1. finance
2. personal finance
3. lending
4. home financing

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 19104 and the
NBER. Email: [email protected]
‡
Division of Research and Statistics, Board of Governors of the Federal Reserve System.
Email:
[email protected].
§
Department of Research, Federal Reserve Bank of Philadelphia. Email: [email protected].
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 2014.
http://www.fhfa.gov/AboutUS/Reports/Pages/Refinance-Report-February-2014.aspx. Similarly, HAMP was designed to help as many as 4 million borrowers avoid foreclosure by the end of 2012. 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 http://www.treasury.gov/resourcecenter/economic-policy/Documents/HAMPPrincipalReductionResearchLong070912FINAL.pdf
1
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 2005
4
Elul, Souleles, Chomsisengphet, Glennon, and Hunt (2010) also use credit bureau information to study mortgage default decisions in their empirical analysis.
2
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
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
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 2007.5 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 2007.
5
Table 1: Summary Statistics.
Variable
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)
30 days delinquent(%)
60 days delinquent(%)
90 days delinquent(%)
120 days delinquent(%)
150 days delinquent(%)
180 days delinquent(%)
180 days more delinquent(%)
House liquidation (%)
Loan modification (%)
Deviation local unemployment rates (%)
Local house price growth rates (%)
Number of observations
Mean
0
81
0.90
40
52
43
445
79
72
259
7.13
360
1.616
0
0
0
0
0
0
0
0
0
At Origination
Median Std. Dev.
0
0
1
39
1
0.30
0
49
1
50
0
50
445
155
80
11
67
26
228
141
6.99
1.15
360
0
1.429
0.859
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
16,347
6
Whole Sample Period
Mean Median Std. Dev.
16
14
11
76
1
41
0.92
1
0.27
44
0
50
52
1
50
48
0
50
424
432
178
81
78
21
77
76
28
260
228
141
7.35
7.13
1.39
345
347
11
1.679
1.475
0.902
6.86
0.0
25.37
3.10
0.0
17.33
1.62
0.0
12.63
1.40
0.0
11.73
1.25
0.0
11.11
1.14
0.0
10.63
3.86
0.0
19.27
0.64
0.0
8.08
0.26
0.0
5.06
-1.51
-1.81
1.40
-0.32
-0.27
2.15
337,811
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. We elaborate the second observation
regarding lenders’ decisions in more details in the next subsection.
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
Loan Status (beginning of the month)
Current
1 months
2 months
3 months
4 months
5 months
6 months
7 months
8 months
9 months
10 months
11 months
12 months
13 months
14 months
15 months
16 months
More than 17 months
Number of observations
At Liquidation (%)
0.00
0.05
0.05
0.87
2.39
2.71
10.98
26.32
15.48
9.00
7.35
5.19
3.81
4.04
3.12
2.02
2.07
4.46
2,177
At Modification (%)
17.09
18.71
10.74
8.55
6.12
7.39
4.62
4.50
5.31
4.04
2.54
1.96
1.50
0.92
1.73
1.15
0.81
2.31
857
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 2006. About 6 percent of the liquidation occurred
in the first two months of 2009.
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
Variable
Monthly payment (percentage)
Average change in monthly payment ($)
Balance (percentage)
Average change in balance ($)
Interest rate (percentage)
Average change in interest rate (%)
Reduction
83.41
-542
(443)
5.41
-34,030
(39,603)
83.11
-2.980
(1.415)
No Change∗
7.95
1
(19)
30.18
-73
(143)
16.89
0.00
(0.00)
Increase
8.64
287
(1,141)
64.40
12,248
(11,993)
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 xt 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
3.1
Choice set
In each period t, after information xt 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 Pt . If the borrower is one month behind on the payment, then he makes the following
decisions: pay just Pt and stay one-month-delinquent, pay 2Pt 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 Pt , 2Pt , · · · , (d + 1)Pt , 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)Pt , dPt , or (d + 1)Pt 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},



J(d) = {0, 1, 2},




{0, d − 1, d, d + 1},
if d = 0;
if 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(xt ) instead of J(d) because xt includes the
loan delinquency status d. We denote the borrower’s chosen number of payments in period t as
nt ∈ J (dt ) .
3.2
State Transition
The evolution of the state variables is captured by the transition probability F (xt+1 |xt , j),
where, as discussed previously, xt 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
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 i0 denote the initial interest rate and let ir denote the new mortgage
interest rate at the r-th reset. For example, i1 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 libort to denote the index rate.
The lifetime interest rate floor and cap are represented by lf lo and lcap , respectively. The cap
on interest rate change in each period is represented by pcap . 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
fcap .11
Combining all the elements, the new interest rate at the r-th reset in period t is calculated
as follows:

max{ir−1 − fcap , lf lo , min{margin + libort−1 , ir−1 + fcap , lcap }}, if r = 1;
ir =
max{i
r−1 − pcap , lf lo , min{margin + libort−1 , ir−1 + pcap , lcap }}, if r > 1.
(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 libort 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, lf lo , lcap , fcap , pcap } are
fixed throughout the life of the mortgage.
It follows from Equation (1) that ir ∈ [max{ir−1 − fcap , lf lo }, min{ir−1 + fcap , lcap }] if r = 1
and that ir ∈ [max{ir−1 −pcap , lf lo }, min{ir−1 +pcap , lcap }] if r > 1. In other words, {lf lo , 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 − lf lo are
low.
11
Usually, fcap is larger than pcap . That is, the interest rate change is typically larger at the initial reset than
at subsequent resets.
11
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 ir(t+1) = ir(t) . The latter is true when an interest rate is just reset in period t + 1, and
ir(t+1) = ir(t)+1 , where ir(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 balt−1 and interest rate ir(t) . The borrower’s mortgage
monthly payment Pt is calculated so that if the borrower makes a fixed payment of Pt until the
360th period, he will pay off the entire mortgage; specifically,
Pt =
balt−1
1−
1+
ir(t)
12
1
ir(t)
,
(2)
360−t+1
12
and the new balance entering period t + 1 is updated to:



balt = balt−1 1 − 1
1+
ir(t)
12

360−t  .
(3)
Remark: Note that the lenders’ decisions affect the transition of borrowers’ state variables, i.e.,
F (xt+1 |xt , 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
dt , the Libor rate libort , house price ht , changes in local unemployment rate ∆U N Rt , borrower
credit score CSt , and borrower income yt . The evolution of these state variables are as follows:
• Number of late monthly payments: dt+1 = dt − nt + 1, where nt ∈ J (dt ) 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
it follows an AR(1) process in logs
ln(libort+1 ) = λ0 + λ1 ln(libort ) + libor,t ,
where libor,t ∼ N (0, σ 2libor ) 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:
ht+1 = λ2 + λ3 ht + h,t ,
where h,t ∼ N (0, σ 2h ) 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 ∆U N R. We assume that the borrower’s belief regarding
the evolution of ∆U N R is that it follows an AR(1) process:
∆U N Rt+1 = λ4 + λ5 ∆U N Rt + unr,t ,
where unr,t ∼ N (0, σ 2∆U N R ) 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 (CSt+1 ) = λ6 + λ7 ln (CSt ) + λ8 1[d = 1] + λ9 1[d = 2] + λ10 1[d = 3] + λ11 1[d ≥ 4] + cs,t ,
where cs,t ∼ N (0, σ 2CS ) is assumed to be serially independent.
• Income (Yt ): We assume that the borrower’s belief regarding the evolution of his income
is that it follows an AR(1) process:
Yt+1 = λ12 + λ13 Yt + y,t ,
where y,t ∼ N (0, σ 2Y ) 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
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
uj (xt ) + jt ,
where uj (xt ) is a deterministic function of xt and jt is a choice-specific preference shock. The
vector t ≡ 1t , · · · 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:
(
un (xt ) =
Pt β 1 + (n − 1)Pt β 2 + CSt β 3 + Yt β 4 + ∆U N Rt β 5 + X0 β 6 + ξ d + ζ n
if n ≥ 1
ξd
if n = 0,
(4)
where Pt denotes the borrower’s monthly payment in period t. The first term Pt β 1 represents
the disutility from one month’s payment. The second term (n − 1)Pt β 2 is the disutility of
n − 1 months’ payment.14 The next term determines the borrower’s ability or willingness to
make a payment. Specifically, CSt 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 Yt represents the borrower’s current income imputed by
TransUnion. We define ∆U N Rt = U N Rt − U N R, where U N Rt and U N R denote the current
and the average unemployment rates in the borrower’s county of residence, respectively.15 While
U N Rt captures current local macroeconomic conditions, its average captures unobserved timeinvariant differences in macroeconomic conditions across counties. The term X0 is a collection
of the borrower’s initial characteristics at origination which contains original monthly payment
amount (P0 ),inverse loan-to-value ratio at origination (ILT V0 ), 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 d0 for d, d0 ≥ 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 Pt β 1 + (n − 1)Pt β 2 , instead of a single term nPt β 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 2009.
14
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
upaying
of f (xt ) =
T
X
0
δ t β 7 + P P Nt β 8 + CSt β 9 + Yt β 10 + ILT Vt β 11 + ILT V0 β 12 + ζ paying
of f,t ,
t0 =t+1
(5)
Where δ is the discount factor (which we set to be 0.99 in our estimation), P P Nt is whether
the borrower has to pay a prepayment penalty if prepaying in period t, ILT Vt 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 V0 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 Vt (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 + β CST + β ILT VT , if current at T
13
14
15
VT (xT ) =
0,
otherwise.
(6)
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.
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 (xt , t ) = 1 if
a borrower chooses action j given (xt , t ). Recall F (xt+1 |xt , 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
function of state variables which depends on the current state xt and an endogenous choice j.
We can then express the borrower’s problem recursively as follows:

V (xt ; σ) = Et 
(
X
Z
σ j (xt , t ) uj (xt ) + jt + δ
xt+1 ∈Xt
j∈J(xt )
)
V (xt+1 ; σ)dF (xt+1 |xt , j)  .
(7)
The borrower’s optimal decision rule σ ∗ is such that V (xt ; σ ∗b ) ≥ V (xt ; σ) for any possible decision
rule σ in all xt (t = 1, · · · , T ).
4
Estimation
We define the choice-specific value function for action j in period t, vj (xt ) as
Z
V (xt+1 ; σ ∗ )dF (xt+1 |xt , j).
vj (xt ) = uj (xt ) + δ
(8)
xt+1 ∈Xt
The value function can then be written as:


X
V (xt ; σ ∗ ) = Et 
σ ∗j (xt , t ) {vj (xt ) + jt } .
(9)
j∈J(xt )
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:
Z
V (xT )dF (xT |xT −1 , j).
vj (xT −1 ) = uj (xT −1 ) + δ
(10)
xT ∈XT
Note that the value function for period T , V (xT ), does not depend on σ ∗ ; the optimal decision
rule in period T − 1 is then that:
0 (xT −1 ) + j 0 ,T −1 .
σ ∗j (xT −1 , T −1 ) = 1 iff vj (xT −1 ) + j,T −1 ≥ max
v
j
0
j ∈J
(11)
Given the functional form assumption for T −1 , we can show, following Rust (1987), that

V (xT −1 ; σ ∗ ) = ln 

X
j 0 ∈J
where γ is the Euler constant.
16
exp(vj 0 (xT −1 )) + γ
(12)
Now let us consider the borrower’s optimal decision rule in period T −2. In order to calculate
R
vj (xT −2 ), we need to know xT −1 ∈Xt V (xT −1 ; σ ∗ )dF (xT −1 |xT −2 , j), which can be calculated using
equation (12). We then derive σ ∗j (xT −2 , T −2 ) and V (xT −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:
0 (xt ) + j 0 t ,
σ ∗j (xt , t ) = 1 if vj (xt ) + jt ≥ max
v
j
0
j ∈J
and

V (xt ; σ ∗ ) = log 
(13)

X
exp(vj 0 (xt )) + γ.
(14)
j 0 ∈J
Moreover, a borrower’s conditional choice probability for alternative j ∈ J (xt ) is given by:
exp(vj (xt ))
.
j 0 ∈J exp(vj 0 (xt ))
pj (xt ; σ ∗ ) ≡ Et [σ ∗j (xt , t )] = P
(15)
We estimate the model using maximum likelihood. In the data, we observe a path of states
and choices for each individual i: (xi ,ai ) ≡ {(xit ,ait )}Tt=1 , where ait ≡ {aijt }j∈J(xit ) , and aijt is
defined to be a dummy variable equals to one when individual i chose action j in period t. The
likelihood of observing (xi ,ai ) given initial state xi1 and parameter θ for individual i is:
L(xi , ai |xi1 ; θ) =
T
Y
l(ait , xi,t+1 |xit ; θ),
(16)
t=1
where l(ait , xi,t+1 |xit ; θ) is the likelihood of observing (ait , xi,t+1 ) given state xit and parameter
θ:
l(ait , xi,t+1 |xit ; θ) =
Y
[pj (xt ; θ)f (xi,t+1 |xit , j)]aijt .
j∈J(xit )
Parameter estimate θ∗ maximizes the log-likelihood for the whole sample, i.e,
θ∗ = arg max ln L(θ) =
I
X
ln (L(xi , ai |xi1 ; θ))
i=1
=
I X
T
X
X
aijt [ln (pj (xt ; θ)) + ln f (xi,t+1 |xt , j)] .
i=1 t=1 j∈J(xit )
17
(17)
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
dt , makes the payment decision at , 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: (dt = 0, at = 0) . Borrowers who are current in the beginning of the period, but
do not make a payment in the period;
Category 2: (dt = 1, at = 0) . Borrowers who are one month delinquent in the beginning of the
period, but do not make a payment in the period;
Category 3: (dt = 2, at = 0) . Borrowers who are two month delinquent in the beginning of the
period, but do not make a payment in the period;
Category 4: (dt ≥ 3, at = 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
dt ≥ 1, and if he makes at ≥ 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. at < dt + 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
state of the loan is Nevada and the default year of the loan is 2006. 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 2006. 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
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.19
5.2
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
Coefficient
Estimate
Standard Errors
Panel A: Libor ln (libort+1 ) = λ0 +λ1 ln (libor t ) + libor,t
λ0
-0.013
0.010
λ1
0.996***
0.009
σ libor
0.09656***
0.00106
Panel B: House Price ht+1 = λ2 +λ3 ht +h,t
λ2
0.671***
0.010
λ3
0.997***
0.000
σh
2.5419***
0.00979
Panel C: Local Unemp. Rates ∆U N Rt+1 = λ4 +λ5 ∆U N Rt +unr,t
λ4
0.049***
0.007
λ5
0.959***
0.003
σ unr
0.90066***
0.00979
Panel D: Income Yt+1 = λ12 +λ13 Yt +y,t
λ12
0.045***
0.000
λ13
0.945***
0.001
σY
0.09421***
2.24e-05
Panel E: Credit Score:
λ6
λ7
λ8
λ9
λ10
λ11
σ CS
ln (CS t+1 ) = λ6 +λ7 ln (CS t ) + λ8 1[d = 1]
+λ9 1[d = 2] + λ10 1[d = 3] + λ11 1[d ≥ 4] + cs,t
0.149***
0.897***
-0.072***
-0.164***
-0.130***
-0.007***
0.17719***
0.001
0.001
0.001
0.002
0.002
0.000
7.93e-05
Table 4: Coefficient Estimates for Stochastic Processes
21
Coefficient
Estimate
Std. Err.
Panel A: Coefficients in un (xt ) as specified in (4)
Pt : (β 1 )
-0.1660***
(0.0055)
(n − 1)Pt : (β 2 )
-0.0079**
(0.0032)
CSt : (β 3 )
0.0734***
(0.0068)
Yt : (β 4 )
-0.0735***
(0.0087)
∆U N Rt : (β 5 )
-0.0117***
(0.0011)
P0 : (β 6,1 )
-0.1643***
(0.0069)
ILT V0 : (β 6,2 )
0.0313***
(0.0058)
Full Doc: (β 6,3 )
0.0039***
(0.0015)
Orig 2004: (β 6,4 )
-0.0000
(0.0025)
Orig 2005: (β 6,5 )
0.0057**
(0.0024)
Constant: (ξ 0 )
-0.8999***
(0.0350)
Constant: (ξ 1 )
-1.6613***
(0.0344)
Constant: (ξ 2 )
-1.7183***
(0.0375)
Constant: (ξ 3 )
4.9652***
(0.2631)
Constant: (ξ 4+ )
-0.0648***
(0.0093)
0.5480***
(0.0382)
Constant: (ζ 1 )
Constant: (ζ 2 )
-1.4588***
(0.0688)
Constant: (ζ 3 )
-1.7841***
(0.1051)
Constant: (ζ 4+ )
-8.1837***
(0.2547)
Panel B: Coefficients in upaying of f (xt ) as specified in (5)
PT
t0 =t+1 δ
t0
: (β 7 )
P P N t : (β 8 )
CS t : (β 9 )
Yt : (β 10 )
ILT V t : (β 11 )
ILT V 0 : (β 12 )
ζ paying of f
0.0065
(0.0043)
-0.6362***
(0.0782)
0.5488***
(0.0138)
-1.1399***
(0.1064)
8.6215***
(0.1689)
-5.8382***
(0.2902)
-1.0310***
(0.4299)
Panel C: Coefficients in VT (xT ) as specified in (6)
Constant (β 13 )
-16.3973
(18.009)
CS t (β 14 )
0.9895
(1.0759)
ILT V T (β 15 )
10.4758
(8.8002)
Table 5: Coefficient Estimates for Borrowers’ Payoff Functions
22
0
.2
.4
.6
.8
1
Probability of Missing Payments
0
5
Number of Late Monthly Payments
Data
10
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
Probability of Prepayment
0
0
.1
.02
.2
.04
.3
.06
.4
.08
.5
Probability of Missing Payments
0
10
20
30
Loan Age (Months)
Data
40
50
0
Model
10
20
30
Loan Age (Months)
Data
40
50
Model
Figure 2: By Loan Age
.1
.02
.2
.04
.3
.06
.4
.08
.5
.1
Probability of Prepayment
.6
Probability of Missing Payments
1
1.1
1.2
1.3
1.4
1.5
Ratio of Current Payment to Initial Payment
Data
Model
1
1.1
1.2
1.3
1.4
1.5
Ratio of Current Payment to Initial Payment
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
Probability of Prepayment
0
.1
.2
.02
.3
.04
.4
.06
.5
.6
.08
Probability of Missing Payments
40
60
80
100
Loan to Value Ratio
Data
120
40
Model
60
80
100
Loan to Value Ratio
Data
120
Model
Figure 4: By Mortgage Loan-to-Value Ratio
Probability of Prepayment
0
.035
.2
.04
.045
.4
.05
.6
Probability of Missing Payments
2
4
6
8
Updated Credit Score (from TransUnion)
Data
2
4
6
8
Updated Credit Score (from TransUnion)
Model
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
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 2006.
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
Figure 6: Current Monthly Payment Transition by Loan Age and ARM Type
27
Figure 7: Housing Price and Unemployment Rate Trends, by Year of Origination of Loans
28
model under the situation that an interest rate is fixed at the initial fixed rate. In other words,
a mortgage becomes equivalent to a fixed-rate mortgage with an interest rate fixed at the initial
teaser rate.
In Table 6 we report the model’s predictions regarding the fraction of loans in different
status (current, delinquent, foreclosure, or paid off) at different loan ages, for loans originated in
2004, 2005 and 2006 respectively. The panel labeled “Baseline” is the model’s prediction of the
loan status under the actual loan, 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.
Comparing the two panels, we see that the effect of switching the ARMs to FRMs on loan
status depend on the year in which the loans were originated. For those loans that originated in
2004, it seems that the interest rate resets of the ARMs had very little impact on the fractions of
loans that end up in delinquency or foreclosure status. However, interest rate resets significantly
increased the fraction of loans that would be paid off, and reduced the fraction of loans that
would stay current: 48 months after originating in 2004, the fraction of loans that would stay
current in the baseline is 1.8% in the baseline, in contrast to 8.2% under the fixed rate mortgage
counterfactual, while those paid off would be 88% in the baseline, in contrast to 81.6% in the
counterfactual.
For those loans that originated in 2005, the ARM interest rate resets seem to be a much
more important factor for delinquency and foreclosure. At 48-month age, a total 29.2% (20.3%,
respectively) of loans originated in 2005 would be in delinquency (in foreclosure respectively)
under the baseline, while under the fixed rate mortgage counterfactual, 27% (respectively 17.2%)
of the loans would be in delinquency (respectively, in foreclosure). As for loans originated in 2004,
the fraction of current loans would also be significantly higher under the fixed rate mortgage
than under the baseline, and the fraction of loans that are paid off would be smaller under the
FRM than under the baseline.
For those loans that originated in 2006, the interest rate reset seems to have little effect on
the fraction of loans that would be paid off; instead, it has significant effects on the fraction of
loans that are either current or in delinquency (including those in foreclosure). At 48 months,
the fraction of loans in delinquency (respectively, in foreclosure) would be 63% (respectively,
39.7%) under the baseline, much higher than 54.4% and 34.4% respectively predicted under the
FRM. The fraction of loans that stay current at 48 months is 14.6% under FRM, in contrast to
4.6% under the ARM baseline.
The difference in the effect of FRM by the year of the loan origination suggests that the
interaction between the housing market condition at the time loans were originated and whether
the loans are ARM or FRM may be important. We examine these interactions below.
29
30
2005
2005
2006
2006
2006
2006
2006
2006
48
18
24
30
36
42
48
2005
18
42
2004
48
2005
2004
42
2005
2004
36
36
2004
30
30
2004
2005
2004
18
24
24
Year Orig
Loan Age
0.046
0.079
0.148
0.234
0.377
0.525
0.038
0.067
0.110
0.173
0.359
0.505
0.018
0.032
0.630
0.604
0.543
0.468
0.362
0.247
0.292
0.282
0.263
0.231
0.155
0.113
0.104
0.104
0.101
0.097
0.116
0.064
0.078
0.076
0.397
0.339
0.269
0.192
0.120
0.057
0.203
0.181
0.150
0.104
0.065
0.037
0.089
0.084
0.074
0.060
0.045
0.031
% Forcl
Baseline
% Delinq
0.302
0.446
% Current
0.333
0.330
0.322
0.315
0.291
0.243
0.675
0.659
0.640
0.611
0.524
0.408
0.880
0.868
0.845
0.803
0.675
0.508
% Paid off
0.146
0.191
0.258
0.336
0.456
0.583
0.146
0.198
0.261
0.348
0.461
0.582
0.082
0.120
0.178
0.255
0.354
0.498
% Current
0.544
0.509
0.461
0.398
0.309
0.210
0.270
0.250
0.224
0.184
0.139
0.100
0.107
0.106
0.099
0.089
0.076
0.070
% Delinq
0.344
0.291
0.226
0.162
0.099
0.051
0.172
0.149
0.121
0.089
0.056
0.031
0.088
0.079
0.069
0.057
0.045
0.031
% Forcl
Fixed Rate Mortgages
0.322
0.311
0.296
0.276
0.248
0.221
0.596
0.562
0.527
0.480
0.416
0.337
0.816
0.782
0.736
0.671
0.590
0.464
% Paid off
Table 6: Fixed-rate Mortgages and Lifetime floor rates
0.161
0.203
0.258
0.340
0.451
0.567
0.115
0.164
0.229
0.325
0.471
0.584
0.072
0.106
0.160
0.241
0.362
0.501
% Current
0.509
0.483
0.443
0.381
0.296
0.209
0.273
0.255
0.230
0.187
0.129
0.096
0.106
0.108
0.102
0.089
0.076
0.069
0.317
0.271
0.212
0.158
0.104
0.055
0.181
0.155
0.122
0.083
0.052
0.031
0.084
0.077
0.067
0.057
0.045
0.031
% Forcl
Lifetime floor
% Delinq
0.340
0.326
0.311
0.296
0.268
0.238
0.617
0.593
0.555
0.502
0.418
0.339
0.828
0.795
0.747
0.687
0.583
0.457
% Paid off
Housing Price Declines. Many researchers investigated importance of a negative house equity in a borrower’s default decision and found that a negative equity is one of the most importance forces leading to default (references?) In Table 7, we report counterfactual simulation
results to understand the role of substantial housing price declines that first triggered, and then
deepened by, the worst financial crisis since the Great Depression.
We conduct two counterfactuals. 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? In the second counterfactual experiment, we set the
housing price to be at 70% of the housing price at loan origination.
In Panel A where the housing price is set at 70% of the level at loan origination, we see that
the delinquency and foreclosure rates are an order of magnitude higher at all loan ages than
the baseline for loans that were originated in 2004 and 2005. For the mortgages that originated
in 2006, however, the model’s prediction of delinquency rates is not so much different from the
baseline, but the fraction of current loans is much higher and the fraction of paid off loans much
lower under the counterfactual than the baseline. As we showed in Figure 7, loans originated in
2006 eventually did experience a housing price decline of 40% or more, however, the housing price
declines were realized at a slower pace than the 30% decline we introduced in this counterfactual.
As a result, we see more loans that were paid off in the baseline than in the counterfactual when
the loans were still relatively young (when there were a larger discrepancy between the realized
housing price decline and the 30% abrupt price decline in the counterfactual). In fact, most of
the differences in the fraction of paid off loans and current loans between the baseline and the
counterfactual are a result of the differences when the loans were still relatively young.
In Panel B, we report the simulation results under the hypothetical situation that a borrower’s house price stayed constant at its level at the mortgage origination. As should be
expected from Figure 7, setting housing price unchanged at its level of mortgage origination
would have deprived the substantial housing price gains for loans that originated in 2004, 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 rats for 2004 loans (respectively, for 2005 loans) than in the baseline. Analogously,
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. These counterfactual results, taken together, suggest that the effects of the
dynamics of housing prices differ substantially on the loans that originated in different years.
31
32
2004
2004
2004
2004
2005
2005
2005
2005
2005
2005
2006
2006
2006
2006
2006
2006
30
36
42
48
18
24
30
36
42
48
18
24
30
36
42
48
2006
36
2004
2006
30
2004
2006
24
24
2006
18
18
2005
48
2006
2005
42
48
2005
36
2006
2005
30
42
2005
2004
42
24
2004
36
2004
2004
30
2005
2004
24
18
2004
18
48
Year Orig
Loan Age
0.046
0.079
0.148
0.234
0.377
0.525
0.038
0.067
0.110
0.173
0.359
0.505
0.018
0.032
0.064
0.116
0.302
0.446
0.046
0.079
0.148
0.234
0.377
0.525
0.630
0.604
0.543
0.468
0.362
0.247
0.292
0.282
0.263
0.231
0.155
0.113
0.104
0.104
0.101
0.097
0.078
0.076
0.630
0.604
0.543
0.468
0.362
0.247
0.292
0.282
0.067
0.038
0.263
0.231
0.173
0.110
0.155
0.113
0.104
0.359
0.505
0.018
0.104
0.101
0.064
0.032
0.097
0.078
0.076
% Delinq
0.116
0.302
0.446
% Current
0.397
0.339
0.269
0.192
0.120
0.057
0.203
0.181
0.150
0.104
0.065
0.037
0.089
0.084
0.074
0.060
0.045
0.031
0.397
0.339
0.269
0.192
0.120
0.057
0.203
0.181
0.150
0.104
0.065
0.037
0.089
0.084
0.074
0.060
0.045
0.031
% Forcl
Baseline
0.333
0.330
0.322
0.315
0.291
0.243
0.675
0.659
0.640
0.611
0.524
0.408
0.880
0.868
0.845
0.803
0.675
0.508
0.333
0.330
0.322
0.315
0.291
0.243
0.675
0.659
0.640
0.611
0.524
0.408
0.880
0.868
0.845
0.803
0.675
0.508
% Current
% Delinq
0.705
0.678
0.644
0.556
0.401
0.302
0.785
0.746
0.666
0.509
0.315
0.236
0.784
0.718
0.604
0.471
0.340
0.250
0.115
0.152
0.205
0.290
0.447
0.585
0.095
0.137
0.211
0.343
0.541
0.647
0.075
0.114
0.204
0.333
0.489
0.600
0.283
0.282
0.280
0.274
0.212
0.160
0.320
0.325
0.314
0.252
0.159
0.127
0.305
0.302
0.269
0.217
0.159
0.121
Panel B: HP It = HP I0
0.209
0.251
0.299
0.400
0.575
0.684
0.149
0.202
0.298
0.455
0.663
0.748
0.141
0.216
0.341
0.486
0.633
0.733
Panel A: HP It = .7 ∗ HP I0
% Paid off
0.196
0.178
0.148
0.115
0.079
0.046
0.221
0.196
0.156
0.106
0.071
0.044
0.230
0.199
0.162
0.116
0.079
0.047
0.462
0.398
0.316
0.221
0.151
0.081
0.497
0.421
0.312
0.212
0.140
0.075
0.530
0.443
0.349
0.251
0.167
0.096
% Forcl
HPI0
Table 7: The Role of Housing Prices
0.612
0.574
0.531
0.462
0.359
0.275
0.593
0.551
0.495
0.424
0.317
0.246
0.630
0.598
0.548
0.468
0.370
0.294
0.098
0.086
0.074
0.061
0.039
0.028
0.078
0.070
0.059
0.049
0.035
0.027
0.091
0.084
0.073
0.060
0.042
0.031
% Paid off
0.126
0.176
0.249
0.337
0.464
0.596
0.177
0.239
0.320
0.420
0.534
0.648
0.155
0.211
0.287
0.379
0.482
0.592
0.228
0.295
0.368
0.456
0.580
0.702
0.261
0.335
0.426
0.549
0.654
0.751
0.262
0.345
0.440
0.541
0.636
0.732
% Current
0.303
0.300
0.288
0.271
0.224
0.168
0.295
0.290
0.265
0.224
0.174
0.134
0.287
0.275
0.246
0.205
0.172
0.135
0.697
0.648
0.584
0.516
0.399
0.287
0.678
0.615
0.536
0.423
0.329
0.238
0.675
0.604
0.517
0.424
0.338
0.252
% Delinq
0.205
0.182
0.147
0.114
0.076
0.042
0.204
0.179
0.146
0.108
0.072
0.042
0.212
0.188
0.156
0.122
0.084
0.048
0.427
0.359
0.290
0.217
0.144
0.079
0.423
0.357
0.280
0.209
0.141
0.073
0.462
0.396
0.325
0.247
0.172
0.097
% Forcl
FRM and HPI0
0.581
0.537
0.479
0.410
0.332
0.255
0.538
0.485
0.430
0.370
0.304
0.237
0.569
0.527
0.480
0.432
0.366
0.290
0.085
0.074
0.060
0.044
0.031
0.023
0.074
0.061
0.052
0.040
0.031
0.023
0.074
0.065
0.056
0.049
0.041
0.033
% Paid off
Fixed Rate Mortgage and House Price. One may also expect that the effect of fixed rate
mortgages on the borrowers’ payment behavior to depend on the housing market conditions. In
Table 7, we also report counterfactual results where we let all the loans to be FRMs, and consider
the same two housing price dynamics as described in the previous section. These counterfactual
results are to be compared with both those in Table 6 and those in Table 7. It suggests that
making the mortgage fixed rate rather than adjustable rates reduces the delinquency rates for
loans of all origination years, at the counterfactual housing price dynamics, but the effects are
not very large.
Labor Market Conditions. In Table 8, we simulate the role of local unemployment rate on
the observed borrowers’ delinquency and foreclosure. We suppose that the local unemployment
rate stayed the same as that at loan origination. The results show that for loans that originated
in 2004, the local unemployment conditions did not change the borrowers’ delinquency and
foreclosure rates much, and slightly increased in the delinquency and foreclosure rates for loans
that originated in 2005. However, for 2006 loans, the worsening labor market condition as
depicted in Figure 7, seems to be a significant contributor to the delinquency and foreclosure
observed in the data. It is worth emphasizing that, in the counterfactual results reported in
Table 8, we are changing the dynamic process for the local unemployment rates while holding
the borrowers’ own income process as estimated.
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
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 are tied to a short-term interest rate. They found that ARM borrowers are less
likely to default (Fuster and Willen, 2014) and that they are more likely to increase consumption
due to a larger disposable income (Keys, Piskorski, Seru and Yao, 2014; Di Maggio, Kermani
and Ramcharan, 2014).
In Table 6, 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
33
34
2005
2005
2006
2006
2006
2006
2006
2006
48
18
24
30
36
42
48
2005
18
42
2004
48
2005
2004
42
2005
2004
36
36
2004
30
30
2004
24
2005
2004
18
24
Year Orig
Loan Age
0.046
0.079
0.148
0.234
0.377
0.525
0.038
0.067
0.110
0.173
0.359
0.505
0.018
0.032
0.630
0.604
0.543
0.468
0.362
0.247
0.292
0.282
0.263
0.231
0.155
0.113
0.104
0.104
0.101
0.097
0.116
0.064
0.078
0.076
0.397
0.339
0.269
0.192
0.120
0.057
0.203
0.181
0.150
0.104
0.065
0.037
0.089
0.084
0.074
0.060
0.045
0.031
% Forcl
Baseline
% Delinq
0.302
0.446
% Current
0.333
0.330
0.322
0.315
0.291
0.243
0.675
0.659
0.640
0.611
0.524
0.408
0.880
0.868
0.845
0.803
0.675
0.508
% Paid off
0.146
0.182
0.244
0.334
0.456
0.576
0.095
0.129
0.193
0.313
0.480
0.590
0.036
0.061
0.113
0.202
0.355
0.493
% Current
0.500
0.475
0.431
0.365
0.266
0.182
0.284
0.274
0.250
0.192
0.122
0.092
0.107
0.107
0.100
0.084
0.065
0.063
% Delinq
0.318
0.264
0.202
0.140
0.084
0.043
0.180
0.153
0.117
0.078
0.052
0.030
0.088
0.079
0.065
0.050
0.038
0.028
% Forcl
∆U N Rt = ∆U N R0
Table 8: The Role of Local Unemployment Rate
0.362
0.351
0.337
0.315
0.288
0.256
0.628
0.606
0.570
0.513
0.411
0.337
0.862
0.838
0.800
0.733
0.600
0.475
% Paid off
provide the best case scenario (or upper bound) on how much monetary policy may reduce the
delinquency and foreclosure rates.
Note, however, setting Libor rate to zero does not necessarily imply that the borrowers’
monthly payment will be lower than their payment in the teaser period. The reason is that for
a vast majority of borrowers, margin rates and life time floor rates are still higher than initial
teaser rates; in fact, borrowers will on average still have their monthly payment increasing by
about 10% even if Libor rate is zero upon the reset of the interest rate. The results in Table 6
suggests that setting Libor rate at zero does not seem to affect the delinquency and foreclosure
rates for 2004 and 2005 loans, though the fraction of paid loans is reduced. However, for 2006
loans, setting Libor rate at zero significantly reduced the delinquency and foreclosure rates, and
significantly reduces the fraction of current loans, though the fraction of paid off loans do not
change much.
7.2.2
Automatic Loan Modification Contingent on Housing Price Index
If a housing price downturn leads to massive default rates, then a way to mitigate this
problem is to tie a mortgage payment to the current house price index. Shiller (?), Mian and
Sufi (??) and Kung (2013) have suggested that such “continuous workout mortgages” might have
reduced the mortgage default and foreclosure. We consider two slightly different automatic loan
modification schemes in this subsection.
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˜t as the modified monthly payment at period t, and Pt as the monthly payment amount
in the absent of modification according to the original loan. Let Ht and H0 denote the housing
price index at period t and at origination respectively. The first counterfactual we consider
assumes that the monthly payment will be automatically modified from Pt to P˜t as follows:
P˜t = Pt × min {1, Ht /H0 } ,
(18)
while the principal balance is not adjusted.
Modification of Principal Balance (and Monthly Payments Too) In the second counterfactual, we assume that
] t = BALt × min {1, Ht /H0 } .
BAL
35
(19)
36
2005
2005
2005
42
48
2006
2006
2006
36
42
48
Revenue per ’06 borrower
2006
30
0.046
0.079
0.148
0.234
203.08K
0.630
0.604
0.543
0.468
0.362
0.247
2006
24
0.377
2006
18
0.525
223.74K
0.292
0.282
0.263
Revenue per ’05 borrower
0.038
0.067
0.110
0.231
0.155
36
0.173
0.359
2005
0.104
2005
0.505
0.018
0.104
30
2004
48
0.032
0.101
0.064
24
2004
42
0.097
0.116
0.113
2004
36
0.078
0.302
2005
2004
30
0.076
0.446
18
2004
24
% Delinq
216.74K
2004
18
% Current
Revenue per ’04 borrower
Year Orig
Loan Age
0.397
0.339
0.269
0.192
0.120
0.057
0.203
0.181
0.150
0.104
0.065
0.037
0.089
0.084
0.074
0.060
0.045
0.031
% Forcl
Baseline
0.333
0.330
0.322
0.315
0.291
0.243
0.675
0.659
0.640
0.611
0.524
0.408
0.880
0.868
0.845
0.803
0.675
0.508
% Paid off
0.324
0.357
0.401
0.450
0.521
0.608
0.110
0.139
0.185
0.265
0.387
0.524
0.032
0.056
0.101
0.184
0.301
0.474
% Current
198.19K
0.369
0.349
0.320
0.290
0.240
0.176
224.07K
0.249
0.243
0.225
0.188
0.138
0.100
216.64K
0.110
0.107
0.101
0.091
0.079
0.065
% Delinq
P˜t = Pt × min
0.227
0.197
0.164
0.125
0.084
0.042
0.173
0.154
0.121
0.087
0.059
0.032
0.089
0.081
0.070
0.057
0.042
0.029
o
HP It
,1
HP I0
% Forcl
n
0.312
0.300
0.285
0.269
0.249
0.222
0.646
0.628
0.603
0.563
0.494
0.400
0.860
0.843
0.809
0.742
0.642
0.495
% Paid off
0.311
0.351
0.392
0.445
0.507
0.594
0.100
0.132
0.177
0.251
0.379
0.529
0.030
0.050
0.096
0.175
0.293
0.452
202.47K
0.203
0.195
0.189
0.178
0.163
0.137
226.04K
0.182
0.180
0.174
0.158
0.125
0.092
216.95K
0.099
0.099
0.092
0.081
0.068
0.062
% Delinq
n
0.172
0.158
0.142
0.122
0.090
0.054
0.150
0.136
0.117
0.088
0.058
0.034
0.082
0.074
0.063
0.051
0.040
0.030
o
0.493
0.464
0.428
0.388
0.341
0.286
0.723
0.696
0.660
0.606
0.519
0.406
0.875
0.855
0.819
0.760
0.660
0.516
% Paid off
HP It
,1
HP I0
% Forcl
^t = BALt × min
BAL
% Current
Table 9: Automatic Modifications of Monthly Payments and Principal Balance
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.
In Table 9 we present the results from these counterfactual simulations. We find that these
automatic modification mortgages do not seem to impact the delinquency and foreclosure rates
for loans that originated in 2004; however, the delinquency and foreclosure rates are significantly
reduced for 2005 and particularly for 2006 loans. Interestingly, we also find that lenders’ revenues do not seem to be lower, and in fact for 2004 and 2005 loans they are higher, than the
baseline. These counterfactual results suggest that automatic modification mortgages, particularly automatic modifications of principal balance contingent on housing price index, could be
a promising alternative mortgage design that can prove to be win-win for both borrowers and
lenders.
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 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.
An important limitation of this paper is that we take lenders’ behavior as given. For the
questions we address, this assumption may be realistic, because lenders’ policy regarding modification and foreclosure do not seem to be too responsive to a variety of government policies that
37
were specifically introduced to increase modification. However, it is important to model lender
behavior explicitly so we can have a better understanding of why lenders’ are not responsive to
government policy. This is a topic we will explore in our companion paper.
References
[1] 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.
[2] Agarwal, Sumit, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet, Tomasz
Piskorski, and Amit Seru, 2012, ”Policy Intervention in Debt Renegotiation: Evidence from
the Home Affordable Modification Program,” University of Chicago Kreisman Working
Paper on Housing Law and Policy No. 7.
[3] Bajari, Patrick, Sean Chu, Denis Nekipelov, and Minjung Park, ”A Dynamic Model of
Subprime Mortgage Default: Estimation and Policy Implications,” 2013, NBER Working
Paper 18850.
[4] Campbell, John Y., and Joao F. Cocco, ”A Model of Mortgage Default,” Journal of Finance,
forthcoming.
[5] 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.
[6] Corbae, Dean, and Erwan Quintin, 1984, ”Leverage and the Foreclosure Crisis,” Journal of
Political Economy, forthcoming.
[7] Demyanyk, Yuliya, and Otto van Hemert, 2011, ”Understanding the Subprime Mortgage
Crisis,” Review of Financial Studies 24(6), 1848-1880.
[8] Deng, Youngheng, John M. Quigley, and Robert van Order, 2000, ”Mortgage Terminations,
Heterogeneity and the Exercise of Mortgage Options,” Econometrica 68(2), 275-307.
[9] Elul, Ronel, Nicholas S. Souleles, Souphala Chomsisengphet, Dennis Glennon, and Robert
Hunt, 2010, ”What ’Triggers’ Mortgage Default?” American Economic Review 100(2), 490494.
[10] Foote, Christopher, Kristopher Gerardi, and Paul Willen, 2008, ”Negative Equity and Foreclosure: Theory and Evidence,” Journal of Urban Economics 64(2), 234-245.
38
[11] Foster, Chester, and Robert van Order, 1984, ”An Option-Based Model of Mortgage Default” Housing Finance Review 3(4), 351-372.
[12] Frame, Scott, Andreas Lehnert, and Ned Prescott, 2008, ”A Snapshot of Mortgage Conditions: Emphasis on Subprime Mortgage Performance,” manuscript.
[13] Garriga, Carlos, and Don Schlagenhauf, 2009, ”Home Equity, Foreclosures, and Bailouts,”
manuscript, Federal Reserve Bank of St. Louis.
[14] Gerardi, Kristopher, Andreas Lehnert, Shane Sherlund, and Paul Willen, 2008, ”Making
Sense of the Subprime Crisis,” Brookings Papers on Economic Activity, 69-145.
[15] Gerardi, Kristopher, and Wenli Li, 2010, ”Mortgage Foreclosure Prevention Efforts,” Federal
Reserve Bank of Atlanta Economic Review 2, 1-13.
[16] Hatchondo, Juan Carlos, Leonardo Martinez, and Juan M. Sanchez, 2011, ”Mortgage Defaults,” Federal Reserve Bank of Saint-Louis Fed Working Paper no. 2011-019A.
[17] 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.
[18] Keane, Michael, and Kenneth Wolpin, 1993, ”Career Choices of Young Men,” Journal of
Political Economy, 105 (3), 473-522.
[19] 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.
[20] Mayer, Christopher, Karen Pence, and Shane M. Sherlund, 2009, ”The Rise in Mortgage
Defaults,” Journal of Economic Perspectives 23(1), 27-50.
[21] Mitman, Kurt, 2012, ”Macroeconomic Effects of Bankruptcy and Foreclosure Policies,”
manuscript, Department of Economics, University of Pennsylvania.
[22] Rust, John (1987). “Optimal Replacement of GMC Bus Engines. An Empirical Model of
Harold Zurcher.” Econometrica, 55(5), 999-1033.
[23] 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.
39
[24] Rust, John (1994b). “Structural Estimation of Markov Decision Processes,” in Robert Engle and Daniel McFadden eds., Handbook of Econometrics, Vol. IV. Amsterdam: NorthHolland.
[25] 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.
40
Variable
Current Credit Score
Income ($1000)
Inverse loan-to-value (%)
Inverse loan-to-value at orig. (%)
Changes in local unemp. rates (%)
Current monthly payment ($1000)
Initial monthly payment ($1000)
Dummy for initial fixed period (%)
Dummy for interest-only period
Loan age (month)
Loan age squared
Months before first reset
Months before first reset squared
Months before interest-only
Months before interest-only squared
Dummy for full documentation
Dummy for loan orig. in 2004
Dummy for loan orig. in 2005
Dummy for loan orig. in 2004 x age
Dummy for loan orig. in 2004 x age2
Dummy for loan orig. in 2005 x age
Dummy for loan orig. in 2005 x age2
Dummy for Arizona
Dummy for California
Dummy for Florida
Constant
Number of observations
Pseudo R2
Category 1 Loans
Category 2 Loans
(dt = 0, at = 0)
(dt = 1, at = 0)
coeff.
s.d.
0.32***
0.04
-0.14
0.42
-1.63***
0.59
0.99
0.62
-0.16**
0.07
0.74***
0.29
-0.90**
0.37
0.77***
0.29
-0.94
0.66
-0.43*
0.24
0.01
0.00
-0.71***
0.12
0.01***
0.00
0.05*
0.03
-0.00*
0.00
0.50***
0.17
-41.58***
17.91
-12.92***
4.23
1.89***
0.82
-0.02**
0.01
0.76***
0.29
-0.01**
0.01
-0.11
0.39
0.12
0.33
-0.22
0.36
3.04
3.53
13,716
0.26
coeff.
s.d.
0.17***
0.05
-0.77
0.49
-1.61***
0.68
-0.46
0.82
-0.02
0.08
0.93***
0.31
-0.98***
0.40
0.59*
0.34
-0.26
0.68
0.56**
0.25
-0.01**
0.00
-0.21***
0.09
0.01*
0.00
0.01
0.03
-0.00
0.00
0.54***
0.19
-12.18
11.13
-6.06
4.91
0.32
0.56
0.00
0.01
0.24
0.39
-0.00
0.01
0.64
0.59
1.05**
0.53
1.05*
0.54
-9.77***
3.42
7,676
0.19
Category 3 Loans
(dt = 2, at = 0)
coeff.
s.d.
0.23***
0.07
-2.57***
0.60
-0.96
0.71
1.44**
0.68
0.01
0.08
0.56
0.43
-0.35
0.52
0.87*
0.47
-0.63
0.91
0.17
0.29
-0.00
0.01
-0.13
0.15
-0.00
0.01
0.01
0.04
-0.00
0.00
0.42*
0.23
-32.32
23.98
-20.36***
7.59
1.31
1.07
-0.01
0.01
1.05**
0.47
-0.01*
0.01
0.04
0.47
-0.05
0.42
-0.23
0l.43
-6.96*
3.93
5,740
0.1966
Table A1: Lenders’ Decisions for Loans in Categories 1-3.
Notes: Results are from logit Regressions where the dependent variable is a dummy for loan modification. ***
indicates significance at 1 percent confidence level; ** at 5 percent; and * at 1 percent.
41
Variable
Current Credit Score
Income ($1000)
Inverse loan-to-value (%)
Inverse loan-to-value at orig. (%)
Changes in local unemp. rates (%)
Current monthly payment ($1000)
Initial monthly payment ($1000)
Dummy for initial fixed period (%)
Dummy for interest-only period
Loan age (month)
Loan age squared
Months before first reset
Months before first reset squared
Months before interest-only
Months before interest-only squared
Dummy for full documentation
Dummy for loan orig. in 2004
Dummy for loan orig. in 2005
Dummy for loan orig. in 2004 x age
Dummy for loan orig. in 2004 x age2
Dummy for loan orig. in 2005 x age
Dummy for loan orig. in 2005 x age2
Dummy for Arizona
Dummy for California
Dummy for Florida
Number of late payments
Number of late payments squared
Inverse ltv x number of late payments
Inverse ltv x number of late payments squared
Change in unemp rates x number of late payments
Change in unemp rates x number of late payments2
Modification
coeff.
s.d.
-0.05
0.05
-0.56** 0.24
-0.20
0.99
-0.66*
0.38
0.25**
0.12
-0.06
0.19
0.04
0.22
-0.09
0.22
-0.70
0.40
0.29*** 0.09
-0.00*** 0.00
0.07
0.05
-0.00
0.00
0.02
0.02
-0.00
0.00
0.20**
0.10
-11.67** 5.66
-3.12*
1.72
0.49*
0.28
-0.00
0.00
0.11
0.12
-0.00
0.00
-0.79
1.60
2.38*
1.20
2.20*
1.22
0.70*
0.38
-0.03** 0.02
0.04
0.18
0.00
0.11
-0.02
0.02
0.00
0.00
Liquidation
coeff.
s.d.
0.04*
0.02
0.21*
0.12
3.57***
0.55
-0.89***
0.53
0.20*
0.11
-0.01
0.09
-0.15
0.11
-0.14
0.11
0.36***
0.14
-0.01
0.04
-0.00
0.00
0.01
0.02
0.00
0.00
-0.01
0.01
0.00*
0.00
-0.09*
0.05
1.90***
0.71
0.94
0.57
0.02
0.05
-0.00
0.00
0.02
0.04
-0.00
0.01
1.99***
0.65
-0.51
0.57
-4.68***
0.64
0.72***
0.18
-0.02*** 0.0.01
-0.52***
0.10
0.02***
0.00
-0.05***
0.02
0.00**
0.00
Table A2: Lenders’ Decisions on Category 4 Loans (to be continued in Table A3).
Notes: Results are from multinomial logit Regressions where the alternatives are modification, liquidation and
waiting (omitted). *** indicates significance at 1 percent confidence level; ** at 5 percent; and * at 1 percent.
42
Variable
Dummy for 4 months deliq.
Dummy for 5 months deliq.
Dummy for 6 months deliq.
Dummy for 7 months deliq.
Dummy for 8 months deliq.
Dummy for 9 months deliq.
Dummy for 10 months deliq.
Dummy for 11 months deliq.
Arizona x months of deliq.
Arizona x months of deliq. squared
California x months of deliq.
California x months of deliq. squared
Florida x months of deliq.
Florida x months of deliq. squared
Originated in 2004 x months of deliq.
Originated in 2004 x months of deliq. squared
Originated in 2005 x months of deliq.
Originated in 2005 x months of deliq. squared
Constant
Number of observations
Pseudo R2
Modification
Liquidation
coeff.
s.d.
coeff.
s.d.
1.45
0.97
-3.74*** 0.69
0.96
0.84
-2.47*** 0.48
1.01
0.71
-2.19*** 0.40
0.47
0.60
-0.59*
0.32
0.54
0.50
0.67*** 0.25
0.75*
0.41
0.42**
0.20
0.60*
0.35
0.10
0.16
0.23
0.32
0.13
0.13
0.00
0.37
-0.20** 0.10
-0.01
0.02
0.00
0.00
-0.46*
0.25
0.13
0.09
0.02
0.01
-0.01** 0.00
-0.50**
0.25
0.48*** 0.10
0.02
0.01
-0.01*** 0.00
-0.21
0.16
-0.29*** 0.10
0.01
0.01
0.01*** 0.00
-0.07
0.10
-0.19*** 0.07
0.00
0.01
0.01*** 0.00
-10.98*** 2.66
-5.44*** 1.29
33,449
0.15
Table A3: Lenders’ Decisions on Category 4 Loans (continued from A2).
Notes: Results are from multinomial logit Regressions where the alternatives are modification, liquidation and
waiting (omitted). *** indicates significance at 1 percent confidence level; ** at 5 percent; and * at 1 percent.
43