The generalized cash balance problem: optimization‐based one step ahead optimal control

Abstract

AbstractThis paper generalizes the Sethi–Thompson model of cash balance dynamics to the case of multiple current and investment accounts, with the cash flows or demands incident on the current accounts. Cash transfers can occur between any pair of current–current or current–investment accounts, with transaction costs proportional to the amount transferred, always deducted from current accounts. Performance indices are defined and maximization of an index for the next instant (day), by suitable choice of cash transfers that satisfy desired constraints on account balances, defines the basic optimal control problem referred to as one step ahead optimal control. It is shown that this one step ahead optimal control can be written as a mixed integer linear program of small size, leading to an implementable real time strategy for the generalized cash balance problem. In addition, for the simple case of a single cash and investment account, the proposed strategy is derived analytically and compared with a strategy of Miller–Orr type, called a three‐level strategy. Omniscient optimal control over a horizon is defined as the globally optimal control strategy, assuming the demands over the entire horizon to be known: although computable a posteriori, it is not implementable a priori, and only serves as a benchmark for comparison. The proposed one step ahead optimal or myopic approach makes no assumptions about the distribution of cash demands, and examples show that it attains performance indices which are superior to those attained by three‐level control strategies, and compare favorably with the omniscient benchmark performance indices.