STOCHASTIC

in general, a monotonic function of time. In behavioral terms, the optimal
sequential policy dictates that more of the asset is sold when the price drops
than when it rises. A recent paper by Bawa (1973) discusses the optimal minimax
regret policy for selling an indivisible asset (i.e., an asset which can only be
sold as a whole). This model and its solution are presented in Exercise CR-14.
IV. Cash Balance Management
The applications of stochastic dynamic programming in finance constitute
a large and growing literature. There are a number of important papers in this
literature which could not be reprinted in this book because of space limitations.
In particular, several papers dealing with the important problem of cash
balance management have necessarily been excluded. We have attempted
instead to cover the essential features of these works in the Exercises. The
exercises are limited primarily to mathematical questions and readers who are
especially interested in the cash balance problem are urged to consult the
original articles.
The problem of cash balance management is similar in many respects to
the problem of inventory management for a physical good. The firm retains
an inventory of cash to meet its daily requirements for money. The actual level
of cash on hand fluctuates due to deposits and withdrawals associated with
the firm's operations. The fluctuations in cash balance are, to a certain extent,
unpredictable. Thus uncertainty exists regarding future cash requirements. In
the literature on cash balance management, it is universally assumed that
deposits to and withdrawals from the cash balance are describable by a sequence
of iid random variables with known distribution (or else are known with
certainty).
Associated with the firm's cash balance are certain costs. First, there is an
opportunity cost for holding cash in a nonproductive capacity. Second, there
are penalty costs associated with delay in meeting requirements for cash. As
in any "inventory" problem, the specification of penalty costs in actual
situations might be far from simple, but it is surely true that the costs are real
nevertheless, and may have important effects on optimal operating policies.
Finally, there are costs associated with the deliberate altering of the cash
balance level, such as the costs of "paperwork," and brokerage fees. The
objective is assumed to be the minimization of discounted or average cost of
the firm's cash balance.
Although the cash balance problem, as outlined above, is formally quite
similar to the problem of inventory management, it also differs from the latter
in a number of important ways. First, since both withdrawals and deposits
occur, the instantaneous "demand" for cash can be negative as well as positive.
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