Determinants of automobile loan default and prepayment.
Agarwal, Sumit ; Ambrose, Brent W. ; Chomsisengphet, Souphala 等
Introduction and summary
Automobiles, meaning cars and light trucks, are the most commonly
held nonfinancial assets among Americans. In 2001, the share of families
that owned automobiles was over 84 percent--higher than the share that
owned primary residences at 68 percent. Further, automobile ownership
statistics are fairly stable across various demographic characteristics,
such as income, age, race, employment, net worth, and homeownership. So
how do we pay for all these automobiles? Roughly three-quarters of
automobile purchases are financed through credit, and loans for
automobile purchases are one of the most common forms of household
borrowing. (1) In 2003, debt outstanding on automobile loans was over
$1,307 billion. (2) According to past studies on auto sales, third party
financing (direct loans) accounts for the largest portion of the
automobile credit market, with dealer financing (indirect loans) second
and leasing third. (3)
What are the risks that lenders in the automobile market face? The
first, most obvious risk is default--that is, the person who took out a
loan to buy a car or truck fails to pay it back. A second significant
risk for lenders in this market is prepayment risk--that is, the car or
truck purchaser pays off the loan early, reducing the lender's
stream of interest payments. (Hereafter we use the terms automobiles,
autos, and cars, as well as vehicles, interchangeably.)
At present, the third party auto loan market relies on a
"house rate" for pricing loans, such that all qualified
borrowers with similar risk characteristics pay the same rate. The
lender does not rely on any information about the automobile's make
and model to price the loan. Rather, the lender simply underwrites the
loan based on the borrower's credit score and required down
payment. (4) This contrasts with current practices in the auto insurance
market and the mortgage market. Auto insurers have long recognized that
automobile makes and models appeal to different clienteles and that
these clienteles have heterogeneous risk profiles and accident rates. As
a result, insurers routinely price automotive insurance based on auto
make and model. Also, before mortgage lenders originate loans, typically
they have information on the underlying assets (for example, a house) as
well as the borrowers' personal characteristics. Thus, information
about the underlying assets often plays a role in determining mortgage
contract rates. Given the current practices in the auto insurance market
and mortgage market, the question naturally arises as to whether
incorporating information on automobile make and model would help third
party lenders refine their loan pricing models. Specifically, if we
assume that the choice of auto make and model reveals individual
financial (or credit) risk behavior of the borrower, what does this tell
us about the borrower's propensity to prepay or default on his
loan?
Studying individual risk behavior in the auto loan market may be
important for investors, as well as lenders. Over the years, a growing
percentage of the stock of automobile debt has been held in
"asset-backed securities." Pricing these contracts is
complicated by the borrower's options to default and prepay, which
are distinct but not independent. Thus, one cannot calculate accurately
the economic value of the default option without simultaneously
considering the financial incentive to prepay.
In perfectly competitive markets, we expect well-informed borrowers
to make decisions about whether to pay their auto loans early or late
(or on time) in a way that increases their wealth. For example,
individuals can increase their wealth by defaulting on an auto loan when
the market value of the auto debt equals or exceeds the value of the
automobile. Alternatively, individuals can prepay their auto loan to
take advantage of declining interest rates. (5)
In this article, we use a competing risks framework to analyze the
prepayment and default options on auto loans, using a large sample of
such loans. To the best of our knowledge, there are two other studies,
Heitfield and Sabarwal (2003) and Agarwal, Ambrose, and Chomsisengphet
(2007), that provide competing risks models of default and prepayment of
automobile loans.
Here, we document several interesting patterns. For example, a loan
on a new car has a higher probability of prepayment, whereas a loan on a
used car has a higher probability of default. In addition, we find that
a decrease in the credit risk of an auto loan holder, as measured by the
FICO (Fair Isaac Corporation) score, lowers the probability of default
and raises the probability of prepayment. We also find that an increase
in the loan-to-value ratio (LTV) increases the probability of default
and lowers the probability of prepayment. An increase in income raises
the probability of prepayment, whereas a rise in unemployment increases
the probability of default. And a decrease in the market rate (the
three-year Treasury note rate) increases both the probabilities of
prepayment and default. These findings are roughly in line with what we
would expect.
Interestingly, we also find that loans on most luxury automobiles
have a higher probability of prepayment, while loans on most economy
automobiles have a lower probability of default. This indicates that
consumer choices regarding automobile make and model provide information
about the probabilities of default and prepayment, even holding
traditional risk factors (FICO score, LTV, and income) constant.
In the next section, we describe our data. Then, we discuss our
methodology and describe the regression results from the model for auto
loan prepayment and default.
Data
The proprietary data that we analyze are from a large financial
institution that originates direct automobile loans. (6) We focus on
direct loans in this article because this is the market where lenders
compete. Direct loans are issued directly to the borrower, and indirect
loans are issued through the dealer. In the case of indirect loans,
financial institutions have agreements with automobile dealerships to
provide loans at fixed interest rates. However, they have to compete
with automobile finance companies that can provide the loans at a much
cheaper rate, even if they have to bear a loss on the loans. For
example, a General Motors Corporation (GM) finance company can afford to
take a loss on the financing for a GM automobile while making a profit
on the automobile sale. Hence, financial institutions cannot compete in
the market for indirect automobile loans.
Our original sample consists of over 24,384 direct auto loans. Auto
loans are issued with four-year and five-year maturities as well as
fixed rates. We observe the performance of these loans from January 1998
through March 2003, such that a monthly record of each loan is
maintained until the automobile loan is either paid in full (at loan
maturity), prepaid, defaulted, or stays current. Certain accounts are
dropped from the analysis for the following reasons: Loans were
originated after March 2002; loans were written for the financial
institution's employees; and loans were associated with fraud or
with stolen automobiles. We also drop loans that were paid in full. In
addition, once the loan has been defaulted or has been prepaid,
subsequent monthly records are removed from the data set. Finally, we
have a total of 20,466 loans with 4,730 prepayments (23.11 percent) and
534 defaults (2.61 percent) during the study period. (7)
Loan characteristics include automobile value, automobile age, loan
amount, LTV, monthly payments, contract rate, time of origination (year
and month), and pay off year and month for prepayment and default. We
also have access to the automobile make, model, and year. Finally, we
know whether the loan was issued toward the purchase of a used or new
automobile. Borrower characteristics include credit score (FICO score),
(8) monthly disposable income, and borrower age. The market rate used in
this analysis is the three-year Treasury note rate. We also include the
unemployment rate in the county of residence of the borrower. A majority
of the loans originated in eight northeastern states--Connecticut,
Maine, Massachusetts, New Hampshire, New Jersey, New York, Pennsylvania,
and Rhode Island.
Table 1 presents summary statistics for all loans. The median loan
amount is $14,027, with a median LTV of 78 percent and a median annual
percentage rate (APR) of 8.99 percent. The median FICO score is 723 in
our sample, which also happens to be the national median score in 2005
(see note 8). The median monthly disposable income is $3,416. Finally,
the median owner, loan, and car ages are 40 years, 54 months, and 4
years, respectively. The blue book value (the car's market value)
(9) at loan origination ranges from $4,625 to $108,000. These statistics
are comparable with the overall statistics for a typical auto loan
portfolio.
Next, table 2 compares these median statistics on all auto loans
with the median statistics for loans on used cars, as well as loans on
new cars. The median FICO scores are 722 and 726 for loans on used and
new vehicles, respectively. The median LTV ranges from 74 percent for
loans on used automobiles to 87 percent for loans on new automobiles.
Finally, the median loan amount is about two and a half times for new
cars as compared with that for used cars. These statistics reveal the
differences between the borrowers who buy new and used automobiles.
Despite these differences, the credit risk characteristics between the
borrowers for new versus used autos are not significantly different, as
reflected by the similar FICO scores.
Table 3 presents the distribution of loans on used and new
automobiles by loan outcome. The first row shows the number of loans
that are current at the end of the sample period--that is, those that
are not defaulted or prepaid. While 20 percent of loans on used autos
and 32 percent of loans on new autos are prepaid, only 2.77 percent of
loans on used vehicles and 2.13 percent of loans on new vehicles are
defaulted. (10) Overall, 75 percent of all loans are originated for used
cars and 25 percent are originated for new ones. The descriptive
statistics show that a higher percentage of borrowers who have loans for
new automobiles prepay, while a slightly higher percentage of borrowers
who have loans for used automobiles default.
Table 4 presents the distribution of the auto loans across the
various states. Thirty-three percent of the loans originated in New
York, 22 percent in Massachusetts, and 1 percent in Florida, while 3
percent originated across the 41 states (and the District of Columbia)
not listed individually in the table.
Table 5 presents the distribution of the loan origination by
quarter. Since most U.S. and European automobile manufacturers typically
introduce the new versions of their established models (as well as brand
new models) in the third quarter, 41 percent of all auto loans in the
sample originated in that quarter. Next, 26 percent of the loans
originated in the first quarter. The earned income tax credit (EITC)
refunds, which typically become available to recipients in the first
quarter, might help explain why 26 percent of the loans originated then.
(11) Finally 18 percent of all auto loans originated in the second
quarter, and 15 percent originated in the fourth quarter. (12) Since a
majority of the loans in our sample are for used car purchases, this
suggests that consumers even tie their used automobile buying decisions
to the introduction of the new automobiles. This is evident from the
distribution of the loans for used car purchases by quarter. The
distribution is fairly similar to that of the loans for new car
purchases. Finally, table 6 provides a distribution of the auto loans by
auto make. Loans on Chevy automobiles constitute the largest percentage,
and those on Jaguar and Porsche automobiles constitute the smallest
shares.
Variables
In our regression results for default and prepayment, the dependent
variable can take on the following values: Current = 0, prepay = 1, and
default = 2. We regress this variable against a variety of independent
variables that control for the economic environment as well as various
borrower risk factors.
We first isolate variables to capture the prepayment option. To
approximate the prepayment option, we follow the approach outlined in
Calhoun and Deng (2002) and construct an auto loan prepayment premium
that is defined as [PPOption.sub.t-6] = ([r.sub.ct-6]-
[r.sub.mt-6])/([r.sub.mt-6]), where [r.sub.ct-6] is the coupon rate on
the existing auto loan and [r.sub.mt-6] is the three-year Treasury note
rate. (13) We expect [PPOption.sub.t-6] to be positively related to
prepayment behavior--that is, consumers are more likely to prepay and
trade in their cars with the decline in the prevailing three-year
Treasury note rate relative to the original loan coupon rate.
To determine the impact of differences in auto depreciation rates
on loan termination probabilities, we estimated the depreciation
schedule for each auto manufacturer based on the five-year market values
for autos reported by the National Automobile Dealers Association (NADA)
on its website (www.nada.com). For example, to determine the average
expected depreciation for Subaru cars, we collected the estimated market
value during the fall of 2003 for Subaru's base-level Forester,
Impreza, and Legacy models from the 1998 model year through the 2002
model year. This provides a rough estimate of the yearly change in value
for a base-level model experiencing an average driving pattern (as
determined by the NADA). For each model, we then calculate the simple
yearly depreciation experienced by the base car model (without
considering possible upgrades or add-ons), and we average the expected
depreciation by manufacturer. Unfortunately, given the heterogeneous
nature of the models from year to year, we are unable to match all
models to a set of used car values. Thus, we assumed that all models for
each manufacturer follow a similar depreciation schedule. Obviously, our
valuation algorithm is only an approximation, since the values of
individual cars will vary based on the idiosyncratic driving habits of
the borrowers.
Based on these estimated changes in car prices, we construct the
monthly loan-to-value ratio (CLTV). We expect the monthly loan-to-value
ratio to be positively related to default probability because the higher
depreciation in the auto value (holding other things constant) serves to
increase the loan-to-value ratio. Given the significant depreciation in
auto value upon purchase, many borrowers have an auto loan balance
greater than the current car value. Thus, including CLTV allows for a
direct test for the link between auto quality and credit performance.
That is, if an auto manufacturer produces a disproportionate number of
low-quality cars, then the secondary market value for the
manufacturer's cars will reflect this lower quality.
In addition to changes in the auto value relative to the debt
burden, we also capture changes in borrower credit constraints via the
time-varying borrower credit score (FICO). Borrower credit history is
one of the key determinants of auto loan approval. Thus, we expect the
FICO score to be negatively related to default probability, implying
that borrowers with lower current FICO scores are more likely to default
on their auto loans. (14)
Local economic conditions may also affect borrower loan termination
decisions. For example, borrowers facing possible job losses are more
likely to default because they may be unable to continue making loan
payments. We use the county unemployment rate (Unemployment), updated
monthly, as a proxy for local economic conditions: the unemployment rate
is for the county of residence of the borrower. Finally, we include a
series of dummy variables that denote the borrower's location
(state) to control for unobserved heterogeneity in local economic
conditions.
We also control for other variables, such as the age of the
borrower, state-specific effects, account seasoning (time since loan
origination), and calendar time effects. Lastly, we also control for the
make, model, and year of the automobile. It is well documented that
different auto makes and models have different depreciation functions,
so an auto make dummy wilt help isolate the auto make's specific
depreciation. For example, Aizcorbe, Corrado, and Doms (2000) and
Corrado, Dunn, and Otoo (2003) use fixed effects models by assigning
dummy variables for each automobile make, which can be used as a proxy
for the measurement of the physical characteristics of the automobile
make. Since the characteristics of an automobile are fixed, the dummy
variables capture the cross-sectional variation in the auto's
market values.
Methodology
Using a loan-level model, we empirically evaluate the effect of
market changes in interest rate exposure on prepayment risk for an
automobile loan portfolio. We also do this for the effect of liquidity
constraints--as measured by FICO scores--and the effect of unemployment
on default risk. Previous empirical prepayment and default models using
loan-level data are typically based on techniques of survival analysis
(originally used in biological studies of mortality). (15) Kalbfleisch
and Prentice (1980) and Cox and Oakes (1984) provide a classic
statistical treatment of the topic. For further details, see the
appendix.
Since our primary purpose is to determine how borrower consumption
decisions can affect loan performance, we follow Gross and Souleles
(2002) and separate x into components representing borrower risk
characteristics, economic conditions, and consumption characteristics.
Specifically, we assume that
1) [x'.sub.j][[beta].sub.j] = [[beta].sub.0][[tau].sub.t] +
[[beta].sub.l][State.sub.i] + [[beta].sub.2][risk.sub.u] +
[[beta].sub.3][econ.sub.it] + [[beta].sub.4][car.sub.it]
where [[tau].sub.t] represents a series of dummy variables
corresponding to calendar quarters that allow for shifts over time in
the propensity to default or prepay; [State.sub.i] represents a series
of dummy variables corresponding to the state of residence of the
borrower; [risk.sub.it] represents a set of borrower characteristics,
including credit score, that reflect the lender's underwriting
criteria; [econ.sub.it] is a set of variables capturing changes in local
economic conditions; and [car.sub.it] s a set of variables identifying
information concerning the type of car purchased.
Empirical results
We look at the results from the competing risks model that capture
the determinants of auto loan prepayment and default. Table 7 presents
the results. (16) We control for state dummies, loan age, owner age, and
quarter time dummies.
The results (estimated coefficients) in the first column of data
show that the probability of default is higher in the first, second,
third, and fourth quarters of 2000. However, the probability of default
is lower in the first and second quarters of 1999. Also, the results in
the fourth column show the probability of prepayment is higher in the
first, second, third, and fourth quarters of 2002, but the probability
of prepayment is lower in the fourth quarter of 2000. These results
high-light the effects of macroeconomic conditions on default and
prepayment probabilities. Because of weakening macroeconomic conditions
in 2000, there were more defaults and fewer prepayments. However, with
dropping interest rates and subsequent attractive automobile
offers--some of which featured no closing costs, zero percent financing,
and no down payment--prepayment and trade-in rates in 2002 were much
higher. These results are consistent with the literature on consumer
durable goods purchases, transactions costs, and liquidity constraints.
(17)
Next, we look at the automaker control variables. The competing
risks model contains 31 dummy variables denoting the various automakers.
The estimated coefficients provide interesting insights into the
prepayment and default behavior of the borrowers with respect to the
makes of the automobiles they eventually purchase. Specifically, we find
that loans for most luxury automobile makes, such as Lexus, BMW, and
Cadillac, have a higher probability of prepayment, while loans for most
economy automobile makes, such as Geo, Buick, and Honda, have a lower
probability of default. It is interesting that some luxury automobiles
(for example, Jaguar and Saab) have higher probabilities of default and
prepayment. This implies that certain luxury automobiles have a premium
in the used car market; luxury vehicles in the used car market are
preferred by liquidity-constrained consumers.
We interpret the results from the ninth and tenth rows (Owner age
and Owner [age.sup.2]) of table 7, and find that younger borrowers
(those below the median age of 40) have a higher probability of default
than the older borrowers (those at the median age of 40 and above). We
also find that the older borrowers have a higher probability of
prepayment than their younger counterparts. The results also confirm
that younger borrowers are liquidity constrained and thus more likely to
own a used automobile. Account seasoning (time since loan origination)
increases both the probabilities of default and prepayment---our
interpretation of the results from the eleventh and twelfth rows (Loan
age and Loan [age.sup.2]) of table 7. These results are intuitive.
Finally, we look at some of the important determinants of default
and prepayment as indicated by the option value theory. First, the
results show that the auto loan prepayment premium ([PPOption.sub.t6])
is positive and statistically significant for the probability of
prepayment and also, surprisingly, for the probability of default. The
first result indicates that the higher the difference between the auto
loan rate and the market rate is, the higher the probability of
prepayment and trade-in. Again, this result is consistent with the
literature on consumer durable goods purchases. A trade-in at lower
interest rates both lowers the monthly payments out of disposable income
and increases the share of durable goods in household wealth. However,
it is a little surprising that a bigger difference in the loan rate and
the market rate also increases the probability of default. One possible
explanation is that liquidity-constrained consumers, who have bad credit
risk profiles, are priced out of the low market rates, but the option to
default remains valuable.
Monthly payments, or the debt service burden (Payment t-6), are
also positively related to both the probability of prepayment and
probability of default. We expect that a higher debt service burden for
liquidity-constrained consumers could lead to a higher probability of
default; however, it could also lead to a higher probability of
prepayment for consumers who do not have liquidity constraints. (18)
Monthly income (Monthly [income.sub.t0]) is negatively related to
default but positively related to prepayment. This result is consistent
with theory. The county unemployment rate ([Unemployment.sub.t-6]) is
positively related to both the probabilities to default and prepay. Once
again we expect a higher unemployment rate to lead to a higher default
probability, but higher unemployment could also lead some to prepay and
cash out equity from their automobiles. These results are largely
consistent with Heitfield and Sabarwal (2003).
Next, we look at the monthly loan-to-value ratio ([CLTV.sub.t-6]] ,
the FICO score ([FICO.sub.t-6]), and the new auto indicator. All three
of these are measures of liquidity constraints. As expected,
liquidity-constrained consumers are more likely to have a high LTV and a
low FICO score, and they are more likely to buy used automobiles. The
results show that the FICO score is negatively related to default
probability, LTV is positively related to default probability, and the
new auto indicator is negatively related to default probability.
Moreover, a higher FICO score and a new auto indicator lead to a higher
probability of prepayment, and a higher LTV leads to a higher
probability of prepayment. (Heitfield and Sabarwal [2003] do not control
for LTV, FICO, automobile age, automobile make, and income, so we cannot
compare our results with theirs.)
Marginal effects
Table 8 presents the marginal effect of a borrower owning a new
automobile on prepayment and default rates of auto loans over a 30-month
period. This table also shows the marginal effects of changes in FICO
score, LTV, auto loan prepayment premium, income, and county
unemployment rate on the prepayment and default rates of automobile
loans over a 30-month span. The results show that a borrower owning a
new automobile reduces the probability of default by as much as 15
percent but raises the probability of prepayment by 13 percent. An
increase of 20 points in the FICO score lowers the probability of
default by 12 percent but raises the probability of prepayment by 8
percent. These results suggest that an increase in the credit risk
profile or an ease in liquidity constraints reduces one type of hazard
(default) but increases another type of hazard (prepayment). A 5 percent
drop in LTV reduces the probability of default by 4 percent but
increases the probability of prepayment by 7 percent. This would
indicate that a drop in LTV raises the overall wealth of the household.
Next we note that a 10 percent increase in income raises the probability
of prepayment by 8 percent. These results are consistent with the
theoretical literature on consumer durable goods purchases and liquidity
constraints (Eberly, 1994). A 1 percent increase in the county
unemployment rate significantly increases the probability of default by
as much as 9 percent. This is a fairly striking result and suggests that
liquidity constraints can significantly increase default rates. Finally,
a 1 percent decrease in the market interest rate in relation to the auto
loan annual percentage rate--that is, a 1 percent increase in the auto
loan prepayment premium--increases prepayment probability by 6 percent.
The results also suggest that the decrease in the market rate will
increase the probability of default by 3 percent. One possible
explanation for these results could be that liquidity-constrained
consumers may not be able to get favorable interest rates on their
loans.
Conclusion
Automobiles are highly visible consumption goods that are often
purchased on credit. In this article, we use a unique proprietary data
set of individual automobile loans to assess whether borrower
consumption choice reveals information about future loan performance.
Given that individual self-selection is evident in the automobile market
(as in the auto insurance market and the mortgage market), a natural
question arises as to whether this self-selection also reveals
information about the consumer's propensity to prepay or default on
an auto loan. We adopt the competing risks framework to analyze these
auto loan prepayment and default risks empirically, using a sample of
20,466 individual loans that were issued toward the purchases of both
new and used automobiles.
Our results can be summarized as follows. A loan on a new car has a
higher probability of prepayment, whereas a loan on a used car has a
higher probability of default. A decrease in the credit risk of a loan
holder, as measured by the FICO score, lowers the probability of auto
loan default and raises the probability of prepayment. An increase in
the LTV increases the probability of default and lowers the probability
of prepayment. An increase in income raises the probability of
prepayment, whereas a rise in unemployment increases the probability of
default. A decrease in the market rate (the three-year Treasury note
rate) increases both the probabilities of prepayment and default. And
perhaps most interestingly, we also find that loans on most luxury
automobiles have a higher probability of prepayment, while loans on most
economy automobiles have a lower probability of default.
Clearly, this study has some limitations. We are only looking at
direct auto loans that were originated, for the most part, in Northeast
states by a single lender. However, our results imply that lenders could
improve the pricing of automobile loans by considering the type of car
collateralizing the loan. Although the use of auto make/model
information in loan pricing is probably not feasible because of the
multitude of make/ model combinations, the results from this study
suggest that controlling for differences in default and prepayment
patterns based on broader auto types (for example, luxury versus
economy) could improve loan pricing.
APPENDIX
In automobile loan termination analysis, we consider that loans
"die" prior to scheduled maturity from either default or
prepayment. Survival data consist of not only a response variable that
measures the duration of a particular event but also a set of
independent variables that may explain duration of a particular event.
We use duration models to analyze the underlying distribution of the
failure time variable and to assess the effect of various explanatory
variables of the failure time. Duration models estimate the probability
of a particular terminating event of the real world. Hazard models are a
type of duration model that deals with events that may happen at various
times in the future.
Let prepayment or default be the termination events of an
automobile loan. A loan given in period [t.sub.0] has different
probabilities of prepayment and default in one year, two years, ..., t
years. In duration analysis, we are interested in describing the
probability distribution of observed automobile loan duration across an
individual loan. The basic idea behind the hazard model is that it
estimates the conditional probabilities of prepayment and default at
time t, assuming payments are being made from loan inception up to time
t - 1, conditional on the baseline hazard as well as other factors
affecting the prepayment and default behavior of the auto owner. Hence,
we include explanatory variables for factors that could affect the
probabilities of prepayment and default, such as LTV and FICO score.
Let [tau] be a random variable describing time to exit (in months
since origination) due to prepayment or default. Let p([tau] < t) =
F(t), [for all]t [greater than or equal to] 0 be the distribution
function oft at time t. Let f(t) = dF/dt be the probability density
function for [tau]. Then, we can define the hazard function (the
probability of a loan terminating) at time t with the following
equation:
A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Setting 1- F(t) = [bar.F](t) with initial condition [bar.F] (0) =
1, then
A2) h(t) = f(t)/[bar.F](t) = -d[bar.F](t)/dt/[[bar.F](t) = -
1/[bar.F](t) x d[bar.F](t)/dt
represents a differential equation in t with the following
solution,
A3) [bar.F](t) = exp {-[[integral].sup.1.sub.0]h(s)ds}.
This gives the survivor function, [bar.F](t), and the distribution
function. F(t) = 1 - [bar.F](t), oft in terms of the hazard function,
h(t). From equations A1 and A2. we obtain the unconditional density
function of [tau]:
A4) f(t) = h(t)exp {-[[integral].sup.1.sub.0]h(s)ds}.
The parametric specification of the hazard function (log-logistic
functional form) is as follows. Substituting equation A3 into equation
A1 yields:
A5) h(t) = [[lambda].sup.1/[gamma]]
[t.sup.(1-[gamma])/[gamma]]/[gamma](1 + ([lambda]t).sup.1/[gamma].
From equation A2, we have
A6) [bar.F](t) = 1/1 + ([lambda]t).sup.1/[gamma]].
And from equation A3, we have
A7) f(t) = [[lambda].sup.1/[gamma]]
[t.sup.(1-[gamma])/[gamma]]/[gamma](1 +
[([lambda]t).sup.1/[gamma]).sup.2.
Covariates are introduced in the model by setting [lambda]. =
exp(-x'[beta]),where x is a matrix of independent variables
([FICO.sub.t-6], [CLTV.sub.t-6], [PPOption.sub.t-6].
[Unemployment.sub.t-6], etc.) and [beta] is a vector of parameters to be
estimated. Gamma ([gamma]) is an ancillary parameter also estimated from
the data. Estimation is by maximum likelihood allowing for right-side
censoring and left-side truncation.
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NOTES
(1) Aizcorbe, Kennickell, and Moore (2003), pp. 16-17, 19;
Aizcorbe, Starr, and Hickman (2003) report that in 2001 over 80 percent
of new vehicle transactions were financed through loans or leases
(2) See the Federal Reserve's G. 19 statistical release
(www.federalreserve.gov/releases/g19/Current). While this release also
includes debt on mobile homes, education, boats, trailers, or vacations,
a vast majority of the debt is on automobiles
(3) For example, based on a sample of auto sales in southern
California between September 1999 and October 2000, Dasgupta, Siddarth,
and Silva-Risso (20031 report that 24 percent of the transactions were
leased, 35 percent were financed through auto dealers, and the remaining
40 percent were most likely financed from third party lenders (credit
unions or banks)
(4) For example, a borrower with an acceptable credit score may be
offered a loan up to $20,000 conditional on making a 5 percent down
payment Thus. if the borrower purchases an $18,000 car, the lender
provides a $17,100 loan
(5) Over the years, several studies using loan-level data have
investigated the economic drivers of default and prepayment risks on
residential mortgages See Kau et al ( 1992, 1995); Deng ( 1997); Deng
and Quigley (2002); Deng, Quigley, and Van Order 12000); Pavlov (2001);
Calhoun and Deng (2002); and Ambrose and Sanders (2003).
(6) We obtained the sample and permission to use it for our article
from a large financial institution under the condition that we keep the
institution's identity confidential.
(7) In our sample, prepayment is defined as an account that pays
off the loan in lull before loan maturity, while a default is defined as
60 days past due We tried alternative definitions for both prepayment
($2,000, $3,000, and $4,000) and default (90 days past due) However, the
results are qualitatively the same. Since financial institutions try to
repossess the automobile once the account is 60 days past due, our
definition is consistent with current practice.
(8) FICO scores have a range of 300-850 In 2005, the median FICO
score was 723 (see www.businessweek.com/magazine/content/
05_48/b3961124.htm). Typically, a FICO score above 800 is considered
very good, while a score below 620 is considered poor As reported on the
Fair Isaac Corporation website (www.myfico.com), there is a
400-basis-point interest rate spread for a 15-year home equip' loan
between borrowers with FICO scores above 760 and those with scores below
580; those with the higher FICO scores obtain a loan with a lower
interest rate
(9) The Kelley Blue Book, produced by the Kelley Blue Book Company
Incorporated, has become so authoritative and popular that the term
"blue book value" has become synonymous with a car's
market value.
(10) According to the American Bankers Association (ABA), the
national delinquency rate of 30 day's past due for all direct
automobile loans was 2.4 percent in 2002. This statistic is consistent
with the default rates in table 3. It is interesting to note that the
delinquency rate for indirect automobile loans was around 1.9 percent in
2002 The lower delinquency rates for indirect loans can be explained as
follows The ABA does not report the loan performance information for
auto finance companies and financial institutions that compete in the
indirect loans market and that have very stringent origination
guidelines This highlights the point that a study on automobile defaults
should distinguish between direct and indirect loans.
(11) Goodman-Bacon and McGranahan (2008) document that EITC
eligible households receive over 80 percent of the EITC payments. which
averaged $2,113 in 2004. in the first quarter of the year. They also
show that these households tend to spend a sizable portion of their EITC
refunds on automobile purchases
(12) The distributions of loan origination for both new and used
automobiles are similar
(13) We lag the three-year Treasury' note rate by six months
to avoid endogeneity We also conduct similar analyses with both
five-year and one-year Treasury note rates; the results are
qualitatively similar In fact, we lag all other variables by six months
as well.
(14) In a separate regression, we also include the square terms of
CLTV and FICO to control for any nonlinearity in explaining the
prepayment and default rates These results are not reported in this
article
(15) These techniques have also found frequent application in
industrial engineering failure time studies.
(16) We conducted an exhaustive robustness test by including
quadratic specifications for the various risk variables, discrete
dummies for some of the continuous variables, and log transformations.
Though the results are not reported, they are qualitatively similar.
(17) Accordingly, about half of the households adjust their durable
stock to a target share of their total wealth and then allow it to
depreciate until it reaches a critical share of wealth; at this point
they purchase a new durable good so that the stock once again equals the
target share of wealth (Attanasio, 1995; and Attanasio, Goldberg, and
Kyriazidou, 2000)
(18) Heitfield and Sabarwal (2003) find debt service coverage to be
positively related to default but negatively related to prepayment
Sumit Agarwal is a financial economist in the Economic Research
Department at the Federal Reserve Bank of Chicago. Brent W. Ambrose is
the Jeffery L. and Cindy M King Faculty Fellow and professor of real
estate at the Smeal College of Business at the Pennsylvania State
University. Souphala Chomsisengphet is a senior financial economist in
the Risk Analysis Division at the Office of the Comptroller of the
Currency. The authors would like to thank Erik Heitfield, Bert Higgms,
Larry Mielnicki, and Jim Papadonis for helpful comments. They are
grateful to Ron Kwolek for his excellent research assistance. The views
expressed in this article are those of the authors and do not represent
the policies or positions of the Office of the Comptroller of the
Currency or any offices, agencies, or instrumentalities of the United
States government.
TABLE 1
Summary statistics for auto loans
at origination, 1998-2003
75 25
percent percent
level Median level
Blue book value (dollars) 22,125 17,875 14,875
Loan amount (dollars) 20,544 14,027 10,547
Monthly payment (dollars) 318 229 158
Annual percentage rate 9.75 8.99 8.49
Monthly income (dollars) 5,062 3,416 2,357
FICO score 761 723 679
Loan-to-value ratio (percent) 92.86 78.47 70.90
Unemployment rate (percent) 5.40 4.50 2.60
Owner age (years) 50 40 31
Auto age (years) 7 4 1
Loan age (months) 50 54 59
Notes: Blue book value means an auto's market value. FICO score
means fair Isaac Corporation score, which is a credit score with
a range of 300-850 (see note 8 for further details).
TABLE 2 Summary statistics for loans on all,
used, and new autos at origination, 1998-2003
All Used New
autos autos autos
Blue book value (dollars) 17,875 14,283 28,382
Loan amount (dollars) 14,027 10,624 24,583
Monthly payment (dollars) 229 193 324
Annual percentage rate 8.99 9.00 8.74
Monthly income (dollars) 3,416 3,333 3,665
FICO score 723 722 726
Loan-to-value ratio (percent) 78.47 74.37 87.18
Unemployment rate (percent) 4.50 4.50 4.50
Owner age (years) 40 39 40
Auto age (years) 4 6 0
Loan age (months) 54 52 60
Notes: All values are medians. Blue book value means an auto's
market value. FICO score means Fair Isaac Corporation score,
which is a credit score with a range of 300-850 (see note 8 for
further details).
TABLE 3 Loans on all, used, and new autos, by loan outcome, 1998-2003
All autos Used autos
Number Percentage Number Percentage
Good accounts 15,202 74.28 11,843 77.20
Prepayment 4,730 23.11 3,073 20.03
Default 534 2.61 425 2.77
Total 20,466 100.00 15,341 100.00
New autos
Number Percentage
Good accounts 3,359 65.54
Prepayment 1,657 32.33
Default 109 2.13
Total 5,125 100
Note: Good accounts are loans that are current at the end of the
sample period-that is, those that are not defaulted or prepaid.
TABLE 4
Auto loans, by state, 1998-2003
State Number Percentage
Connecticut 3,256 15.91
Florida 199 0.97
Maine 782 3.82
Massachusetts 4,418 21.59
New Hampshire 1,099 5.37
New Jersey 2,536 12.39
New York 6,669 32.59
Pennsylvania 296 1.45
Rhode Island 643 3.14
Other states and
District of Columbia 568 2.78
Total 20,466 100
Note: The percentage column does not total because of rounding.
TABLE 5 Loan originations for all, used,
and new autos, by quarter, 1991-2003
All autos Used autos
Number Percentage Number Percentage
First quarter 5,289 25.84 4,034 26.30
Second quarter 3,714 18.15 3,157 20.58
Third quarter 8,478 41.42 6,053 39.46
Fourth quarter 2,985 14.59 2,097 13.67
Total 20,466 100.00 15,341 100.00
New autos
Number Percentage
First quarter 1,255 24.49
Second quarter 557 10.87
Third quarter 2,425 47.32
Fourth quarter 888 17.33
Total 5,125 100.00
Note: The percentage columns may not total because of rounding.
TABLE 6 Auto loans, by auto make, 1998-2003
Auto make Number Percentage
Acura 608 3.0
Audi 270 1.3
BMW 538 2.6
Buick 475 2.3
Cadillac 573 2.8
Chevy 2,097 10.2
Chrysler 390 1.9
Dodge 1,342 6.6
Geo 467 2.3
General Motors 449 2.2
Honda 1,919 9.4
Hyundai 125 0.6
Infinity 218 1.1
Isuzu 157 0.8
Jaguar 78 0.4
Jeep 1,591 7.8
Lexus 187 0.9
Lincoln 283 1.4
Mazda 400 2.0
Mercedes-Benz 722 3.5
Mitsubishi 433 2.1
Nissan 1,674 8.2
Oldsmobile 386 1.9
Plymouth 358 1.7
Pontiac 628 3.1
Porsche 75 0.4
Rover 147 0.7
Saab 286 1.4
Saturn 293 1.4
Subaru 340 1.7
Toyota 1,963 9.6
Volkswagen 994 4.9
Total 20,466 100.0
Notes: BMW means Bayerische Motoren Werke (Bavarian Motor Works).
The percentage column does not total because of rounding.
TABLE 7
Competing risks model of auto loan
termination through default and prepayment
Default
Coefficient Standard
value error p value
Intercept 6.8050 0.6265 0.0001
New auto dummy -0.0261 0.0113 0.0224
Monthly [income.sub.10]/1,000 -0.0200 0.0170 0.3608
[FICO.sub.t-6] -0.0166 0.0004 0.0001
[Unemployment.sub.t-6] 0.2262 0.0783 0.0039
[CLTV.sub.t-6] 1.0110 0.2958 0.0006
[Paymen.sub.t-6] 0.0002 0.0001 0.0166
[PPOption.sub.t-6] 0.2917 0.0754 0.0001
Owner age -0.0941 0.0137 0.0001
[Owner age.sup.2] 0.0009 0.0002 0.0001
Loan age 0.0316 0.0147 0.0311
[Loan age.sup.2] -0.0013 0.0002 0.0001
1999:Q1 dummy -0.4023 0.2131 0.0591
1999:Q2 dummy -0.5659 0.2080 0.0065
1999:Q3 dummy 0.0793 0.1726 0.6459
1999:Q4 dummy 0.1832 0.1722 0.2876
2000:Q1 dummy 0.3297 0.1727 0.0562
2000:Q2 dummy 0.3799 0.1782 0.0331
2000:Q3 dummy 0.4669 0.1892 0.0136
2000:Q4 dummy 0.5381 0.1905 0.0047
2001:Q1 dummy 0.1727 0.2017 0.3919
2001:Q2 dummy 0.3351 0.1927 0.0821
2001:Q3 dummy 0.1187 0.1909 0.5340
2001:Q4 dummy 0.2523 0.1711 0.1402
2002:Q1 dummy 0.1721 0.1588 0.2784
2002:Q2 dummy -0.0476 0.1628 0.7701
2002:Q3 dummy 0.2841 0.1579 0.0720
2002:Q4 dummy 0.1600 0.1567 0.3072
Connecticut dummy -0.3784 0.1035 0.0003
Florida dummy 0.3926 0.2116 0.0636
Maine dummy -0.3781 0.1885 0.0449
New Hampshire dummy -0.7172 0.1870 0.0001
New Jersey dummy -0.4121 0.1482 0.0054
New York dummy 0.1724 0.1406 0.2201
Pennsylvania dummy -0.5487 0.4691 0.2421
Rhode Island dummy 0.0493 0.1593 0.7570
Acura dummy -0.4570 0.2379 0.0547
Audi dummy -1.7109 0.7163 0.0169
BMW dummy -0.1186 0.2486 0.6334
Buick dummy -1.0463 0.4209 0.0129
Cadillac dummy 0.2226 0.2694 0.4087
Chevy dummy -0.1028 0.1296 0.4275
Chrysler dummy -0.1220 0.3140 0.6976
Dodge dummy 0.3696 0.1240 0.0029
Geo dummy -1.3232 0.7141 0.0639
General Motors dummy -0.1937 0.2665 0.4672
Honda dummy -0.3666 0.1407 0.0092
Hyundai dummy -0.4782 0.4664 0.3052
Infinity dummy -0.4485 0.4590 0.3286
Isuzu dummy 0.2555 0.2619 0.3292
Jaguar dummy 1.1264 0.5201 0.0303
Jeep dummy -0.0876 0.1508 0.5615
Lexus dummy 0.0036 0.2906 0.9902
Lincoln dummy 0.5613 0.2093 0.0073
Mazda dummy 0.1673 0.1734 0.3344
Mercedes-Benz dummy 0.3848 0.1656 0.0201
Mitsubishi dummy 0.0848 0.1833 0.6437
Nissan dummy -0.1012 0.1368 0.4596
Oldsmobile dummy 0.0114 0.2588 0.9647
Plymouth dummy -0.1911 0.2723 0.4828
Pontiac dummy 0.4209 0.1408 0.0028
Rover dummy 0.4033 0.5117 0.4306
Saab dummy 0.6634 0.2367 0.0051
Saturn dummy -0.3285 0.2982 0.2707
Subaru dummy -0.5246 0.3898 0.1784
Toyota dummy -0.0780 0.1376 0.5707
Volkswagen dummy -0.1601 0.1741 0.3579
Log likelihood ratio 1,389
Number of accounts 20,466 534
Prepayment
Coefficient Standard
value error p value
Intercept -5.3690 0.3243 0.0001
New auto dummy 0.0540 0.0258 0.0331
Monthly [income.sub.10]/1,000 0.0280 0.0072 0.0001
[FICO.sub.t-6] 0.0010 0.0003 0.0001
[Unemployment.sub.t-6] 0.1613 0.0414 0.0001
[CLTV.sub.t-6] 1.4485 0.1338 0.0001
[Paymen.sub.t-6] 0.0002 0.0000 0.0001
[PPOption.sub.t-6] 0.0419 0.0178 0.0380
Owner age -0.0338 0.0066 0.0001
[Owner age.sup.2] 0.0003 0.0001 0.0001
Loan age 0.1293 0.0083 0.0001
[Loan age.sup.2] 0.0023 0.0002 0.0001
1999:Q1 dummy 0.1148 0.0796 0.1492
1999:Q2 dummy 0.0029 0.0803 0.9714
1999:Q3 dummy 0.1267 0.0761 0.0962
1999:Q4 dummy -0.1608 0.0825 0.0514
2000:Q1 dummy -0.0198 0.0824 0.8100
2000:Q2 dummy 0.9646 0.0695 0.0001
2000:Q3 dummy 0.0047 0.0905 0.9586
2000:Q4 dummy -0.3303 0.1005 0.0010
2001:Q1 dummy -0.1096 0.0983 0.2650
2001:Q2 dummy 0.0978 0.0942 0.2989
2001:Q3 dummy -0.0554 0.0990 0.5755
2001:Q4 dummy 0.4236 0.0842 0.0001
2002:Q1 dummy 0.1738 0.0933 0.0625
2002:Q2 dummy 0.2261 0.0967 0.0194
2002:Q3 dummy 0.3618 0.0891 0.0001
2002:Q4 dummy 0.4911 0.0863 0.0001
Connecticut dummy -0.5174 0.0505 0.0001
Florida dummy -0.2428 0.1551 0.1175
Maine dummy -0.1795 0.0846 0.0339
New Hampshire dummy -0.1575 0.0677 0.0200
New Jersey dummy -0.1850 0.0672 0.0059
New York dummy -0.2060 0.0785 0.0087
Pennsylvania dummy -0.2002 0.1775 0.2595
Rhode Island dummy -0.2028 0.0962 0.0350
Acura dummy 0.0951 0.1089 0.3828
Audi dummy 0.3795 0.1553 0.0145
BMW dummy 0.4202 0.0969 0.0001
Buick dummy -0.1134 0.1158 0.3272
Cadillac dummy 0.3811 0.1233 0.0020
Chevy dummy 0.1509 0.1586 0.3241
Chrysler dummy 0.2540 0.2335 0.3121
Dodge dummy 0.1048 0.0691 0.1295
Geo dummy -0.2126 0.2185 0.3305
General Motors dummy 0.2865 0.2008 0.3234
Honda dummy -0.0533 0.0686 0.4369
Hyundai dummy -0.1337 0.2423 0.5812
Infinity dummy 0.3301 0.1558 0.0404
Isuzu dummy -0.0585 0.1777 0.7419
Jaguar dummy 0.7451 0.3425 0.0296
Jeep dummy 0.0910 0.0711 0.2008
Lexus dummy 0.6604 0.1302 0.0001
Lincoln dummy 0.1187 0.1241 0.3388
Mazda dummy -0.1009 0.1149 0.3798
Mercedes-Benz dummy 0.0950 0.0908 0.2953
Mitsubishi dummy 0.1854 0.0998 0.0633
Nissan dummy -0.0020 0.0730 0.9779
Oldsmobile dummy -0.0152 0.1196 0.8988
Plymouth dummy -0.0039 0.1213 0.9744
Pontiac dummy 0.0680 0.0933 0.4665
Rover dummy 0.2235 0.2367 0.3451
Saab dummy 0.3294 0.1153 0.0043
Saturn dummy -0.0927 0.1454 0.5235
Subaru dummy 0.0388 0.1343 0.7726
Toyota dummy -0.1041 0.0688 0.1305
Volkswagen dummy 0.1278 0.0759 0.0922
Log likelihood ratio
Number of accounts 4,730
Notes: FICO score means Fair Isaac Corporation score, which is a
credit score with a range of 300-850 (see note 8 for further
details). LTV means loan-to-value ratio. BMW means Bayerische
Motoren Werke (Bavarian Motor Works). Porsche is excluded from
the regression analysis because there are no defaults on loans
for Porsches in the sample.
TABLE 8
Marginal effects on auto loan termination through
default and prepayment over a 30-month period
Default Prepayment
(percent) (percent)
New auto -15 13
FICO score increase by 20 points -12 8
Loan-to-value ratio
decrease by 5 percent -4 7
Auto loan prepayment premium
increase by 1 percent 3 6
Income increase by 10 percent 0 8
County unemployment rate
increase by 1 percent 9 3
Notes: FICO score means Fair Isaac Corporation score, which is
a credit score with a range of 300-850 (see note 8 for further
details). For details on the calculation of the auto loan
prepayment premium, see p. 20. The county unemployment rate
is for the county of residence of the borrower.
