  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.
