A mixture model for credit card exposure at default using the GAMLSS framework
Introduction
The Basel regulatory accords have set out risk-sensitive regulatory capital requirements stipulating the minimum level of capital that banks must hold as a function of various types of risk. Under the advanced internal ratings-based (A-IRB) approach, authorised banks are permitted to use their own methods to calculate three parameters that are central to one such source of risk—credit risk. These are: probability of default (PD), loss given default (LGD), and exposure at default (EAD). In retail credit risk, PD and LGD have thus far received the bulk of attention by credit risk researchers, whilst EAD has been studied far less extensively. This paper is motivated by this fact and aims to close the gap by focusing on EAD modelling.
EAD is defined as the outstanding debt at the time of default and measures the potential loss the bank would face in the absence of any further repayments. The A-IRB approach requires producing suitable EAD estimates for all loans that are not yet in default. For some types of loans, those estimates can be relatively straightforward; for example, the EAD for term loans such as residential mortgages and personal loans could be inferred simply from the current exposure amount plus potential subsequent interest and fees (Witzany, 2011). In contrast, for revolving retail exposures such as credit cards and overdrafts, the estimation is more complex, as customers are allowed to draw up to a specified limit and can repay any amount at any time (as long as the minimum level is met). As a result, each borrower’s account balance may change substantially in the run-up to default, and using the current balance may severely underestimate the true exposure risk. For these types of credit, the Basel Accords have suggested estimating a credit conversion factor (CCF), which is usually defined as the proportion of the undrawn amount (i.e.credit limit minus drawn amount) that will be drawn by the time of default. This CCF should reflect the likelihood of additional drawings between estimation and default time. From the predicted CCF, the estimated EAD then follows as: Even though statistical methods to estimate the CCF have been proposed, several drawbacks were soon identified. For example, the CCF distribution is highly bimodal, estimates must be restricted to the [0,1] range, and models may struggle to cope with the contracting denominator when the current drawn amount is already close to the limit. Therefore, in the literature, alternative methods have been suggested to avoid the undesired properties of CCF models, which include predicting EAD directly (Tong et al., 2016).
In this paper, we focus on EAD modelling for credit cards, which has received limited attention in the literature. Most of the studies on EAD modelling have thus far focused on corporate credit, whilst fewer address retail customers (Gürtler et al., 2018). This is partly explained by the greater availability of public data on the corporate sector, and by the fact that the financial status and health of corporate customers can be inspected from share and market-traded products (Leow & Mues, 2012), enabling easier access to data. Nonetheless, credit cards make up the largest share of revolving retail credit for most A-IRB banks and contribute the largest number of defaults compared to other revolving line products (Qi, 2009). This should contribute sufficient information about the characteristics of defaulted accounts to enable statistical modelling.
To avoid the problems associated with CCF estimation, we choose the EAD amount itself as the response variable. This choice, however, poses other challenges. For example, the observed value range of realised EAD levels could be very wide and thus difficult to capture statistically (Yang & Tkachenko, 2012). To cope with its right-skewness, Tong et al. (2016) therefore proposed a gamma distribution for (non-zero) EAD and built a direct EAD model under the generalised additive model for location, scale, and shape (GAMLSS) framework (Stasinopoulos et al., 2017), which was shown to outperform several benchmark models (including for CCF) on a dataset from a UK lender. In this paper, we take a similar approach, but we further extend it by distinguishing between two subgroups of credit card borrowers—those whose balance hit the limit at least once in the run-up to default, versus those who never maxed out their card over that same outcome period—introducing two mixture components to our models. The rationale for doing so is that we hypothesise that not just the EAD but also its risk drivers (and that of its dispersion) could differ substantially between the two groups. A similar mixture element was previously proposed by Leow and Crook (2016), along with their panel models for card balance (and limit), but besides us using a different modelling framework applied to (cross-sectional) default cohort data, our approach differs from theirs in that we allow for non-parametric terms, and we do not assume that the balance of maxed-out accounts has to match the limit value exactly.
To empirically validate the effectiveness of the GAMLSS model (versus OLS), the proposed mixture approach, and its combined application, we construct a set of benchmark models against which we compare the predictive performance of our newly proposed model. All models are fitted using a large dataset of credit card defaults from a large Asian lender, which has not been previously used in the EAD literature.
To summarise, the contributions of our new model and analysis are that we: (1) estimate EAD directly, instead of using the conventional CCF approach; (2) analyse EAD in the hitherto under-researched area of retail credit cards; (3) apply the idea of EAD mixture models under the GAMLSS framework and compare its performance to a series of benchmark models; (4) identify the factors that significantly impact the mean and dispersion of EAD, giving further insights into the risk drivers of EAD; and (5) inspect any differences in the risk drivers, depending on whether the account hits the limit prior to default.
The paper is structured as follows. In Section 2, the existing literature on EAD modelling is reviewed. Section 3 explains the data and variables used, and Section 4 illustrates how statistical models are constructed. The results are presented and discussed in Section 5. Section 6 concludes.
Section snippets
Literature review
In order to model the EAD of revolving exposures, the Basel II and III Accords have implicitly suggested estimating a credit conversion factor (CCF), which is the proportion of the undrawn amount at the time of estimation (i.e.credit limit minus current balance) that will be drawn by the time of default: where denotes the amount of money owed by credit card borrowers at the present time (), is the credit limit or maximum amount that the
Data and variables
The original dataset provides monthly account-level data on the consumer credit cards of a large Asian bank from January 2002 to May 2007. We define EAD as the outstanding balance at default time, taking the amount owed by the borrower, excluding any subsequent interests and additional fees; any debt incurred after default is not included in the EAD calculation. We say that an account goes into the default state when a borrower: (1) misses or cannot make the minimum repayment amount required by
Statistical models
The following subsections outline our newly proposed model, GAMLSS.Mix, and three benchmark models, GAMLSS, OLS.Mix, and OLS.
Results and discussion
In this section, we present the results of our newly proposed model and the performance comparisons with the benchmark models. In addition, we inspect the significant relationships between explanatory variables and response parameters.
Conclusions and future research
Exposure at default (EAD) is one of the key parameters used to calculate the regulatory capital requirements under the advanced internal ratings-based (A-IRB) approach. To estimate EAD, credit conversion factor (CCF) models were implicitly suggested by the Basel Accords and have been studied in the literature, but several drawbacks of such models can prove problematic. In this paper, we therefore mainly focused on estimating EAD via a direct model, rather than applying CCF or other related
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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