If we simply include all the state variables, behavioural scores and credit limits, into (3.1), the size of the state space will be substantial. Therefore we divided these two variables into a number of separate groups or bands in order to ensure our model’s robustness.
This procedure is called coarse-classifying. A suitable classification is able to maximize difference from one group to next and minimize the difference within a group. As we are modelling the credit card accounts’ transition as a Markov chain, we can use the Chi- square test to examine whether the split is good enough. With a good split one can get a good approximation to the Markovian assumption. To check whether the Markov chain satisfies this assumption, for every state, we are interested in whether the hypothesis that the probability of moving from (lt, it) to it+1 is independent of the state at t−1, i.e.
(lt−1, it−1). Define nt(lt−1, it−1;lt, it;it+1) to be the number of times that a credit account
was in state (lt−1, it−1) at time t−1 followed by moving to (lt, it) at time t and it+1 at
timet+ 1. Similarly definent(lt, it;it+1) to be the number of times that a customer was in
state (lt, it) at timet and then moved to behaviour score it+1 at time t+ 1. If we assume
the chain is stationary, the estimator for p(it+1|lt−1, it−1, lt, it) is:
ˆ p(it+1|lt−1, it−1, lt, it) = T−2 P t=0 nt(lt−1, it−1;lt, it;it+1) T−2 P t=0 nt(lt−1, it−1;lt, it) (3.5)
The Markovity of the chain corresponds to the hypothesis that p(it+1|1,1, lt, it) =
p(it+1|2,1, lt, it) = . . . = p(it+1|L,1, lt, it) = p(it+1|1,2, lt, it) = . . . = p(it+1|L, I, lt, it)
, for lt, it, it+1. To check on the Markovity of state (lt, it), we use the chi-square test
(Anderson and Goodman, 1957). Let
χ2(l t,it)= X (lt−1,it−1) X it+1 n∗(lt−1, it−1;lt, it)[ˆp(it+1|lt−1, it−1, lt, it)−p(iˆ t+1|lt, it)]2 ˆ p(it+1|lt, it) (3.6)
where ˆ p(it+1|lt, it) = T−1 P t=1 nt(lt, it;it+1) T−1 P t=1 nt(lt, it) (3.7) and n∗(lt−1, it−1;lt, it) = T−1 X t=1 nt(lt−1, it−1;lt, it) (3.8)
Anderson and Goodman (1957) showed that if the chain is Markov (3.6) has a chi-square distribution with (I−1)(L−1)2 degree of freedom, whereL is the number of credit limit band and I is the number of behavioural score band.
A traditional approach is to start with a fine classification i.e. with more bands then one really wants and then check if one can combine adjacent bands. Alternatively, one can split the best split into two classes and then splitting one of these into two more until it is not worth splitting further.
In this study, we coarse-classified the behavioural score and credit limit simultaneously. That is we arbitrarily classified behavioural score into five categories. Then we coarse- classified the credit limit. After we found some improvement on the chi-square value (i.e. a split that generates a small chi-square value), we then stopped coarse-classify the credit limit. We then used the latest credit limit split and then classified the credit limit accordingly. Then, we start to coarse-classified the behavioural score. After obtaining a good behavioural score split, we then stopped coarse-classifying the behavioural score and then looked at the split of the credit limit again. This process repeated many times. Here we only reported a summary of the coarse-classify process for illustration.
Initially, our attempt was to classify the credit limit into five bins: £0/missing, £1 to £2000, £2001 to £4000, £4001 to £6000 and £6001+. The chi-square value of this segmentation was extremely large. We believe this is due to the grouping across consumers who have a significant difference in the financial and consumption behaviour. We thus
decided to further break down the credit limit to ten groups: £0/missing, £1 to £500,
£501 to 1000,£1001 to£1500,£1501 to£2500, £2501 to£3500,£3501 to£4500,£4501 to £5500, £5501 to £7500, £7501+. Upon generating the optimal policy by using this credit limit, we found an unusual pattern on the last two credit limit groups. Studying the transition probability and the profit value, shows that the behaviour of the last two groups are very close to each other, with the optimal values almost the same. Thus we decided to merge the last two credit limit groups to end up with the credit limit states defined in Table 3.1.
Index Credit limit (in £) Account Description 0 Closed Closed 1 1-500 Limit 1 2 501-1000 Limit 2 3 1001-1500 Limit 3 4 1501-2500 Limit 4 5 2501-3500 Limit 5 6 3501-4500 Limit 6 7 4501-5500 Limit 7 8 5501 or above Limit 8
Table 3.1: List of credit limit status
The monthly generated behavioural score from the lender’s internal system ranged from 200 to 780. Accounts with score lower than 365 were labeled as in risk. Apart from this, there is no standard rule on classifying behavioural score into discrete bins. We thus first counted the frequency of behavioural scores across four years, divided scores into ten categories, and then allocated every behavioural score record to one behavioural score category (so there are 50,797 ∗48 counts in the frequency table in total). As in finding bins for the credit limit, we monitored the improvement on the chi-square test. The performance with ten behavioural scores was poor even if we aggregated some of the behaviour categories. So we began with just 2 states - one that the account has a behavioural score and the other that it has no behavioural entry. Then we split the behavioural score into categories, in order to improve the model fit of the Markov chains.
We found this was best when the behavioural score was split into four bands. There were three non-behavioural score states -closed, inactiveandbad. The behavioural score status are presented in Table 3.2.
Index Behavioural score Account Description 0 - Closed
1 - Inactive
2 - Bad (bankruptcy or charge-off) 3 200-570 Risk Account
4 571-721 Score 1 5 722-742 Score 2 6 743-758 Score 3 7 759+ Score 4
Table 3.2: List of behavioural score status
We found using the second approach is more suitable for the behavioural score binning.