Calculating optimal limits for transacting credit card customers

We present a model of credit card profitability, assuming that the card-holder always pays the full outstanding balance. The motivation for the model is to calculate an optimal credit limit, which requires an expression for the expected outstanding balance. We derive its Laplace transform, assuming that purchases are made according to a marked point process and that there is a simplified balance control policy in place to prevent the credit limit being exceeded. We calculate optimal limits for a compound Poisson process example and show that the optimal limit scales with the distribution of the purchasing process and that the probability of exceeding the optimal limit remains constant. We establish a connection with the classic newsvendor model and use this to calculate bounds on the optimal limit for a more complicated balance control policy. Finally, we apply our model to real credit card purchase data.
Comment: 17 pages. Submitted to the Journal of the Operational Research Society on 20th May 2015. This version updated with minor corrections: corrected typo on pg. 3; fixed incorrect index in equation 1; added a definition for the profit function on pg. 3