A Model of Mortgage Default

• Author: JOÃO F. COCCO,JOHN Y. CAMPBELL
• Published date: 01 August 2015
• Date: 01 August 2015
• DOI: http://doi.org/10.1111/jofi.12252

THE JOURNAL OF FINANCE •VOL. LXX, NO. 4 •AUGUST 2015

A Model of Mortgage Default

JOHN Y. CAMPBELL and JO ˜

AO F. COCCO∗

ABSTRACT

In this paper, we solve a dynamic model of households’ mortgage decisions incorpo-

rating labor income, house price, inflation, and interest rate risk. Using a zero-profit

condition for mortgage lenders, we solve for equilibrium mortgage rates given bor-

rower characteristics and optimal decisions. The model quantifies the effects of ad-

justable versus fixed mortgage rates, loan-to-value ratios, and mortgage affordability

measures on mortgage premia and default. Mortgage selection by heterogeneous bor-

rowers helps explain the higher default rates on adjustable-rate mortgages during

the recent U.S. housing downturn, and the variation in mortgage premia with the

level of interest rates.

THE EARLY YEARS OF THE 21ST CENTURY were characterized by unprecedented

instability in house prices and mortgage market conditions, both in the United

States and globally.After the housing credit boom in the mid-2000s, the housing

downturn of the late 2000s saw dramatic increases in mortgage defaults. Fore-

closures appear to have had negative feedback effects on the values of neigh-

boring properties, worsening the decline in house prices (Campbell, Giglio, and

Pathak (2011)). Losses to mortgage lenders stressed the financial system and

contributed to the larger economic downturn. These events have underscored

the importance of understanding household incentives to default on mortgages,

and the way in which these incentives vary across different types of mortgage

contracts.

In this paper, we study the mortgage default decision using a theoretical

model of a rational utility-maximizing household. We solve a dynamic model of

a household that finances the purchase of a house with a mortgage, and must

in each period decide how much to consume and whether to exercise options to

default, prepay, or refinance the loan. Several sources of risk affect household

∗John Y.Campbell is at the Department of Economics, Harvard University, and NBER, and Jo˜

ao

F. Cocco is at the Department of Finance, London Business School, CEPR, CFS, and Netspar. We

would like to thank seminar participants at the AEA 2012 meetings, Bank of England, Berkeley,

Bocconi, Essec, the 2012 SIFR Conference on Real Estate and Mortgage Finance, the 2011 Spring

HULM Conference, Illinois, London Business School, Mannheim, the Federal Reserve Bank of

Minneapolis, Norges Bank, and the Rodney L. White Center for Financial Research at the Whar-

ton School, as well as Stefano Corradin, Andra Ghent, Francisco Gomes, Jonathan Heathcote,

John Heaton, Pat Kehoe, Ellen McGrattan, Steve LeRoy, Jim Poterba, Nikolai Roussanov, Sam

Schulhofer-Wohl, Roine Vestman, Paul Willen, and Albert Zevelev for helpful comments on an

earlier version of this paper. Weare particularly grateful to three anonymous referees and to Ken

Singleton (Editor) for comments that have significantly improved the paper.

DOI: 10.1111/jofi.12252

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decisions and the value of the options on the mortgage, including house prices,

labor income, inflation, and real interest rates. We use multiple data sources

to parameterize these risks.

Importantly, we study household decisions for endogenously determined

mortgage rates. We model the cash flows of mortgage providers, including a

loss on the value of the house in the event the household defaults. We then use

risk-adjusted discount rates and a zero-profit condition to determine the mort-

gage premia that in equilibrium should apply to each contract. Since household

mortgage decisions depend on interest rates and mortgage premia, and these

decisions affect the profits of banks, we must solve several iterations of our

model for each mortgage contract to find a fixed point. Thus, our model is

not only a model of mortgage default, but also a micro-founded model of the

determination of mortgage premia.

The literature on mortgage default emphasizes the role of house prices and

home equity accumulation for the default decision. Deng, Quigley, and Van

Order (2000) estimate a model, based on option theory, in which a household’s

option to default is exercised if the option is in the money by some specific

amount. Borrowers do not default as soon as home equity becomes negative;

they prefer to wait since default is irreversible and house prices may increase.

Earlier empirical papers by Vandell (1978) and Campbell and Dietrich (1983)

also emphasize the importance of home equity for the default decision.

As in this literature, in our model mortgage default is triggered by negative

home equity, which tends to occur for a particular combination of the shocks

that the household faces: house price declines in a low-inflation environment

with large nominal mortgage balances outstanding. Also as in previous litera-

ture, households do not default as soon as home equity becomes negative.

A novel prediction of our model is that the level of negative home equity

that triggers default depends on the extent to which households are borrowing

constrained. As house prices decline, households with tightly binding borrowing

constraints will default sooner than unconstrained households, because they

value the immediate budget relief from default more highly relative to the

longer-term costs. The degree to which borrowing constraints bind depends on

the realizations of income shocks, the endogenously chosen level of savings, the

level of interest rates, and the terms of the mortgage contract. For example,

adjustable-rate mortgages (ARMs) tend to default when interest rates increase,

because high interest rates increase required mortgage payments on ARMs,

tightening borrowing constraints and triggering defaults.

We use our model to illustrate these triggers for mortgage default and to

explore several interesting questions about the effects of the mortgage system

on defaults and mortgage premia.

First, we use our model to examine how the adjustability of mortgage rates

affects default behavior, comparing default rates for ARMs and fixed-rate mort-

gages (FRMs). Not surprisingly, both ARMs and FRMs experience high default

rates when there are large declines in house prices. However, for aggregate

states with moderate declines in house prices, ARM defaults tend to occur

A Model of Mortgage Default 1497

when interest rates are high—because high rates increase the required pay-

ments on ARMs—whereas the reverse is true for FRMs.

Second, we determine mortgage premia in the model and compare the results

to the data. For most parameterizations and household characteristics the

model predicts that mortgage premia should increase with the level of interest

rates. In U.S. data, this appears to be the case for FRMs, but not for ARMs. The

model is able to generate ARM premia that decrease with interest rates when

we assume that ARM borrowers have labor income that is not only riskier on

average, but also correlated with the level of interest rates. Such a correlation

arises naturally if interest rates tend to be lower in recessions. We use our

model to perform welfare calculations and show that households with this

type of income risk benefit more from ARMs relative to FRMs, supporting the

hypothesis that such households disproportionately borrow at adjustable rates.

Even though our model can generate the qualitative patterns of mortgage

premia observed in the data, it is harder to match those patterns quantitatively.

Our model does not easily explain the large ARM premia observed in U.S. data

when interest rates are low.Furthermore, our model generally predicts a larger

positive effect of interest rates on FRM mortgage premia than that observed

in U.S. data. Our model can deliver FRM mortgage premia that better match

the data if there is refinancing inertia (Miles (2004), Campbell (2006)), so that

households do not refinance their FRMs as soon as it is optimal to do so.1

Third, we use our model to investigate how ratios at mortgage origination

such as loan-to-value (LTV), loan-to-income (LTI), and mortgage-payment-to-

income (MTI) affect default probabilities. The LTV ratio measures the house-

hold’s initial equity stake, while LTI and MTI are measures of initial mortgage

affordability. A clear understanding of the relation between these ratios and

mortgage defaults is particularly important in light of the recent U.S. experi-

ence. Figure 1plots aggregate ratios for newly originated U.S. mortgages over

the last couple of decades, using data from the Monthly Interest Rate Survey

(MIRS) of mortgage lenders conducted by the Federal Housing Finance Agency

(FHFA).2This figure shows that there was an increase in the average LTV in

the years before the crisis, but to a level that does not seem high by historical

standards. A caveat is that the survey omits information on second mortgages,

which became far more common during the 2000s.3Even looking only at first

mortgages, however, there is a striking increase in the average LTI ratio, from

an average of 3.3 during the 1980s and 1990s to as high as 4.5 in the mid-

2000s. This pattern in the LTI ratio is not confined to the United States; in the

United Kingdom the average LTI ratio increased from roughly two in the 1970s

and 1980s to above 3.5 in the years leading up to the credit crunch (Financial

Services Authority (2009)). Interestingly, as can be seen from Figure 1,the

1Guiso and Sodini (2013) provide a survey of the household finance literature.

2The LTV series is taken directly from the survey, and the LTI series is calculated as the ratio

of the average loan amount obtained from the same survey to the median U.S. household income

obtained from census data. The survey is available at www.fhfa.gov.

3In addition, the figure shows the average LTV, not the right tail of the distribution of LTVs,

which may be relevant for mortgage default.