Credit Risk of Portfolio of Consumer Loans
Modeling Credit Risk of Portfolio of Consumer Loans
Madhur Malik* and Lyn Thomas
School of Management, University of Southampton, United Kingdom, SO17 1BJ
One of the issues that the Basel Accord highlighted was that though techniques for estimating the probability
of default and hence the credit risk of loans to individual consumers are well established, there were no
models for the credit risk of portfolios of such loans. Motivated by the reduced form models for credit risk in
corporate lending, we will seek to exploit the obvious parallels between behavioural scores and the ratings
ascribed to corporate bonds to build consumer lending equivalents. We incorporate both consumer specific
ratings and macroeconomic factors in the framework of Cox Proportional Hazard models. Our results show
that default intensities of consumers are significantly influenced by macro factors. Such models then can be
used as the basis for simulation approaches to estimate the credit risk of portfolios of consumer loans.
Keywords: Finance, Credit Risk, Survival Analysis, Credit Scoring.
Introduction
In credit scoring the main interest is in developing a scoring system which can correctly
rank the customers in terms of their relative default risk so that the customers above some
cut-off score are more or less riskier than those who are below. Credit scoring models can
broadly be classified into two types, application scoring and behavioural scoring. The
objective of both is to classify whether a customer will default (Bad) or not default
(Good) in a given time period, which leads to estimates of probability of default (PD) of
the customer in that period. Application scores are used to predict customers’ default risk,
say 12 months in future, at the time of application made for the loan. In application
scoring, past customers are classified as Good or Bad based on whether they defaulted,
which usually means 90+ days delinquent, during the first 12 months of the starting of
the loan. The information available at the time of application in the form of application
variables and credit bureau records is then used to estimate the probability of being
good/bad in the given time period. Behavioural scoring
is similar in principal to
application scoring except that in behavioural scores we observe the recent, say last one
year, payment and purchase behaviour of customers who have been granted loan and use
this information in addition to the information available for application scoring to predict
the probability of default in next twelve months or some other fixed time horizon. As the
name suggests in behavioural scoring the individuals behaviour with a particular lender
*
E-mail: madhur@soton.ac.uk
1
Credit Risk of Portfolio of Consumer Loans
and on a specific product is considered in addition to the information the lender has
through credit bureaus. The above estimates of default probabilities are then transformed
into scores, which are used as a basis to accept or reject a customer for credit, depending on
the cut-off decided by the banks for application scorecards or to make lending decisions on
current customers, like increasing/decreasing credit limit, offering new financial products,
offering new interest rates, based on behavioural scores. Lenders update their behavioural
scores monthly by using the most recent information on their customers.
With the advent of the Basel II banking regulation (BCBS, 2004) it is not just enough to
correctly rank customers according to their default risk but also one needs to measure
accurately the PD, as the predicted PDs are used to calculate the minimum capital needed
to set aside for the portfolio of consumer loans. Moreover PD has to be predicted not just
at an individual level but also for segments of the loan portfolio) The limitation of the
above approach of developing scorecards is that it uses a snapshot of customers who joined
say during certain months in calendar time (for application score) or who are on books
during certain months in calendar time (for behavioural scores), which does not take care of
the changes in economic conditions and the quality of loans over time. Motivated by the
reduced form models for portfolio credit risk in corporate lending (Lando, 1994; Duffie,
Saita and Wang, 2007) we will seek to exploit the obvious parallels between behavioural
scores and the ratings ascribed to corporate bonds to build consumer lending equivalents.
Similar recent studies conducted in corporate credit risk include (Duffie, Saita, and Wang,
2007) who studied multi-period corporate default prediction with time varying covariates.
They exploit the time-series dynamics of the macroeconomic and firm-specific covariates
and combine these with a short-horizon default model to estimate the likelihood of default
over several future periods. (Campbell, Hilscher, and Szilagyi, 2007) in their recent study
do not model the time-series evolution of the predictor variables but instead estimate
separate logit models for firm default probabilities at short and long horizons. (Figlewski,
Frydman and Liang, 2007) fit Cox intensity models for credit events, including defaults or
major upgrades and downgrades in credit rating. Their models incorporate both firmspecific factors related to a firm’s credit rating history and a broad range of
macroeconomic variables. Their results show that, in addition to being strongly influenced
2
Credit Risk of Portfolio of Consumer Loans
by ratings related factors, intensities of occurrence of credit events are significantly affected
by macroeconomic factors. (Shumway, 2001) uses a discrete duration model with time
dependent covariates and demonstrates that hazard models are statistically superior to static
models that do not take into account the fact that a firm is exposed to the risk of a credit
event over multiple periods. However there has been no work on building duration models
for the credit risk of portfolios of consumer loans.
In this paper, we incorporate consumer specific ratings (behavioural score) and
macroeconomic factors in the framework of Cox Proportional Hazard to build a model for
customer’s default probability in the next twelve months, given all the current information
available on an individual along with the values of macroeconomic factors for one year
ahead. The results of our analysis show that default intensities of consumers are
significantly influenced by macroeconomic factors and the time of origination of the loan.
This shows that the information contained in behavioural score, which is developed on the
history of loans who started during a certain period in calendar time, could not capture in
full the driving force behind the dynamics of default behaviour. Finally, we will
demonstrate how our model of individual consumers default risk can be used to simulate
the distribution of defaults in a portfolio of consumer loans.
The development of our model will not affect the rules governing Basel II but could be
expected to have an impact on the way banks segment and stress test their portfolios under
its regulations. Such models will also be of considerable use in the segmentation and
pricing of portfolios of consumer loans for securitization purposes - an area where the
theories of corporate and consumer risk management should, but as yet do not, meet.
The idea of employing survival analysis for building credit-scoring models was first
introduced by (Narain, 1992) and then developed further by (Thomas et al, 1999 and 2002).
Thomas et al. (1999) compared performance of exponential, Weibull and Cox’s
nonparametric models with logistic regression and found that survival-analysis methods are
competitive with, and sometimes superior to, the traditional logistic-regression approach.
3
Credit Risk of Portfolio of Consumer Loans
In the next section, we shall briefly discuss the notion of hazard rate and survival
probability, and the theory associated with the Cox proportional hazard rate model. We
then develop a hazard rate model to predict future hazard rates of customers conditional on
the information on customers available today. These predicted hazards are then combined
to predict the probability of default for twelve months ahead. Finally, we discuss the results
of the simulations to construct the default distribution of portfolio of loans and draw some
conclusions.
Preliminaries
Let T be nonnegative continuous random variable representing the time to default of an
individual from a homogeneous population in which all individuals experience the same
probability laws governing their default. In survival analysis the probability distribution of
T is described in the following three most popular ways: the survivor function denoted S(t),
which gives the probability of surviving beyond time t, the probability density function
denoted f(t) and the hazard function (hazard rate) denoted h(t), which is the risk of
defaulting at time t, given the person has survived till time t. Mathematically,
P ( t ≤ T < t + δt T ≥ t )
h ( t ) = lim
δt →0
δt
t
P ( T ≤ t ) =1− S ( t ) =1− e
− ∫ h ( x ) dx
0
The hazard function fully specifies the distribution of T and so determine both the density
and the survivor function.
Let x(t) denote the time covariate vector at time t and X(t)={x(u): 0 ≤ u < t} specify the
path or history of the covariate process up to time t. The components of x(t) may include
fixed covariates measured at time 0 as well as measurements of risk factors on an
individual or on the environment that varies over time. Given the covariate path up to time t
the conditional hazard function is defined as
4
Credit Risk of Portfolio of Consumer Loans
P ( t ≤ T < t + δt X ( t ) ,T ≥ t )
h ( t ; X ( t ) ) = lim
δt →0
δt
In the presence of time dependent covariates the relative risk model (Cox, 1972) describes
the hazard rate process as
h ( t ; X ( t ) ) = h0 ( t ) exp βT Z ( t )
where Z(t) are functions of X(t) and t and have left-continuous sample paths, and h0(t) is
an unknown function (baseline) giving the hazard for the standard set of conditions, when x
= 0. From the above, it is easy to observe that the Cox model assumes, at any time t, the
ratio of the hazard rates of two different individuals does not involve the baseline hazard. In
particular, the ratio of hazard rates stay constant over time if the covariates are all timeindependent. The Cox regression model is therefore often referred to as the proportional
hazards model. In our analysis we employ time-dependent covariates, and hence the ratio
of hazard rates of two individuals will change over time.
Let t1 < t2 < ... < tn be n distinct default times where the individual who defaults at time ti
has characteristics Zi(ti) at default. Let Ri be the set of individuals who are at risk of default
just before time ti , 1 ≤ i ≤ n. Then the likelihood function of the observed data is:
n
L (β ) = ∏
i =1
exp βT Z i ( ti )
∑ exp β
T
Zl ( ti )
l ∈Ri
In our data, an individual can leave the risk set either due to default or censored event,
because the account got closed or reached the end of the sample period. In estimating
h(t;X(t)), any event other than default is treated as right-censored. An individual that is
censored at time t will contribute to the risk set Ri if ti ≤ t, and will be excluded if t <ti.
5
Credit Risk of Portfolio of Consumer Loans
The ith term in the partial likelihood function above is the conditional probability that an
individual with covariates Zi(ti) experiences a default event at ti given all individuals in Ri
and that exactly one individual default at ti.
Cox model assumes that the hazard functions are continuous. However, credit performance
data are normally recorded only monthly so that multiple defaults at one time can be
observed. These are tied default times, and the likelihood function must be modified
because it is now unclear which individuals to include in the risk set at each failure time t1,
t2, t3, etc.
Let di denote the number of defaults at time ti and Di be the set of individuals defaulted. Let
S denote any subset of di individuals drawn from the risk set Ri and let Si be the collection
of all these subsets. The generalized version of the likelihood function arising from the
Cox’s model is
n
L (β ) = ∏
i =1
exp βT
∑Z
j
( ti )
∑ exp β ∑ Z
T
S ∈Si
j∈Di
j∈S
j
( ti )
The summation in the denominator is difficult to calculate, so easier approximations were
proposed by (Breslow, 1974) and (Efron, 1977).
Modelling Methodology
In this article we propose to build a hazard rate model for predicting probability of default
of a customer in the next twelve months, given all the current information available on a
customer along with the values of macroeconomic factors for one year ahead. The results
of our analysis will show that default intensities of consumers are significantly influenced
by macro factors and by the (calendar) time when the loan started. It will show that the
information contained in macro economic variables and the origination quality of loans are
major drivers of the dynamics of default probability.
6
Credit Risk of Portfolio of Consumer Loans
Data Description
The original dataset contains records of customers of a major UK bank who were on the
books as of January 2001 together with all those who joined between January 2001 and
December 2005. The data set consists of customers monthly behavioural score along with
the information on their time to default or censored and default or censored status. We
have no information on time to default of those individuals who have joined and defaulted
before Jan 2001. Hence, for model fitting purpose we consider approx. 50,000 records of
customers behavioural scores for each month from Jan 2001 – Dec 2004, along with
information on age of loan and default status. We tested our results using customers
performance during 2005 from a holdout sample. In our analysis, individuals who closed
the account during the above time period or who got truncated at the end of observation
period, i.e., Dec 2004, are considered censored. Anyone, who become 90 days delinquent,
charged off or got bankrupt between Jan 2001-Dec 2004 is considered as having defaulted.
The time to default/censored is measured in months from the start of the loan.
Macro Economic Variables
Traditionally behavioural score models are built on customers performance with the bank
over a twelve month or some other fixed time performance period, using characteristics
like average account balance, number of times 30+ days delinquent, etc. In that sense
behavioural score can be considered as capturing the idiosyncratic risk of customers.
However, it is observed in corporate credit risk models (Lando, 1994; Duffie et al, 2007),
that though idiosyncratic risk is a major risk factor, during economic slowdowns systemic
risk factors emerge and have had a substantial effect on the default risk in a portfolio of
loans. In context of UK economy, we analyse the following four macro economic variables
which have been found to effect the default behaviour of consumers in the consumer
finance literature (Tang et al, 2006) and represents the general economic and investment
climate. The variables considered are
(a) Percentage Change in Consumer Price Index over 12 Months: reflects the inflation
felt by customers and high levels may cause rise in customer default rate.
7
Credit Risk of Portfolio of Consumer Loans
(b) Monthly average Sterling Inter-bank lending rate: higher values might correspond
to general tightness in the economy and increased difficulty in raising cash to make
debt service payments.
(c) Annual Return on Log of FTSE 100: gives the yield from stock market and also
reflects the fact that a buoyant stock market may encourage purchasing of financial
products.
(d) GDP Quarter on Quarter Previous Year.
A positive sign of the coefficient of a macroeconomic variable in the hazard equation is
associated with the increase in risk of default and vice versa. We did try to include
unemployment rate in our model but it always entered with a wrong sign and also affected
the sign and significance of other macroeconomic covariates. This could be because there
is no significant variation in UK’s unemployment rate during 2001 to 2004 so spurious
effects are occurring. Therefore we decided to drop unemployment rate from our models.
There is a general perception in economics (Figlewski et al, 2007) that change in economic
conditions does not have an instantaneous effect on default rate. To allow for this effect we
considered lagged values of macroeconomic covariates in the form of weighted average
over a six months period with an exponentially declining weight of 0.88. This choice is
motivated by a recent study made by (Figlewski et al, 2007).
Hazard Rate Models
In our analysis, we shall model the hazard rate of customers by incorporating information
on:
a) Age of Loan
b) Behavioural scores
c) Macro Economic Variables
d) Time Loan Taken Out (Vintage)
8
Credit Risk of Portfolio of Consumer Loans
If t is time since loan started, the hazard of default at time t for a person i from Vintagei
under current economic conditions EcoVari(t) and having behavioural score BehScri(t) is
hi ( t ) = h0 ( t ) e
aBehScri ( t )+bEcoVari ( t )+ cVintagei
In the above equation h0(t) is the baseline hazard which represents the risk due to the age of
loan. On could think of the idiosyncratic risk, systemic risk and the risk due to the vintage
quality of loans as being represented by BehScr(t), EcoVar(t), and Vintage, respectively.
Vintages correspond to each quarter from 2001 to 2004 and defined as binary variables
with a value 1 if an individual starts a loan in a particular vintage otherwise 0. The partial
likelihood method, first proposed by Cox (1972), estimates the parameters a, b and c in the
hazard function without knowledge of the baseline hazard h0(t). Since in our analysis, we
need to predict the probability of default, say, in next twelve months from now, we use
Nelson-Aalen form of estimator (Kalbfleisch, 2002), to recover baseline hazard function
h0(t) given by
dt
h0 ( t ) =
l
exp β
∑
∈
T
Zl ( t )
Rt
where dt represents the people defaulted at time t and Rt denotes the risk set at time t.
Figure 1 shows the movement of default rate with age of loan for 10 years of duration in
our dataset. It is evident from the graph that propensity of default is high during the early
stage of a loan. This goes well with the market assumption that default rates are high during
the first 12-24 months of the starting of the loan, which is the reason why many banks use
12-24 months of performance window in their scorecards to classify good/bad customers.
However, we also notice from the graph that the default rate does not drop immediately
after the first year. Instead, it gradually comes down, which is the reason why age of loan
should not be ignored as one of the factors while developing credit risk models.
9
Default Rate
Credit Risk of Portfolio of Consumer Loans
9
16
22
29
35
42
48
55
61
67
74
80
87
93 100 106 113
Age of Loan (Months)
Fig. 1. Default Rate v Age of Loan
Coming to the second and most important factor in our model, behavioural score represents
the risk due to the individual’s performance with the bank. Behavioural score is related to
the probability of default by a linear relationship. In our dataset behavioural score predicts
the probability of default in the next 6 months. Banks use behaviour score to rank their
customers and decide on their policies accordingly. One of the problem in using
behavioural score as the sole representation of individual’s probability of default is that the
relationship of behavioural score to probability of default may change over time. It means
that the behavioral score of two individuals in different months or the behavioural score of
same individual in different months may not represent the same probability of default. This
is the problem of recalibration of scorecards arising from the change in log of odds to score
relationship in credit scoring and a major motivation behind introducing macroeconomic
and vintage variables in our hazard equation.
To understand the relevance of vintage variables and macro economic variables in our
model we consider the following example. Suppose two individuals A and B start their
loan during different periods in calendar time say the first quarter of 2001 and the first
quarter of 2003 respectively. Suppose they both have the same score x after six months into
10
Credit Risk of Portfolio of Consumer Loans
the loan. It could be possible that the score x may not represent the same default probability
for two customers. Figure 2 shows the default rates of customers based on the year they
joined. It seems that customers who joined during 2003 are at higher risk (have higher
default rates) than those who joined during 2001-2002. This gives us the intuition to
incorporate year of origination of loan to test the dependency of hazard rate on the time the
customers joined. To capture this effect we can define binary variables corresponding to
whether the loan was started in a specific quarter or not. If these variables come out to be
nonzero and significant in the model then it is a clear indication that the hazard rate of two
individuals n months into the loan with the same score in month n would depends not only
on the score but also on the calendar time during which the individuals started the loan.
Similarly, to explain the affect of macro variables on hazard rate consider an individual
who has got the same score x during two different months on books. Since the value of
economic variable might be different in different months, the hazard rate of an individual at
Default Rate
the same score level in different months might be different.
2001
2002
2003
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
Age of Loan (Months)
Forecasting Multi-Period Hazard Rate and Default Probability
To predict the probability of default of customers for one year ahead in calendar time we
first predict their hazard rate in time since loan started and then transform it into a
11
Credit Risk of Portfolio of Consumer Loans
calendar time for twelve months ahead. This is achieved by developing twelve hazard
rate models each with the lagged behavioural score variable in the hazard equation. In
particular, for customer i who survived up to t months on books, the hazard rate for k
months ahead is given by
hi ( t+k ) = h0 ( t+k ) e
ak BehScri ( t )+ bk EcoVari ( t + k )+ ck Vintagei
.
Each hazard model has economic variables taken as a weighted average over the last six
months. Take, for example, a customer with behavioural score s in Dec 2003 and who
started his loan in Jan 2003. To predict his hazard rate in Feb 2004, based on the score in
Dec 2003, we can select parameter estimates from hazard rate model with a behavioural
score of lag 2 and use weighted average of the values of economic variables for the six
months from Aug 2003-Jan 2004. (If one really were in December 2003 then the January
2004 value in this weighted average would be a forecast of the future economic
situations). We can then predict the hazard rate for 14 months into the loan, which gives
us the predicted hazard rate for a customer in Feb 2004. We coarse classify the
behavioural scores into five bins in order of decreasing risk with people having low
behavioural scores, i.e., who are at higher risk of default, in the first bin. We also define
one special value bin which represents people who do not currently have a behavioural
score which means they have always been inactive or only recently become active. We
then define new time varying covariates BehScore-i (i=1,…,5) which are binary (0,1)
variables corresponding to the behavioural score bins defined above in terms of worst to
best ranking.
Table 3 present parameter estimates from the two sets of models Model A and Model B
for each month up to twelve months ahead. Model A predicts hazard rate using just
behavioural score as a time covariate however, Model B uses time varying
macroeconomic variables and time of origination of the loan, which is captured by
vintage variables, in addition to the behavioural scores. We employ "stepwise selection"
procedure to only keep variables that contribute significantly to the explanatory power of
the model. The likelihood ratios and the associated p-values show that incorporating
macroeconomic and vintage level information provides a significant improvement across
all periods. However, the predictive power of Model B over Model A decrease with the
12
Credit Risk of Portfolio of Consumer Loans
increase in prediction horizon. In Model B, the coefficients of behavioural score bins,
leaving aside the special value bin, go from high to low for all periods, which agrees with
the risk associated with each bin. The prediction power of behavioural score decreases
with the increase in prediction horizon which is evident from the decrease in the
difference of coefficient values between behavioural score bins with each increasing
month. However, similar to what was observed for corporate defaults by (Figlewski et al,
2007), there has been little change in the value of the estimated coefficients for
behavioural score bins for each month as we move from Model A to Model B. This
means that the information contained in macroeconomic and vintage variables though
significant is incremental to that what is captured by the behavioural score alone. This
could be possibly because economic conditions remained approximately static during
2001-2004. Three macroeconomic variables namely Interest rate, GDP growth and CPI
(rate of inflation) proved significant at various horizons. The sign of the macroeconomic
variables remain consistent throughout the twelve models which forecast forward default
rates 1, 2, 3 up to 12 months ahead. The sample size decreases as the covariate time lag
increases because the minimum period of observation is increasing. Since each vintage
represents a quarter between 2001 to 2004, observations from recent vintages will be
excluded from the dataset as we increase the forecast horizon. So for example, to estimate
12-month predictive hazard rate model people who joined between 2001-2003 will
appear in the dataset. However, it appears that overall the effect of vintages gradually
diminish as we increase the forecast horizon. Suppose h(1), h(2), h(3), … h(12) represent
the hazard rates of an individual, who has neither defaulted or left till this month, for next
one month, two month, three month, etc., respectively in calendar time. From the
definition of hazard rate and the law of conditional probability, the probability of default
in next 12 months (PD12) can be estimated by PD12 = 1 − Π 12
n =1 ( 1 − h ( n ) ) . We tested
our results on a hold out sample for 2005. We considered people who joined between Jan
2002-Dec 2004 and have neither defaulted nor closed their accounts till Dec 2004. We
then estimate their PD12 for 2005. The results of our analysis are shown in table 1 below.
13
Credit Risk of Portfolio of Consumer Loans
Table 1
Test Sample Actual No. of Expected No. of
Model
Defaults in 2005 Defaults in 2005
Size
Model A
565
14091
Model B
959
1022
Figure 3 shows the ROC curves for 2005 on a holdout sample based on PDs obtained from
Model A and Model B. It shows that the risk ranking of customers based on information
provided by behavioural score is almost the same as that obtained by adding information on
macroeconomic changes and the time of origin. However, from Table 1 above, it is
apparent that the actual PD estimates from the two models are not the same and that Model
B certainly provides better estimates of expected number of defaults. This shows that the
relationship of log (Default : Not Default) to score does vary over time and may be
explained by adding extra information on macroeconomic changes and time of origination
of the loan.
Portfolio Default Distribution
In a recent paper Duffie, Saita, and Wang (2007) studied the time series dynamics of the
covariate processes in the context of multi-period corporate default prediction. Motivated
by this we model interest rate, GDP and CPI as an independent autoregressive time series
of order 1 and simulate the default distribution of loans from a holdout sample for the year
2005. We assume that given the value of economic variables for a fixed time period in
calendar time, the defaults are conditionally independent. Table 3 shows the parameter
estimates for each of the macroeconomic variable in our model which uses the following
specification
yt − µ = ρ ( yt −1 − µ ) + σet ,
where
is the estimated mean of the series yt and et are independent standard normal
errors.
14
Credit Risk of Portfolio of Consumer Loans
Table 2 Estimates from and AR(1) Model for Macroeconomic Covariates.
Variable
Mean ( )
Coefficient (ρ)
Interest Rate
5.656389
0.98704
0.209164
GDP
2.925
0.93775
0.212757
CPI
1.465741
0.87823
0.237134
Standard Error (σ)
The simulation steps to draw the default distribution of portfolio of loans are explained
below.
1) Generate a sample path of the three macroeconomic variables above for twelve
months from January 2005 taking initial value Y0 in AR(1) equation from
December 2004 and independently drawing twelve random variables corresponding
to error terms from standard normal distribution.
2) Calculate the hazard rate and hence PD12 using Model B for each customer in the
hold out sample for 2005 using the sample path of macroeconomic values generated
in step 1 above.
3) Generate random numbers from a uniform distribution between 0 and 1
corresponding to each observation.
4) Define an individual as default (1) if PD12 > Random Value and Non-default (0) if
PD12 < Random Value.
5) Calculate the total number of defaults in step 4.
6) Repeat Step 1, 2 ,3 and 4 with random seed values and draw the distribution of
proportion of defaults.
Figure 4 shows the smoothed default distribution of proportion of defaults for 2005 in a
holdout sample of approximately 14,000 customers. The peak of the distribution is at
a default rate of approximately 6.5%. The distribution is though somewhat asymmetric
with higher probabilities of high default rates. The simulation suggest that there is a
0.1% probability of having default rates above 10%.
15
Credit Risk of Portfolio of Consumer Loans
One can use more sophisticated models of the time series dynamics of macroeconomic
variables and still apply the above simulation procedure using the hazard rate model we
have developed. Our objective here is just to show such simulations can be undertaken
since there is a rich literature on such macroeconomic time series models (
)
Conclusions
In this article, we build a proportional hazard rate model for customers default
probability for up to twelve months ahead using recent information on customers
behavioural scores along with the values of general macroeconomic factors. We do not
predict customers’ future behavioural scores since we believe the current behavioural
score contains the best estimate of future default risk based only on customer specific
information.
Instead we can use existing models for predicting the time series
dynamics of macroeconomic factors which are the drivers of the dynamics of default
behaviour. We estimated twelve hazard rate models for each month up to twelve
months ahead by lagging behavioural score covariate. Our major conclusion is that
inclusion of macroeconomic factors and time of origination improves the default
predictions significantly. However, it does not have a major impact on the
discriminating ability of the model beyond that given by the behavioural score. Such
models lend themselves to building credit risk estimates of portfolios of consumer loans
where the correlations between defaults of different loans are given by changes in the
macroeconomic conditions.
References:
Basel Committee on Banking Supervision (BCBS) (2004). International
Convergence of Capital Measurement and Capital of Capital Standards: a revised
framework. Bank of International Settlements, Basel.
Breslow N E (1974). Covariance analysis of censored survival data. Biometrics 30:
89-99.
Campbell J, Hilscher J and Szilagy J (2007). In Search of Distress Risk.
Forthcoming Journal of Finance.
Cox D R (1972). Regression models and life-tables (with discussion). J Royal
Statist Society, Series B 74: 187-220.
16
Credit Risk of Portfolio of Consumer Loans
Duffie D, Saita L and Wang K (2007). Multi-Period Corporate Default Prediction
with Stochastic Covariates. Journal of Financial Economics 83 (3): 635-665.
Efron B (1977). The efficiency of Cox’s likelihood function for censored data. J
Amer Statist Assoc 72: 557-565.
Figlewski S, Frydman H and Liang W (2007). Modelling the Effect of
Macroeconomic Factors on Corporate Default and Credit Rating Transitions.
Working Paper No. FIN-06-007, NYU Stern School of Business.
Jarrow R, Lando D and Turnbull S (1997). Pricing Derivatives on Financial
Securities Subject to Credit Risk. Journal of Finance. 50: 53-86.
Kalbfleisch J D and Prentice R L (2002). The Statistical Analysis of Failure Time
Data. Wiley: New York.
Lando D (1994). Three essays on contingent claims pricing. PhD thesis, Cornell
University, Ithaca, NY.
Narain B (1992). Survival analysis and the credit granting decision. In: Thomas L
C, Crook J N, Edelman D B (eds). Credit Scoring and Credit Control. OUP:
Oxford, pp 109-121.
SAS Online Doc 9.1.2. The PHREG Procedure. SAS Institute: Cary NC, USA.
Shumway T (2001). Forecasting Bankruptcy More Accurately: A Simple Hazard
Model. Journal of Business. 74 (1): 101-124.
Stepanova M and Thomas L C (2002). Survival Analysis Methods for Personal
Loan Data. Operations Research. 50: 277-289.
Tang L, Thomas L C, Thomas S and Bozzetto J-F (2007). It's the economy stupid:
Modelling financial product purchases. International Journal of Bank Marketing.
25: 22-38.
Thomas L C, Banasik J and Crook J N (1999). Not if but when loans default. J Opl
Res Soc 50: 1185-1190.
17
Credit Risk of Portfolio of Consumer Loans
Model A
Model B
1
0.9
P (s | Non-default)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
P (s |Default)
!
() *
(
) ! !
+
,
"
)
#
$%
!
&
$
() )
%
'
, "- .
"
)
$!
"- .0
/
$
+
1
14.00%
12.00%
Probability
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
4.00%
5.00%
6.00%
7.00%
8.00%
Proportion of Defaults
,
$
! !
!
18
9.00%
10.00%
11.00%
/
Credit Risk of Portfolio of Consumer Loans
BehScore Bins
BehScorer1
BehScorer2
BehScorer3
BehScorer4
BehScorer5
Special Value
MacroEconomic Factors
Interest Rate
GDP
CPI
Vintages
VintQrt1
VintQrt2
VintQrt3
VintQrt4
VintQrt5
VintQrt6
VintQrt7
VintQrt8
VintQrt9
VintQrt10
Vintage11
Vintage12
Vintage13
Vintage14
VintQrt15
Likelihood Ratio Test: Model B vs Model A
p-value
1-Month Predictive
Model
Parameter Estimates
2-Month Predictive 3-Month Predictive
Model
Model
Parameter Estimates Parameter Estimates
4-Month Predictive 5-Month Predictive 6-Month Predictive
Model
Model
Model
Parameter Estimates Parameter Estimates Parameter Estimates
Model A
Model B
Model A
-2.24671
-8.54738
-9.60406
-10.03671
-11.5715
-6.01966
Model B
Model A
Model B
Model A
Model B
Model A
Model B
Model A
Model B
-2.3323
-8.66029
-9.71268
-10.14294
-11.66931
-6.11543
-2.6242 -2.70791 -3.09697
-6.98778 -7.09739 -5.62776
-8.08176 -8.19148 -6.08841
-9.68832 -9.79372 -6.96915
-11.9021 -11.99937 -7.60843
-6.47088 -6.56873 -5.80204
-3.18273
-5.73832
-6.20220
-7.07702
-7.71063
-5.90250
-2.75823
-4.9138
-5.57769
-5.72644
-6.20157
-5.1372
-2.84724
-5.02491
-5.69440
-5.83734
-6.30834
-5.24298
-2.48733
-4.08235
-4.84958
-5.03777
-5.42494
-4.58765
-2.56478
-4.18073
-4.95517
-5.13704
-5.52223
-4.68248
-2.61022
-3.90613
-4.50302
-4.83499
-5.22294
-4.48938
-2.67749
-3.99306
-4.60184
-4.92699
-5.31600
-4.57994
0.26229
-0.42539
0.25069
-0.38256
0.24416
-0.31511
0.22181
-0.24414
0.22013
-0.27600
0.23729
-0.34237
-1.21870
-1.13194
-1.01896
-0.75628
-0.81972
-0.73978
-0.51075
-0.38299
-0.26098
-0.13124
-1.15786
-1.07397
-0.92263
-0.68562
-0.77152
-0.69347
-0.44722
-0.31696
-0.22958
-0.99363
-0.96448
-0.77844
-0.59465
-0.71633
-0.62936
-0.39576
-0.26262
-0.18531
-0.79594
-0.80352
-0.65098
-0.50605
-0.65304
-0.62144
-0.35188
-0.24582
-0.21149
-0.74695
-0.74484
-0.64334
-0.46615
-0.60409
-0.60719
-0.34361
-0.21872
-0.19159
-0.79301
-0.79143
-0.70042
-0.49181
-0.66492
-0.64168
-0.36271
-0.23177
-0.19938
0.19690
0.77901
218.5007
<.0001
1.00105
213.7275
<.0001
19
175.9484
<.0001
139.9434
<.0001
122.2438
<.0001
109.3105
<.0001
Credit Risk of Portfolio of Consumer Loans
BehScore Bins
BehScorer1
BehScorer2
BehScorer3
BehScorer4
BehScorer5
Special Value
MacroEconomic Factors
Interest Rate
GDP
CPI
Vintages
VintQrt1
VintQrt2
VintQrt3
VintQrt4
VintQrt5
VintQrt6
VintQrt7
VintQrt8
VintQrt9
VintQrt10
Vintage11
Vintage12
Vintage13
Vintage14
VintQrt15
Likelihood Ratio Test: Model B vs Model A
p-value
7-Month Predictive
Model
Parameter Estimates
8-Month Predictive 9-Month Predictive 10-Month Predictive 11-Month Predictive 12-Month Predictive
Model
Model
Model
Model
Model
Parameter Estimates Parameter Estimates Parameter Estimates Parameter Estimates Parameter Estimates
Model A
Model B
Model A
Model B
Model A
Model B
Model A
Model B
Model A
Model B
Model A
Model B
-2.3721
-3.60232
-4.13394
-4.36663
-4.85296
-4.1005
-2.43000
-3.67689
-4.21920
-4.44559
-4.93306
-4.17712
-2.45326
-3.54707
-3.94328
-4.35139
-4.81405
-4.06151
-2.49524
-3.60116
-4.00594
-4.40785
-4.87324
-4.11792
-2.4375
-3.44737
-3.94487
-4.27954
-4.71469
-3.99174
-2.48732
-3.50893
-4.0159
-4.34375
-4.78202
-4.05697
-2.42873
-3.33606
-3.78145
-4.05897
-4.61785
-3.89062
-2.48829
-3.40538
-3.85942
-4.13097
-4.69305
-3.96378
-2.33096
-3.25984
-3.59847
-4.05529
-4.46676
-3.78074
-2.38991
-3.32759
-3.67288
-4.12304
-4.54106
-3.85201
-2.26133
-3.12304
-3.51835
-3.9288
-4.32225
-3.66479
-2.32921
-3.19961
-3.60081
-4.00519
-4.40504
-3.74210
0.24696
-0.28462
-0.52755
-0.54694
-0.50450
-0.27505
-0.46768
-0.45223
-0.18336
0.51457
98.2929
<.0001
0.27168
0.26628
0.39202
0.4977
0.33990
-0.10516
0.63936
0.10792
-0.14904
-0.18930
-0.16601
-0.21061
-0.16383
-0.17845
0.24378
-0.29266
0.37177
-0.14673
0.13935
-0.19260
-0.17424
0.32766
0.33385
0.35307
0.51792
0.45821
0.66400
60.3232
<.0001
20
51.8505
<.0001
52.2771
<.0001
59.3396
<.0001
54.3668
<.0001