Since the advents of credit cards in the 1960s, lenders have used credit scoring, both application and behavioural scoring to monitor and control default risk. However in the last decade their objective has changed from minimising default rates to maximising profit. Lenders have recognized that operating decisions are crucial in determining how much profit is achieved from a card. This paper focuses on the most important decision in an operating policy: the management of credit limit. Soman and Cheema (2002) conducted a study on the use of credit limit policies in encouraging spending and found that the availability of additional credit does promote card usage in some consumers. Consumers assumed lenders have some sophisticated models, which was used to determine appropriate credit limits, but that is not the case in reality.
So how do lenders currently decide on what credit limit to offer a credit card customer? Most use subjective policies based on a risk/return matrix, i.e. they agree credit limits for each combination of risk band and average balance, which is considered a surrogate for the return to the lender from that customer. This approach is static in that it does not consider whether or how the customers default risk and profitability to the lender will change over time. Nor is there any model to guide what are the optimal credit limits to choose.
We therefore propose using Markov Decision Processes (MDP) to improve the credit limit decision. A MDP model provides a way of making sequential decision by considering the evolution of a customer’s behaviour over time. It also allows one to calculate the profitability of a credit card customer under the optimal dynamic credit limit policy. Lenders keep a wealth of historical credit card data, in particular the monthly values of a customer’s behavioural score which is their way of assessing the default risk of the customer in the next year. Building the Markov decision process model on behavioural scores has the advantage that most lenders have been keeping this data on customers for a number of years. With the advent of the Basel Accord in 2008, lenders are required to keep such data for five years and are encouraged to keep it through a whole economic cycle.
MDPs have been used in a number of different contexts (Heyman and Sobel, 1982; Ross, 1983; White, 1985, 1988, 1993; Kijma, 1997). The first application of MDPs in consumer credit was by Bierman and Hausman (1970) who looked at the repayment of a loan where no further borrowing was allowed. The model assumed the repayment of the customer followed a prior probability distribution. Using a Bayesian approach, the model revised the probability of repayment in the light of the collection history. Modifications of the basic model were made both in the accounting rules (Dirickx and Wakeman, 1976) and in the form of the Markov chain (Frydman et al., 1985) followed. The use of MDP models to manage the characteristics of consumer lifetime value can be found in Trench et al. (2003) where MDP models were used with the objective of adjusting a consumer’s credit card limit or annual percentage rate (APR). The objectives in that paper are similar to the one in this paper. However their state space did not involve behavioural scores nor were they concerned with the problems that occur in estimating the transition probabilities if there are low default rates. Instead they used a six dimensional state space each dimension having only two or three categories describing the recency and frequency of purchases and payments. They developed mechanisms for reducing the size of the transition matrix through merging states. Ching et al. (2004) used MDPs to manage the customer lifetime value generated from telecommunication customers. The state space in that study again used marketing measures not risk measures and the decision was whether to implement promotions.
This paper is the first to use behavioural score bands as the basis for MDP models. The advantage of basing the model on behavioural score are considerable. Almost all lenders calculate such scores every month for every individual both as a basis for their Basel Accord probability of default calculations, and as a way of segmenting the population on risk see our previous discussion on risk/reward matrices.
When modelling real problems using Markov Decision Processes, the curse of dimensionality (Puterman, 1994) can mean the state space is very large and that one would need a large amount of data to obtain robust estimates of the transition probabilities. Using behavioural scores helps to overcomes this first difficulty because it itself is a ”sufficient statistic” of the risk of the account and already contains information from a number of different characteristics. Also by aggregating states one can obtain a simple but meaningful state space. In our case we make part of each state an interval of behavioural scores and similarly combine possible credit limits into bands, to make up the other part of the state. We also take each of these credit limit bands to be one of the possible actions that can be chosen.
The difficulty with the quantity of data needed to calculate robust estimators of the transition probabilities is less severe in the consumer credit context because of the size of the data sets available to lenders. The only problem is that with some portfolios of loans, the number of movements directly into default from some states is so low (quite possibly zero) that the resultant estimates of zero transition probability of default may affect the structure of the Markov chain, making it non-robust. This problem of estimating default probabilities in low default portfolios also occurs in the Basel Accord mentioned earlier. We therefore use an approach suggested in that context by Pluto and Tasche (2006) and extended by Benjamin et al. (2006) which ensures the resulting Markov chain model is robust and conservative. The conservativeness is reasonable as one would prefer the model to underestimate rather than over estimate the profitability of a credit card account.
The main contribution of this paper is to show how one can use Markov decision process models based on states consisting of behavioural score bands scores which most lenders calculate on a monthly basis to determine optimal credit limit policies in terms of profitability. The rest of the paper is organized as follows: Section 2 describes the MDP model formulation. Section 3 discusses the estimation of the transition probabilities including the probabilities of defaulting immediately. Section 4 presents the practical issues in applying the MDP model to the real credit card data and the results of the case study. The final section draws some conclusion on the model and the resultant case study.
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