Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Mortgage Innovation and the Foreclosure Boom Dean Corbae and Erwan Quintin University of Wisconsin - Madison
April 18, 2013
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Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Motivation • Between 2003 and 2006, there were important changes in the
composition of residential mortgages in the US: 1. More nontraditional (low downpayment/delayed amortization) mortgages, 2. More subprime borrowers. • Since the end of 2006, house prices have dropped by about
25%, nontraditional mortgages have dried up, and foreclosure rates have more than doubled. Question: How much did nontraditional mortgages contribute to the foreclosure boom?
2 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Purchase Loans with CLTV≥97% as a fraction of all loans 45.00%
40.00%
35.00%
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
2007
2006
2005
2004
2001
2003
2002
1999
2000
1998
1997
1996
1995
1994
1993
1991
1992
1990
1989
1988
1987
1986
1985
1984
1983
1981
1982
1980
0.00%
Source: Pinto, E. (2010) “Government Housing Policies in the Lead-up to the Financial Crisis: A Forensic Study”, mimeo.
Definition 3 / 69
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Recent trends in US mortgages and foreclosures 2
20
1.75
1.5
1
10
Subprime fraction
1.25 Foreclosure rate
Facts
0.75
0.5
0.25
0 1998
2000
2002
2004
2006
2008
2010
0 2012
Sources: Haver analytics, National Delinquency Survey (Mortgage Bankers Association). Quarterly foreclosure rates are the fraction of all loans that enter the foreclosure process in a given quarter.
Definition 4 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
A model of housing
• Heterogeneous agents choose to own or rent, how to finance
house purchases, and how to terminate mortgage contracts • Mortgage holders may default because: 1. their home equity is negative 2. they can’t afford current payments • Mortgage terms reflect default risk, hence vary with initial
income/asset position, as well as loan size.
5 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Key features
• Two types of mortgages: 1. Traditional (FRM): 20% downpayment, constant payments 2. Non-traditional (Low Initial Payments, LIP): Zero down, delayed amortization • LIPs cause default rates to rise because: 1. High default-risk agents (those with low earnings and assets) enter the mortgage market (selection effect) 2. Home-equity is slow to build
6 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Quantitative experiment Stage 1: Only traditional mortgages are available (pre-2003); Stage 2: Non-traditional mortgages are introduced (2003-2006); Stage 3: Unanticipated price collapse and no new LIP originations (post-2006). • All parameters are calibrated to stage 1 only. • Model can explain 98.6% of the rise of foreclosures in the
data between 2007Q1 and 2009Q1. • In the counterfactual where new mortgages are not
introduced, the same price shock accounts for 57.3% of the increase in foreclosures. • Thus, the origination of nontraditional mortgages for two
model periods can explain over 40% of the rise in foreclosures. 7 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Some Literature 1. Empirical: Gerardi et. al. (2009): • Documents that subprime loans have high CLTV • Negative net equity is in general necessary but not sufficient
for foreclosure. More on empirical approaches
2. Structural: • Campbell and Coco (2011) - Mortgage decision problem with
• • •
•
multiple sources of uncertainty (e.g. earnings, house prices, etc.) and default. Chatterjee and Eyigungor (2011) - Infinite maturity IOM mortgages. Garriga and Schlagenhauf (2009) - Pooling within mortgage types so cannot separate prime vs subprime within a contract. Herkenhoff and Ohanian (2012) - Infinite maturity IOM mortgages. Since period is one month (for job matching purposes), must consider delinquency before default. Tables 1 and 2 document that foreclosures arise before 2 years. Mitman (2011) - One period mortgages with costless refinance. 8 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Outline a) Environment b) Equilibrium c) Parameterization d) SS Results • • • • • •
Selection into nontraditional contracts Default Hazards across contracts Distribution of Interest Rates Pooling vs Separating Equilibria Welfare Gains from introducing contracts Antideficiency Policies
e) Transition Results
9 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Environment • Time is discrete and infinite. • Continuum of agents. • Young agents become mid-aged with probability ρM , mid-aged
agents become old with probability ρO , old agents die with probability ρD . • Young or mid-aged agents earn stochastic income yt drawn
from a three state {ysL , ysM , ysH } Markov chain πs where s ∈ {Y , M}.
• Old agents earn y O with certainty. • Agents are born with no assets and with an income level
drawn from πY .
10 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
• Agents value consumption and housing services according to:
E0
∞ X
β t u (ct , ht ) .
t=0
• Agents can save at gross rate 1 + rt > 0 in period t in youth
and mid-age, and in annuities that pay off (1 + rt )/(1 − ρD ) in old age if alive.
11 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Housing 1
• Agents can rent quantity h of housing capital at rate Rt . • When agents become mid-aged they can purchase h0 ∈ {h2 , h3 } of housing for unit price qt , where h3 > h2 > h1 . • Agents enjoy a fixed ownership premium θ > 0 as long as they own a quantity ht ∈ {h2 , h3 } of housing capital. Therefore, u(ct , ht ) = U(ct , ht ) + θ1{ht ∈{h2 ,h3 }}
• Homeowners face uninsurable idiosyncratic shocks (e.g. neighborhood effects) to their housing capital. Specifically, their housing capital follows a Markov Process over {h1 , h2 , h3 } with transition matrix: 1 0 0 λ , where λ > 0. P(ht+1 |ht ) = λ 1 − 2λ 0 λ 1−λ
• Houses of size ht carry maintenance costs δht . • Agents can sell/foreclose on their house in any period, but are then
constrained to be renters for at least one period after which they obtain the option to buy a house with probability γ. • Old agents must sell their house. 12 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Financial intermediary • Can borrow and lend at world riskless return rt at date t. • Linear technology: can transform quantity kt ≥ 0 of deposits
into quantity At kt of housing capital, where At > 0, and vice-versa. • Rents and sells housing capital. Rented capital bears
maintenance cost δ in each period. • Issues all mortgages. Mortgages carry administrative cost φ. • In the event of default, intermediary loses fraction χ > 0 of
the portion of the principal it collects.
13 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Mortgages
• Newly mid-aged agents with assets a0 and income y0 can
purchase a house of size h0 by selecting a mortgage type ζ = {FRM, LIP} with yield r ζ (a0 , y0 , h0 ). • FRMs require down payments νh0 qt , where ν ∈ (0, 1), and
fixed payments for T periods. • LIPs require no down-payment, interest-only payments for
nLIP periods, and fixed payments for T − nLIP periods. mortgage payment func.
14 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Shape of payments Mortgage Payment Schedule 0.175 2
m(n;κ=(FRM,14.5%,h )) 2
m(n;κ=(LIP,14.5%,h ))
0.17
0.165
0.16
0.155
0.15
0.145
0.14
0.135
0.13
0.125
0
5
10
14
Mortgage age (n)
15 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Timing 1. Youth: • Receive age shock and signal of income realization. • Make savings decision.
2. Middle-age: • Receive age shock and signal of income realization. • New mid-aged agents make home-buying and mortgage choice
decision. • Existing homeowners may receive a devaluation shock and
decide whether to default or sell. • Make mortgage or rental payments as well as savings decisions.
3. Old: • Newly old agents sell their house if they own one. • Receive death shock or income. • Make (dis)saving decision. 16 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Definition Steady State Equilibrium 1. Household savings, house purchase/sale, contract choice, default decisions are optimal given pricing functions. 2. Intermediaries behave competitively: • q=
1 A
IP
(i.e. linear tech pins down prices).
• R = rq + δ (i.e. PDV of rental payments equals price). • for each ζ ∈ {FRM, LIP}, r ζ (a0 , y0 , h0 ) is such that
W κ (ω0 ) − (1 − ν1{ζ=FRM} )qh0 = 0 (i.e. EPDV of mortgage payments equals principal).
3. The housing capital market clears.
MC
NIPA
4. The distribution of household states is invariant given agent decisions. dist 17 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Parameterization
• We choose parameters so that, when only FRMs are available,
our economy matches the relevant features of the US economy prior to 2003. FRMonly • One period = 2 years. • Preferences are given by
U(c, h) = ψ ln(c) + (1 − ψ) ln(h).
18 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Income process • From the PSID 1999 and 2001 • Split households into terciles and age groups (< or > 34). • Transition matrix for each age group calibrated to match
mobility patterns across terciles between 1999 and 2001. • The incomes of mid-aged agents yM ∈ {0.3129, 1, 2.5164} with
the median normalized to 1. The transition matrix is 0.8032 0.1804 0.0164 0.1545 0.6901 0.1554 0.0423 0.1295 0.8282
• The incomes of young agents yY ∈ {0.2937, 0.7855, 1.7452}
with transition matrix 0.6828 0.2581 0.0591 0.2690 0.5103 0.2207 0.0481 0.2317 0.7202 19 / 69
Facts
Model Environment
Equilibrium
Parameter Description Parameters determined independently ρM Fraction of young agents who become mid-aged ρO Fraction of mid-aged agents who become old ρD Fraction of old agents who die r Storage returns δ Maintenance rate ν Downpayment on FRMs T Mortgage maturity nLIP Interest-only period for LIPs Parameters determined jointly θ Owner-occupied premium λ Housing shock probability A Housing technology TFP β Discount rate φ Mortgage service cost χ Foreclosing costs ψ Utility share on consumption h1 Size of rental unit h2 Size of regular house h3 Size of luxury house
Parameterization
Value
Target
1/7
SS Results
1/10
14 years of earnings on average prior to home purchase 30 years on average between home purchase and retirement 20 years of retirement on average
8% 5% 0.20 15 3
2-year risk-free rate Residential housing gross depreciation rate Average Loan-to-Value Ratio 30 years 6-years interest-only
3.220 0.120 0.571 0.833 0.042 0.440 0.800 0.640 0.850 1.300
Homeownership rates Foreclosure rates Average Loan-to-income ratio at origination Average ex-housing asset-to-income ratio Average mortgage yields Loss-incidence estimates Average housing spending share Rent-to-income ratio for low-income agents Owner’s housing spending share Foreclosure discount
1/15
Transition
20 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Steady state statistics
Homeownership rate Avg. ex-housing asset/income ratio Avg. loan to income ratio Avg. housing expenditure share Rents to income ratio for renters Avg. housing spending share for homeowners Avg. mortgage yields (FRMs, LIPs) Loss-incidence estimates Foreclosure rates Foreclosure discount
Data 67.00 0.93 1.36 0.20 0.40 0.20 (14.50,NA) 0.50 3.00 0.75
Benchmark 66.78 0.96 1.36 0.19 0.39 0.22 (14.35,NA) 0.50 2.97 0.71
FRM +LIP 72.12 0.94 1.51 0.20 0.39 0.23 (14.06,17.51) 0.46 3.70 0.70
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Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Young agents’ problem
• State: ω = (a, y )
VY (a, y ) s.t. c + a
′
= =
max ′
c≥0,a ≥0
U c, h1 + βEy ′ |y 1
+
(1 − ρM )VY (a′ , y ′ ) ρM VM (a′ , y ′ , 1, h′ , 0; ∅)
y + a(1 + r ) − Rh .
Mid-aged agents contract choice problem Mid-aged agents default decision problem
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Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Distribution of assets upon entering mid-age Benchmark
−3
6
x 10
L
y =y 0
5
M
y =y 0
4
H
y =y 0
3 2 1 0
0
0.5
1
1.5
2
2.5
Initial assets (a ) 0
Change in distribution for FRM + LIP economy
−3
8
x 10
y0=yL 6
y0=yM y0=yH
4 2 0 −2
0
0.5
1
1.5
2
2.5
Initial assets (a0)
• Lose spike at small house downpayment. • Average savings of newly mid-aged hhs fall by 27% with LIPs
included. 23 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Selection: Distribution of contract choice Benchmark
−3
3.5
x 10
y0=yL,FRM
3
y0=yM,FRM y0=yH,FRM
2.5 2 1.5 1 0.5 0
0
0.5
1
1.5
2
2.5
Initial assets (a0)
FRM + LIP
−3
4
x 10
y0=yL,FRM
3.5
y0=yM,FRM
3
y0=yH,FRM
2.5
y0=yL,LIP
2
y0=yM,LIP
1.5
y0=yH,LIP
1 0.5 0
0
0.5
1
1.5
2
2.5
Initial assets (a0)
Asset poor agents select into LIPs while asset rich agents opt for FRMs. Sel Tab 24 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Selection: Contract type by average age
Housing decision Small house Big house
FRM 30.83 32.70
LIP 30.92 27.17
Younger first time home buyers are more likely to choose a low downpayment either by buying a small house or using a LIP.
25 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
LIPs imply slower home equity accumulation Principal balance over time 1.5
b(n;κ=(FRM,14.5%,h2)) b(n;κ=(LIP,14.5%,h2))
1
0.5
0
0
2
4
6
8
10
12
14
Home equity 1.5
qh2−b(n;κ=(FRM,14.5%,h2)) qh2−b(n;κ=(LIP,14.5%,h2))
1
qh1−b(n;κ=(FRM,14.5%,h2)) qh1−b(n;κ=(LIP,14.5%,h2))
0.5
0
−0.5
0
2
4
6
8
10
12
14
26 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Termination hazard rates by contract type Default Hazard 0.12 FRM LIP 0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
12
14
Mortgage Age
Sale Hazard 0.2 FRM LIP
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
0
2
4
6
8
10
12
14
Mortgage Age
• •
Construct hazard rate (fraction of terminations due to default or sale conditional on staying in the home up to date n) from a pseudopanel of 50,000 mortgages drawn from the steady state distribution of our model economy. Default hazards are uniformly higher for LIPs than for FRMs due to selection and equity effects.
27 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Determinants of hazard risks • We run Cox proportional hazard regressions with competing
events (default vs. sale) on our pseudopanel. Covariate LIP indicator Loan to Income Ratio Assets to Loan Ratio
Default 0.6812∗∗∗ (0.0243) 0.1182∗∗∗ (0.0064) -0.2147∗∗∗ (0.0358)
Detail
Sale (0.0154) 0.3287∗∗∗ (0.0050) -0.6309∗∗∗ (0.0215) -0.2515∗∗∗
Notes: Standard errors are in parenthesis; Log Likelihood : -76943.461; *** significant at 1% level
• LIP selection, higher loan-to-income ratios, and lower
assets-to-loan ratio increase the probability of default. • Results are consistent with Gerardi, et. al. (2009) who find
higher initial loan-to-value and higher subprime purchase indicator lead to more default. 28 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Default frequencies by mortgage type
Benchmark FRM FRM + LIP FRM LIP
voluntary
involuntary
total
2.96
0.00
2.97
2.78 5.61
0.01 0.22
2.79 5.83
Default rates are twice as high on LIPs than on FRMs.
defn of default
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Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Determinants of foreclosure • 98.5% of defaults involve negative equity. • However, 70.7% of agents with negative equity choose to
continue meeting payments. • 92% of agents who have positive net equity and face a
positive probability of being in involuntary default next period choose to sell. • Since agents with positive net equity have the most to lose,
it’s not surprising that most agents who end up in involuntary default also have negative net equity.
30 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Interest rate offerings 3
2
FRM interest rate schedule for h
FRM interest rate schedule for h
0.165
y0=yL
0.165
y0=yL
0.16
y0=yM
0.16
y0=yM
0.155
y0=yH
0.155
y0=yH
0.15
0.15
0.145
0.145
0.14
0.14
0.135
0.135 0
2
4
6
8
0
2
4
6
8
Initial assets (a0)
Initial assets (a0)
LIP interest rate schedule for h3
LIP interest rate schedule for h2
y =yL
0.3
y =yL
0.3
0
0
y =yM
y =yM
0 0
0.25
0
y0=yH
0.25
y =yH
0.2
0.2
0.15
0.15 0
2
4
6
Initial assets (a ) 0
Truncated Rates
8
0
2
4
6
8
Initial assets (a ) 0
31 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Equilibrium distribution of interest rates Distribution of FRM interest rates 0.014
Separating Pooling
0.012 0.01 0.008 0.006 0.004 0.002 0 0.12
0.14
0.16
0.18
0.2
0.22
0.24
Distribution of LIP interest rates
−3
5
x 10
Separating Pooling
4 3 2 1 0 0.12
0.14
0.16
0.18
0.2
0.22
0.24
• Define subprime as bottom 30% of hhs with highest mortgage
interest rates. • Avg (2-year) return is 14.09% on prime and 18.04% on subprime mortgages. • 14.03% on prime FRMs and 15.13% on subprime FRMs. • 14.60% on prime LIPs and 18.27% on subprime LIPs. 32 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Variation in equilibrium returns by contract
CV (yield) for FRMS CV (yield) for other
Data 0.153 0.341
Benchmark 0.0355 NA
FRM + LIP 0.0168 0.1800
• Model underpredicts variation of mortgage rates in the data.
33 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Determinants of log mortgage yield (from pseudopanel)
Covariate Assets at origination Income at origination Loan size
Coefficient (0.0004) -0.1184∗∗∗ (0.0005) 0.1089∗∗∗ (0.0013) -0.0867∗∗∗
Notes: s.e. in parenthesis; R 2 = 0.7231; *** sig. at 1% level
• Higher assets and income at origination receive lower
mortgage rates, but bigger loans receive higher rates.
34 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Separation matters
Homeownership rate Ex-housing asset/income ratio Loan to income ratio Avg. housing expenditure share Rents to income ratio for renters Housing spending share for homeowners Avg. mortgage yields (FRMs) Loss-incidence estimates Foreclosure rates Foreclosure discount
FRM+LIP 72.12 0.94 1.51 0.20 0.39 0.23 (14.06,17.51) 0.46 3.70 0.70
FRM+LIP, pooling 75.47 1.04 1.66 0.19 0.39 0.23 (13.99,17.75) 0.49 5.16 0.69
• Foreclosure rates are 40% higher with pooling contracts, with
low-risk borrowers subsidizing even more high-risk borrowers. Rent-own decisions
35 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Value of innovation • Q: Starting from the ss cross-sectional earnings and wealth
distribution of FRM only economy, how much would people be willing to pay to be in the FRM+LIP economy? L , k M , k H } be the consumption-equivalent welfare • Let {kage age age
changes associated with the introduction of IOMs for agents of a given age and y ∈ {y L , y M , y H }. • Without refinancing, only relevant for young and mid-age about to buy a house. Age Y M(n=0) Total
L kage 0.68% 0.05% 0.73%
M kage 0.30% 0.35% 0.65%
H kage 0.04% 0.37% 0.41%
Overall 0.34% 0.26% 0.60%
• Average welfare gain associated with availability of the LIP
option is 0.6% in consumption-equivalent terms.
Calculation 36 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Policy: recourse imposes harsher punishment • Anti-deficiency (Non-recourse) laws: borrower is not
responsible for any deficiency. Banks cannot attach to the household’s assets. • Some states have them (AZ,CA,FL . . . ), others don’t. • What if all states had recourse?
Non-recourse Recourse
Intermediary min{(1 − χ)qh, b} min{(1 − χ)qh + a, b}
Hhs a + max{(1 − χ)qh − b, 0} max{(1 − χ)qh + a − b, 0}
• Harsher punishment lowers extensive default margin. • Higher repayment lowers intensive loss incidence.
37 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
The role of recourse
Homeownership rate Avg. ex-housing asset/income ratio Avg. loan to income ratio Avg. homeowner housing expenditure share Rents to income ratio for renters Avg. housing spending share for homeowners Avg. mortgage yields (FRMs, LIPs) Loss-incidence estimates Foreclosure rates Foreclosure discount
Benchmark (no recourse) 66.78 0.96 1.36 0.19 0.39 0.22 (14.35,NA) 0.50 2.97 0.71
Full recourse 69.41 0.96 1.35 0.20 0.39 0.21 (12.81,NA) 0.78 1.55 0.73
• Foreclosure rates are 48% lower with recourse. • Ghent and Kudlyak (2009) estimate that at average borrower
characteristics, the likelihood of default is 20% lower with recourse. 38 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
The role of recourse (cont.) • Q: Starting from the ss cross-sectional earnings and wealth
distribution of FRM+IOM economy, how much would people be willing to pay to be in an economy with recourse? Age Y M(n=0) M(n>0) Total
L kage 0.11% 0.05% -0.07% 0.09%
M kage 0.15% 0.04% -0.11% 0.07%
H kage 0.09% 0.01% -0.13% -0.03%
Overall 0.12% 0.03% -0.10% 0.04%
• Young and home purchasers benefit from lower interest rates
but older mortgage holders without the refinance option face the harsher penalty. Overall gain is small. 39 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Main experiment
Stage 1: Only traditional mortgages are available; Stage 2: Non-traditional mortgages introduced for two model periods; Stage 3: Unanticipated shock to A causes average home prices to collapse by 25%, no originations of nontraditional mortgages. Intermediary losses following unexpected aggregate shock are paid for through lump sum taxes.
40 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Summary of transition results
Data Frac. of LIPs in orig. in stage 2 Increase in foreclosures 2007Q1-2009Q1
[20-35%] 150%
LIPs in stage 2 33% 148%
No LIPs in stage 2 0% 86%
nLIP = 0 in stage 2 37% 189%
• Model can explain 98.6% of the rise of foreclosures in the
data between 2007Q1 and 2009Q1. • In the counterfactual where new mortgages are not
introduced, the same price shock accounts for 57.3% of the increase in foreclosures. • Thus, the origination of nontraditional mortgages for two
model periods can explain over 40% of the rise in foreclosures.
41 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Fraction of LIPs
LIPs in stage 2 LIP LIPs with n =0 in stage 2 No LIPs in stage 2
0.12
0.1
0.08
0.06
0.04
0.02
0
1
5
10
15
20
25
• Data and model yield similar fraction of LIPs in originations
between Q1 of 2003 to Q4 of 2006.
42 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Foreclosure crisis
0.09
LIPs in stage 2 LIP LIPs with n =0 in stage 2 No LIPs in stage 2
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0
5
10
15
20
43 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Summary
• Question: How much did nontraditional mortgages contribute
to the foreclosure boom? • Answer: Nontraditional mortgages increased the magnitude of
the foreclosure crisis by over 40%. • Other Findings: • If financial intermediaries had not tried to separate borrowers on observable characteristics, then steady state foreclosure rates would be 40% higher. • Strengthening antideficiency policies could lower steady state foreclosures by roughly 50%.
44 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Coming soon: “High Leverage Loans and the Foreclosure Boom”
1. Introduce aggregate risk in home values: q ∈ {qL , qN , qH }. 2. Make low-downpayment available throughout
45 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Real home values (CS) in the long-run
250
200
150
100
50
0 1880
1900
1920
1940
1960
1980
2000
2020 46 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
The experiment 1. Calibrate price process to match long-term data 2. Calibrate parameters so that, following a long period of q = qN and using PTI limits of 25% (as they are in the data), the use of low-downpayment mortgages is around 5% 3. Relax underwriting standards for 4 model periods with q = qH 4. Then qH and underwriting standards return to pre-1998 values Preliminary results: • The model captures the rise in low-downpayment after 98, the
rise in HO rates, and the the foreclosure boom • Counterfactual 1: PTI standards not relaxed after 98 • Counterfactual 2: No middle stage, price falls from qN to qL • Foreclosure rates peak 30% to 50% below benchmark 47 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Mortgage payment function
• Fixed-rate mortgages (FRMs) m
FRM,t
(a0 , y0 , h0 ) =
r FRM,t (a0 , y0 , h0 ) 1 − (1 + r FRM,t (a0 , y0 , h0 ))−T
(1 − ν)h0 qt , ∀n ∈ {0, T − 1}
• Low-initial payment mortgages (LIPs) LIP,t (a0 , y0 , h0 ) h0 qt r
LIP,t mn (a0 , y0 , h0 ) =
r LIP,t (a0 ,y0 ,h0 ) LIP ) h0 qt 1−(1+r LIP,t (a0 ,y0 ,h0 ))−(T −n
if n < nLIP if n ≥ nLIP
Back
48 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Housing Market Clearing Condition The market for housing capital clears provided Z Z h′ 1{H ′ =1} P(h′ |ω)dµM = Ak h1{H ′ =1,h(ω)=h} dµM − ΩM
ΩM
• In equilibrium the production of new housing capital must
equal the housing capital lost to devaluation. • Both the rental and owner-occupied markets clear since the
intermediary is willing to accommodate any allocation of total housing capital by the arbitrage condition. Back
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Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Intermediary’s problem • Arbitrage between renting and selling houses implies
q=
+∞ X R −δ , (1 + r )t t=1
which determines rental payments R. • Housing capital investment k maximizes profits: Akq − k ⇒ q =
1 . A
• Intermediary must expect zero net profits on all mortgages.
W κ (ω0 ) − (1 − ν1{ζ=FRM} )qh0 = 0. where W κ (ω0 ) is the expected present discounted value of a loan contract κ = (ζ, r ζ , h0 ). 50 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Intermediary decision making
• The intermediary’s value function is denoted W (ω). It is
given by 1 If the mid-age household is currently a homeowner and the mortgage is not paid off so that ω = (a, y , 1, h, n; κ) with n ∈ (0, T − 1] W κ (ω) = D I (ω) + D V (ω) min{(1 − χ)qh, b(n; κ)} + S(ω)b(n; κ) m(n; κ) W κ (ω ′ ) I V + Eω′ |ω + 1 − D (ω) − D (ω) − S(ω) 1+r +φ 1+r +φ where S(ω) = 1 if household sells the house.
51 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
2 If the household has just turned mid-age and its budget set is not empty so that ω0 = (a0 , y0 , 0, h1 , . . .) and ( y0 + (a0 + ι − 1{ζ=FRM} νh0 q)(1 + r ) − m(0; κ) − δh0 ≥ 0 a0 − 1{ζ=FRM} νh0 q ≥ 0 then m(0; κ) W κ (ω ′ ) W (ω0 ) = + Eω′ |ω0 1+r +φ 1+r +φ κ
3 In all other cases, W (ω) = 0. Back
52 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Truncated Rates • The rate is truncated since the household default probability is
too high for the bank to break-even at any mortgage rate below the rate at which the mortgage payment in the first period is so high that the budget set is empty. • The left truncation can be thought of as an endogenous borrowing constraint associated with different borrower characteristics. • In that period (i.e. when n = 0), the budget set is empty when c = a′ = 0 and m(0; ζ, r ζ ) > y0 + (a0 + ι − vqh · 1{ζ=FRM} )(1 + r ). Since m(0; ζ, r ζ , h0 ) is strictly increasing in r ζ , we know there is an interest rate r ζ that depends on y0 and a0 such that for any r > r ζ the bank cannot break even. Back 53 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Newly middle-aged agents n = 0 • State: ω = (a, y , H, h, n; κ) • The value function VM (a, y , 0, h, 0; ∅) for a newly middle-aged
agent solves
VM (a, y , 0, h, 0; ∅) = + +
max
c≥0,a′ ≥0,H ′ ∈{0,1},κ∈K (ω0 )
u(c, (1 − H ′ )h1 + H ′ h0 )
(1 − H ′ )βEy ′ |y (1 − ρO )VM (a′ , y ′ , 0, h1 , 1; ∅) + ρO VO (a′ ) (1 − ρO )VM (a′ , y ′ , 1, h′ , 1; κ) H ′ βE(y ′ ,h′ )|(y,h0 ) +ρO VO (a′ + max {qh0 − b(1; κ), 0})
subject to: c + a′
=
y + (1 + r )(a − H ′ ν1{ζ=FRM} qh0 ), −H ′ (m(0; κ) + δh0 ) − (1 − H ′ )Rh1 ,
a
≥
H ′ ν1{ζ=FRM} qh0 ,
where K (ω0 ) is the set of mortgage contracts available with typical element κ = (ζ, r ζ , h0 ). Back 54 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Value function for a mid-aged agents with mortgage
= + +
VM (a, y , 1, h, n; κ) max u(c, (1 − H ′ )h1 + H ′ h) c≥0,a′ ≥0,(H ′ ,D I ,D V ,S)∈{0,1}4 (1 − H ′ )βEy ′ |y (1 − ρO )VM (a′ , y ′ , 0, h1 , n + 1; ∅) + ρO VO (a′ ) (1 − ρO )VM (a′ , y ′ , 1, h′ , n + 1; κ) ′ H βE(y ′ ,h′ )|(y,h) +ρO VO (a′ + max {qh − b(n + 1; κ), 0})
subject to: c + a′
=
y + (1 + r )(a + (1 − H ′ ) max((1 − (D I + D V )χ)qh − b(n; κ), 0)) −H ′ (m(n; κ) + δh) − (1 − H ′ )Rh1
DI
=
1 if and only if y + a(1 + r ) − m(n; κ) − δh < 0
DV
=
1 if H ′ = 0 and qh − b(n; κ) < 0
S
=
1 − H′ − DI − DV
Back to young’s prob. 55 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Definition of default 1. Involuntary default D I (ω) = 1 ( H=1 y + (a + ι)(1 + r ) − m(n; κ) − δh < 0 2. Voluntary default D V (ω) = 1 H = 1 y + (a + ι)(1 + r ) − m(n; κ) − δh ≥ 0 qh − b(n; κ) < 0 H ′ = 0 Back to default freq.
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Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Selection
Table: Rent-or-own decision rules by asset and income group Contract House size Benchmark yL yM yH FRM + LIP yL yM yH
Rent h1
h2
LIP h3
h2
FRM h3
a0 < 0.32 a0 < 0.30 a0 < 0.30
– – –
– – –
0.32 ≤ a0 < 3.14 0.30 ≤ a0 < 1.24 0.30 ≤ a0 < 0.53
3.14 ≤ a0 1.24 ≤ a0 0.53 ≤ a0
a0 < 0.20 – –
0.20 ≤ a0 < 0.94 a0 < 0.30 –
– – a0 < 0.53
0.94 ≤ a0 < 3.14 0.30 ≤ a0 < 1.24 –
3.14 ≤ a0 1.24 ≤ a0 0.53 ≤ a0
Back
57 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Welfare calculation for consumption equivalents
U
FRM+LIP
i
(y )
= = =
E0 E0
" "
∞ X
β
t
β
t
u(cti ,bench (1
t=0
∞ X t=0
+k
i
), hti ,bench )
#
ψ ln(cti ,bench ) + ψ ln(1 + k i ) +(1 − ψ) ln(hti ,bench ) + θ1{ht ∈{h2 ,h3 }}
#
ψ ln(1 + k i ) (1 − β) (1 − β) FRM+LIP i =⇒ k i = exp [U (y ) − U bench (y i )] − 1 ψ U bench (y i ) +
Back
58 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Distribution of young agents
Let (nL , nM , nH ) be the invariant income distribution implied by the income process. The invariant distribution µY on ΩY solves, for all y ∈ {yL , yM , yH } and A ⊂ ℜ+ : Z 1{aY′ (ω)∈A} Π(y |ω)dµY (ω) µY (A, y ) = µ0 1{0∈A,y =yj } nj +(1−ρM ) ω∈ΩY
59 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Middle-aged agents
µM (A, y , H, h, n; κ)
=
ρM
Z
ΩY
1{(H,h,n)=(0,h1 ,0)} 1{a′
Y
Z
(ω)∈A} Π(y |ω)dµY (ω) (ω)∈A} Π(y |ω)P(h|ω)dµM (ω)
+
(1 − ρ0 )
×
1{n(ω)=0,Ξ(ω)=κ} + 1{n(ω)>0,κ=κ(ω)}
ΩM
1{(H ′ (ω)=H,n(ω)=n−1,a′
M
60 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Old agents
Z 1{aO′ (ω)∈A} dµO (ω) µO (A) = (1 − ρD ) ΩO Z 1{aM′ (ω)+max{H ′ (ω)[qh(ω)−b(n+1,κ)],0}∈A} dµM (ω) +ρO ΩM
SS def
61 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Rent-or-own decision rules in pooling and separating equilibria
Contract House size FRM + LIP yL yM yH FRM + LIP, pooling yL yM yH
Rent h1
h2
LIP h3
h2
FRM h3
a0 < 0.20 – –
0.20 ≤ a0 < 0.94 a0 < 0.30 –
– – a0 < 0.53
0.94 ≤ a0 < 3.14 0.30 ≤ a0 < 1.24 –
3.14 ≤ a0 1.24 ≤ a0 0.53 ≤ a0
a0 < 0.08 – –
0.08 ≤ a0 < 0.68 a0 < 0.30 –
– – a0 < 0.53
0.68 ≤ a0 < 3.41 0.30 ≤ a0 < 1.34 –
3.41 ≤ a0 1.34 ≤ a0 0.53 ≤ a0
Back
62 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Cox proportional hazard specification
e as the hazard rate at date n for homeowner i due • Define γn,i
to event of default or sale, e ∈ {D, S}.
e e e = H (γ e × exp {β e 1 • γn,i n LIP LIP,i + βLTY LTYi + βATL ATLi })
where H : R ⇒ [0, 1] is an increasing function. • The coefficients are estimated via MLE with the following log likelihood function: log L(γ, β) =
N X i =1
+ −
Di · log 1 − exp(− exp(γ D (ki ) + Xi (ki )′ β D) D ′ S Si · log 1 − exp(− exp(γ (ki ) + Xi (ki ) β P ki −1 D ′ D S ′ S n=0 exp(γ (t) + Xi (n) β ) + exp(γ (t) + Xi (n) β )
Back
63 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
On calibrating to FRMs only before 2003
• In Figure 1, we can see the fraction of non-FRMs accounts for
about 15 percent of all mortgages before 2003. • However, 2/3 of that fraction of non-FRMs were standard
nominally indexed ARM, which look more like traditional mortgages than LIPs, until 2002. Back
64 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Some Steady State Accounting C + H · (R + δ) = Y + r · S + H · R + X where • C is goods consumption • R · H is housing services consumption • δ · H is investment • Y is the aggregate endowment • r · S is return to storage (or interest payments abroad if S < 0) • R · H + X is imputed rents plus “rental income of persons”
(i.e. X is the difference between imputed rents and what people actually pay for their housing consumption like mortgage payments plus maintenance for owners) Back
65 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Gerardi et. al.’s approach 1. Estimate a default/refi competing hazard model with panel mortgage data that includes a proxy for home values (home equity) as an explanatory variable 2. Ask: if 2002 vintage of loans had experienced the same average price shock as 2005 vintage, at what average rate would they have defaulted? 3. Idea: 2002 vintage was written under more typical/stringent leverage and income tests standards 4. Answer: 2002 loans would have defaulted at about half the rate 2005 loans did back
66 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
How our approach differs from and complements the econometric approach • These numbers are predicated on 1. a specific econometric model, 2. the quality of controls (zip-codes vs actual home values), and 3. the assumption that the 2002 borrower pool is what the 2005 pool would have been with 2002 underwriting standards (no sample selection effects) • Our calculations do not require these assumptions but, of
course, are conditional on our modeling choices • Further, our model can be used to simulate the role of policy,
such as recourse statutes back
67 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
Definition of high-CLTV fraction
Fraction of loans with CLTV≥ 97% =
Volume of loans with CLTV ≥ 97% Total volume of loans
Back
68 / 69
Facts
Model Environment
Equilibrium
Parameterization
SS Results
Transition
National Delinquency Survey definitions
Fraction of subprime mortgages is the stock of loans lenders report as subprime in NDS divided by the total stock of loans The foreclosure rate is the number of foreclosure starts in the course of a given quarter divided by the total stock of mortgages at the start of the quarter Back
69 / 69
