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This paper uses numerical methods to model the demand for excess reserves by a representative bank in a framework that includes many realistic features of the current reserve market structure in the United States. In particular, the model incorporates a 14-day maintenance period and an accurate representation of carryover provisions. We use the model to evaluate the effect of various changes to the operating environment (increased uncertainty, modified penalties) and changes to policy (paying interest on reserves).

The demands for reserve balances and excess reserves figure in a number of issues in monetary economics. For example, the current operational strategy of the Federal Reserve is to target the federal funds rate. To do this, the open market desk attempts to supply the amount of reserves demanded by banks, less the expected borrowing from the discount window, at the targeted interest rate. This policy of targeting the interest rate on a day-to-day basis requires high frequency estimates of the demand for reserve balances in order to guide the actions of the desk. This is done daily by the staff at the Board of Governors of the Federal Reserve and the New York Federal Reserve Bank. In this framework, understanding the forces affecting the daily demand for reserves and forecasting this demand becomes central to the implementation of monetary policy. The demand for excess reserves is playing a particularly important role in monetary policy now given the ongoing shift away from reserve requirements around the world. Several countries have already moved to monetary systems without reserve requirements so that their demand for reserves is entirely a demand for ‘excess’ reserves (Borio, 1997). The United States is also moving in that direction due to the adoption of ‘retail sweep programs’ by commercial banks. These programs, which transfer deposits from reserveable transactions accounts to non-reservable ‘savings’ accounts, have resulted in significantly lower levels of required reserves, leaving some banks in the position of not needing to hold reserve balances to meet their reserve requirements. Because of this, the demand for reserves has become more sensitive to levels of payment flows and the risk of overdraft penalties. Understanding how this demand behaves is becoming an important issue in applied monetary economics. Because of the importance of understanding the demand for reserves, bank reserve management strategy has been the subject of a significant body of research. Following Poole (1968), the literature on reserve demand has generally focussed on a representative bank's precautionary motive for holding reserves. In such models, the prototypical bank faces uncertainty about its end-of-day reserve position at the Federal Reserve and must choose a targeted level of reserves that balances the costs of holding non-interest bearing excess reserves against the costs associated with not meeting reserve requirements. Poole developed the basic single period version of this reserve demand that lies at the heart of most of the subsequent work. He also extended his model to a multi-day maintenance period, but for convenience abstracted from daily overdraft penalties along with carryover provisions (which did not apply at the time). Several other papers have extended this optimizing model of reserve demand, although they all abstract from important aspects of the reserve structure. Clouse and Dow (1999) examine optimal reserve demand in a two period maintenance period with heterogeneous banks and both fixed and variable costs, but without carryover options. Longworth (1989) examines optimal reserve demand in a multi-day maintenance period following the Canadian system but does not include carryover provisions (which do not apply in Canada) and used restrictive assumptions (in particular, uniform distributions) in order to come up with an approximate solution. Most recently, Furfine (1998) uses this approach to examine the effect of variability of reserves on reserve demand over the maintenance period. He includes all days of the period but simplifies the reserve demand problem by abstracting from carryover, differential overdraft penalties and assumes certainty equivalence in a key Euler equation, which allows the model to be more easily estimated. His estimates suggest that increased variability will result in higher reserve demand. While yielding many valuable insights, these analyses have nonetheless ignored certain aspects of the reserve market that are quite important to bank reserve managers in practice. For example, reserve managers have to be concerned with possible penalties for account overdrafts, with the rather complicated set of rules (to be discussed in Section 2) allowing for carryover of reserve surpluses and deficiencies from one maintenance period to the next, with penalties for reserve requirement deficiencies, and with complications in reserve accounting associated with weekends and holidays. Applied papers, which need to incorporate these features, have abandoned dynamic optimization and have generally suggested plausible ‘rules of thumb’ which can be compared with empirical evidence, for example, Spindt and Hoffmeister (1988), Spindt and Tarhan (1984) and Griffith and Winters (1995). In this paper we develop a detailed optimizing model of the demand for reserves that includes a full 14-day maintenance period along with accurate treatments of weekend accounting and carryover rules. In order to determine optimal reserve management policies we use numerical methods, specifically dynamic programming, to solve the decision problem of a representative bank (discussed in Section 3). The use of numerical methods to calculate a solution to a very general model of reserve demand allows us to extend previous results and to investigate phenomena that are difficult to address by other means. One of the principle interests of the open market desk is the pattern of reserve demand over the days of the maintenance period, which can be inferred either directly through the level of demand or indirectly through the behavior of the federal funds rate (which is determined by the demand and supply of reserves). Four papers that look at interest rates to provide empirical evidence on this demand are Hamilton (1996) which examines the average levels of the federal funds rate over the period, Furfine (1998), Spindt and Hoffmeister (1988) and Griffith and Winters (1995) which look at variability of the federal funds rate across the maintenance period. While there are differences in exact patterns found, there is much common ground suggesting lower reserve demand on Fridays and higher demand on the last days of the maintenance period, particularly on the last day. All of these features are produced by our model and can be explained as an optimal response to the current institutional structure. In addition, we are able to show how this pattern is affected by changes in the level of reserves carried in by banks, and the level of uncertainty they face. How banks respond to carryover provisions is another concern for the Federal Reserve's open market desk. The carryover provisions allow banks to apply some of the ‘excess’ reserves maintained in the current period to meet reserve requirements in the subsequent period, or alternatively, to make up a current reserve deficiency next period. While carryover is commonly used as a buffer against expected changes in a bank's reserves, it also acts as a ‘loan’ to banks which they can take advantage of by alternating between targeting positive and negative carryover. Spindt and Tarhan (1984) discuss how banks would plausibly treat carryover and provide estimates that show that carryover behavior does depend on expectations of future costs, particularly changes in interest rates. Friedman and Roberts (1983) construct a simple model of the carryover decision and provide empirical evidence of an alternating carryover target. However, they treat the value of reserves carried over to the next maintenance period as the interest cost of reserves next period, abstracting from the effect of uncertainty and the fact that whether carry-out is positive or negative changes the state the bank is in next period. Section 5 of this paper investigates optimal carryover strategies. This is one place where numerical methods are essential; the rules covering carryover are quite complex and not easily handled by analytical approaches. It also forces a dynamic approach; the value of carry-out today depends on reserves held in the following period, which will depend on holdings in the subsequent period, and so on. Our use of dynamic programming allows us to solve this infinite horizon problem. We find that it is indeed optimal for banks to adjust their reserve targeting strategy because of carryover provisions, and generally to alternate between positive and negative positions. However, we also show that uncertainty can substantially limit their willingness to take advantage of carryover provisions, which may affect the dynamic pattern of reserve demand. The effect of carryover provisions on the intra-maintenance period pattern of reserve demand, though its interaction with overdraft risks, is also examined (in Section 4). Our model can be also be used to examine the influence of various possible structural changes to the reserve market on reserve demand. For example, sweep programs (discussed in Section 4) have resulted in a substantial decline in required reserve balances in the last few years, which might have an effect on the pattern of reserves held over the maintenance period. We find that it is indeed optimal for banks to adjust their timing of reserves holdings in response to lower average required balances. Apart from shedding light on optimal reserve management policies given the current reserve market structure, the general framework we develop is quite flexible and can be readily applied to study reserve demand in alternative reserve market settings. For example, Congress and the Administration have recently considered proposals that would allow the Federal Reserve to pay interest on required reserve balances or on excess reserves. In Section 6, we report how the model predicts optimal reserve management policies should change in response if such policies were adopted.

This paper has examined the daily demand for reserves in an optimizing model of bank reserve management that takes into account the full 14-day character of maintenance periods, weekend effects, and carryover provisions. The previous literature has developed a model of the precautionary demand for reserves that has emphasized the tradeoff between the risk of overdrafts and reserve deficiencies and the opportunity cost of holding reserves. But in order to solve for the predictions of the model, the literature has been forced to abandon much of the institutional detail of reserve requirements; or alternatively, abandon the assumption of optimization or the incorporation of future costs into that decision. Our use of numerical methods to solve the optimization problem and to simulate the dynamic demand for reserves allows us to address a wide variety of issues. We have shown the effect of uncertainty, the level of reserve requirements, and overdraft penalties on the pattern of reserve holding over the maintenance period. We have extended the analysis of carryover behavior by taking into account the effect of uncertainty on targeted carry-out, and by showing how reserves carried into the maintenance period affect reserve demand. We have also used our model to examine the effect of proposed institutional changes to reserve requirements, such as paying interest on excess reserves. We are optimistic that this model can be used to address other issues of interest in the operation of monetary policy. A practical concern for the open market desk are ‘work-off rates’: how fast banks will reduce their reserve holdings in response to an unexpected increase in reserves. More generally, we want to know how an unexpected change in a bank's reserve position on one day will affect its demand for reserves on subsequent days. Several papers have provided approximate answers: Spindt and Hoffmeister (1988) assume that banks will work off the entire amount of extra reserves evenly over the maintenance period, while Longworth (1989) argues that a bank will generally want to offset some, but not all, of the effect of the shock because of the increased risk of overdrafts on subsequent days of the maintenance period. Our model is able to provide a detailed analysis of the optimal response, including showing how it will depend on the day of the maintenance period and the amount of carry-in. Further topics of research may include studying the shape of the demand schedule, particularly its responsiveness to change in interest rates, and the effect of interest rate uncertainty on bank demand. This would require including a stochastic interest rate variable in the model, which would add an additional level of computational complexity, but no change in the method of analysis. Perhaps most importantly, our model is quite flexible in addressing the effect of potential changes in reserve policy. What form these policy initiatives will take remains to be seen; they will depend on future innovations in bank reserve management (such as sweep programs) and the response by the Federal Reserve.