at the relative importance of these different drivers of default, we analyze our structurally estimated model under various counterfactual scenarios. Our structural estimation suggests that the factors that drive the borrower delinquency and foreclosure differ substantially by the year of loans origination. For loans that originated in 2004 and 2005, which precedes the severe downturn of the housing and labor market conditions, the interest rate resets associated with ARMs, as well as the housing and labor market conditions do not seem to be important factors for borrowers delinquency behavior, though they are important factors that determine 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 importantly to their high delinquency rates. Our counterfactual policy simulations suggest that monetary policies in the most optimistic scenario might have limited effectiveness in reducing the delinquency rates of 2004 and Elul, Souleles, Chomsisengphet, Glennon, and Hunt (2010) also use credit bureau information to study mortgage default decisions in their empirical analysis. 2

4 loans, but could be much more effective for 2006 loans. Interestingly, we found that automatic modification loans in which the monthly payment and principal balance of the loans are automatically reduced when housing prices decline can reduce delinquency and foreclosure rates, and significantly so for 200 loans, without having much a negative impact on lenders expected income. There are several structural models on mortgage defaults and foreclosures. None of them, however, captures lenders decisions beyond setting interest rates and liquidation despite the use of other loss-mitigation tools such as mortgage modification in practice. Furthermore, they all treat default as a terminal event that leads to liquidation with certainty. We briefly review several closely related papers. Bajari, Chu, Nekipelov, and Park (2013) is the closest in spirit and methodology to our paper. Both papers provide and estimate using micro data dynamic structural models to understand borrowers behavior and to conduct policy analyses. There are, however, key differences in addition to the two mentioned previously. First, we estimate borrowers decisions structurally, i.e., our approach resolves borrowers optimal behavior whenever policy changes. By contrast, Bajari, et al. (2013) can only accommodate policy interventions that result in state-variable realizations that are actually observed for a subset of borrowers in the data and does not change the state transition process. In other words, the new optimal behavior is correctly captured by the estimation decision rules of some of the borrowers in the data. We view this as a serious limitation of their model. Additionally, they focus on fixed-rate subprime mortgages which are much less prevalent than the adjustable-rate subprime mortgages. Our model, by contrast, nest both cases. Finally, we bring in income and credit score from credit bureau files which afford us additional information not available in the mortgage data. Campbell and Cocco (2014) study a dynamic model of households mortgage decisions incorporating labor income, house price, inflation, and interest rate risk to quantify the effects of adjustable versus fixed mortgage rates, mortgage loan-to-value ratio, and mortgage affordability measures on mortgage premia and default. Corbae and Quintin (2013) 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 a broader recourse on default rates and welfare. Mitman (2012) considers the interaction of recourse and bankruptcy on mortgage defaults. Chatterjee and Eyigungor (2015) analyze default of long-duration collateralized debt. None of these works make use of mortgage loan level data as in our paper and that of Bajari et al. (2013). There are several recent empirical papers that adopt regression techniques to study lenders loss mitigation practices and the impact of government intervention policies on these practices. For example, Haughwout, Okah, and Tracy (2010) estimate a competing risk model using 3

5 modifications of subprime loans originated between December 2004 and March 2009 excluding capitalization modifications. They find a substantial impact of payment reduction on mortgage re-default rates. Agarwal, Amromin, Ben-David, Chomsisengphet, and Evanoff (2010) analyze lenders loss mitigation practices including liquidation, repayment plans, loan modification, and refinance of mortgages originated between October 2007 and May 2009 from OCC-OTS Mortgage Metrics data and find a much 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. We innovation over these papers lies in our structural modeling of borrowers incentives to default to lenders loss-mitigation practices and to policies that affect these practices. Finally, the paper also adds to the increasing literature on the recent subprime mortgage crisis, including, among many others, Foote, Gerardi, and Willen (2008), Demyanyk and van Hemert (2011), Keys, Benjamin, Tanmoy Mukherjee, Amit Seru, and Vikrant Vig (2010), and Gerardi, Kristopher, Andreas Lehnert, Shane Sherlund, and Paul Willen (2008). 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 some summary and 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, Libor interest rates, and incomes. 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 implications 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 Our data come from three differences sources, the CoreLogic Private Label Securities data ABS, the CoreLogic Loan Modification data, and the TransUnion-CoreLogic Credit Match Data. The CoreLogic ABS data consist of loans originated as subprime and Alt-A loans and represents about 90 percent of the market. The data include loan level attributes generally required of issuers of these securities when they originate the loans as well as historical performance, which are updated monthly. The attributes include borrower characteristics (credit scores, owner occupancy, documentation type, and loan purpose); collateral characteristics (mortgage loan-tovalue ratio, property type, zip code); and loan characteristics (product type, loan balance, and loan status). 4

6 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 principals are forgiven, whether the mortgage terms are changed, etc. The merge of the two data sets are straightforward as each loan is uniquely identified by the same loan id in both data sets. The TransUnion-CoreLogic Credit Match Data provide consumer credit 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, number of credit accounts, credit balances, and delinquency history. We then merge our data with CoreLogic monthly zip code level house price index based on repeated sales and county level unemployment rates from the Bureau of Labor Statistics. Thus our constructed data have several advantages over most of those used in the literature. First, the match with the mortgage modification data allow us to identify lenders actions more closely and therefore separate delinquent mortgages that are self-cured from delinquent mortgages that become current after lender modification. Second, the TransUnion data enable us to capture borrowers other liabilities as well as the payment history of these liabilities. This information is important for borrowers mortgage payment decision. 2.2 Data Description We focus on subprime adjustable-rate mortgage loans originated in the four crisis states, Arizona, California, Florida, and Nevada, between 2004 and In particular, we take a 1.75 percent random sample of adjustable-rate mortgages with an initial interest rate fixed period of two or three years and a mortgage maturity of 30 years that are for borrowers primary residence, first lien, and not guaranteed by government agencies such as Fannie Mae, Freddie Mac, the Federal Housing Administration, and Veterans Administration. 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 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 dynamic sample period. The average age of the loan is 16 months in the sample and the median 5 The subprime mortgqage market dried up after the mortgage crisis broke out in

7 Table 1: Summary Statistics. At Origination Whole Sample Period Variable Mean Median Std. Dev. Mean Median Std. Dev. Age of the loan (months) Share of 2-yr fixed period (%) Prepayment penalty (%) Interest-only mortgages (%) Full document at orig. (%) Purchase loan (%) Risk score Inverse-LTV ratio at orig. (%) Annual income ($1000) Principal balance ($1000) Current interest rate (%) Remaining mortgage terms (months) Monthly payment ($1000) days delinquent(%) days delinquent(%) days delinquent(%) days delinquent(%) days delinquent(%) days delinquent(%) days more delinquent(%) House liquidation (%) Loan modification (%) Deviation local unemployment rates (%) Local house price growth rates (%) Number of observations 16, ,811 6

8 is 14 months. At origination, 81 percent of the sample are loans with two-year fixed-rates. Through the sample period, however, 76 percent of the sample are loans originated with twoyear initial fixed-rate period indicating that more of those loans have terminated. 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 for the whole sample. About half of the mortgages have full documentation both at the origination and through the sample period. While about 43 percent of the mortgages are purchase loans at the origination, the ratio increases to about 48 percent indicating that purchase loans are less likely to default than refinance loans. The risk scores are estimated by TransUnion. They range between 150 and 950 with a high score indicating low risk. Consistent with being subprime, mortgage borrowers in the sample all have relatively low risk scores, averaging about 445 at origination, and the scores deteriorate somewhat as the loans age suggesting that the relatively less riskier borrowers may have refinanced their loans and therefore left our sample. Additionally, both the average and the mean mortgage loan-to-value ratios exceed 100 at origination and they do not change much as the loans aged. 6 The annual household income estimated by TransUnion average about $72,000 at origination and $77,000 dynamically. The fact that both mean and median income are higher in the dynamic sample than at origination suggests that mortgage loans from low income households are terminated earlier in our sample. Loan balances average $259,000 at origination with a median of $228,000. These numbers are not very different from their dynamic counterparts suggesting that borrowers do not make much loan payments during our sample period. The mortgage interest rates average about 7.13 percent at origination with a median of 6.99 percent. Interestingly, 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 to higher rates after the initial fixed-rate period expires. Unemployment rates tend to be lower than their local averages. Local house prices, on the other hand, all depreciate. The two most striking observations emerge from Table 1. First, some mortgages stayed in delinquency status for a long time without being liquidated. Particularly, in our sample, close to 7 percent of loans are 30-day delinquent, 3 percent are 60-day delinquent, 2 percent are 90-day delinquent, etc. What is most surprising is that close to 4 percent of the loans are actually over half a year delinquent. The house liquidation rate, by contrast, is only 0.64 percent. Second, about 0.26 percent of all mortgage loans are modified by their lenders. This ratio is obviously much higher if we consider loans that are delinquent. regarding lenders decisions in more details in the next subsection. We elaborate the second observation 6 We report inverse mortgage-loan-to-value ratio in Table 1. The reason is because in our estimation we assume that house prices follow an AR(1) process with a normal distribution. The mortgage loan-to-value ratio, which is the inverse of a normal random variable, does not have a mean. See the model section for more details. 7

9 Loan Status (beginning of the month) At Liquidation (%) At Modification (%) Current months months months months months months months months months months months months months months months months More than 17 months Number of observations 2, Table 2: Loan Status at the Beginning of the Month when Liquidation or Modification Occurs. 2.3 Lenders Choices: Descriptive Statistics From Table 1 we know that lenders do not always respond to borrowers mortgage delinquency immediately by liquidating them. We study lenders decisions in more details in this subsection. We start by presenting the months of delinquency at liquidation and mortgage modification in Table 2. As can be seen, mortgage liquidation typically occurs when the borrower is between 6 month and 9 month delinquent. While houses with loans less than 3 months delinquent rarely gets liquidated, many houses are liquidated when the mortgage is over one year delinquent. As a matter of fact, about 4.46 percent of the loans liquidated is over 17 months delinquent. As a side note, the average loan age is 27 months at liquidation. About half of the liquidation occurred in 2008, 30 percent in 2007, and 8 percent in About 6 percent of the liquidation occurred in the first two months of Turning to loan modifications, they 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 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 8

10 Variable Reduction No Change Increase Monthly payment (percentage) Average change in monthly payment ($) (443) (19) (1,141) Balance (percentage) Average change in balance ($) -34, ,248 (39,603) (143) (11,993) Interest rate (percentage) Average change in interest rate (%) (1.415) (0.00) n/a Table 3: Terms of Modification. Notes: No change refers to monthly payment change less than 50andtotalloanbalancechangelessthan500. Standard deviations are in parenthesis. of interest rate reset. These suggest that servicers are aware that these borrowers will default imminently without mortgage modification. 7 Table 3 presents 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. However, about 9 percent of the modifications produce higher payments of about $287 on average. Capitalization of modification is very common with arrearage added to total principal balance. Indeed over 64 percent of the modified loans have an increase of principal balance of $12,248 on average. Only 5 percent of the loans have a principal reduction averaging $34,030. Nonetheless, more 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 any interest rate increase. All of the loans are brought into the current status after modification. 3 The Model The model describes a borrower s behavior from the time his mortgage is originated until period T which we specify later. We do not model lenders decision but estimate it parameterically, which borrowers take as given. Time is discrete and finite with each period representing one month. Let x t denote the state vector in period t, which includes time-invariant borrower and mortgage characteristics such as information collected at mortgage origination and house location as well as time-varying characteristics such as a mortgage s delinquency status, interest rates, local housing market conditions, local unemployment rates, etc. 7 Haughwout (2010) documented similar observations but their sample are 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). 9

11 3.1 Choice set In each period t, after information x t is realized, a borrower chooses an action j. He has three choices: make the monthly mortgage payment, skip the payment, or pay off the mortgage. The option to pay off the mortgage, however, is only available to borrowers who are current on mortgage payment. 8,9 Moreover, the borrower has different options of making mortgage payments, depending on the number of late monthly payments denoted by d 0 he has. If the borrower is current on his mortgage payment, then he decides whether to make one monthly payment P t. If the borrower is one month behind on the payment, then he makes the following decisions: pay just P t and stay one-month-delinquent, pay 2P t to be current again, or do not pay anything. To generalize, if the borrower has d unpaid monthly payments at the beginning of time t, he can make the following decisions: pay P t, 2P t,, (d + 1)P t, or nothing. To simplify the problem, for d 2, we assume that if the borrower decides to pay he only has the options to pay (d 1)P t, dp t, or (d + 1)P t to become two-month delinquent, one-month delinquent, or current, respectively. 10 Formally, a borrower s choice set with d unpaid payments is denoted by J(d): {0, 1, paying off}, if d = 0; J(d) = {0, 1, 2}, if d = 1; {0, d 1, d, d + 1}, if d 2, where the number zero refers to the action of not making any payment. For the remaining paper, we will sometimes denote the choice set by J(x t ) instead of J(d) because x t includes the loan delinquency status d. We denote the borrower s chosen number of payments in period t as n t J (d t ). 3.2 State Transition The evolution of the state variables is captured by the transition probability F (x t+1 x t, j), where, as discussed previously, x t represents the state vector, and j represents the borrower s action at time t. We now discuss each of the state variables. 8 In the data, some borrowers pay off their mortgages even when they are delinquent. Based on our conversation with CoreLogic, we believe this is mostly because of reporting lag as borrowers typically stop making payments on their current mortgage during mortgage refinance or house sale. 9 In reality, a borrower can pay off the mortgage by refinancing or by selling the house. Our data, unfortunately, do not allow us to make such a distinction. 10 It is rare in the data for borrowers to make payments after they are more than 2 months late that would still leave them 60 days or more delinquent. Additionally, recall that a borrower in our model can still choose not to pay and hence be more than 3 months late on his mortgages. 10

12 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. Given the data, we focus on loans that have a two or three years of initial fixed period and 30 years maturity. Almost all of the loans have a six-month adjustment frequency after the initial fixed period. In terms of notation, let i 0 denote the initial interest rate and let i r denote the new mortgage interest rate at the r-th reset. For example, i 1 denotes the interest rate at the first reset right after the fixed-rate period. The term margin represents the margin rate. Since most ARM in our data are indexed to the six-month Libor rate, we use libor t to denote the index rate. The lifetime interest rate floor and cap are represented by l flo and l cap, respectively. The cap on interest rate change in each period is represented by p cap. 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, therefore, denote the cap for interest rate change at the first reset by f cap. 11 Combining all the elements, the new interest rate at the r-th reset in period t is calculated as follows: max{i r 1 f cap, l flo, min{margin + libor t 1, i r 1 + f cap, l cap }}, if r = 1; i r = (1) max{i r 1 p cap, l flo, min{margin + libor t 1, i r 1 + p cap, l cap }}, if r > 1. 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 life long 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, life time mortgage interest rate cap. Note that libor t evolves stochastically. The borrower, therefore, needs to form expectations about future values for Libor in order to predict the interest rate he will have to pay. The values for the other mortgage parameters {margin, l flo, l cap, f cap, p cap } are fixed throughout the life of the mortgage. It follows from Equation (1) that i r [max{i r 1 f cap, l flo }, min{i r 1 + f cap, l cap }] if r = 1 and that i r [max{i r 1 p cap, l flo }, min{i r 1 +p cap, l cap }] if r > 1. In other words, {l flo, l cap, f cap, p cap } 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 f cap, p cap and l cap l flo are low. 11 Usually, f cap is larger than p cap. That is, the interest rate change is typically larger at the initial reset than at subsequent resets. 11

13 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 number of resets that occurred up to period t. 12 Note that either r(t + 1) = r(t) or r(t + 1) = r(t) + 1. The former is true when both period t and t + 1 are in between two resets, and i r(t+1) = i r(t). The latter is true when an interest rate is just reset in period t + 1, and i r(t+1) = i r(t)+1, where i r(t)+1 is calculated using the formula in (1). Once the new interest rate is determined, the new monthly payment can be calculated based on the interest rate and the beginning of the period mortgage balance. Consider a borrower in period t with remaining mortgage balance bal t 1 and interest rate i r(t). The borrower s mortgage monthly payment P t is calculated so that if the borrower makes a fixed payment of P t until the 360th period, he will pay off the entire mortgage; specifically, P t = i bal r(t) t ( 1+ i r(t) 12 ) 360 t+1, (2) and the new balance entering period t + 1 is updated to: 1 bal t = bal t 1 1 ( 1 + i r(t) 12 ) 360 t. (3) Remark: Note that the lenders decisions affect the transition of borrowers state variables, i.e., F (x t+1 x t, j) incorporates the lenders responses. If the lender chooses to modify the loan, it will lead to possible changes of the borrower s loan status, interest rate, monthly payment and mortgage balance; 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 t, house price h t, changes in local unemployment rate UNR t, borrower credit score CS t, and borrower income y t. The evolution of these state variables are as follows: Number of late monthly payments: d t+1 = d t n t + 1, where n t J (d t ) is the number of monthly payments a borrower makes at time t. Libor: We assume that the borrower s belief regarding the evolution of Libor rates is that 12 For example, if the initial fixed-rate is at least as long as t periods, r(t) = 0. If an interest rate is reset for the second time in period t, r(t) = 2. 12

14 it follows an AR(1) process in logs ln(libor t+1 ) = λ 0 + λ 1 ln(libor t ) + ɛ libor,t, where ɛ libor,t N(0, σ 2 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 t+1 = λ 2 + λ 3 h t + ɛ h,t, where ɛ h,t N(0, σ 2 h ) is assumed to be serially independent. Local unemployment rate: We focus on the deviation of the current unemployment rate in a county from the average of monthly unemployment rates from 2000 to 2009 in the same county, which we denote by UNR. We assume that the 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 + ɛ unr,t, where ɛ unr,t N(0, σ 2 UNR ) is assumed to be serially independent. Credit score (CS): We assume that the borrower s belief regarding the evolution of the log of his credit score is that it follows the following process: ln (CS t+1 ) = λ 6 + λ 7 ln (CS t ) + λ 8 1[d = 1] + λ 9 1[d = 2] + λ 10 1[d = 3] + λ 11 1[d 4] + ɛ cs,t, where ɛ cs,t N(0, σ 2 CS ) is assumed to be serially independent. Income (Y t ): We assume that the borrower s belief regarding the evolution of his income is that it follows an AR(1) process: Y t+1 = λ 12 + λ 13 Y t + ɛ y,t, where ɛ y,t N(0, σ 2 Y ) 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 13

15 various characteristics and are invariant to policy changes. 13 Borrowers take these decision rules as given. We provide details in the estimation section. 3.4 Payoff Function We specify a borrower s current-period payoff from taking action j in period t as u j (x t ) + ɛ jt, where u j (x t ) is a deterministic function of x t and ɛ jt is a choice-specific preference shock. The vector ɛ t ( ɛ 1t, ɛ J(xt)t) is drawn from the Type I Extreme Value distribution that is independently and identically distributed over time. 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: u n (x t ) = { Pt β 1 + (n 1)P t β 2 + CS t β 3 + Y t β 4 + UNR t β 5 + X 0 β 6 + ξ d + ζ n if n 1 ξ d if n = 0, (4) where P t denotes the borrower s monthly payment in period t. The first term P t β 1 represents the disutility from one month s payment. n 1 months payment. 14 make a payment. The second term (n 1)P t β 2 is the disutility of The next term determines the borrower s ability or willingness to Specifically, CS t is the borrower s updated current credit score provided by TransUnion. It captures not only the borrower s past payment history but also his ability to obtain future credit. The term Y t represents the borrower s current income imputed by TransUnion. We define UNR t = UNR t UNR, where UNR t and UNR denote the current and the average unemployment rates in the borrower s county of residence, respectively. 15 While UNR t captures current local macroeconomic conditions, its average captures unobserved timeinvariant differences in macroeconomic conditions across counties. The term X 0 is a collection of the borrower s initial characteristics at origination which contains original monthly payment amount (P 0 ),inverse loan-to-value ratio at origination (ILT V 0 ), the year of loan origination, and whether the borrower s income is fully documented. ξ d is a dummy variable for the borrower s payment status d at the beginning of the period. We assume that ξ d = ξ d for d, d 3. Finally, ζ n is a constant for taking action n. We normalize ζ 0 = 0 because only relative utility is identified 13 This characterization of lender behavior seems to be consistent with the data. In a companion paper, we endogenize lenders decisions and investigate why they did not change much after the government introduced various policies to reduce foreclosures and encourage loan modifications. 14 We use P tβ 1 + (n 1)P tβ 2, instead of a single term np tβ 1 to allow for the possibility that paying more than a single monthly payment amount could have a different utility cost than making only one payment. 15 The average is taken over the periods of 2000 to

16 in a discrete choice model. When a borrower, who is current on the mortgage (d = 0), chooses to pays off the mortgage (j = payoff), the deterministic part of the flow payoff T u paying off (x t ) = δ t β 7 + P P N t β 8 + CS t β 9 + Y t β 10 + ILT V t β 11 + ILT V 0 β 12 + ζ paying off,t, t =t+1 Where δ is the discount factor (which we set to be 0.99 in our estimation), P P N t is whether the borrower has to pay a prepayment penalty if prepaying in period t, ILT V t is the ratio of the borrower s current house price to the remaining balance, i.e., the inverse of mortgage loan-tovalue ratio, and ILT V 0 is the inverse mortgage loan-to-value ratio at origination. 16 We assume that the model is terminated when the borrower pays off the mortgage. 17 If the house is liquidated, then V t (liquidated) = 0. 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. 18 The model is then terminated, and the borrower receives the terminal payoff β 13 + β 14 CS T + β 15 ILT V T, if current at T V T (x T ) = (6) 0, otherwise. Remark: In our framework, we assume that the lender can affect a borrower s flow utility only if the lender forecloses (or liquidates) the house. If the lender chooses to modify the loan terms, or wait, we assume that the borrower s flow utility is affected only to the extent that the modified loan term affects the borrower s monthly payment. Of course, dynamically, the lender s choices affect the borrowers ability to stay current in the mortgage and subsequently the probability of being foreclosed. (5) 3.5 Value Function The borrower sequentially maximizes the sum of expected discounted flow payoffs in each period t = 1,..., T. Let us define σ to be a borrower s decision rule such that σ j (x t, ɛ t ) = 1 if a borrower chooses action j given (x t, ɛ t ). Recall F (x t+1 x t, j) denotes a transition probability 16 We assume that the house price follows an AR(1) process with the shock drawn from a normal distribution. The inverse of a normal random variable, however, does not have mean. In the analysis, we therefore use the inverse loan-to-value ratio ILT V instead of the mortgage loan-to-value ratio. 17 We make this assumption because the mortgage loan exits our data base once the borrower pays off or refinance the mortgage. 18 To 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

17 function of state variables which depends on the current state x t and an endogenous choice j. We can then express the borrower s problem recursively as follows: V (x t ; σ) = E ɛt j J(x t) { } σ j (x t, ɛ t ) u j (x t ) + ɛ jt + δ V (x t+1 ; σ)df (x t+1 x t, j). (7) x t+1 X t The borrower s optimal decision rule σ is such that V (x t ; σ b ) V (x t; σ) for any possible decision rule σ in all x t (t = 1,, T ). 4 Estimation We define the choice-specific value function for action j in period t, v j (x t ) as v j (x t ) = u j (x t ) + δ V (x t+1 ; σ )df (x t+1 x t, j). x t+1 X t (8) The value function can then be written as: V (x t ; σ ) = E ɛt j J(x t) σ j(x t, ɛ t ) {v j (x t ) + ɛ jt }. (9) In order to solve for the optimal decision rule σ, we use backward induction following the standard methods on dynamic discrete choice model with a finite number of period (see, for example, Rust (1987, 1994a, 1994b) and Keane and Wolpin, 1993). We start from period T 1. The choice-specific value function in period T 1 is given by: v j (x T 1 ) = u j (x T 1 ) + δ V (x T )df (x T x T 1, j). x T X T (10) Note that the value function for period T, V (x T ), does not depend on σ ; the optimal decision rule in period T 1 is then that: σ { } j(x T 1, ɛ T 1 ) = 1 iff v j (x T 1 ) + ɛ j,t 1 max vj (x T 1 ) + ɛ j j J,T 1. (11) Given the functional form assumption for ɛ T 1, we can show, following Rust (1987), that V (x T 1 ; σ ) = ln exp(v j (x T 1 )) + γ (12) j J where γ is the Euler constant. 16

18 Now let us consider the borrower s optimal decision rule in period T 2. In order to calculate v j (x T 2 ), we need to know x T 1 X t V (x T 1 ; σ )df (x T 1 x T 2, j), which can be calculated using equation (12). We then derive σ j (x T 2, ɛ T 2 ) and V (x T 2 ; σ ) similarly as we did in period T 1. We repeat this process until we reach the initial period. In general, the borrower s optimal decision rule in period t is: σ { } j(x t, ɛ t ) = 1 if v j (x t ) + ɛ jt max vj (x t ) + ɛ j j J t, (13) and V (x t ; σ ) = log exp(v j (x t )) + γ. (14) j J Moreover, a borrower s conditional choice probability for alternative j J (x t ) is given by: p j (x t ; σ ) E ɛt [σ j(x t, ɛ t )] = exp(v j (x t )) j J exp(v j (x t)). (15) We estimate the model using maximum likelihood. In the data, we observe a path of states and choices for each individual i: (x i,a i ) {(x it,a it )} T t=1, where a it {a ijt } j J(xit ), and a ijt is defined to be a dummy variable equals to one when individual i chose action j in period t. The likelihood of observing (x i,a i ) given initial state x i1 and parameter θ for individual i is: T L(x i, a i x i1 ; θ) = l(a it, x i,t+1 x it ; θ), (16) t=1 where l(a it, x i,t+1 x it ; θ) is the likelihood of observing (a it, x i,t+1 ) given state x it and parameter θ: l(a it, x i,t+1 x it ; θ) = [p j (x t ; θ)f(x i,t+1 x it, j)] a ijt. (17) j J(x it ) Parameter estimate θ maximizes the log-likelihood for the whole sample, i.e, I θ = arg max ln L(θ) = ln (L(x i, a i x i1 ; θ)) I T = i=1 t=1 j J(x it ) i=1 a ijt [ln (p j (x t ; θ)) + ln f(x i,t+1 x t, j)]. 17

19 5 Estimation Results 5.1 Lenders Decisions As previously discussed, we estimate lenders policy functions parametrically using Logit or multinomial logit regressions. In any period t, we assume that the timing of interaction between the borrower and the lender is as follows. The borrower enters period t with a delinquent status d t, makes the payment decision a t, after which the lender makes the decisions regarding whether to modify, liquidate, or do nothing about the loan based on the delinquent status of the loan at the end of the period t. 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 t = 0, a t = 0). Borrowers who are current in the beginning of the period, but do not make a payment in the period; Category 2: (d t = 1, a t = 0). Borrowers who are one month delinquent in the beginning of the period, but do not make a payment in the period; Category 3: (d t = 2, a t = 0). Borrowers who are two month delinquent in the beginning of the period, but do not make a payment in the period; Category 4: (d t 3, a t = 0). Borrowers who are three-or-more-month delinquent at the beginning of 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 d t 1, and if he makes a t 1 payment, the lender s only choice is waiting even though the status of the loan at the end of the period is still one or more month delinquent (i.e. a t < d t +1). In our specification of the lenders decisions, we note that lenders 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. For loans in category 4, we assume that lenders decides among three options: modification, liquidation, and waiting. We specify a multinomial logit function to represent the lenders probabilities of choosing the three alternatives. The estimation results for lenders decisions are reported in Appendix Tables A1 and A2. For all regressions, the default 18

20 state of the loan is Nevada and the default year of the loan is In all regressions, the default lender decision is waiting. Category 1 Loans. For category 1 loans, lenders are more likely to modify if the borrower has a high credit score, high loan-to-value ratio, high monthly payment but low initial monthly payment, and full documentation. The loan is also more likely to be modified if it is still within the initial fixed period though the probability of modification decreases with the number of months left in the fixed-rate period. An older loan is slightly less likely to be modified. Compared to loans made in 2006, loans originated in 2004 or 2005 are much less likely to be modified perhaps reflecting the quality of those loans as they were made during the peak of the housing boom and borrowers were of less quality. However, loans originated in 2004 and 2005 are more likely to be modified as they age than those originated in Higher than historical local average unemployment rates reduce lenders incentive to modify. Category 2 Loans. For category 2 loans, the factors that explain modification probability are similar to those that are current at the beginning of the period with a few exceptions. Older loans now are more likely to be modified. There are no longer cohort effects, but geographic pattern appears. Loans in California and Florida are more likely modified than loans in Nevada. Category 3 Loans. For category 3 loans, a borrower is more likely to receive modification if he has high a credit score, low income, low initial loan-to-value ratio, still in the initial fixed period, and with full documentation. Loans originated in 2005 are less likely to be modified though are more likely to be modified as they age. Category 4 Loans. For category 4 loans, we include many more explanatory variables to our multinomial logit regressions. A loan is more likely modified if income is low, initial loan-to-value ratio is high, local unemployment rate goes up, the borrower has more missed payments, and the loan is relatively seasoned with full documentation. As in the previous cases, loans made in 2004 and 2005 are less likely modified. Loans in California and Florida are more likely modified. Furthermore, most loans are modified when they are 9 or 10 months delinquent. In terms of liquidation, interestingly, a high credit score and high income make borrowers marginally more likely to be liquidated. Lower current mortgage loan-to-value ratio but higher initial loan-to-value ratio increase the liquidation probability. Loans that are still in the interest-only period and loans made in 2004 are also more likely liquidated. Full documentation marginally reduces liquidation probability. Arizona is more likely to liquidate than Nevada but Florida less likely. The more missed payments, especially when mortgage loan-to-value is high, 19

21 the more likely the loan will be liquidated. However, the effect is weaker when local unemployment rates also go up. Finally, the most liquidation occurs when the loan misses 8 or 9 months of payment. Remark. Note that in the data section we documented that the most popular modification is recapitalization coupled with interest rate reset. After modification, borrowers payment status is brought to current. For simplification, we assume in our analysis that the new reset interest rate is the initial teaser interest rate during the fixed-interest period of ARM. We also assume that the modified loan is a fixed rate mortgage with the maturity equal to the remainder of the initial loan. This simplification allows us to avoid having to estimate a separate lender decision rule on the new reset interest rate upon modification Estimates of the Stochastic Processes In Section 3.2, we also described that borrowers and lenders have beliefs about some stochastic processes such as the evolution of Libor rates, the local housing prices, local unemployment rates, income and credit scores. We assume that the borrowers have rational expectations about these processes and estimate them using the ex post realizations of these processes. The estimates for these stochastic processes 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. Table 4 shows that 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. 5.3 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 see that a borrower derives negative utilities from high mortgage payments, and more so if he makes more than one payment in a given month. Additionally, he is more likely to make payments when his credit score is high but less likely to make payments when the local unemployment rate is high as his payment ability is positively correlated with his credit score but negatively correlated with the local unemployment rate. Interestingly, the higher the current income, the less likely the borrower will make the mortgage payment. This counter intuitive result may stem from the imprecise nature of the income estimate by TransUnion. In terms of conditions at origination, a borrower s payment ability 19 In the data, the mean differnce between the new interest rate upon modification and the initial teaser rate is 16 basis points and the median is 37 basis points. Therefore this assumption is a rough approximation. 20

22 Coefficient Estimate Standard Errors Panel A: Libor ln (libor t+1 ) = λ 0 +λ 1 ln (libor t ) + ɛ libor,t λ λ *** σ libor *** Panel B: House Price h t+1 = λ 2 +λ 3 h t +ɛ h,t λ *** λ *** σ h *** Panel C: Local Unemp. Rates UNR t+1 = λ 4 +λ 5 UNR t +ɛ unr,t λ *** λ *** σ unr *** Panel D: Income Y t+1 = λ 12 +λ 13 Y t +ɛ y,t λ *** λ *** σ Y *** 2.24e-05 Panel E: Credit Score: ln (CS t+1 ) = λ 6 +λ 7 ln (CS t ) + λ 8 1[d = 1] +λ 9 1[d = 2] + λ 10 1[d = 3] + λ 11 1[d 4] + ɛ cs,t λ *** λ *** λ *** λ *** λ *** λ *** σ CS *** 7.93e-05 Table 4: Coefficient Estimates for Stochastic Processes 21

23 Coefficient Estimate Std. Err. Panel A: Coefficients in u n (x t ) as specified in (4) P t : (β 1 ) *** (0.0055) (n 1)P t : (β 2 ) ** (0.0032) CS t : (β 3 ) *** (0.0068) Y t : (β 4 ) *** (0.0087) UNR t : (β 5 ) *** (0.0011) P 0 : (β 6,1 ) *** (0.0069) ILT V 0 : (β 6,2 ) *** (0.0058) Full Doc: (β 6,3 ) *** (0.0015) Orig 2004: (β 6,4 ) (0.0025) Orig 2005: (β 6,5 ) ** (0.0024) Constant: (ξ 0 ) *** (0.0350) Constant: (ξ 1 ) *** (0.0344) Constant: (ξ 2 ) *** (0.0375) Constant: (ξ 3 ) *** (0.2631) Constant: (ξ 4+ ) *** (0.0093) Constant: (ζ 1 ) *** (0.0382) Constant: (ζ 2 ) *** (0.0688) Constant: (ζ 3 ) *** (0.1051) Constant: (ζ 4+ ) *** (0.2547) Panel B: Coefficients in u paying off (x t ) as specified in (5) T t =t+1 δt : (β 7 ) (0.0043) P P N t : (β 8 ) *** (0.0782) CS t : (β 9 ) *** (0.0138) Y t : (β 10 ) *** (0.1064) ILT V t : (β 11 ) *** (0.1689) ILT V 0 : (β 12 ) *** (0.2902) ζ paying off *** (0.4299) Panel C: Coefficients in V T (x T ) as specified in (6) Constant (β 13 ) (18.009) CS t (β 14 ) (1.0759) ILT V T (β 15 ) (8.8002) Table 5: Coefficient Estimates for Borrowers Payoff Functions 22

24 Probability of Missing Payments Number of Late Monthly Payments Data Model Figure 1: By Beginning-of-Period Delinquency Status is greatly reduced by the amount of the payment. High house value relative to mortgages (or low mortgage loan-to-value ratio) and full document increase the propensity to make payments. There is no strong cohort effect. Finally, turning to the constants associated with each payment status at the beginning of the period captured by ξ 0 to ξ 4+, the model requires a very high value associated with 3 months delinquent in order to explain the payment rate for such borrowers. For constants associated with payment decisions, the high disutility the borrower suffers from making large number of payments indicates their reluctance or inability to do so. From Panel B, we see that the borrower s repayment decisions are positively correlated with the tenure left with the mortgage contract, but negatively correlated with prepayment penalty. A borrower with higher current credit score, high current house value relative to mortgage, but low house value relative to mortgage at origination is more likely to payoff his mortgage. As before, the estimated income generates a counter-intuitive sign. Finally, from Panel C, we see that at the terminal period T, as expected a borrower s continuing payoff is positively correlated with the updated credit score and the current house valueto-mortgage ratio. 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 probability of missing payment conditional the delinquency status at the beginning of the period in the data and that predicted by our estimated model. Note that a borrower cannot prepay the mortgage or sell the house when he is behind in mortgage payment. The model does an excellent job in capturing the patterns in the data. The more payments a borrower misses, the more likely that he will miss payments again. More important, once the 23

25 Probability of Missing Payments Probability of Prepayment Loan Age (Months) Data Model Loan Age (Months) Data Model Figure 2: By Loan Age Probability of Missing Payments Probability of Prepayment Ratio of Current Payment to Initial Payment Ratio of Current Payment to Initial Payment Data Model Data Model Figure 3: By Relative Monthly Payment borrower is three months or more behind his payment schedule, he will stay delinquent with almost certainty. Figure 2 compares the probability of missing payments and the probability of prepayment by loan age in the data and those predicted by our model. Note that while we capture the probably of default by loan age well, the match with the probability of prepayment is less so 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 have existed their fixed-teaser-rate period. The peak of prepayment, by contrast, occurs at 24 months, the time when the majority of the loans fixed-rate period expires. Figure 3 charts the probability of default and prepayment by the ratio of current monthly mortgage payment to initial monthly payment. The fits are reasonably 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, consistently with the observations we documented 24

26 Probability of Missing Payments Probability of Prepayment Loan to Value Ratio Loan to Value Ratio Data Model Data Model Figure 4: By Mortgage Loan-to-Value Ratio Probability of Missing Payments Probability of Prepayment Updated Credit Score (from TransUnion) Data Model Updated Credit Score (from TransUnion) Data Model Figure 5: By Credit Score earlier that a borrower has a higher probability of default shortly after his mortgage payment resets to a higher value. After that, the probability of default declines somewhat and then hovers at around 50 percent. The prepayment probability, on the other hand, increases consistently with the increase in the current mortgage payment relative to the initial mortgage payment. Figure 4 depicts the default probability and the prepayment probability by the current mortgage loan-to-value ratio. The model does a good job at capturing both series. As expected, the large the mortgage loan-to-value ratio is, the more likely the borrower will default and less likely he will prepay or make a payment at all. Finally, Figure 5 charts the default probability and the prepayment probability by 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 or prepay. In other words, a borrower with a high credit score will make his mortgage 25

27 payments on time. 7 Counterfactual Simulations In this section, we report counterfactual simulation results that are aimed to address two sets of questions. The first set of simulations are 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 are aimed at the policies, particularly monetary policy, that may help reduce defaults. It is useful to start out with some basic facts about the changes in monthly payments, housing 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 In Figure 7, we plot the percentage changes of local housing prices and local unemployment rates at the loans age for loans originated in 2004, 2005 and 2006 respectively. It shows that for loans that originated in 2004, the local housing prices experienced on average more than 30% gains before it declined at around these loans reached about 24 months of loan age; for loans that originated in 2005, there was also a modest (about 10%) and short-lived hosing price gains up to loan age of 12 months before the housing market crash. In contrast, the loans that originated in 2006 seemed to immediately experience housing price declines as deep as close to 45%. 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. The differences by loan origination year on these dimensions explain why the effects of a variety of counterfactual changes differ by loan origination years we discuss below. 7.1 Understanding the Factors for Defaults and Prepayments Adjustable-Rate Mortgages. An amount of the mortgage payment in an ARM is fixed for a few years initially and then resets every six month. The initial fixed rate is typically lower than typical mortgage payments after an interest rate starts to reset. 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 of ARM interest rates. To quantify how much the initial reset of ARMs contributed to the subprime borrower s default and prepayment rates observed in the data, we simulate the 26

28 Figure 6: Current Monthly Payment Transition by Loan Age and ARM Type 27

29 Figure 7: Housing Price and Unemployment Rate Trends, by Year of Origination of Loans 28