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The Persistence of Financial Distress

WP 17-14

Kartik Athreya
Federal Reserve Bank of Richmond
Jose Mustre-del-Rio
Federal Reserve Bank of Kansas City
Juan M. Sanchez
Federal Reserve Bank of St. Louis

The Persistence of Financial Distress∗
Kartik Athreya†

José Mustre-del-Rı́o‡

Juan M. Sánchez§

November 9, 2017

Working Paper No. 17-14

Abstract
Using recently available proprietary panel data, we show that while many (35%) US
consumers experience financial distress at some point in the life cycle, most of the events
of financial distress are primarily concentrated in a much smaller proportion of
consumers in persistent trouble. Roughly 10% of consumers are distressed for more than
a quarter of the life cycle, and less than 10% of borrowers account for half of all distress
events. These facts can be largely accounted for in a straightforward extension of a
workhorse model of defaultable debt that accommodates a simple form of hetero-geneity
in time preference but not otherwise.

JEL classification: D60, E21, E44.
Keywords: default, financial distress, consumer credit, credit card debt.

∗

For comments and suggestions, we thank seminar participants at the 2016 SED Meetings, 2016 NBER
Summer Institute, and 2017 Stockman Conference. We thank Hee Sung Kim for outstanding research
assistance. All errors are our own. The views expressed herein are those of the authors and should not be
attributed to the FRB of Kansas City, Richmond, St. Louis, or the Federal Reserve System.
†
Federal Reserve Bank of Richmond; e-mail: kartik.athreya@rich.frb.org
‡
Federal Reserve Bank of Kansas City; e-mail: jose.mustre-del-rio@kc.frb.org
§
Federal Reserve Bank of St. Louis; e-mail: juan.m.sanchez78@gmail.com.

Introduction

What are the empirics of financial distress in the United States, and to what extent can we
understand them through the lens of state-of-the-art quantitative models of defaultable debt?
The goal of this paper is to answer these two questions. We tackle the first question using
newly available proprietary panel data and tackle the second question estimating multiple
models of defaultable consumer debt over the life cycle.
Our empirical work provides estimates of both the extensive and intensive margins of financial distress over the life cycle. We show that while many (35%) US consumers experience
financial distress at some point in the life cycle, most distress events are primarily accounted
by a much smaller proportion of consumers in persistent trouble: distress incidence is nearly
double its unconditional rate even a decade after the initial distress event, and roughly 10%
of the consumers are distressed for more than a quarter of the life cycle. In turn, just 10%
of borrowers account for half of all distress. We then show that these facts can be largely
accounted for in a straightforward extension of a workhorse model of defaultable debt that
accommodates a simple form of heterogeneity in time preference but not otherwise. Indeed,
the workhorse model of unsecured consumer debt without heterogeneity generates very little
persistence.
Our focus on the persistence of financial distress arises because it is the latter that
provides essential guidance to the appropriate interpretation of the risks of encountering
distress over one’s lifetime. For example, if financial distress is highly transitory, a given
incidence for it over the life cycle would suggest that most or all households face similar
risks over their lives. If, on the other hand, financial distress is highly persistent, the same
incidence would be disproportionately accounted for by a much smaller number of borrowers
who repeatedly, or in a sustained fashion, encounter distress.
Our work contributes in two ways. First, to our knowledge, our work is novel in providing
a complete description of the incidence, concentration, and dynamics of financial distress.
Our findings are striking: As for incidence, at least a third of all households will experience
financial distress over their lifetime. As for dynamics, financial distress is very persistent.
For instance, even a decade after experiencing a period of severe delinquency, borrowers are
at roughly twice the risk of being severely delinquent as the unconditional person their age.
Moreover, we find that this measure of persistence is essentially invariant over the life cycle,
despite unconditional persistence having a clear life cycle hump and decline beyond middle
age.1 As for concentration, financial distress is so unequally distributed that the top 10% of
borrowers account for fully half of all financial distress events.
The second contribution of this paper is to show that the facts on financial distress are
suggestive of persistent differences in household preferences related to the use of credit and
debt repayment. Specifically, we demonstrate that one is led fairly naturally to a model that
retains most of the features of a baseline model of defaultable consumer debt but allows for a
parsimonious form of heterogeneity in terms of household time preference. This makes clear
that the risk of financial distress is one that is resolved early in life, with most borrowers
knowing that they will face very little risk in life and a much smaller few knowing that they
1
In addition to these facts, and as with the facts on incidence described above, the persistence of financial
distress is very similar across all 50 states. Results are discussed in the Appendix.

will face a bleak outlook.
The difficulty that households appear to have in clearing their debt obligations is suggestive of at least a subset facing significant constraints on access to relatively cheap credit.
After all, why would any consumer repeatedly be seen in the data with debt in arrears, if
not for their inability to refinance those debts or shift them to creditors with whom they are
not delinquent?
One motivation for our work, particularly our empirical efforts, is that in recent work,
the extreme events of bankruptcy or outright repudiation play an important role in helping
discipline the quantitative strength of limited commitment on allocations. Specifically, it is
the observable event of personal bankruptcy that provides a main target for the calibration
or parameterization of the models. Recent work uses such models to analyze the implications
of regulations (especially bankruptcy law) on outcomes. For example, recent reforms like
the Bankruptcy Abuse Prevention and Consumer Protection Act of 2005 (BAPCPA), or the
effects of competing social insurance policies on credit use have been studied through the
lens of what is now a “standard default model” (e.g., Livshits et al. (2007), Chatterjee et al.
(2007)). A consistent finding in this work is that debt relief makes credit expensive and
so sensitive to borrower circumstances that the overall ability to smooth consumption (and
hence ex-ante welfare) is substantially worsened. A caveat is that sudden large shocks that
force households to consume, or spend (e.g., legal judgments or uninsured medical expenses),
restore the ability of default to provide net benefits in an ex-ante sense. But as noted above,
absent clear evidence that the baseline models used in these analyses capture well the time
path of overall financial conditions, such as when measured by the presence of financial
distress, there is reason for concern about the sensitivity of that finding.
While our analysis suggests the presence of heterogeneity in discounting, we stress that
such variation is still a stand-in for a variety of other forces–notably unobserved demands
for consumption within the household arising from a variety of sources. The appropriate
interpretation of our findings is therefore not that individuals are necessarily widely varying
in their personal levels of patience, but rather that a sizable subset of consumers are persistently rendered effectively impatient, potentially by a host of additional factors not modeled
here. Future work that allows for more detail on household-level shocks, intra-household
bargaining, and other (persistent) within-household resource variation is therefore essential
before reaching conclusions that individuals are to be “implicated” in their fates.2 Indeed,
it is for this reason that we avoid any normative analysis in this paper.

1.1

Related work

Financial distress or household financial “fragility” has received significant attention in recent
work and has been the topic of interest with the general public.3 Interest in the ability of the
household to shield itself from susceptibility to shocks through the use of financial markets
2

Indeed, the important work of Becker and Mulligan (1997) shows the list of deep forces shaping timepreference is long. More specifically, their analysis shows how income, wealth, mortality, addictions, uncertainty, and other variables affect the degree of time preference. Our work underscores the need for empirical
work more capable of allowing researchers to unpack the particular circumstances facing households, especially those of the subset whose consumption needs are, evidently, very persistently urgent.
3
http://www.cbsnews.com/news/the-financial-fragility-of-the-american-household/

is, of course, longstanding. However, recent work has been aided by the arrival of more
detailed data on household balance sheets (Lusardi et al. (2011), Lusardi (2011), Jappelli
et al. (2013), Ampudia et al. (2016), Brunetti et al. (2016)) and aims to gauge borrowing
capacity and resilience to sudden, unforeseen expenditures. Specifically, this work primarily
focuses on measuring the ability of households to remain current on incurred debts, as well as
the question of how much borrowing the household could feasibly engage in, in a short term
period, e.g., 30 days—especially to cover an unforeseen “expense” (as opposed to a change
in income, say). A rough summary of this work might be this: A substantial proportion
of households in the US as well as in the EU are, by various measures, “fragile” or in—or
near—financial distress.
Our work is also clearly related to the far larger body of work concerned with the measurement of liquidity constraints across consumers. Substantively, this work tries to measure the
proportion of US households who are liquidity constrained, and therefore, not well-positioned
to deal with adverse shocks. These include papers of Jappelli and Pagano (1999), Hall and
Mishkin (1982), Zeldes (1989) and others. More recently, Gross and Souleles (2002) use
exogenous variation in credit line extensions to gauge the fraction who increase their debt in
response (and hence can be viewed as having been constrained). They find (unsurprisingly)
that those close to their limits increased borrowing by most but so did even those further
away from their credit limit. The consensus seems to be that roughly 20% are “constrained”
either in terms of excess sensitivity to income or in terms of how they respond to survey
questions. Compared to this previous literature, our study uncovers the persistence of financial distress. This has important implications for welfare analysis and policy design, as we
will show.
Our work contributes to the research programs above in two ways. First, to our knowledge, we are the first to focus on the empirical dynamics of consumer financial distress,
which we broadly define to be situations in which the household remains susceptible to any
deviation of income from its ex-ante expectation. In this sense, our measures are informed by
the line of work emphasizing household insurance, particularly Kaplan and Violante (2010),
and the “insurance coefficient” approach of Blundell et al. (2008). Our emphasis, relative
to the preceding work, is on direct measures of financial conditions that have empirical
counterparts.
Second, our work extracts a previously unknown implication from the “standard default
model.” We have already noted above that benchmark models of unsecured consumer debt
and default over the life cycle do not imply anything near the observed level of persistence.
These include models based primarily on those of Livshits et al. (2007) and Athreya (2008).
For example, when distress is measured by severe delinquency (i.e., having a debt 120 days
or more past due), the model-implied gaps between the unconditional and conditional probabilities of distress over the life cycle are (i) far too small, at only 15 percentage points at the
one-year mark, compared to far larger gap in the data of 60 percentage points, and (ii) far too
transitory: At even the three-year mark, the model fails completely to generate separation
between the conditional and unconditional probabilities of being in financial distress.
Our empirical results, and the inability of workhorse models to account for them, suggest
that underlying persistent heterogeneity may be an important force in consumer behavior.
We pursue this intuition and demonstrate that an underlying environment in which households may differ systematically from each other in their “type”, as defined by their patience,
3

allows for much greater explanatory power.
Our emphasis on understanding household financial distress, and our use of the facts of
distress to learn about underlying time preference, are both novel. However, our findings
inform a larger body of recently emerging work that uses consumer credit to conclude that
permanent heterogeneity in time-discounting is an important feature of the data.4 Closest of
all is the work of Fulford and Schuh (2017), who demonstrate that household credit utilization
and life cycle consumption and savings (credit-use) patterns clearly suggest important heterogeneity in time preference. Indeed, these authors estimate that nearly two-thirds (64%)
of all households are effectively impatient, enough so to live essentially hand-to-mouth. Our
work strongly complements theirs by showing that the facts of financial distress—a state
that is clearly and unambiguously observable—drive one to reach very similar conclusions.
In particular, our model differs from theirs, and all other previous work, by deriving financial distress from a model that incorporates default as an option for borrowers. This in turn
allows our work to capture the complications posed by default risk for credit pricing and
availability. Notably, terms across borrowers will vary (both over time for a given borrower
and across different borrowers at any given time) in response to the evolution of their balance
sheet and future earnings prospects.
Two other recent papers also use credit market data to conclude that there is nontrivial
variation in patience across borrowers. First, Gorbachev and Luengo-Prado (2016) use National Longitudinal Survey of the Youth (NLSY) data to conclude—from the observation of
variation in individuals in the extent to which they borrow and save simultaneously—that
US households vary substantially in time preference. Second, Meier and Sprenger (2017)
conclude in favor of discount-rate heterogeneity from data obtained in a field experiment
on credit use. Lastly, while not about consumer credit use, Parker (2017) finds that US
households are better described as varying systematically in their preferences than in terms
of the shocks they receive based on household consumption responses to random variation in
receipt of lump-sum cash transfers (arising from stimulus payments during the Great Recession).5 Indeed, he argues that the observed lack of consumption smoothing in those data are
“associated with a measure of impatience” among other persistent differences. Overall, the
fact that credit use data, financial distress data, and data on consumer response to transfer
payments, all point to variation in discounting is noteworthy and suggests that this may be
a genuine, and genuinely important, form of heterogeneity.
The remainder of the paper is organized as follows. Section 2 defines our preferred
measures of financial distress, and provides an empirical analysis of the behavior of these
measures. Section 3 then lays out a variant of a standard life cycle model of consumption
and defaultable debt that largely accounts for the empirics of financial distress. Section 4
provides the main comparisons of models with data. Section 5 illustrates that settings that
4

A much larger literature has used data on consumption and income, and sometimes wealth as well, to
estimate models that imply preference heterogeneity more generally. These include early work of Lawrance
(1991), Gourinchas and Parker (2002) and Cagetti (2003). Other work on the presence of discount-factor
heterogeneity includes: Hausman (1979), Samwick (1998), Warner and Pleeter (2001), and Belzil and Hansen
(1999). Lastly, see Frederick et al. (2002) for a survey.
5
Relatedly, Mustre-del Rı́o (2015) finds that persistent employment differences across males in the US
cannot be explained by differences in wealth or wages and hence are indicative of persistent differences in
the disutility of work.

do not allow for discount-factor heterogeneity simply cannot capture what would appear to
be critical features of observed financial distress. Section 6 concludes.

Financial Distress in the US

The first goal of this paper is to establish the empirics of financial distress. As indicated
above, we exploit recently available account-level panel data from Federal Reserve Bank of
New York Consumer Credit Panel/Equifax. These data cover an 18 year window for a large
number of account holders.
We focus on individuals with complete credit histories between 1999Q1 to 2017Q2. Additionally, we restrict our attention to the cohort that enters 1999Q1 between the ages of
25-55. Thus, by the end of our observation period the oldest individuals in our sample are
73, while the youngest are 43. Because our model will focus on default and delinquency
behavior prior to retirement we further restrict our measurements to individuals through the
age of 65.6 While our analysis focuses on a specific cohort over a particular time period,
we note that our observations on the incidence of financial distress are robust to looking at
repeat cross-sections over the 1999Q1-2017Q2 period. Additionally, our observations on the
persistence of financial distress do not seem to be driven simply by behavior during and after
the Great Recession. We will define an individual to be in financial distress (FD) in a given
year if, in that period, they are recorded as having at least one severely delinquent account:
an account for which payment is at least 120 days past due. Additional details about our
data appear in the Appendix.
To start, consider first the “extensive” margin of distress: How broadly shared an experience is financial distress? Figure 1 takes a life cycle perspective. The left panel shows the
fraction of individuals in delinquency. What emerges is central to what follows: financial
distress, while relevant for consumers of all ages, is not widespread. The line in the figure
begins near 10-14% among the young, and falls below 10% later in life.
As for the “intensive” margin of financial distress, consider the right panel of Figure
1. We see that the fraction of debt in delinquency follows a very similar pattern, with the
youngest having the largest fraction of debt in delinquency (e.g., roughly 13% at age 25)
with this proportion falling substantially to around 6% by age 55.
Next, to begin assessing the persistence of financial distress, we compare the unconditional probability of falling into delinquency with the conditional probability. Specifically,
we condition on the time elapsed since a transit into financial distress by an individual. In
Figure 2 we see very clearly just how persistent the state of distress is for US consumers.
Conditional on being in distress today, the likelihood of being distressed in six years (the orange dotted line) is nearly triple that of the unconditional rate (the black line) over the entire
life cycle. As we show further below, this particular feature will elude the standard model of
defaultable consumer debt and will fairly decisively suggest the importance of heterogeneity
in individual time preference.
One might also ask whether an alternative “extensive” margin measure might indicate
something different. In particular, instead of defining distress to be a situation in which
an individual has severely delinquent debt, one could measure the proportion of consumers
6

In all figures we plot data through age 55 because we measure default up to 10 years in the future.

.14
.06

.08

Fraction of debt in FD, unconditional
.08
.1
.12

Fraction of individuals in FD, unconditional
.1
.12
.14

.16

Figure 1: The Incidence of Financial Distress Over the Life Cycle

40
Age

Source: See Appendix

Probability of being in FD
.2
.3
.4

Figure 2: The Persistence of Financial Distress Over the Life Cycle (debt)

40
Age

Unconditional
4 years after being in FD
8 years after being in FD

2 year after being in FD
6 years after being in FD
10 years after being in FD

Source: See Appendix

who have depleted their available credit (e.g, those who have “maxed out” their credit
cards). Figure 3 (which excludes those with 0 credit limit) shows that the answers remain
very similar: limited borrowing capacity remains a prevalent issue for a small, but far from
negligible, group of borrowers, throughout the life cycle, and just as when measured by
delinquency, indicates substantial persistence.7
We now provide additional detail on the persistence of financial distress (defined unless
otherwise indicated by the presence of severely delinquent accounts on a consumer’s balance
sheet). One particular point to keep in mind is that the more transitory distress is, the less
7

In our dataset, on average, about 50% of individuals in delinquency have also depleted their credit.

Probability of reaching credit limit
.2
.3
.4
.5

Figure 3: The Persistence of Financial Distress Over the Life Cycle (credit limit)

40
Age

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

Source: See Appendix

one might view it as relevant to household well-being. In particular, one might conclude
that highly fleeting distress indicates optimal use by borrowers of the “real option” to force
their creditors to implicitly refinance their loans (subject to the costs associated with being
severely late on payments).
Figure 4 provides further evidence that distress is not highly fleeting, but instead persistent over long periods. It displays the distribution of time spent (as a proportion of the 18
years in sample) in financial distress for those who have been delinquent at any time during
the sample window. It is clear that while it is indeed fleeting for some (roughly 30%), for
70% of consumers, distress is much more routine state of affairs. Indeed, more than 30% of
all those who experience financial distress spend at least a quarter of their lives in it!
A second way to evaluate the persistence of financial distress is to examine the number
of distinct spells of delinquency that an individual will experience. Figure 5 shows how,
for those who have experienced financial distress at least once, the number of spells they
experience is often substantial, with roughly a tenth of the sample experiencing four or more
spells.

Percent
30

Figure 4: The Duration of Financial Distress (share of life)

.4
.6
Share of life (in the sample) in FD

Source: See Appendix

Percent
30

Figure 5: The Duration of Financial Distress (spells)

4
5
6
7
Number of spells by individuals

Source: See Appendix

A more direct way of showing that a small share of the population accounts for most of
the financial distress is to look at the concentration of financial distress. In Figures 6 and 7,
we show the Lorenz curves for our two measures of financial distress. They show that most
of the 80% of the financial distress is accounted by less than 20% of people.
The measures presented thus far are “extensive margin” measures: They are based on
measures of financial distress that are binary—whether or not someone has severely delinquent debt, or whether or not someone has reached their credit limit. While the data suggest
8

Share of FD
.4
.6

Figure 6: The Concentration of Financial Distress (debt)

.4
.6
Share of Population

Source: See Appendix

Share of individuals reaching credit limit
.2
.4
.6
.8

Figure 7: The Concentration of Financial Distress (credit limit)

.4
.6
Share of Population

Source: See Appendix

that by these metrics, financial distress is not only frequent, but also persistent, it might
still not be an economically important phenomenon if the debts that consumers are severely
delinquent on are themselves trivial. We now demonstrate that they are not.
A natural intensive-margin measure is one that relates the volume of debt in delinquency
to the total debt owed by individuals over the life cycle. Figure 8 clearly indicates that
financial distress among those facing it is “intense,” measured either in terms of the average
(across individuals) proportion of debt or average number of accounts severely delinquent.
9

When debt is the measure (left panel), we see not only do distressed borrowers have almost
all (roughly 80%) their debts in delinquency, but also that there is virtually no life cycle
component to the intensity of distress, as the intensity of distress falls from nearly 88% early
in the life cycle to 83% at older ages.

.83

.76

Fraction of debt in FD, conditional
.84
.85
.86
.87

Fraction of cards in FD, conditional
.78
.8
.82

.84

.88

Figure 8: The Intensity of Financial Distress Over the Life Cycle

40
Age

Source: See Appendix

Figure 9 summarizes the amount of delinquent debt by 50th, 75th and 90th percentile
by age.8 It shows that the median debt in delinquency is fairly stable over the life cycle, but
the upper tail (measured either by the 75th or 90th percentiles) grows substantially over the
life cycle. Of course, the incidence of distress is lower late in the life cycle, but it is clear
that among the distressed, the highest distress occurs among older individuals.
Collecting the extensive- and intensive-margin empirics, we can summarize our findings as
follows: Financial distress among US individuals is driven by a relatively small proportion of
individuals who experience significant and persistent debt repayment problems. These facts
are novel and certainly suggest that financial distress may well be an important phenomenon,
especially from the point of view of a subset of individuals looking out over a life cycle.9
Discerning this importance, however, requires a model, which we turn to next.

Understanding Financial Distress

We now address the question of how well the facts documented above, particularly the
persistence of financial distress, can be accounted for in a setting where households make
choices over consumption, credit, and repayment. Our model builds on important work of
Livshits et al. (2007) because it provides a benchmark life cycle consumption savings model
in which debt may be repudiated. Our model will feature two tractable extensions of this
8

Each year has been deflated by January’s Current Price Index for Urban Consumers (CPI-U) as published
by the Bureau of Labor Statistics (BLS).
9
In the Appendix we also establish that the incidence of distress (as measured by severe delinquency) is
prevalent in very similar ways across all 50 states. This occurs despite what might seem at first glance to be
potentially salient differences in consumer default regulations.

FD debt (in 2017 dollars)
5000
10000
15000

20000

Figure 9: Distressed Debt by Age

40
Age

50th percentile
90th percentile

75th percentile

Source: See Appendix

environment: (1) Allowance for informal default, and (2) heterogeneity in time preference.
We will show that these extensions largely, though not perfectly, account for the facts.
We then turn to demonstrating these extensions are, in a sense, “identified” by the data.
Specifically, we show that a fairly comprehensive set of alternative models, including ones
that allow for income risk to be extremely persistent, and ones that do not distinguish
between informal and formal default, each fail to deliver simultaneously the incidence of
financial distress, its persistence, or concentration.

3.1
3.1.1

A Benchmark Model
Model

There is a continuum of finitely-lived individuals who are risk-averse and discount the future
exponentially. Individuals survive to the next period with probability %n , which depends on
age n. Each agent works for a finite number of periods and then retires at age W . In each
period, agents choose consumption c and assets (or debt) a0 . Debt may be repudiated in
one of two ways. First, the agent may simply cease payment. This is known as delinquency.
With delinquency, a household’s debt is not necessarily forgiven, however. Instead, debts
are forgiven with probability γ. The probabilistic elimination of debts is meant to capture
the presence of creditors periodically giving up on collections efforts. With probability 1 − γ,
then, a household’s rolled over debt is not discharged, and in this case, the household pays a
“penalty” rate, η, of interest higher than the average rate paid by borrowers.10 Moreover, in
any period of delinquency, consumption equals income up to a threshold τ .11 Second, and as
10

This representation captures the main ingredients in Athreya et al. (2017). They show evidence that
penalty rates modeled like this are able to capture key features of delinquency.
11
The remaining income is lost, for instance, when dealing with debt collectors.

is standard in models of unsecured debt, agents may invoke formal default via a procedure
that represents consumer bankruptcy. If this is the path chosen, all debts are erased, and in
the period of filing for bankruptcy, consumption equals income net of the monetary cost f
of filing for bankruptcy. Unlike delinquency, there is no income garnishment in bankruptcy.
While all agents are assumed to have identical attitudes toward risk, they will be allowed
to vary in their willingness to substitute consumption across time. Specifically, we assume
individuals can be divided into two types via their subjective discount factors, with the
calibration (described below) determining (1) the magnitudes and (2) the proportion of the
population, carrying each of the two values. More precisely, let pL denote the proportion of
individuals who have a discount factor βL . The remaining 1 − pL share of individuals are
potentially more patient and thus have a discount factor βH ≥ βL . Denote an individual’s
discount type by j.
In this framework lifetime utility is written as
Gi,j,n (z, ε, a) = max{Vi,j,n (z, ε, a), Bi,j,n (z, ε), Di,j,n (z, ε, a)}

(1)

where V , B and D are lifetime utilities for households paying back their debt, filing for
bankruptcy, and being delinquent on their debt, respectively.
Next, the lifetime utility of bankruptcy is
Bi,j,n (z, ε) = u(yi,n (z, ε) − f ) + %n βj E [Gi,j,n+1 (z 0 , ε0 , 0)|z] .

(2)

Recall from above that if bankruptcy is chosen, then in that period, household consumption
equals income net of bankruptcy filing costs f , while in the period following bankruptcy, the
household has no debt.
Now suppose the household decides to be delinquent on its debt. In this case, lifetime
utility reads as:
Di,j,n (z, ε, a) = u(min{yi,n (z, ε), τ })
+%n βj E [(1 − γ)Gi,j,n+1 (z 0 , ε0 , (1 + η)a) + γGi,j,n+1 (z 0 , ε0 , 0)|z] .

(3)

This reflects the features described above. In particular, it makes clear that in the period of
delinquency, household consumption equals income up to a threshold τ , and in the period
after choosing to be delinquent, two states can occur: with probability (1−γ) the household’s
debt is rolled over at an interest rate of η and hence a0 = (1 + η)a. Alternatively, with
probability γ the household’s debt is fully discharged and hence the household enters the
period with no debt (i.e. a0 = 0).
Finally, suppose the household decides to pay back its debt. This is simply the case
of a pure consumption and savings model, with only the continuation value imparting any
difference between it and something entirely standard. The consumer who repays debt as
promised receives lifetime utility of
Vi,j,n (z, ε, a) = max{a0 ,c} u(c) + %n βE [Gi,j,n+1 (z 0 , ε0 , a0 )|z] ,
subject to

(4)
0

c + a qi,j,n (z, a ) = a + yi,n (z, ε),
c ≥ 0.
12

where qi,j,n (z, a0 ) is the price of debt a0 and is defined below.
In what follows, the policy function R indicates whether the household pays back its debt
(repay), becomes delinquent, or files for bankruptcy:

if
Vi,j,n (z, ε, a) = max{Vi,j,n , Bi,j,n , Di,j,n }
 1
2
if
Di,j,n (z, ε, a) = max{Vi,j,n , Bi,j,n , Di,j,n }
Ri,j,n (z, ε, a) =

0 otherwise.
Because default is an option borrowers hold, lenders must be compensated for the risk
they bear, at least on average. Specifically, we require that lenders break even in expectation
on each loan, given the information they have on borrowers. Specifically, lenders must
forecast, based on the borrower’s current state, the probability that their income one period
hence (when debt comes due) will fall into a set where default (either via delinquency or
bankruptcy) becomes more valuable than repayment. The probability of default will, of
course depend on the probability distribution of income one period hence, and also on the
discount factor of the borrower in question. Let the price of a debt issuance by a given
borrower of type i, j, n be given as qi,j,n (z, a0 ). This price function is then taken as given by
all borrowers and by virtue of the diversification assumed in the continuum, breaks lenders
to such a household type even with probability one. It satisfies the following condition:
h
i
1
0
0
00
%n E IRi,j,n+1 (z0 ,ε0 ,a0 )=1 + IRi,j,n+1 (z0 ,ε0 ,a0 )=2 (1 − γ)(1 + η)qi,j,n+1 (z , a )|z (5)
qi,j,n (z, a ) =
1+r
with a00 = (1 + η)a0
The first term on the right-hand side represents the probability that the household repays
its debt. The second term represents the probability that the household chooses to become
delinquent when given the option to repay, file for bankruptcy, or become delinquent. This
term takes into account that delinquent debt tomorrow is fully discharged at a rate γ.
An equilibrium in this economy is a set of value functions, optimal decision rules for the
consumer, default probabilities, and bond prices, such that equations (1) to (4) are satisfied
and prices satisfy the zero-profit condition (5).
3.1.2

Calibration

We begin, in Table 1, with the parameters we set externally. The penalty rate for delinquent
debt is set to 20% annually, following Livshits et al. (2007). Bankruptcy filing costs are to
2.8% of average income, or roughly $1,000, again following Livshits et al. (2007). Turning
to the income process parameters we follow Kaplan and Violante (2010). During working
ages log income equals yi,n = ln + zi,n + ei,n . Here ln is a life cycle component, ei,n is an
i.i.d transitory component, and zi,n is a persistent component that follows a random walk:
zi,n = zi,n−1 +εi,n . We assume εi,n and ei,n are normally distributed with variances σε2 and σe2 ,
respectively. To set retirement income, we follow Hatchondo et al. (2015). While in retirement, the household receives a fraction of the last realization of the persistent component of
its working-age income using the replacement ratio formula: max{A0 + A1 exp(zi,W −1 ), A2 }.
In order to be consistent with US replacement ratios, we calibrate A0 , A1 and A2 such that
13

the replacement ratio declines with income, from 69 to 14%, with an average replacement rate
of 47%. The age-specific survival probabilities follow Kaplan and Violante (2010). Lastly,
the initial distribution of wealth and earnings matches wealth holdings for 25- to 26-year-olds
from the Panel Study of Income Dynamics (PSID).
Table 1: Benchmark Model Parameters Determined Externally
Parameter
σ, Coefficient of relative risk aversion
r, Risk free interest rate
W , Retirement age
η, Roll-over interest rate on delinquent debt
f , Bankruptcy filing cost (as a share of average income)
σε2 , Variance of permanent shocks
σe2 , Variance of transitory shocks
A0 , Replacement ratio
A1 , Replacement ratio
A2 , Replacement ratio

Value
2.000
3.000%
65
20%
0.028
0.05
0.01
0.71
-0.045
0.14

In addition to the parameters set without reference to model outcomes, our model features
other parameters to be determined in calibration. These are the following: the threshold
for income garnishments in delinquency, τ , the probability that delinquent debt is fully
discharged, γ, and the parameters governing the distribution of discount factor types, βL , βh ,
and pL . For simplicity, we anchor the discount factor of the patient borrower at βH = 1.00.12
The remaining parameters are chosen such that the model matches the moments in bold as
seen in Table 2. In particular, the model is calibrated to replicate the incidence of financial
distress at ages 30 and 50, the average incidence of bankruptcy between the ages of 25 and
65, and the share of individuals who are never in financial distress (over the sample period
we observe them).
The implied parameter values are presented in Table 3. In terms of the calibration
performance, we see from Table 2 that overall, the baseline model captures well what it
targets.
3.1.3

Results from the Benchmark Model

The goal of this part of the paper is to gauge the ability of standard theory to account for
the facts of financial distress: Its incidence, its concentration, and its persistence. In Table
2 we see that both in terms of concentration, and in terms of persistence, the model does
quite well. The top 20% of model borrowers account for the vast majority (85%) of all
financial distress, something very close to what is observed. Similarly, the model captures
well, though with a less steady decline and initial level, the persistence of financial distress.
Turning next to a more general summary of concentration, Figure 10 demonstrates excellent congruence between model and data. As we will see below, alternatives to this baseline
12

Allowing this parameter to be flexibly calibrated does not affect our results as the calibration converges
to value very close to 1.0.

Table 2: Benchmark Model Calibration
Moment
Data Model
FD rate at age 30 (in %)
15.16 15.19
FD rate at age 50 (in %)
10.06 7.54
Average BK rate ages 25-65 (in %)
0.74
0.74
Pop. that never defaults (in %)
64.23 66.09
FD share of top 20% of pop. (in %) 88.07 84.86
Avg. cond. FD rate at 2 years
0.52
0.40
Avg. cond. FD rate at 4 years
0.34
0.33
Avg. cond. FD rate at 6 years
0.23
0.30
Avg. cond. FD rate at 8 years
0.17
0.28
Avg. cond. FD rate at 10 years
0.15
0.26
Note: Numbers in bold denote targeted moments in the calibration of the model.

Table 3: Calibrated Parameters for Benchmark Model
Parameter
Earnings threshold in DQ τ
Discharge shock to DQ debt γ
Low discount factor βL
High Discount factor βH
Share of pop. of type L
Note:

†

Value
4.79
0.34
0.74
1.00†
0.56

denotes a parameter fixed by assumption.

model have substantially more difficulty in matching this, and other, aspects of financial
distress.
The overall concentration of financial distress across households is one dimension of interest, but it does not directly inform us on the persistence of financial distress over the
life cycle. To understand the extent to which the baseline setting predicts how long distress
lasts, in probabilistic terms, consider the following findings. In Figure 11 we see that the
model captures well the near-term persistence of financial distress, measured here by the
conditional probability of being in distress at various leads beyond distress at an initial age,
across the entire life cycle. It also largely, but not perfectly captures persistence at most
ages. However, we see that the model predicts an even slower decay than observed in these
conditional probabilities. In other words, at longer leads, persistence is somewhat too high
in the model. When averaged over all ages, this yields the probabilities in Table 2 being
higher than their empirical counterparts.
To see this more clearly, consider Figure 12. This figure compares model with data along
the dimension of the probability (weighted by the age distribution) of distress at various
leads beyond an initial distress period. As before, we see that while the model produces
less persistence at extremely short leads, it predicts slightly greater persistence at longer
horizons. As we will show, despite the imperfect fit here, it is far better than a wide class of
alternatives that might initially be seen as candidates.
An aspect of financial distress that sheds light on the source of its concentration across
15

Share of FD
.4 .5 .6 .7

Figure 10: The Concentration of Financial Distress: FRBNY/Equifax data vs Benchmark
model

.4
.5
.6
Share of Population
Data

Benchmark

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 11: The Persistence of Financial Distress Over the Life Cycle: FRBNY/Equifax data
(left) and Benchmark model (right)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

households is the distribution of the number of spells of distress across households (for those
who have experienced it). The left panel of Figure 13 shows that in the data, the bulk of
people who experienced distress also experienced only one spell. The right panel of Figure 13
suggests the model has a similar implication, though it understates the share of individuals
with only one spell.
The preceding intuition can be sharpened by referring to Figure 14. This displays the
fraction of an individual’s life spent in financial distress. As is clear the data place considerable mass on outcomes involving large proportions of life spent in distress. Indeed, about
30% of individuals spend a quarter or more of their lives in financial distress. As seen in the
right-hand panel, the model does fairly well in accounting for this, though underpredicting
somewhat the proportion of those spending very small proportions of their lives in distress.
16

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 12: Average Persistence of Financial Distress: FRBNY/Equifax data (left) and
Benchmark model (right)

50
Percent
30
10
0

Percent
30

Figure 13: The Duration of Financial Distress (spells): FRBNY/Equifax data (left) and
Benchmark model (right)

4
5
6
7
Number of spells by individuals

50
Percent
30
10
0

Percent
30

Figure 14: The Duration of Financial Distress (share of life): FRBNY/Equifax data (left)
and Benchmark model (right)

.4
.6
Share of life (in the sample) in FD

3.2

Why are Informal Default and Discount-factor Heterogeneity
Essential?

Having demonstrated that a relatively simple extension of the canonical approach of Livshits
et al. (2007) is sufficient to allow that workhorse model to account well for the dynamics of
financial distress as a nontargeted outcome, we now detail results that suggest that these
extensions are also likely to be necessary. In this sense, the facts we establish on financial
distress suggest that borrowers may differ very fundamentally from each other—in ways
simply not captured by income risk or by the precise specification of options for debt default.
3.2.1

Distress as Bankruptcy, RIP Income Risk

We begin by focusing on a simpler model that coincides very closely with that of Livshits
et al. (2007). In this setting, default is modeled as bankruptcy (BK), and income is, as
in that benchmark, of the standard “RIP” variety. Critically, the model is one in which
default removes debt obligations altogether and imposes costs in the current period. Table
4 presents the parameters that are externally chosen for this model. Many are the same as
in our benchmark model, with the exception of the roll-over interest rate on delinquent debt
and the filing cost of bankruptcy, which are both mechanically set to zero given the structure
of this model. Given the simplicity of this model, the parameters that remain to be pinned
down are the discount factor β (which is common across all individuals) and the income
garnishment threshold when in bankruptcy τ . For ease of comparison with the benchmark
model, this model is calibrated to match similar facts on bankruptcy, as seen in Table 5
below, while the implied parameter values are displayed in Table 6. The reader is referred
to the Appendix for a detailed statement of the environment.
Table 4: Bankruptcy Model Parameters Determined Externally
Parameter
σ, Coefficient of relative risk aversion
r, Risk free interest rate
W , Retirement age
A0 , Replacement ratio
A1 , Replacement ratio
A2 , Replacement ratio

Value
2.000
3.000%
65
0.71
-0.045
0.14

RIP process
σε2 , Variance of permanent shocks
σe2 , Variance of transitory shocks

0.05
0.01

HIP process
{σα2 , σβ2 , σε2 , σe2 }
ρz
corrαβ

{0.022, 0.00038,0.047,0.029}
0.821
-0.23

Table 5: Bankruptcy Model Calibrations
Moment

Data

BK
RIP
14.95
6.53
44.00
59.02
0.261
0.152
0.116
0.099
0.089

FD rate at age 30
14.24
FD rate at age 50
9.30
Pop. that never defaults
64.23
FD share of top 20% of pop.
88.07
Avg. cond. FD rate at 2 years 0.520
Avg. cond. FD rate at 4 years 0.340
Avg. cond. FD rate at 6 years 0.231
Avg. cond. FD rate at 8 years 0.171
Avg. cond. FD rate at 10 years 0.153

BK
HIP
14.17
9.30
35.19
54.0
0.275
0.199
0.167
0.145
0.133

Note: Numbers in bold denote targeted moments in the calibration of each model.

Table 6: Calibrated Parameters for Bankruptcy Models
Parameter
Discount factor β
Earnings threshold in default τ

BK
RIP
0.839
4.276

BK
HIP
0.687
3.294

The results are fairly self-explanatory. Not only does this model fail to generate the
proportion of those who never experience financial distress, those who do remain in it for far
too short a time. One way to see this immediately is to look at Figure 15.

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 15: The Persistence of Financial Distress Over the Life Cycle: FRBNY/Equifax data
(left) and BK RIP model (right)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

Similarly, we can see from the bar chart in Figure 16 that while by construction this
model misses on persistence at the one year mark (since repeat bankruptcy is not allowed),
its implications for persistence are counterfactual beyond that.
Lastly, but importantly, we see in Figure 17 that this simpler alternative fails substantially
at generating the concentration of financial distress.
19

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 16: Average Persistence of Financial Distress: FRBNY/Equifax data (left) and BK
RIP model (right)

Share of FD
.4 .5 .6 .7

Figure 17: The Concentration of Financial Distress: FRBNY/Equifax data vs BK RIP model

.4
.5
.6
Share of Population
Data

3.2.2

BK+RIP

Distress as Bankruptcy, HIP Income Risk

The inability of the preceding model to capture the facts need not lead one to the much richer
setting that we ultimately defined as the baseline model. Indeed, a natural first step before
broadening the notion of financial distress, and certainly before allowing for the discount
factor heterogeneity that we will eventually allow for is to ask: Have we represented income
risk well? Specifically, it is relevant to gauge if the model’s failure is driven by an income
persistence that is a poor representation of variation in income prospects in the cross-section
(within the bankruptcy-only model). To do this, we now extend the previous model to allow
for heterogeneity in earnings profiles, i.e. the so-called HIP specification of Guvenen (2009).
This modification imparts permanent differences in borrowers’ lifetime income growth rates.
Thus, log income now is of the form: yi,n = ln + αi + βi n + zi,n + ei,n . As before, ln
denotes the life cycle component common to all households of age n and ei,n is a transitory
20

component. The term αi + βi n is the life cycle component that is household-specific. Next,
zi,n follows an AR(1) process:zi,n = ρz zi,n−1 + εi,n . As in Guvenen (2009), we assume the
random vector (αi , βi ) is distributed across households with zero mean, variances of σα2 and
σβ2 , and correlation of corrαβ . Finally, εi,n , and ei,n are normally distributed with variances
σε2 , and σe2 , respectively. We calibrate this model in the same fashion as the RIP version. The
third column of Table 5 displays the calibration of this model and some basic nontargeted
moments. The second column of Table 6 displays the implied parameter values consistent
with these empirical targets. As with the previous models, this model matches several basic
cross-sectional observations on financial distress, but does less well on nontargeted moments
than the baseline model.
As seen in Figure 18, the baseline model with the HIP process generates more persistence,
but only later in life. Figure 19 shows that again, by construction, this model misses on the
persistence at the one year mark, but also severely underestimates the persistence several
years out. Lastly, and just as under RIP income risk, Figure 20 shows the share of individuals
who never default over 15 years is counterfactually low in this model.
The stark inability of either variant of the benchmark consumer default model to generate
the proportion of nondefaulters is a key observation, and will inform the development of
extensions that do better. The reason is this: Any model that matches the unconditional
incidence of financial distress over the life cycle but fails on persistence will necessarily—
and grossly, in the cases above—overstate the probability that an individual experiences
financial distress. After all, 65% of the agents in the model will hit distress, compared with
just 35% in the data! On the other hand, this observation clarifies that capturing persistence
is absolutely central to providing a reasonable description of people’s experience over the life
cycle as they relate to financial distress.

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 18: The Persistence of Financial Distress Over the Life Cycle: FRBNY/Equifax data
(left) and BK HIP model (right)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

So far, we have presented results that show that under the two major categories of income
risk considered in the literature, financial distress is predicted to be far more transitory than
in the data. The forces underlying this turn out to be the ones at work in general life cycle
consumption/savings models. Specifically, in the model above, under either income process,
life cycle consumption leads households, all else equal, to borrow. Income risk, however,
21

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 19: Average Persistence of Financial Distress: FRBNY/Equifax data (left) and BK
HIP model (right).

Share of FD
.4 .5 .6 .7

Figure 20: The Concentration of Financial Distress: FRBNY/Equifax data and BK HIP
model.

.4
.5
.6
Share of Population
Data

BK+HIP

pushes consumers away from borrowing (as long as any precautionary motive is present, as
it is in our parametrization of preferences). As long understood, in models without default,
such as the classic work of Gourinchas and Parker (2002), the net result is for consumption
to track income early in the life cycle—just as in the data and for it to then fall below
it as retirement nears—just as in the data. The question is then, how does the ability to
default alter this basic finding? Athreya (2008) finds that for plausible (RIP) specifications
of income risk, the ability to default, when quantitatively disciplined, restricts credit access
substantially. This results in the model behaving similarly to one in which individuals are
simply more constrained. (In other words, the state-contingency provided by the ability to
obtain debt relief is not very valuable.). This behavior rules out very large defaultable debts
from building up frequently in early life and virtually eliminates debts altogether as agents
22

near retirement. By contrast, the data clearly show that the incidence of financial distress
does not vanish even very late in (pre-retirement) life.
The previous two models, taken together, suggest that the particulars of individual income risk, while of course important in ways established by the work cited above, is not
at the heart of explaining the persistence of financial distress. This is, a priori, not an obvious conclusion. After all, consumption smoothing motives have long been understood to
hinge on the particulars for income risk. Most notably, a setting (with infinite-lived agents,
especially) where income shocks are very persistent is one in which borrowing to smooth
consumption makes less sense than simply lowering consumption to reflect a new, and substantially worse, reality. Conversely, the more purely transitory the income risk, the more
that individuals will incur debts to maintain smooth consumption. And yet, we have seen
that fairly disparate representations of income risk, once models are calibrated to match
basic cross-sectional observations, fail completely to describe the persistent state of affairs
that characterizes financial distress in US data.
3.2.3

Distress as Delinquency and Bankruptcy

However, neither of the models just considered distinguish between nonpayment of debt and
bankruptcy, while our measure of distress in the data is that of severe delinquency. It is
therefore important to assess whether this omission is important in the failure of the models
to match the persistence (and concentration) of financial distress. A main reason, at least
up front, to worry about this omission is that bankruptcy carries fixed costs arising from
income loss or other fees. Thus, one would not expect it to be used repeatedly. By contrast,
financial distress as delinquency does not impose such costs. A model that allows for such
an option therefore seems warranted.
The model we now consider follows that of Athreya et al. (2017) and allows for informal
delinquency where, as in the baseline model, borrowers may opt to cease repayment and
pay a “penalty” interest rate on existing debt. This model (DQBK) is disciplined by three
facts: The rate of financial distress at age 30, at age 50, and the formal bankruptcy rate.
The Appendix presents this model’s performance in matching these facts and its calibrated
parameters.
Focusing on the empirical performance of this model, from Figure 21 we see that this
model matches the concentration of financial distress quite well, albeit missing slightly concentration of the top 20% of the population. As for persistence across the life cycle, we
see from Figure 22 that while persistence is reasonably captured during middle age, it is
far away from the data at both young and old ages. The miss at old ages highlights the
importance of discount factor heterogeneity, whereby effectively impatient older individuals
still find themselves systematically in distress (note that Figure 11 displays no such decline
in persistence).

Share of FD
.4 .5 .6 .7

Figure 21: The Concentration of Financial Distress: FRBNY/Equifax data and DQBK
model

.4
.5
.6
Share of Population
Data

DQBK

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 22: The Persistence of Financial Distress Over the Life Cycle: FRBNY/Equifax data
(left) and DQBK model (right)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

35
Pr( FDn)
Pr(FDn+4 | FDn)
Pr(FDn+8 | FDn)

40
Age

45
Pr(FDn+2 | FDn)
Pr(FDn+6 | FDn)
Pr(FDn+10 | FDn)

.8
Probability of being in FD
.2 .3 .4 .5 .6 .7
.1
0

Probability of being in FD
.2 .3 .4 .5 .6 .7

Figure 23: Average Persistence of Financial Distress: FRBNY/Equifax data (left) and
DQBK model (right)

The observant reader will have noticed that this model (DQBK) seems to match many
of the facts as well, or nearly as well, as our benchmark model, which features substantial
heterogeneity in discounting. But the comparisons thus far provide an incomplete picture.
To see this, in Table 8 of the Appendix we show that the best calibration of the DQBK model
implies a discount rate for all agents of roughly 17% (i.e. β=0.83). This is implausible for
two main reasons. First, and most importantly, this value is at odds with those used in the
vast majority of the vast literature on consumption and savings. Indeed, in that work, the
modal discount factor has been equal, or close, to 0.95. Even models like ours that aim to
account for default, and hence may be driven to allow lower values the discount factor, do not
imply values anywhere near as low as 0.83. For example, the benchmark model of Livshits
et al. (2007) the discount factor uses a discount factor of 0.94. It is, of course though not
random why baseline models with a single discount factor generally impose discount factors
much larger than in the DQBK model: Essentially any model that aims to match wealth
accumulation over the life cycle will fail completely at lower values. Figure 24 compares
the performance of both the DQBK and Benchmark models relative to data from the 1998
Survey of Consumer Finances (SCF).13 This figure makes clear just how drastically the
former underpredicts mean wealth accumulation (expressed relative to income) over the life
cycle relative to both the data and the Benchmark model. While not shown here, uniformly
low discount factors are similarly problematic for the “BK” model that defined financial
distress as bankruptcy, illustrating yet another shortcoming of that model for understanding
financial distress.14 While failures of this magnitude are perhaps inconsequential in settings
not trying to understand balance sheet positions, they are unlikely to be so for addressing
the question of interest here. Second, as documented at the outset, analysis of a wide variety
of empirical evidence (most recently that of Fulford and Schuh (2017) and Parker (2017))
provides strong support for the view that that the null hypothesis should be in favor of
models that allow heterogeneity in discounting rather than not.
3.2.4

Why don’t models that disallow variation across agents in discount rates
capture persistence well?

The answer lies in the equilibrium incentives to use any form of default. In smoothing
consumption over both the life cycle and across contingencies agents in the model are, as
usual, seeking to equalize marginal utilities along both dimensions. In the presence of the
option to default, the marginal cost of moving consumption across dates and states varies
with the agents current state. Whenever income is persistent, current income helps predict
future income and hence default incentives. As a result those faring poorly at present face
high marginal costs of credit all else equal. This in turn makes borrowing unattractive all else
equal. Nonetheless, all models are calibrated to yield plausible overall default rates. This, in
turn, demands that borrowers be willing to pay high costs for credit, and is why in the case
with no discount factor heterogeneity, the single discount rate produced in the calibration
(β =0.83) is far higher than implied by environments that abstract from consumer debt, and
especially ones that abstract from consumer default. A common, but high, discount rate,
however runs into difficulty because it almost immediately implies a widespread willingness
13
14

We chose this cross-section as it most closely aligns with the 1999 cohort from our Equifax sample.
Results are available upon request

Wealth to income ratio
2
4
6

Figure 24: Wealth to income ratios: SCF 1998 data, Benchmark model, and DQBK model

40
Age
Data
DQBK

Benchmark

to borrow and default, all else equal. Yet the data on the concentration of distress suggests
almost the opposite: a few borrowers account for most of distress seen. At the same time,
the presence of costs of default make this option not one to be entered into repeatedly. In the
case of formal bankruptcy, there are fixed costs, and in the case of delinquency, borrowers
need to be induced into paying the penalty rate (20%) repeatedly if the model is to predict
correctly the persistence of financial distress. Neither of those is likely under a baseline
setting in which there is a single agent-type. By contrast, once this assumption is relaxed,
we can account both for the persistence (at most ages over the life cycle) and concentration
of financial distress. In particular, the allowance for heterogeneity in discounting yields
the intuitive change (relative to the baseline) of a two tier outcome whereby a majority
(56%) are characterized as being effectively impatient (β=0.74) while a minority (44%) are
extremely patient (β=1.00). In sum, the data on the persistence of financial distress and its
concentration within a small group of borrowers make it essentially necessary (and sufficient
too) to allow for discounting to vary in the population.

Conclusion

This paper establishes first that using recently available proprietary panel data, while many
(35%) US consumers experience financial distress at some point in the life cycle, most of
financial distress events are primarily accounted by a much smaller proportion of consumers
in persistent trouble. For example, about 10% are distressed for more than a quarter of the
life cycle, and less than 10% of borrowers account for half of all distress. Second, we show that
these facts can be largely accounted for in a straightforward extension of a workhorse model
of defaultable debt that accommodates a simple form of heterogeneity in time preference but
not otherwise. Specifically, the data are strongly consistent with the presence of a subset of
effectively impatient consumers, rendered so potentially by a host of additional factors not
27

modeled here. Future work that allows for more detail on household-level economic dynamics
is therefore essential to more deeply understand the sources of this apparent heterogeneity,
certainly before reaching any conclusions that “implicate” individuals in their fates simply
via differences in their preferences.

References
Ampudia, M., H. van Vlokhoven and Z. Dawid, “Financial Fragility of Euro Area
Households,” Journal of Financial Stability 27 (2016), 250–262.
Athreya, K., “Default, Insurance, and Debt over the Life-Cycle,” Journal of Monetary
Economics 55 (May 2008), 752–774.
Athreya, K., J. M. Sánchez, X. S. Tam and E. R. Young, “Bankruptcy and delinquency in a model of unsecured debt,” Technical Report, Federal Reserve Bank of St.
Louis, 2017, forthcoming International Economic Review.
Becker, S. G. and C. B. Mulligan, “The Endogenous Determination of Time Preference,” The Quarterly Journal of Economics 112 (August 1997), 729?758.
Belzil, C. and J. Hansen, “Subjective Discount Rates, Intergenerational Transfers and
the Return to Schooling,” Technical Report, IZA Discussion paper series, No. 60, October
1999.
Blundell, R., L. Pistaferri and I. Preston, “Consumption Inequality and Partial
Insurance,” American Economic Review 98 (December 2008), 1887–1921.
Brunetti, M., G. Elena and T. Costanza, “Is Financial Fragility a Matter of Illiquidity? An Appraisal for Italian Households,” Review of Income and Wealth 62 (December
2016), 628–649.
Cagetti, M., “Wealth Accumulation Over the Life Cycle and Precautionary Savings,”
Journal of Business and Economic Statistics 28 (2003), 339–353.
Chatterjee, S., D. Corbae, M. Nakajima and J.-V. Rı́os-Rull, “A Quantitative
Theory of Unsecured Consumer Credit with Risk of Default,” Econometrica 75 (June
2007), 1525–1591.
Chatterjee, S., D. Corbae and J. V. Rios-Rull, “A Finite-Life Private-Information
Theory of Unsecured Consumer Debt,” Journal of Economic Theory 142 (2008), 149–177.
Frederick, S., G. Loewenstein and T. O’Donoghue, “Time Discounting and Time
Preference: A Critical Review,” Journal of Economic Literature XL (June 2002), 351–401.
Fulford, S. L. and S. Schuh, “Credit Card Utilization and Consumption over the Life
Cycle and Business Cycle,” Technical Report, Federal Reserve Bank of Boston, September
2017.

Gorbachev, O. and M. J. Luengo-Prado, “The Credit Card Debt Puzzle: The Role
of Preferences, Credit Risk, and Financial Literacy,” Technical Report, Federal Reserve
Bank of Boston, 2016, research Department Working Paper 16-6.
Gourinchas, P.-O. and J. A. Parker, “Consumption Over the Life Cycle,” Econometrica 70 (January 2002), 47–89.
Gross, D. B. and N. S. Souleles, “Do Liquidity Constraints and Interest Rates Matter for Consumer Behavior? Evidence from Credit Card Data.,” Quarterly Journal of
Economics 117 (February 2002), 149–185.
Guvenen, F., “An Empirical Investigation of Labor Income Processes,” Review of Economic Dynamics 12 (2009), 58–79.
Hall, E., Robert and F. Mishkin, “The Sensitivity of Consumption to Transitory
Income: Estimates from Panel Data on Households,” Econometrica 50 (March 1982),
461–481.
Hatchondo, J. C., L. Martinez and J. M. Sánchez, “Mortgage Defaults,” Journal
of Monetary Economics 76 (November 2015), 173–190.
Hausman, J. A., “Individual Discount Rates and the Purchase and Utilization of EnergyUsing Durables,” Bell Journal of Econmics 10 (Spring 1979), 33–54.
Jappelli, T. and M. Pagano, “The Welfare Effects of Liquidity Constraints,” Oxford
Economic Papers 51 (1999), 410–430.
Jappelli, T., M. Pagano and M. Di Maggio, “Households? Indebtedness and Financial
Fragility,” Journal of Financial Management Markets and Institutions (2013), 26–35.
Kaplan, G. and G. L. Violante, “How Much Consumption Insurance beyond SelfInsurance?,” American Economic Journal: Macroeconomics 2 (October 2010), 53–87.
Lawrance, E. C., “Poverty and the Rate of Time Preference: Evidence from Panel Data,”
Journal of Political Economy 99 (February 1991), 54–77.
Livshits, I., J. MacGee and M. Tertilt, “Consumer Bankruptcy: A Fresh Start,” The
American Economic Review 97 (March 2007), 402–418.
Lusardi, A., “Americans’ Financial Capability,” NBER Working Paper No. 17103, June
2011.
Lusardi, A., D. Schneider and P. Tufano, “Financially Fragile Households: Evidence
and Implications,” Brookings Papers on Economic Activity (Spring 2011), 83–134.
Meier, S. and C. Sprenger, “Present-Biased Preferences and Credit Card Borrowing,”
Technical Report, Federal Reserve Bank of Boston, 2017, research Department Working
Paper 07-3. Forthcoming American Economic Journal: Applied Economics.

Mustre-del Rı́o, J., “Wealth and Labor Supply Heterogeneity,” Review of Economic
Dynamics 18 (2015), 619–634.
Parker, J. A., “Why Don’t Households Smooth Consumption? Evidence from a $25
Million Experiment,” American Economic Journal: Macroeconomics 9 (2017), 153–183.
Samwick, A. A., “Discount Rate Heterogeneity and Social Security Reform,” Journal of
Development Economics 57 (October 1998), 117?146.
Warner, J. T. and S. Pleeter, “The Personal Discount Rate: Evidence from Military
Downsizing Programs,” The American Economic Review 91 (March 2001), 33?53.
Zeldes, S., “Consumption and Liquidity Constraints: An Empirical Investigation,” Journal
of Political Economy 97 (April 1989), 305–346.

Data and Moment Construction

This appendix provides a description of the data used. All our empirical work leverages information from Federal Reserve Bank of New York Consumer Credit Panel/Equifax, unless
otherwise noted. We trimmed our sample such that individuals missing in any quarter from
1999Q1 to 2017Q2 are dropped. Additionally, we restrict attention to individuals who enter
the sample in 1999Q1 between the ages of 25 and 55.
Unconditional Fraction of Individuals in DQ. Unconditional fraction of individuals
in delinquency (DQ), also called unconditional probability of being in DQ, is calculated
by finding the ratio of DQ debt to total number of individuals. DQ debt is computed as
the sum of balances of all delinquent accounts if an individual is more than 120 DPD, or
Severe Derogatory, i.e., DQ debti,j = crtr attr111 + crtr attr112, for individual i at age j.
A dummy variable 1DQi,j is defined for all individuals, where 1DQi,j = 1 if DQ debti,j > 0.
Note that if individual is in delinquency at least one quarter on a particular age, 1DQi,j = 1.
Then unconditional fraction of individuals in DQ is calculated as
Nj
P

Unconditional fraction of individuals in DQ =

i=1

1DQi,j

Nj
Unconditional Fraction of Debt in DQ. Similarly, unconditional fraction of debt
in DQ is computed by finding the ratio of DQ debt to total debt. Total debt is computed
as the sum of balances of all accounts, i.e., Total debti,j = crtr attr107 + crtr attr108 +
crtr attr109 + crtr attr110 + crtr attr111 + crtr attr112. Then unconditional fraction of
debt in DQ is
Nj
P
DQ debti,j
Total debti,j
Unconditional fraction of debt in DQ = i=1
Nj
Conditional Probability of Being in DQ. We compute the probability of being in
DQ conditional on being in DQ h years ago as
Nj
P

Conditional probability of being in DQ debt =

i=1

1DQi,j · 1DQi,j+h
Nj
P
i=1

1DQi,j

It is important to note that 1DQi,j does not contain everyone who is in delinquency at age
j when computing conditional probability. In fact, any individual whose age is j + h > j ∗
where j ∗ is maximum age in the sample period is dropped. For example, if an individual
i is in delinquency at age 40 in year 2014, j ∗ = 43, and this individual is excluded from
1DQi,40 in the computation for conditional probability for age greater than 43, since we do
not have data beyond 2017. This individual is not excluded when computing unconditional
probability.

Unconditional Probability of Reaching Credit Limit. Unconditional probability
of reaching credit limit is calculated by finding the ratio of individuals reaching credit limit to
total number of individuals. Another dummy variable 1Crediti,j is defined for all individuals,
where 1Crediti,j = 1 if bank balance ≥ credit limit, i.e., crtr attr169 ≥ crtr attr180. Similarly,
if the individual has reached credit limit at least one quarter on a particular age, 1Crediti,j = 1.
Then unconditional probability of reaching credit limit is calculated as
Nj
P

Unconditional probability of reaching credit limit =

i=1

1Crediti,j
Nj

Conditional Probability of Reaching Credit Limit. Similarly, probability of reaching credit limit, conditional on reaching credit limit h years ago, is computed as
Nj
P

Conditional probability of reaching credit limit =

i=1

1Crediti,j · 1Crediti,j+h
Nj
P
i=1

1Crediti,j

Average Life of DQ. Average life of DQ for individual i is computed as the ratio of
total number of quarters i is in DQ (1DQi,j = 1) to the total number of quarters in the
sample period for i. Let DQnumi be the total number of quarters i is in DQ, and let Ti
denote the total number of quarters in the sample period for i. Then
DQnumi
Ti
Note that Figure 4 excludes individuals who do not spend any quarter in DQ because the
large proportion of the population that does not enter DQ distorts the scale of the histogram.
Average life in DQ for i =

Delinquency Spell Number. A delinquency spell begins when the individual is in
DQ (1DQi,j = 1) in the current quarter but was not in DQ the preceding quarter. Similarly,
a delinquency spell ends when the individual is not in DQ in the current quarter but was
in DQ preceding quarter. If the first and last observation is in DQ, we take that quarter
to be the start or end of the DQ respectively. Note that an individual can have multiple
delinquency spell throughout his life. Also note that x-axis of Figure 5 has been trimmed
to 10 for illustrative purpose. The original scale spans to 14, but the cumulative density
between 11 to 14 spells accounts for less than 0.1%.
Lorenz Curves. Lorenz curves are calculated using two measures: being in DQ, and
reaching credit limit. After sorting out the individuals in a nondecreasing order by DQnumi ,
share of DQ (y-axis of Lorenz curve) is computed as the following
î
P

Share of DQ for î =

i=1
N
P
i=1

DQnumi
DQnumi

î
Share of DQ for î is then plotted against the share of population that is given by .
N
Similar computation applies for credit limit.
Delinquency Intensity. Delinquency intensity is computed as the average ratio of debt
in DQ to total debt among people that have entered DQ. Hence it is
Nj
P
DQ debti,j
Total debti,j
Delinquency Intensity = i=1 N
Pj
i=1

1DQi,j

Alternative measure of delinquency intensity is calculated by taking number of bankcard
120 DPD or Severe Derogatory. Let Num cardi,j = crtr attr17 + crtr attr38, while Total cardi,j = crtr attr33 + crtr attr34 + crtr attr35 + crtr attr36 + crtr attr37 + crtr attr38.
Define 1Cardi,j = 1 if Num cardi,j > 0. Then it is computed as
Nj
P
Num cardi,j
Total cardi,j
Delinquency Intensity = i=1 N
Pj
i=1

1Cardi,j

Delinquent Debt Figure 9 is computed by taking 50th, 75th and 90th percentile of
DQ debti,j by age. Note that the amount of DQ debt has been inflation-adjusted to 2017
January dollars using seasonally adjusted CPI from US Bureau of Labor Statistics.

Cross-state Comparisons

To ensure that our findings are not simply driven by the vagaries of any single state of
the union in the data, Figures 25 and 26 present the life cycle incidence and persistence of
financial distress across the six most populous states in the data. As is clear, not only are
the qualitative patterns extremely similar across states, so are the quantities. Thus, we see
that across the US, financial distress patterns are very similar, and this plausibly amenable
to analysis within a model framework that abstracts from what might have seemed, a priori,
relevant differences across states.

Figure 25: The Persistence of Financial Distress Over the Life Cycle and Across States (debt)

Probability of being in FD
.2
.3
.4
.5
.1

Probability of being in FD
.2
.3
.4
.5

Texas

California

40
Age

Unconditional
4 years after being in FD
8 years after being in FD

2 year after being in FD
6 years after being in FD
10 years after being in FD

40
Age

Unconditional
4 years after being in FD
8 years after being in FD

2 year after being in FD
6 years after being in FD
10 years after being in FD

Probability of being in FD
.2
.3
.4
.5
.1

Probability of being in FD
.2
.3
.4
.5

Florida

New York

40
Age

Unconditional
4 years after being in FD
8 years after being in FD

2 year after being in FD
6 years after being in FD
10 years after being in FD

40
Age

Unconditional
4 years after being in FD
8 years after being in FD

2 year after being in FD
6 years after being in FD
10 years after being in FD

Ohio

Probability of being in FD
.2
.4

Probability of being in FD
.2
.3
.4
.5

Pennsylvania

Unconditional
4 years after being in FD
8 years after being in FD

40
Age

2 year after being in FD
6 years after being in FD
10 years after being in FD

Unconditional
4 years after being in FD
8 years after being in FD

40
Age

2 year after being in FD
6 years after being in FD
10 years after being in FD

Source: Federal Reserve Bank of New York Consumer Credit Panel/Equifax

Figure 26: The Persistence of Financial Distress Over the Life Cycle and Across States
(credit limit)
Texas

Probability of reaching credit limit
.1
.2
.3
.4
.5

Probability of reaching credit limit
.2
.3
.4
.5
.6

California

40
Age

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

40
Age

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

Florida

Probability of reaching credit limit
.1
.2
.3
.4
.5

Probability of reaching credit limit
.2
.3
.4
.5
.6

New York

40
Age

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

40
Age

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

Ohio

Probability of reaching credit limit
.1
.2
.3
.4
.5

Pennsylvania

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

40
Age

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

Unconditional
4 years after reaching credit limit
8 years after reaching credit limit

40
Age

2 year after reaching credit limit
6 years after reaching credit limit
10 years after reaching credit limit

Source: Federal Reserve Bank of New York Consumer Credit Panel/Equifax

A Model of Distress as Bankruptcy

In this section we provide details of the model of distress as bankruptcy from Section 3.2.1
in the main text.
We assume that each period households may default on existing debt. Like in our benchmark model in the main text, households trade-off the advantages and disadvantages of
bankruptcy. The key advantage is the discharge of debts: current period expense obligations
are eliminated and in the period after bankruptcy, debt is set at zero. Thus, a household
with too much debt may find it beneficial to file for bankruptcy. There are two disadvantages of doing so, however. In the period of bankruptcy, a proportion of income, τ , is lost.15
Additionally, in that period, consumption equals income—neither saving nor borrowing are
allowed. In this environment, lifetime utility can be written as
Gi,n (z, ε, a) = max{Vi,n (z, ε, a), Bi,n (z, ε)}
| {z } | {z }
Pay

(6)

Bankruptcy

where V and B (defined below) are lifetime utilities for households paying back the debt
and filing bankruptcy, respectively. This means that a household has the choice of filing
bankruptcy. The policy function R indicates whether the household pays back the debt
(repay) or not,

1 if Vi,n (z, ε, a) ≥ Bi,n (z, ε),
Ri,n (z, ε, a) =
0 otherwise.
Suppose the receives the opportunity to file for bankruptcy and chooses to do so. Then,
lifetime utility is
Bi,n (z, ε) = u(min{yi,n (z, ε), τ }) + %n βE [Gi,n+1 (z 0 , ε0 , 0)|z] .

(7)

During the bankruptcy period, the household’s consumption equals earned income up to a
threshold τ > 0. In the period after bankruptcy, the household will have no debt.
Now suppose the household pays back its debt. Then it faces the debt price qn (z, a0 ) and
lifetime utility
Vi,n (z, ε, a) = max{a0 ,c} u(c) + %n βE [Gi,n+1 (z 0 , ε0 , a0 )|z] ,
subject to

(8)
0

c + a qi,n (z, a ) = a + yi,n (z, ε),
c ≥ 0.
Equilibrium prices must imply zero-expected profits. In general, a price function qi,n (z, a0 )
implies zero profits if the following equation is satisfied.
qi,n (z, a0 ) =

1
%n E [Ri,n (z 0 , ε0 , a0 )|z] .
1+r

(9)

Chatterjee et al. (2008) build a model where no punishment is required after filing bankruptcy. There,
asymmetric information is crucial to create incentives for debt repayment, because households signal their
type by paying back their debt.

Looking at this equation it is very clear why the price function (or interest rates) depends on
(a0 , z). It depends on a0 because it affects the bankruptcy decision, R, in each possible state.
It depends on z because it determines the transition probability to each z 0 and therefore
next period’s earned income, y.
An equilibrium in this economy is a set of value functions, optimal decision rules for the
consumer, default probabilities, and bond prices, such that equations (6) to (8) are satisfied
and prices satisfy the zero-profit condition (9).

Calibration of Model with Distress as Delinquency
and Bankruptcy

In this section we present the model implied moments and calibrated parameters for the
model presented in Section 3.2.3. The targeted moments are presented in Table 8, while the
implied parameter values are presented in Table 8.
Table 7: DQBK Model Calibration
Moment
Data
FD rate at age 30 (in %)
15.16
FD rate at age 50 (in %)
10.06
Average BK rate ages 25-65 (in %)
0.74
Pop. that never defaults (in %)
64.23
FD share of top 20% of pop. (in %) 88.07
Avg. cond. FD rate at 2 years
0.52
Avg. cond. FD rate at 4 years
0.34
Avg. cond. FD rate at 6 years
0.23
Avg. cond. FD rate at 8 years
0.17
Avg. cond. FD rate at 10 years
0.15

Model
15.16
5.73
0.74
64.50
81.84
0.33
0.23
0.19
0.16
0.15

Note: Numbers in bold denote targeted moments in the calibration of the model.

Table 8: Calibrated Parameters for DQBK model
Parameter
Discount factor β
Earnings threshold in DQ τ
Discharge shock to DQ debt γ

Value
0.83
3.85
0.34

Full text of Working Papers (Federal Reserve Bank of Richmond) : The Persistence of Financial Distress, Working Paper 17-14

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