Simple Optimal Policy for Cash Management: The Average Balance Requirement Case

This paper treats a problem of stochastic cash management under an average compensating-balance requirement. It develops a dynamic programming formulation of the problem in which the relevant state is a unidimensional quantity equivalent to the forecasted average balance at the end of the averaging period. Under usably broad conditions, it establishes the optimality of a transient policy of simple type, similar to the two-sided inventory type policy familiar from certain earlier studies of stationary cash balance problems having absolute balance requirements. The results apply to cases in which the transactions costs contain both fixed and proportional components. The paper discusses also a numerical example drawn from the literature of the cash balance problem and shows by simulation of the optimal (and simply modified forms of the optimal) policy, that good protection is afforded against negative balances, even though the model does not explicitly constrain the negative-balance probabilities.