Operations in Financial Services—An Overview (4 of 6)


5.1. Cash Inventory Management Under Deterministic and Stochastic Demand

Organizations, households, and individuals need cash to meet their liquidity needs. In the era of checks and electronic transactions, an amount of cash does not have to be in physical currency, but may correspond only to a value in an account that has been set up for this purpose. To meet short-term liquidity needs, cash must be held in a riskless form, where its value does not fluctuate and is available on demand, but earns little or no interest. Treasury bills and checking accounts are considered riskless. Cash not needed to meet short-term liquidity needs can be invested in risky assets whereby it may earn higher returns, but its value may be subject to significant fluctuations and uncertainty, and could become wholly or partially unrecoverable. Depending on the type of risky asset, its value may or may not be quickly recoverable and realizable at a modest cost (as with a public equity that is listed in a major stock exchange). Determining the value of certain types of risky assets (e.g., private equity, real estate, some hedge funds, and asset-backed fixed income securities) may require specialized valuation services, which could involve significant time and cost. Risky assets can also be subject to default, in which case all or part of the value becomes permanently unrecoverable.

Researchers have produced over the last few decades a significant body of work by applying the principles of inventory theory to cash management. We review the cash management literature from its beginnings so we can put later work in context, and we have not identified an earlier comprehensive review that accomplishes this purpose. Whistler (1967) discussed a stochastic inventory model for rented equipment that wasformulatedasadynamicprogram; thiswork served as a model for the cash management problem. One of the early works produced an elegant result that became known as the Baumol–Tobin economic model of the transactions demand for money, independently developed by Baumol (1952) and Tobin (1956). The model assumes a deterministic, constant rate of demand for cash. It calculates the optimal “lot sizes” of the risky asset to be converted to cash, or the optimal numbers of such conversions, in the presence of trans-action and interest costs. Tobin’s version requires an integer number of transactions and therefore approximates reality more closely than Baumol’s, which allows that variable to be continuous.

The concept of transactions demand for money, addressed by the Baumol–Tobin model, is related to, but subtly different from, precautionary demand for cash that applies to unforeseen expenditures, opportunities for advantageous purchases, and uncertainty in receipts. Whalen (1966) developed a model with a structure strikingly similar to the Baumol–Tobin model, capturing the stochastic nature of precautionary demand for cash. Sprenkle (1969) and Akerlof and Milbourne (1978) observed that the Baumol–Tobin model tends to under-predict demand for money, partly because it fails to capture the stochastic nature of precautionary demand for cash. Sprenkle’s paper elicited a response by Orr (1974), which in turn prompted a counter-response by Sprenkle (1977).

Robichek et al. (1965) propose a deterministic short-term financing model that incorporates a great degree of realistic detail involved in the financial officer’s decision-making process, which they formulate and solve as a linear program. They include a discussion on model extensions for solving the financing problem under uncertainty. Sethi and Thompson (1970) proposed models based on mathematical control the-ory, in which demand for cash is deterministic but does vary with time. In an extension of the Sethi– Thompson model, Bensoussan et al. (2009) allow the demand for cash to be satisfied by dividends and uncertain capital gains of the risky asset, stock.

In what became known as the Miller–Orr model for cash management, Miller and Orr (1966) extended the Baumol–Tobin model by assuming the demand for cash to be stochastic. The cash balance can fluctuate randomly between a lower and an upper bound according to a Bernoulli process, and transactions take place when it starts moving out of this range; units of the risky asset are converted into cash at the lower bound, and bought with the excess cash at the upper bound. Transaction costs were assumed fixed, i.e., independent of transaction size. In a critique of the Miller–Orr model, Weitzman (1968) finds it to be “robust,” i.e., general results do not change much when the underlying assumptions are modified.

Eppen and Fama (1968, 1969, 1971) proposed cash balance models that are embedded in a Markovian framework. In one of their papers, Eppen and Fama (1969) presented a stochastic model, formulated as a dynamic program, with transaction costs proportional to transaction sizes. Changes in the cash balance can follow any discrete and bounded probability distribution. In another one of their papers, Eppen and Fama (1968) developed a general stochastic model that allowed costs to have a fixed as well as a variable component. They showed how to find optimal policies for the infinite-horizon problem using linear programming. In their third paper, Eppen and Fama (1971) proposed a stochastic model with two risky assets, namely “bonds” and “stock”; the stock is more risky but has a higher expected return. They also discussed using “bonds” (the intermediate-risk asset) as a “buffer” between cash and the more risky asset. Taking a similar approach, Daellenbach (1971) proposed a stochastic cash balance model using two sources of short-term funds. Girgis (1968) and Neave (1970) presented models with both fixed and proportional costs and examined conditions for policies to be optimal under different assumptions. Hausman and Sanchez-Bell (1975) and Vickson (1985) developed models for firms facing a compensating-balance requirement specified as an average balance over a number of days.

Continuous-time formulations of the cash management problem were based on the works of Antelman and Savage (1965), and Bather (1966), who used a Wiener process to generate a stochastic demand in their inventory problem formulations. Their approach was extended to cash management by Vial (1972), whose continuous-time formulation had fixed and proportional transaction costs, and linear holding and penalty costs, and determined the form of the optimal policy (assuming one exists). Constantinides (1976) extended the model by allowing positive and negative cash balances, determined the parameters of the optimal policy, and discussed properties of the optimal solution. Constantinides and Richard (1978) formulated a continuous-time, infinite-horizon, discounted-cost cash management model with fixed and proportional transaction costs, linear holding and penalty costs, and the Wiener process as the demand-generating mechanism. They proved that there always exists an optimal policy for the cash management problem, and that this policy is of a simple form. Smith (1989) developed a continuous-time model with a stochastic, time-varying interest rate.

5.2. Supply Chain Management of Physical Currency

Physical cash, i.e., paper currency and coins, remains an important component of the transactions volume even in economies that have experienced a significant growth in checks, credit, debit and smart cards, and electronic transactions. Advantages of cash include ease of use, anonymity, and finality; it does not require a bank account; it protects privacy by leaving no transaction records; and it eliminates the need to receive statements and pay bills. Disadvantages of cash include ease of tax evasion, support of an “underground” economy, risk of loss through theft or damage, ability to counterfeit, and unsuitability for online transactions.

Central banks provide cash to depository institutions, which in turn circulate it in the economy. There are studies on paper currency circulation in various countries. For example, Fase (1981) and Boeschoten and Fase (1992) present studies by the Dutch central bank on the demand for banknotes in the Netherlands before the introduction of the Euro. Ladany (1997) developed a discrete dynamic programming model to determine optimal (minimum cost) ordering policies for banknotes by Israel’s central bank. Massoud (2005) presents a dynamic cost minimizing note inventory model to determine optimal banknote order size and frequency for a typical central bank.

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The production and distribution of banknotes, and the required infrastructure and processes, have also been studied. Fase et al. (1979) discuss a numerical planning model for the banknotes operations at a central bank, with examples from pre-Euro Netherlands. Bauer et al. (2000) develop optimization models for determining the least-cost configuration of the US Federal Reserve’s currency processing sites given the trade-off between economies of scale in processing and transportation costs. In a study of costs and economies of scale of the US Federal Reserve’s currency operations, Bohn et al. (2001) find that the Federal Reserve is not a natural monopoly. Opening currency operations to market competition and charging fees and penalties for some services provided for free by the Federal Reserve at that time could lead to more efficient allocation of resources.

The movement of physical cash among central banks, depository institutions, and the public must be studied as a closed-loop supply chain (see, e.g., Dekker et al. 2004). It involves the recirculation of used notes back into the system (reverse logistics), together with a flow of new notes from the central bank to the public through depository institutions (forward logistics). The two movements are so intertwined that they cannot be decoupled. Rajamani et al. (2006) study the cash supply chain structure in the United States, analyze it as a closed-loop supply chain, and describe the cash flow management system used by US banks. They also discuss the new cash recirculation policies adopted by the Federal Reserve to discourage banks’ overuse of its cash processing services, and encourage increased recirculation at the depository institution level. Among the practices to be discouraged was “cross shipping,” i.e., shipping used currency to the Federal Reserve and ordering it in the same denominations in the same week. To compare and contrast new and old Federal Reserve policies for currency recirculation, Geismar et al. (2007) introduce models that explain the flow of currency between the Federal Reserve and banks under both sets of guidelines. They present a detailed analysis that provides optimal policies for managing the flow of currency between banks and the Federal Reserve, and analyze banks’ responses to the new guidelines to help the Federal Reserve understand their implications. Dawande et al. (2010) examine the conditions that can induce depository institutions to respond in socially optimal ways according to the new Federal Reserve guidelines. Mehrotra et al. (2010), which is a paper in this special issue of Production and Operations Management, address the problem of obtaining efficient cash management operating policies for depository institutions under the new Federal Reserve guidelines. The mixed-integer programming model developed for this purpose seeks to find “good” operating policies, if such exist, to quantify the monetary impact on a depository institution operating according to the new guidelines. Another objective was to analyze to what extent the new guidelines can discourage cross shipping and stimulate currency recirculation at the depository institution level. Mehrotra et al. (2010a) study pricing and logistics schemes for services such as fit-sorting and transportation that can be offered by third-party providers as a result of the Federal Reserve’s new policies.

5.3. Other Cash Management Applications in Banking and Securities Brokerage

US banks are required to keep on reserve a minimum percentage (currently 10%) of deposits in client transaction accounts (demand deposits and other checkable deposits) at the Federal Reserve. Until very recently, banks had a strong incentive to keep funds on reserve at a minimum, because these funds were earning no interest. Even after the 2006 Financial Services Regulatory Relief Act became law, authorizing payment of interest on reserves held at the Federal Reserve, banks prefer to have funds available for their own use rather than have them locked up on reserve. Money market deposit accounts (MMDA) with checking allow banks to reduce the amounts on reserve at the Federal Reserve by keeping deposits in MMDA accounts and transferring to a companion checking account only the amounts needed for transactions. Only up to six transfers (“sweeps”) per month are allowed from an MMDA to a checking account, and a client’s number and amount of transactions in the days remaining in a month is unknown. Therefore, the size and timing of the first five sweeps must be carefully calculated to avoid a sixth sweep, which will move the remaining MMDA balance into the checking account. Banks have been using heuristic algorithms to plan the first five sweeps. This specialized inventory problem has been examined by Nair and Anderson (2008), who propose a stochastic dynamic programming model to optimize retail account sweeps. The stochastic dynamic programming model developed by Nair and Hatzakis (2010) introduces cushions added to the minimum sweep amounts. It determines the optimal cushion sizes to ensure that sufficient funds are available in the transaction account in order to cover potential future transactions and avoid the need for a sixth sweep.

The impact of the sequence of transaction postings on account balances and resultant fees for insufficient funds, similar to the cost of stock-outs in inventory management, has been studied by Apte et al. (2004). They investigate how overdraft fees and non-sufficient funds (NSF) fees interact in such situations.

Brokerage houses make loans to investors who want to use leverage, i.e., to invest funds in excess of their own capital in risky assets, and can pledge securities that they own as collateral. In a simple application of this practice, known as margin lending, the brokerage extends a margin loan to a client of up to the value of equity securities held in the client’s portfolio. The client can use the loaned funds to buy more equity securities. Calculating the minimum value required in a client’s account for a margin loan can become complex in accounts holding different types of securities including equities, bonds, and derivatives, all with different margin requirements. The complexity increases even more with the presence of long and short positions and various derivative strategies practiced by clients. Rudd and Schroeder (1982) presented a simple transportation model formulation for calculating the minimum margin, which represented an improvement over the heuristics used in practice. A significant body of subsequent work has been published on this problem, especially by Timkovsky and collaborators, which is more related to portfolio strategies and hedging. We believe that the approach in the paper by Fiterman and Timkovsky (2001), which is based on 0–1 knapsack formulations, is methodologically the most relevant to mention in this overview.


6.1. Queueing Environments and Modeling Assumptions

In financial services, in particular in retail banking, retail brokerage, and retail asset management (pension funds, etc.), queueing is a common phenomenon that has been analyzed thoroughly. Queueing occurs in the branches of retail banks with the tellers being the servers, at banks of ATM machines with the machines being the servers, and in call centers, where the operators and/or the automated voice response units are the servers. These diverse queueing environments turn out to be fairly different from one another, in particular with regard to the following characteristics:

(i) the information that is available to the customer and the information that is available to the service system,

(ii) the flexibility of the service system with regard to adjustments in the number of servers dependent on the demand,

(iii) the order of magnitude of the number of servers.

Even though in the academic literature the arrival processes in queueing systems are usually assumed to be stationary (time-homogeneous) Poisson processes, arrival processes in practice are more appropriately modeled as non-homogeneous Poisson processes. Over the last couple of decades, some research has been done on queues that are subject to non-homogeneous Poisson inputs (see, e.g., Massey et al. 1996). The more theoretical research in queueing has also focused on various aspects of customer behavior in queue, in particular abandonment, balking, and reneging. For example, Zohar et al. (2002) have modeled the adaptive behavior of impatient customers and Whitt (2006) developed fluid models for many-server queues with abandonment. In all three queueing environments described above, the psychology of the customers in the queue also plays a major role. A significant amount of research has been done on this topic, see Larson (1987), Katz et al. (1991), and Bitran et al. (2008). As it turns out, reducing wait times may not always be the best approach in all service encounters. For example, in restaurants and salons, longer service time may be perceived as better service. In many cases, customers do not like waiting, but when it comes their turn to be served, would like the service to take longer. In still others, for example, in grocery checkout lines, customers want a businesslike pace for both waiting and service. The latter category, which we may call dispassionate services, are more common in financial service situations, though the former, which we may call hedonic services, are also present—for example, when a customer visits their mortgage broker or insurance agent, they would not like to be rushed. In the following subsections, we consider the various different queueing environments in more detail.

6.2. Waiting Lines in Retail Bank Branches and at ATMs

The more traditional queues in financial services are those in bank branches feeding the tellers. Such a queue is typically a single line with a number of servers in parallel. There are clearly no priorities in such a queue and the discipline is just first come first served. Such a queueing system is typically modeled as an M/M/s system and is discussed in many standard queueing texts. One important aspect of this type of queueing in a branch is that management usually can adjust the number of available tellers fairly easily as a function of customer demand and time of day. (This gives rise to many personnel scheduling issues that will be discussed in the next section.)

In the early 1980s retail banks began to make extensive use of ATMs. The ATMs at a branch of a bank behave quite differently from the human tellers. In contrast to a teller environment, the number of ATMs at a branch is fixed and cannot be adjusted as a function of customer demand. However, the teller environment and the ATM environment do have some similarities. In both environments, a customer can observe the length of the queue and can, therefore, estimate the amount of time (s)he has to wait. In neither the teller environment, nor the ATM environment, can the bank adopt a priority system that would ensure that more valuable customers have a shorter wait. Kolesar (1984) did an early analysis of a branch with two ATM machines and collected service time data as well as arrival time data. However, it became clear very quickly that a bank of ATMs is capable of collecting some very specific data automatically (e.g., customer service times and machine idle times), but cannot keep track of certain other data (e.g., queue lengths, customer waiting times). Larson (1990), therefore, developed the so-called queue inference engine, which basically provides a procedure for estimating the expected waiting times of customers, given the service times recorded at the ATMs as well as the machine idle times.

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Emmanuel D. (Manos) Hatzakis, Suresh K. Nair, and Michael Pinedo


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