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International Journal of Forecasting

journal homepage: www.elsevier.com/locate/ijforecast

Transition matrix models of consumer credit ratings

Madhur Malik1, Lyn C. Thomas *

Quantitative Financial Risk Management Centre, School of Management, University of Southampton, Southampton SO17 IB], UK

article info

abstract

Keywords: Markov chain Credit risk Logistic regression Credit scoring

Although the corporate credit risk literature includes many studies modelling the change in the credit risk of corporate bonds over time, there has been far less analysis of the credit risk for portfolios of consumer loans. However, behavioural scores, which are calculated on a monthly basis by most consumer lenders, are the analogues of ratings in corporate credit risk. Motivated by studies of corporate credit risk, we develop a Markov chain model based on behavioural scores for establishing the credit risk of portfolios of consumer loans. Although such models have been used by lenders to develop models for the Basel Accord, nothing has been published in the literature on them. The model which we suggest differs in many respects from the corporate credit ones based on Markov chains — such as the need for a second order Markov chain, the inclusion of economic variables and the age of the loan. The model is applied using data on a credit card portfolio from a major UK bank. © 2011 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction

Since the mid 1980s, banks' lending to consumers has exceeded that to companies (Crouhy, Galai, & Mark, 2001). However, not until the subprime mortgage crisis of 2007 and the subsequent credit crunch was it realised what an impact such lending had on the banking sector, and also how under-researched it is compared to corporate lending models. In particular, the need for robust models of the credit risk of portfolios of consumer loans has been brought into sharp focus by the failure of the ratings agencies to accurately assess the credit risks of the Mortgage Backed Securities (MBS) and collateralized debt obligations (CDO) which are based on such portfolios. Many reasons for the subprime mortgage crisis and the subsequent credit crunch have been put forward (Demyanyk & van Hemert, 2008; Hull, 2009), but, clearly, one reason why the former led to the latter was the lack of an easily updatable model of the credit risk of portfolios of consumer loans. This lack of a suitable model of portfolio level consumer risk

* Corresponding author. Tel.: +44 238059 7718; fax: +44 238059 3844. E-mail addresses: Madhur.Malik@fsa.gov.uk (M. Malik), l.Thomas@soton.ac.uk (L.C. Thomas).

1 Tel.: +44 238059 7718; fax: +44 238059 3844.

was first highlighted during the development of the Basel Accord, when a corporate credit risk model was used to calculate the regulatory capital for all types of loans (Basel Committee on Banking Supervision, 2005), even though the basic idea of such a model — that default occurs when debts exceed assets — is not the reason why consumers default.

This paper develops a model for the credit risk of portfolios of consumer loans based on the behavioural scores of the individual consumers whose loans make up that portfolio. Such a model is attractive to lenders, since almost all lenders calculate behavioural scores for all of their borrowers on a monthly basis. The behavioural score is usually translated into the default probability over a fixed time horizon (usually one year) in the future for that borrower, but one can also consider it as a surrogate for the unobservable creditworthiness of the borrower. We build a Markov chain credit risk model based on behavioural scores for consumers which has similarities with the reduced form mark to market corporate credit risk models based on the rating agencies' grades (Jarrow, Lando, & Turnbull, 1997). Such behavioural score based Markov chain models have been developed by lenders for their Basel modelling, but no analysis has appeared in the literature; in this paper, we discuss the features which

0169-2070/$ - see front matter © 2011 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2011.01.007

should be included in such models and compare a standard and a more sophisticated version of the model. The methodology constructs an empirical forecasting model for deriving a multi-period distribution of the default rate for long time horizons based on migration matrices built from a historical database of behavioural scores. Although it is possible to calibrate the scores to the long run probability of default if one has data for a sufficiently long outcome period, such data are not available in practice. The transition matrix approach allows one to undertake such a calibration using much shorter data series. In our case study we use the lenders' behavioural scores, but we can also use the same methodology on generic bureau scores.

The approach also helps lenders to make long term lending decisions by estimating the risk associated with the change in the quality of portfolio of loans over time. Since the model includes economic conditions, the approach allows banks to stress test their retail portfolios, as required by the Basel Accord and other banking regulations. In addition, the model provides insights on portfolio profitability, the determination of appropriate capital reserves, and the creation of estimates of portfolio value by generating portfolio-level credit loss distributions.

There have recently been several papers which have looked at modelling the credit risk in consumer loan portfolios. Rosch and Scheule (2004) take a variant of the one factor Credit Metrics model, which is the basis of the Basel Accord. They use empirical correlations between different consumer loan types and try to build in economic variables to explain the differences during different parts of the business cycle. Perli and Nayda (2004) also take the corporate credit risk structural model and seek to apply it to consumer lending, assuming that a consumer defaults if his assets are below a specified threshold. However, consumer defaults are usually more about cash flow problems, financial naivete or fraud, and thus such a model misses some aspects of consumer defaults.

Musto and Souleles (2005) use equity pricing as an analogy for changes in the value of consumer loan portfolios. They use behavioural scores, but take the monthly differences in behavioural scores as the return on assets when applying their equity model.

Andrade and Thomas (2007) describe a structural model for the credit risk of consumer loans, where the behavioural score is a surrogate for the creditworthiness of the borrower. A default occurs if the value of this reputation for creditworthiness, in terms of access to further credit, drops below the cost of servicing the debt. Using a case study based on the Brazilian credit bureau, they found that a random walk was the best model for the idiosyncratic part of creditworthiness. Malik and Thomas (2010) developed a hazard model of the time to default for consumer loans, where the risk factors were the behavioural score, the age of the loan, and economic variables, and used it to develop a credit risk model for portfolios of consumer loans. Bellotti and Crook (2009) also used proportional hazards to develop a default risk model for consumer loans. They investigated which economic variables might be the most appropriate, though they did not use behavioural scores in their model. Thomas (2009b)

reviewed the existing consumer credit risk models and pointed out the analogies with some of the established corporate credit risk models.

Since the seminal paper by Jarrow et al. (1997), the Markov chain approach has proved popular in modelling the dynamics of the credit risk in corporate portfolios. The idea is to describe the dynamics of the risk in terms of the transition probabilities between the different grades the rating agencies award to the firm's bonds. There are papers which look at how both economic conditions and the industry sector of the firm affects the transition matrices (Nickell, Perraudin, & Varoli, 2001), while others generalise the original idea of Jarrow et al. by using Affine Markov chains (Hurd & Kuznetsov, 2006) or continuous time processes (Lando & Skodeberg, 2002). However, none of these suggest increasing the order of the Markov chain or considering the age of the loan, which are two of the features which we introduce here, in order to model the consumer credit risk using Markov chains. This is surprising, because there has been work on downgrading by rating agencies which suggests that there is a momentum effect where, once a company has been downgraded, it is more likely to be further downgraded than to be subsequently upgraded (Bangia, Diebold, & Schuermann, 2002).

Markov chain models have been used in the consumer lending context before, but none of the published papers have used the behavioural score as the state space, nor has the objective of the models been to estimate the credit risk at the portfolio level. The first such application was by Cyert, Davidson, and Thompson (1962), who developed a Markov chain model of customers' repayment behaviours. Subsequently, more complex models have been developed by Ho (2001), Thomas, Ho, and Scherer (2001) and Trench et al. (2003). Schniederjans and Loch (1994) used Markov chain models to model the marketing aspects of customer relationship management in the banking environment.

Behavioural score based Markov chain models are sometimes used in the industry (see Scallan, 1998), but mainly as ways of assessing provisioning estimates, and they do not include the economic drivers and months on the books effects presented in this paper. Moreover, the introduction of economic factors into the model allows one to deal with the correlations between defaults on individual loans in a portfolio, since they are affected by common economics. One can obtain the mean default rate in a portfolio from the long run distributions, while a Monte Carlo simulation using the transitions of individual loans would give the distribution of the default rate.

In Section 2, we review the properties of behavioural scores and Markov chains, while in Section 3 we describe the Markov chain behavioural score based consumer credit risk model developed. This is parameterised by using cumulative logistic regression to estimate the transition probabilities of the Markov chain. The motivation behind the model and the accuracy of the model's forecasts are shown by means of a case study, and Section 4 describes the data used in the case study. Sections 5-7 give the reasons why the model includes higher order transition matrices (Section 5); economic variables for explaining the non-stationarity of the chain (Section 6); and the age of

the loan (Section 7). Section 8 describes the full model used, while Section 9 reports the results of out-of-sample forecasts, and out-of-time and out-of-sample forecasts, using the model. The final section draws some conclusions, including how the model could be used. It also identifies one issue — which economic variables drive consumer credit risk — where further investigation would benefit all models of consumer credit risk.

2. Behaviour score dynamics and Markov chain models

Consumer lenders use behavioural scores updated every month to assess the credit risk of individual borrowers. The score is considered to be a sufficient indication of the probability that a borrower will be "Bad", and so default within a certain time horizon (normally taken to be the next twelve months). Borrowers who are not Bad are classified as ''Good''. Thus, at time t, a typical borrower with characteristics x(t) (which may describe the recent repayment and usage performance, the current information available on the borrower at a credit bureau, and socio-demographic details) has a score s(x(t), t), so

p(Blx(t), t) = p(Bls(x(t), t)).

Some lenders obtain a Probability of Default (PD), as required under the Basel Accord, by taking a combination of behavioural and application scores. New borrowers are scored using only the application score to estimate PD, then once there is sufficient history for a behavioural score to be calculated, a weighted combination of the two scores is used to calculate PD; eventually, the loan is sufficiently mature that only the behavioural score is used to calculate PD. The models described hereafter can also be applied to such a combined scoring system.

Most scores are log odds score (Thomas, 2009a), and thus the direct relationship between the score and the probability of being Bad is given by

/P(G|s(x(t), t)) \ s(x(t), t) = log 1 ^ P(B|s(x(t), t))

lww 5\P (B|s(x(t), t)) J luw "

1 + es(x(t),t) '

though in reality this may not hold exactly. Applying the Bayes theorem to Eq. (2) gives the expansion where ifpG(t) is the proportion of the population who are Good at time t (pB (t) is the proportion who are Bad), one has

s(x(t), t) = log

P(Gls(x(t), t)) P(Bls(x(t), t))

= log( p*)) \PB(t )J

,, (P(s(x(t), t)|G, t)

+ log -

S\P(s(x(t), t)|B, t)

= Spop(t) + woet(s(x(t), t)).

The first term is the log of the population odds at time t and the second term is the weight of evidence for that score (Thomas, 2009a). This decomposition may not hold exactly in practice, and is likely to change as a scorecard ages. However, it shows that the term spop(t), which is common to the scores of all borrowers, can be thought to play the role of a systemic factor which affects the default risk of all

of the borrowers in a portfolio. Normally, though, the time dependence of a behavioural score is ignored by lenders. Lenders are usually only interested in ranking borrowers in terms of risk, and they believe that the second term (the weight of evidence) in Eq. (3), which is the only one which affects the ranking, is more stable over time than spop(t), particularly over horizons of two or three years. In reality, the time dependence is important because it describes the dynamics of the credit risk of the borrower. Given the strong analogies between behavioural scores in consumer credit and the credit ratings used for corporate credit risk, one obvious way of describing the dynamics of behavioural scores is to use a Markov chain approach similar to the reduced form mark to market models of corporate credit risk (Jarrow et al., 1997). To use a Markov chain approach with behavioural scores, we divide the score range into a number of intervals, each of which represents a state of the Markov chain; hereafter, when we mention behavioural scores we are thinking of this Markov chain version of the score, where the states are intervals of the original score range.

Markov chains have proved ubiquitous models of stochastic processes because their simplicity belies their power to model a variety of situations. Formally, we define a discrete time {t0, t1,..., tn,...: n e N} and a finite state space S = {1, 2,..., s} first order Markov chain as a stochastic process {X(tn)}neN, with the property that for any s0, s1,..., sn-1, i, j e S:

P [X (tn+1) = j | X (t0) = s0, X (t1) = s1,..., X (tn-1)

= sn-1, X (tn) = i] = P [X (tn+1) = j | X (tn) = i]

= Pij (tn, tn+1) , (4)

where pij (tn, tn+1) denotes the transition probability of going from state i at time tn to state j at time tn+1.The S x S matrix of elements pij (.,.), denoted P(tn, tn+1), is called the first order transition probability matrix associated with the stochastic process {X(tn)}neN. If n (tn) = (n1(tn), ..., ns(tn)) describes the probability distribution of the states of the process at time tn, the Markov property implies that the distribution at time tn+1 can be obtained from that at time tu by n (tn+1) = n (tn) P (tn, tn+1). This extends to a m-stage transition matrix, so that the distribution at time tn+m for m > 2 is given by

n (tn+m) = n (tn) P (tn, tn+1) ... P (tn+m-1, tn+m) .

The Markov chain is called time homogeneous or stationary, provided that

Pij(tn, tn+1) = Pij Vn e N.

Assume that the process {X(tn)}neN is a nonstationary Markov chain, which is the case with the data we examine later. If one has a sample of the histories of previous customers, let ni(tn), i e S, be the number who are in state i at time tn, whereas let nij(tn, tn+1) be the number who move from state i at time tn to state j at time tn+1. The maximum likelihood estimator of pij (tn, tn+1) is then

pij (^ tn+1) =

nij (^ tn+l) ni (tn) '

If one assumes that the Markov chain was stationary, then, given the data for T + 1 time periods n = 0, 1, 2,..., T, the transition probability estimates become

J2 nij(tn, tn+1)

n=0 ,-s

p ij = —T—1-. (7)

E ni (tn)

Note that the Markov property means that previous transitions do not affect the current probabilities of transition, and thus in these calculations we do not need to be concerned that transitions coming from the same customer are dependent. All transitions are essentially independent, even those from the same customer. One can weaken the Markov property so that the information required to estimate the future of the chain is the current state and the previous state of the process. This is called a second order Markov chain, which is equivalent to the process being a first order Markov chain, but with state space S x S. The concept can be generalized to defining kth order Markov chains for any k, though of course, the state space and the size of the transition probability matrices go up exponentially as k increases.

3. Behavioural score based Markov chain model of consumer credit risk

The behavioural score Bt of a borrower is an observable variable given by a scorecard. It is related to the underlying unobservable "creditworthiness" of the borrower, Ut, which also depends on the length of time the loan has been running and the current economic situation. Our model is constructed by assuming that the borrower's behavioural score is in one of a finite number of states, namely {s0 = D, s1,..., sn, C}, where si (i > 0) describes an interval in the behavioural score range; s0 = D means that the borrower has defaulted, and C is the state when the borrower closed his loan or credit card account, having repaid everything (an absorbing state). The Markov property means that the dynamics of the behavioural score from time t onwards are conditional on the realization of the score state at time t — 1, Bt-1, or at least that its movement between the score range intervals depends only on which interval it is currently in. Given that the behavioural score is in state si, i = 1,..., n, at time t — 1, we write the latent variable Ut at time t as Uti. For the active accounts, Uti is defined so that the relationship between Bt and Uti is

Bt = Sj & pj < U < ^j+1, j = 0, 1,..., n

with p0 = pn+1 = to, (8)

where pj are the values in the unobservable creditworthiness which correspond to the end points of the behavioural score intervals si.Moreover, one chooses pi1 so that if the consumer defaults, one must have U't < pi1. The dynamics of the underlying variable Uti are assumed to be related to the explanatory variable vector xt—1 by a linear regression of the form Ult = —fi'xt—1 + s't, where $ is a column

vector of regression coefficients and s't are random error terms. If the s't are standard logistic distributions, then this is a cumulative logistic regression model, and the transition probabilities of Bt are given by

Prob (Bt = D|Bt_1 = Si) = logit (/ + ß'x—) , Prob (Bt = S1 |Bt-1 = Si) = logit (/2 + ß'xt-1)

- logic (/n + frx—), (9)

Prob (Bt = Sn\Bt—1 = Si) = 1 — logit p + P'iXt—1).

Estimating the cumulative logistic model using usual maximum likelihood means that, conditional on the realization of the time dependent covariate vector xt— 1, transitions to various states for different borrowers in the next time period are independent, both cross-sectionally and through time. Thus, the dynamics of the behavioural scores are driven by the explanatory variable xt— 1. In the model presented here, we assume three types of drivers: economic variables, the age of the loan and the previous behaviour of the score. We justify these choices in Sections 5-7 by looking at their effect on the simple first order Markov chain model. Note that states C and D are absorbing states, and thus there are no transitions from them; we will discuss the modeling of movements to the closed state, C, in Section 8

This has parallels with some of the corporate credit risk models. In credit metrics, for example (Gordy, 2000), the transitions in corporate ratings are given by changes in the underlying "asset" variables in a similar fashion, but with quite different drivers.

Since the behaviour scores are only calculated monthly, the calendar time t needs to be discrete; then, the credit-worthiness at time t of a borrower, whose creditworthi-ness at time t — 1 was in state i, is given by the latent variable U't, which satisfies the relationship

Ut = aikStatet—k — bi EcoVart—1

— ciMoBt—1 + s't (10)

where Statet—k is a vector of indicator variables denoting the borrower's state at time t — k, EcoVart—1 is a vector of economic variables at time t — 1, and MoBt—1 is a vector of indicator variables denoting the length of time (in months) the loan has been on the books (Months on Books) at time t — 1. One could smooth this latter effect by using a continuous variable of the age of the loan, but we instead describe the effect using more predictive binary variables for different age bands. a, b, and c are coefficients in the expression, and s't is a random variable representing a logit error term. Since Uti depends on i, the underlying creditworthiness at time t depends on the state at t — 1, and thus the behavioural score at time t will also depend on the state, and hence the behavioural score, at time t — 1. If aik = 0, then the creditworthiness at time t also depends on the state at time t — k, and thus the Markov chain model of the corresponding behavioural scores Bt will be of order k.

Table 1

First order average transition matrix.

Initial state Transition state

13-680 681-700 701-715 716-725 726-high Closed Default

13-680 49.0 22.1 9.6 4.0 4.0 4.7 6.7

(0.2) (0.2) (0.1) (0.1) (0.1) (0.1) (0.1)

681-700 15.7 34.7 25.1 9.6 11.2 2.8 0.8

(0.1) (0.2) (0.2) (0.1) (0.1) (0.1) (0.0)

701-715 6.0 13.6 35.9 18.1 23.4 2.6 0.5

(0.1) (0.1) (0.2) (0.1) (0.1) (0.1) (0.0)

716-725 3.0 6.1 15.7 28.3 44.1 2.5 0.3

(0.1) (0.1) (0.1) (0.2) (0.2) (0.1) (0.0)

726-high 0.7 1.2 2.7 4.3 88.4 2.4 0.2

(0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0)

The transitions also depend on economic variables and on the length of time the loan has been being repaid. Since the coefficients depend on i, the impact of these other factors will vary from state to state. If the score band intervals were ofequal length and the decomposition in Eq. (3) really held, then one would expect aik = 0, ci = 0, bi = b, and thus this model allows for more complex dynamics in the behavioural scores.

The Months on books term does not occur in any corporate credit model, but is of real importance in consumer lending (Breeden, 2007; Stepanova & Thomas, 2002). Similarly, it is rare to have higher order Markov chains models in corporate credit, although the state space is sometimes extended to include whether there have recently been upgrades or downgrades in the ratings. Thus, although corporate credit models may have more complex factors affecting their dynamics, such as the industry type, geographical area and seniority of the debt, they are not affected so much by recent changes of state or the age of the loan, which are important in consumer credit risk models.

4. Data description

The data set used for the case study in this paper contains records of the credit card 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 scores, along with the information on their time since account opened, time to default or time when the account was closed within the above period. We randomly selected approximately 50,000 borrowers to form a training data set which contained their histories over the period January 2001-December 2004. We tested our Markov models using the customer's performance during 2005 from a subsample of the 50,000 and also from a holdout sample of approximately 15,000 customers. Anyone who became 90 days delinquent (even if this was subsequently cured), was charged off, or was declared bankrupt, is considered as having defaulted.

The bank reported that there were no major changes in credit limit setting or minimum repayment levels during the period under consideration, nor were there any changes to the scorecard or intentional attempts to change the mix of the portfolio of borrowers through portfolio acquisition or marketing campaigns. To analyse

the changes in the distribution of behavioural score, we first coarsely divide the behavioural scores into various segments. Initially, we segment the behavioural score into deciles of the distribution of the score among all of the borrowers in the sample over all of the months in the sample. We use the chi-square statistic to decide whether to combine adjacent deciles if their transition probabilities are sufficiently similar. This technique of coarse classifying is standard in scorecard building (Thomas, 2009a) for dealing with continuous variables where the relationship with default is nonlinear. In this case, it led to a reduction to five scorebands, namely s1 = {13-680}, s2 = {681700}, s3 = {701-715}, s4 = {716-725} and s5 = {726 and above}. In addition to these five states, there are two more special states corresponding to Default and Account Closed. If there are too many states in the chain, the parameter estimates lose robustness, while if there are too few one loses structure and does not have enough segments to validate the model according to the Basel Accord requirements.

Behavioural scores are generated or updated every month for each individual, so it would be possible to estimate a 1-month time step transition matrix. Since transitions between some states will have very few 1 month transitions, such a model may lead to less than robust estimates of the parameters. Hence, we use 3-month time steps. Longer time steps, say six or twelve months, make it harder to include the impact of the changes in economics and the months on books effect. In the following sections we will justify the use of higher order Markov chains and provide an analysis of the effects of the time varying macroeconomic and months on books covariates on behavioural score transitions.

5. Order of the transition matrix

We first estimate the average transition matrix, assuming that the Markov chain is stationary and first order, using the whole duration of the sample from January 2001 to December 2004. Table 1 shows the 3-month time step transition matrix for that sample, where the figures in brackets are the standard sampling errors. As one might expect, once a borrower is in the least risky state (s5), there is a high probability, 88%, that they will stay there in the next quarter. More surprisingly, the state with the next highest probability of the borrower staying there is s1, the riskiest behavioural score state, while the borrowers in the

Table 2

Second order average transition matrix.

(Previous state, current state) Terminal state

13-680 681-700 701-715 716-725 726-high Closed Default

(13-680, 13-680) 58.0 19.2 6.9 2.3 1.6 5.0 7.0

(681-700, 13-680) 42.2 27.8 12.2 4.2 3.2 3.8 6.6

(701-715, 13-680) 36.7 28.3 13.0 6.5 5.2 4.2 6.1

(716-725, 13-680) 34.7 23.8 15.4 8.4 7.0 3.8 6.9

(726-high, 13-680) 22.8 18.9 16.0 9.5 19.9 5.2 7.7

(13-680, 681-700) 24.5 36.7 21.3 7.0 6.6 3.1 0.8

(681-700, 681-700) 14.0 40.4 25.7 8.2 7.9 3.1 0.7

(701-715, 681-700) 12.4 34.4 29.4 10.1 10.3 2.7 0.7

(716-725, 681-700) 13.8 27.7 26.8 12.9 15.5 2.5 0.8

(726-high, 681-700) 9.3 20.9 23.0 15.0 28.5 2.4 1.0

(13-680, 701-715) 14.2 19.0 28.2 17.6 17.0 3.6 0.5

(681-700, 701-715) 7.6 19.8 36.6 15.8 17.1 2.5 0.6

(701-715, 701-715) 4.7 12.2 45.7 17.7 16.7 2.6 0.4

(716-725, 701-715) 4.2 11.0 36.6 22.5 22.6 2.6 0.5

(726-high, 701-715) 4.3 8.9 24.1 18.3 41.3 2.6 0.6

(13-680,716-725) 9.9 11.8 16.7 20.9 37.1 3.2 0.6

(681-700,716-725) 4.9 11.3 19.8 22.6 37.7 3.4 0.2

(701-715,716-725) 3.0 7.5 21.6 28.9 36.0 2.7 0.3

(716-725,716-725) 2.4 4.5 15.5 42.1 32.9 2.4 0.3

(726-high, 716-725) 1.8 4.1 12.3 23.6 55.4 2.5 0.3

(13-680, 726-high) 5.5 5.6 7.9 8.5 69.3 3.1 0.2

(681-700, 726-high) 3.1 6.4 10.2 12.1 64.7 3.2 0.3

(701-715, 726-high) 2.1 4.1 9.6 12.2 68.8 2.9 0.3

(716-725, 726-high) 1.5 3.0 6.6 12.1 73.8 2.8 0.2

(726-high, 726-high) 0.5 0.8 2.0 3.4 90.7 2.4 0.2

other states move around more. The probabilities of defaulting in the next quarter are monotonic, with, as one would expect, 13-680 being the most risky state, with a default probability of 6.7%, and 726-high the least risky state, with a default probability of 0.2%. Note that there is obvious stochastic dominance Q2j>k Pj < 52j>k P+j) for all of the active states, which shows that the behavioural score correctly reflects future score changes, as well as future defaults.

This first order Markov chain model assumes that the current state has all of the information needed in order to estimate the probability of the transitions next quarter, and thus these are not affected by the borrower's previous states. If this is not true, one should use a second or higher order Markov chain model. This might seem surprising, in that a behavioural score is considered to be a sufficient statistic of the credit risk. However, this is a very specific credit risk — the chance of default in the next 12 months — whereas the Markov chain describes the dynamics of the credit risk estimates over a different 12 month interval each period. Thus, it is quite possible that the score does not include all of the information needed to estimate how this risk is likely to change. Table 2 displays the estimates of the transition matrix for such a second order chain, obtained in a similar way as Table 1. Analysing Table 2 shows that there are substantial changes in the transition probabilities based on the previous state of the borrower. Consider, for example, if the current state is the risky one s1 = {13-680}. If the borrowers were also in the risky state last quarter, then their chance of either staying in it or defaulting in the next quarter is 58% + 7% = 65%; if they were in the least risky state in the last quarter {726+} but are now in s1 , the chance of being in s1 or defaulting next quarter is 22.8% + 7.7% = 30.5%.

Thus, there is a propensity to reverse direction and return in the direction one came. This effect is seen in all five of the behavioural score interval states in the model. These results do not support the "momentum" idea, that borrowers whose score has dropped are more likely to drop further (see Bangia et al., 2002, for examples in corporate credit), but suggests there may be an event of very short duration which appears and then is reversed in the next quarter, such as being put in arrears due to some misunderstanding. This effect, seen in all five states, could be due to using score bands rather than the scores themselves, and thus the previous score band might suggest where in the interval the score is. However, the same result was seen when a finer classification, i.e. more states with smaller intervals, was used. One could investigate whether higher order models are even more appropriate, but for third and higher order Markov chains, data sparsity and robustness of predictions become problems, and so we use a second order chain to model the dynamics of the behavioural scores.

6. Macroeconomic variables

Traditionally, behavioural score models are built on customers' performances with the bank over the previous twelve months, using characteristics like the average account balance, number of times in arrears and current credit bureau information. Thus, the behavioural score can be considered as capturing the borrower's specific risk. However, it was shown for corporate credit risk models (Das, Duffie, Kapadia, & Saita, 2007) that although the borrower-specific risk is a major factor, systemic risk factors emerge during economic slowdowns, and have had a substantial effect on the default risk in a portfolio of loans.

Table 3

Correlation matrix of macroeconomic factors.

Interest rate

% change in CPI

% change in GDP

% change in net lending

Unemployment rate

Return on FTSE 100

Interest rate % change in CPI % change in GDP % change in net lending

Unemployment rate Return on FTSE 100

-0.51 0.34

0.01 0.39

-0.11 -0.23

-0.11 1

-0.71 0.87

0.14 -0.23 0.85 1

-0.49 0.70

0.01 -0.45 -0.71 -0.49

0.39 -0.09 0.87 0.70

Int Rate CPI

FTSE 100 * Net Lending 5.2 ^GDP ■ logOdds

Fig. 1. 3-month observed log(default odds) and macroeconomic variables.

The decomposition of the behavioural score in Eq. (3) suggests that this is also the case in consumer lending, since the population log odds spop(t) must be affected by such systemic changes in the economic environment. The question is, which economic variables affect the default risk of consumers? We investigate five variables which have been suggested in consumer finance as being important (Liu & Xu, 2003; Tang, Thomas, Thomas, & Bozzetto, 2007), together with one variable which reflects market conditions in consumer lending. The variables considered are:

(a) Percentage change in the consumer price index over 12 months: reflects the inflation felt by customers; high levels may cause an increase in the customer default rate.

(b) Monthly average sterling inter-bank lending rate: higher values correspond to a general tightness in the economy, as well as increases in debt service payments.

(c) Annual return on FTSE 100: gives the yield from the stock market and reflects the buoyancy of industry.

(d) Percentage change in GDP compared with the equivalent quarter in the previous year.

(e) UK unemployment rate.

(f) Percentage change in net lending over 12 months: this gives an indication of the funds being made available for consumer lending.

There is a general perception (Figlewski, Frydman, & Liang, 2007) that changes in economic conditions do not have an instantaneous effect on the default rate. To allow for this, we use lagged values of the macroeconomic covariates in the form of a weighted average over a six month period, with an exponentially declining weight of 0.88.

This choice is motivated by the recent study by Figlewski et al. (2007). Since macroeconomic variables represent the general health of the economy, they are expected to show some degree of correlation. Table 3 shows the pairwise correlation matrix for the six macroeconomic variables above, with no lags considered. The entries in bold are the correlations which are considered to be statistically significant at the 5% level. Thus, at the 5% significance level, the interest rate is negatively correlated with the percentage change in CPI and positively correlated with the percentage change in the GDP and the return on the FTSE 100. Similarly, the percentage change in net lending is negatively correlated with the unemployment rate and positively correlated with the percentage change in GDP and return on the FTSE 100 at the 5% significance level. The presence of a non-zero correlation between the variables does not invalidate the model, but the degree of association between the explanatory variables can affect the parameter estimation. Moreover, the variables used are chosen in order to avoid long run trends, and the fact that three of the variables are percentage changes is akin to already taking differences in order to avoid non-stationarity.

Fig. 1 shows the variation of the observed log(Default Odds) over 3-month windows, compared with the lagged macroeconomic factor values used in the analysis for the sample duration of January 2001 to December 2004. The macroeconomic factor values are represented by the primary y-axis, and the log(Default Odds) by the secondary y-axis.

We plot the lagged economic values for each month, though of course we only use the values every quarter in the Markov chain model, since it is quarterly. In the benign

Table 4

Comparison of transition matrices at different calendar times. Initial state Terminal state

13-680 681-700 701-715 716-725 726- Closed Default Number in state

January-December 2001

13-680 52.90 21.77

681-700 17.80 35.56

701-715 8.74 14.84

716-725 3.28 6.99

726- 0.72 1.35

October 03-September 04

13-680 46.24 22.68

681-700 14.79 35.62

701-715 5.42 13.42

716-725 2.68 5.63

726- 0.62 1.14

9.24 3.62 3.67

23.86 9.51 10.40

35.25 17.90 22.72

16.84 27.85 42.64

2.86 4.30 88.39

9.30 4.03 4.18

23.25 9.80 10.99

37.30 18.20 22.89

16.17 29.34 43.79

2.65 4.69 88.80

3.31 5.50 24,015

2.14 0.72 25,235

2.16 0.40 31,477

2.12 0.29 27,781

2.10 0.28 220,981

5.35 8.22 24,060

2.74 0.82 25,235

2.33 0.43 42,200

2.05 0.33 38,932

1.90 0.19 289,814

Table 5

Comparison of transition matrices for loans of different ages. Initial state Terminal state

13-680 681-700 701-715 716-725 726- Closed Default Number in state

1-12 months (new obligors)

13-680 51.0 22.3 8.1 3.1 2.0 5.8 7.6 24,858

681-700 18.2 35.6 24.2 9.3 8.7 3.2 0.8 22,019

701-715 8.1 15.9 30.5 17.8 25.6 2.7 0.5 21,059

716-725 4.5 8.2 14.7 21.4 48.6 2.2 0.3 18,050

726- 1.8 3.0 5.7 7.6 79.3 2.3 0.2 59,767

49-high (mature obligors)

13-680 44.1 23.5 11.3 4.9 7.0 4.0 5.3 28,604

681-700 13.6 32.5 25.6 10.7 14.4 2.5 0.6 39,835

701-715 4.7 11.8 37.2 18.8 24.8 2.5 0.3 66,389

716-725 2.1 5.0 14.9 30.4 44.7 2.6 0.3 67,660

726- 0.4 0.9 2.1 3.7 90.4 2.4 0.2 698,782

environment of 2001-04 there are no large swings in any variable, and the log of the default odds — spop(t) — is quite stable.

To convince ourselves that changes in economic conditions do affect the transition matrix, we look at transition matrices based on data from two different time periods, which have slightly different economic conditions. In order not to complicate matters, we show the differences that occur even in the first order Markov chain. In Table 4, we estimate the first order transition probability matrices for two different twelvemonth periods between January 2001 and December 2004, in order to determine the effect of calendar time on transition probabilities. The first matrix is based on a sample of customers who were on the books during the period January-December 2001, and uses their transitions each quarter during that period, while the second is based on those in the portfolio during the period September 03-October 04 and their performance over that period. Both transition matrices are quite similar to the whole sample average transition matrix in Table 1, with the probability of moving into default decreasing as the behavioural score increases and the stochastic dominance effect still holds. However, there are some significant differences between the transition probabilities of the two matrices in Table 4. For example, borrowers who were in a current state of s1 = {13-680} during the period January-December 2001 have a lower probability of defaulting in the next quarter — 5.5% —

than those who were in the same state during the period September 03-October 04, where the value is 8.22%. We test the differences between the corresponding transition probabilities in the two matrices in Table 4 using the two-proportion z-test with unequal variances. The entries in bold in Table 4 identify the transition probabilities where the differences between the corresponding terms in the two matrices are significant at the 5% level. Note that there are 35 transition probabilities being compared, and thus one might expect 2 significant comparisons at the 5% level if there were really no difference. In actual fact, there are 20 significant differences, which suggests that this calendar effect is real.

7. Months on books effects

As is well known in consumer credit modeling (Bree-den, 2007; Stepanova & Thomas, 2002), the age of the loan (number of months since the account was opened) is an important factor in the default risk. To investigate this, we split the age into seven segments, namely 0-6 months, 7-12 months, 13-18 months, 19-24 months, 25-36 months, 37-48 months, and more than 48 months. The effect of age on behavioural score transition probabilities can be seen in Table 5, which shows the first order probability transition matrices for borrowers who had been on the books for between one and twelve months (upper section) or more than 48 months (lower section). Again, the overall structure is similar to that of Table 1,

Table 6

Parameters for a second order Markov chain with age and economic variables.

Parameter estimates

Initial behavioural score

13-680 Std error 681-700 Std error 701-715 Std error 716-725 Std error 726-high Std error

Interest rate 0.0334 (0.0161) 0.092 (0.0143) 0.0764 (0.0123) 0.0834 (0.0134) 0.0778 (0.00885)

Net lending 0.0129 (0.00489)

Months on books

0-6 -0.027 (0.0351) 0.0161 (0.0347) -0.2182 (0.0368) -0.1637 (0.0448) -0.0849 (0.0315)

7-12 0.2019 (0.0241) 0.1247 (0.0225) 0.2051 (0.0226) 0.2317 (0.0261) 0.3482 (0.018)

13-18 0.2626 (0.0262) 0.2663 (0.0236) 0.2301 (0.0228) 0.2703 (0.0268) 0.2554 (0.0193)

19-24 -0.07 (0.0275) -0.0796 (0.0251) -0.1001 (0.0241) -0.0873 (0.0284) 0.031 (0.0206)

25-36 -0.0015 (0.0244) -0.0521 (0.0223) 0.00191 (0.0198) -0.00487 (0.0229) -0.0254 (0.0162)

37-48 -0.0703 (0.0262) -0.0519 (0.0243) 0.019 (0.0206) -0.0801 (0.0241) -0.00709 (0.0166)

49-high -0.2957 -0.2235 -0.13781 -0.16603 -0.51721

SecState

13-680 0.8372 (0.0165) 0.6762 (0.0168) 0.5145 (0.0222) 0.3547 (0.0337) 0.381 (0.0399)

681-700 0.2365 (0.0201) 0.2847 (0.0139) 0.3598 (0.0146) 0.1942 (0.0224) 0.5168 (0.024)

701-715 -0.0111 (0.0249) 0.0491 (0.0168) 0.1314 (0.0119) 0.1255 (0.0164) 0.2991 (0.0178)

716-725 -0.1647 (0.0345) -0.1764 (0.0239) -0.1795 (0.016) 0.0098 (0.0152) 0.0525 (0.0162)

726-high -0.8979 -0.8336 -0.8262 -0.6842 -1.2494

Intercept/barrier

Default -3.213 (0.0756) -5.4389 (0.0826) -5.8904 (0.1285) -6.011 (0.0967) -5.1834 (0.0506)

13-680 -0.2078 (0.0734) -2.179 (0.0657) -3.2684 (0.1175) -3.6011 (0.0648) -3.8213 (0.0436)

681-700 1.022 (0.0736) -0.3978 (0.0649) -1.9492 (0.1168) -2.461 (0.062) -2.9445 (0.0421)

701-715 1.9941 (0.0746) 0.861 (0.065) -0.1796 (0.1165) -1.2049 (0.0611) -2.06 (0.0415)

716-725 2.7666 (0.0764) 1.6267 (0.0656) 0.7317 0.171 (0.0609) -1.326 (0.0413)

Likelihood ratio p-value 3661.078 <0.0001 3379.459 <0.0001 4137.587 <0.0001 2838.765 <0.0001 20400.65 <0.0001

but there are significant differences between the transition probabilities of the two matrices. Borrowers who are new on the books are at greater risk of defaulting or of having their behavioural score drop than those who have been with the bank for more than four years.

Again, the final block of Table 5 gives the z-statistic, and the bold values indicate transitions where the differences between the new and mature accounts are statistically significant at the 5% level. This occurs in 27 of the 35 transitions calculated.

8. Modeling transition probabilities

Behavioural score segments have a natural ordering structure, with low behavioural scores being associated with high default risks, and vice versa. This is the structure that is exploited when using cumulative (ordered) logistic regression to model borrowers' transition probabilities, as suggested in Section 3 (McElvey &Zavoina, 1975).

The cumulative logistic regression model is appropriate for modelling the movement between the behavioural scorebands and the default state. If one also wished to model whether the borrowers close their accounts or not, one would need to use a two stage model. In the first stage, one would use logistic regression to estimate the probability of the borrower closing the account in the next quarter, given his current state, P(Close|beh.score band). The second stage would be the model presented here, showing the movement between the different scorebands, including default conditional on the borrower not closing the account. To arrive at the final transition probabilities, one would need to multiply the probabilities for each

transition obtained in this second stage by the chance that the account is not closed, as obtained from the first stage (1 - P(Close|beh.score band)). This approach assumes that the residuals of the estimations in the two stages are independent.

Thus, we now fit the cumulative logistic model in order to estimate the transition probabilities of the movement of a borrower's behavioural score, from being in state i at time t — 1 Bt-1 = si to where the borrower will be at time t, Bt. These transitions depend on the current state Bt—1 = si (since they are indexed by i), the previous state of the borrower, Bt—2, the lagged economic variables and the age of the loan (months on books, or MoB). Thus, one uses the model given by Eqs. (6) and (8), but restricted to the second order case, namely

Bt = sj ^ pj < u; < j, j = 0, 1,..., n

with =—то, pn+1 = (11)

U't = — a;Statet—2 — biEcoVart—1 — c;MoBt—1 + s't.

In order to choose which economic variables to include, we recall that Table 3 described the correlations between the variables. To reduce the effects of such correlations (so that the coefficients of the economic variables are understandable), we considered various subsets of the macroeconomic variables as predictors in a cumulative logistic model, where there was little correlation between the variables. In Table 6, we present parameter estimates for the cumulative logistic models for each behavioural score segment with only two macroeconomic variables, namely interest rates and net lending, along with months on books and the previous state. This means that we allow

Table 7

Distribution at the end of each time period for the out-of-sample test period (2005).

Behavioural 1-period

segments

2-period

3-period

4-period

Initial Average Model Observed Average Model Observed Average Model Observed Average Model Observe

distribution matrix predicted matrix predicted matrix predicted matrix predicted

13-680 571 520 560 457 498 561 384 475 566 424 457 573 368

681-700 659 659 696 595 635 702 594 612 711 604 592 719 592

701-715 1094 1011 1066 982 969 1065 918 935 1073 1007 908 1081 938

716-725 973 936 1027 952 902 1036 1038 878 1044 971 859 1049 943

726- 7436 7535 7304 7666 7589 7208 7644 7627 7098 7511 7647 6989 7612

Default 0 72 80 81 140 160 155 206 241 216 270 322 280

the drivers of the dynamics — economic variables and the current duration of the loan — to have different effects on the transitions from different states. The model with these two variables — interest rates and net lending — provided a better fit in terms of the likelihood ratio of the model than other combinations of macroeconomic variables; the next best fit was unemployment and interest rates. We employ stepwise selection, keeping only variables with a 5% significance level for the corresponding regression coefficient to be non-zero. The likelihood ratios and the associated p-values show that, for each current behavioural score segment, transitions to other states in the next time period are significantly influenced by current macroeconomic factors, the current months on books and information on the previous state, as represented by the Secstate variable in Table 6. This model fits the data better than the first order average transition matrix. A positive sign on the coefficient in the model is associated with a decrease in creditworthiness, and vice versa. Thus, the creditworthiness of borrowers will decrease in the next time period, given an increase in interest rates in all current behavioural score segments.

Borrowers who have been on the books for between 7 and 18 months have higher default and downgrading risks than the others. This confirms the market presumption that new borrowers have a higher default risk than older borrowers in any given time period, once they have had sufficient time (i.e. at least 3 months) to default. The coefficients of the Secstate variable, with one exception, decrease monotonically in value from the s1 = {13-680} category to the s5 = {726-high} state. Those with a lower behavioural score last quarter are more likely to have a lower behavioural score next quarter than those with the same current behavioural score but higher previous behavioural scores. Thus, the idea of credit risk continuing in the same direction is not supported.

9. Forecasting multi-period transition probabilities

The model with the parameters given in Table 6 was tested by forecasting the future distributions of the scorebands in the portfolio, including those who have defaulted. The forecast uses the Markov assumption, and thus multiplies the probability transition matrix by itself the appropriate number of times to obtain the forecasts. In the first case, we considered all non-defaulted borrowers in December 2004 and used the model to predict their

distribution over the various behavioral score bands and the default state at the end of each quarter of 2005, where closures were dealt with as described in Section 8. In order to avoid adding extra uncertainty to the forecast, the 2005 values of the two economic variables were used. The results are shown in Table 7. The initial distribution column gives the distribution of borrowers across each behavioural score segment in the test sample in December 2004. The observed column gives the observed distribution of borrowers at the end of each quarter of 2005. The other two columns give the expected numbers of borrowers in each segment at the end of each quarter of 2004, as predicted by the second order average transition matrix in Table 2 and the model in Table 6, respectively.

The second order Markov chain model with economic variables gave predictions, particularly for defaults, which were very close to the actual values for the first two quarters, but began to overestimate the risks thereafter. Thus, by the fourth quarter, the average second order Markov chain model which just takes the average of the transition probabilities is superior.

The analysis was repeated on an out-of-time and out-of-sample portfolio. Again, the distribution of the portfolio at the start of the period (April 2005) was given, and estimates for the next three quarters were obtained using the model in Table 6. The results in Table 8 show that the second order model with economic variables and the months on books effect (Table 6) is better at predicting the actual number of defaults than the second order model without these effects (Table 3), even though both approaches underpredict slightly. The model with the extra drivers is better at predicting the numbers in the default and high risk states, while the second order model which just averages over all transitions is better at predicting the numbers in the low risk categories. In this data set it appears that the second order effect is the most important, followed by the months on books effect. However, this could be due to the relative economic stability throughout the periods represented by both the development sample and the out-of-sample test period.

10. Conclusions

The paper has developed a pilot scheme for how one could use a Markov chain approach based on behavioural scores to estimate the credit risk of portfolios of consumer loans. This is an attractive approach, since behavioural

Table 8

Distribution at the end of each time period for the out-of-time out-of-sample test period (2005).

Behavioural score 1-period 2-period 3-period

segments

Initial Average Model Observed Average Model Observed Average Model Observed

distribution matrix predicted matrix predicted matrix predicted

13-680 1428 949 1040 1199 879 983 1080 769 889 1043

681-700 1278 1054 1117 1096 978 1061 1076 894 996 1001

701-715 1379 1291 1384 1257 1262 1393 1316 1216 1363 1219

716-725 876 1047 1178 812 1051 1228 774 1044 1234 718

726-high 7514 7994 7621 7968 8059 7535 7943 8208 7596 8074

Default 0 139 134 143 245 274 286 344 397 420

scores are calculated monthly by almost all lenders in consumer finance, both for internal decision purposes and for Basel Accord requirements. The paper emphasises that behavioural scores are dynamic, and since they do have a ''systemic'' factor — the population odds part of the score — the dynamics depend on changes in the economic conditions. The paper also suggests that one needs to consider carefully the appropriate order of the Markov chain. Table 2 shows the impact of the previous state and the current state on the subsequent transition, and strongly indicates the need for a second order Markov chain.

Unlike for corporate credit risk, one also needs to include the age of the loan in the modelling, as this affects the credit risk. The out-of-sample comparison of second order models with and without economic factors and age in the model is inconclusive about which model is better, but this was at a time when the economic conditions were very stable. In more volatile conditions, or if one wanted to use the model for stress testing, it would be essential to include the economic effects in the modelling.

Such models are relatively easy for banks to develop, since all of the information is readily available. The model would be useful for a number of purposes: debt provisioning estimation, stress testing in the Basel context, and investigating the relationship between point in time behaviour scores and through the cycle probabilities of default by running the model through an economic cycle. The model could also be used by ratings agencies to update their risk estimates of the securitized products based on consumer loan portfolios. This would require them to obtain regular updates of the behavioural scores of the underlying loans, rather than the present approach of only making one initial rating, based on an application or bureau score. This involves extra work, but might avoid the failure of the ratings of the mortgage backed securities (MBS) seen in 2007 and 2008, and would certainly give early warning of the increasing credit risk of such securities.

There are still issues to be resolved in regard to modelling the credit risk of consumer loan portfolios. One important issue is to identify which economic variables affect consumer credit risk most, and hence should be included in such models. One would expect some differences between such a list and the variables which have been recognised in corporate credit risk modelling, and one may even want to use different variables for different types of consumer lending. For example, house price movements will be important for mortgages but may

be less important for credit cards. One also feels that some of the variables in the models should reflect the market conditions as well as the economic conditions, because the tightening in consumer lending which prevented customers from refinancing did exacerbate the problems of 2007 and 2008. This paper has described how such information on economic and market conditions can be used in conjunction with behavioural scores to estimate portfolio-level consumer credit risks. It points out that although Markov chain models based on behavioural scores have been used by the industry, they have not previously appeared in the literature, and there has certainly been no extension of the model to include the maturity of the loan, the economic factors or the need for higher order Markov chains.

Acknowledgments

We are grateful to the EPSRC for providing funding under the Quantitative Financial Risk Management Centre to support MM. We are also grateful to two referees for their careful reading and helpful suggestions concerning the paper.

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Madhur Malik was a Research Fellow at the University of Southampton. He has recently moved to be a Senior Associate at the Financial Services Authority. He obtained his Ph.D. from the Indian Statistical Institute and has worked as a credit analyst for HSBC and the Lloyds Banking Group.

Lyn C. Thomas is Professor of Management Science at the University of Southampton. His interests are in applying operational research and statistical ideas in the area of finance, particularly in credit scoring and risk modelling in consumer lending. He is a founding member of the Credit Research Centre at the University of Edinburgh and one of the principal investigators for the Quantitative Financial Risk Management Centre based at Southampton. He has authored and co-authored four books in the area, including Consumer Credit models; Pricing, Profit and Portfolios and Credit Scoring and its Applications. He is a Fellow of the Royal Society of Edinburgh and a past President of the Operational Research Society, and was awarded the Beale Medal of that society in 2008.