TECHNICAL FIELD OF THE INVENTION

This invention relates generally to transactions and services management and more particularly to managing available cash in bank branches and /or automatic teller machines (ATMs) The term "ATM" as used herein includes (unless otherwise indicated for a narrower usage in specific instances) the familiar deployed cash dispensing termináis found in bank branch lobbies and also in secure remote locations rented by banks and the underlying infrastructure for servicing such termináis. It also includes similar termináis placed in stores and kiosks. It also includes termináis and infrastructure with other functions such as point-of-sale (POS) termináis, scrip machines and other analogs or extensions of the cash dispensing ATM concept as now known or hereafter revised or supplemented.

BACKGROUND OF THE INVENTION

Liquidity management

Along with risk, liquidity management represents the main rationale for the existence of banks in the classical financial intermediation theory. In the standard framework, the basic management challenge as far as liquidity is concerned is how to cover depositors' random consumption needs and how to set the subsequent deposit insurance mechanisms for these depositors.

During the financial crisis that started in 2007, bank liquidity has become a major issue in a context of uncertainty and instability. Liquidity tensions have made banks develop several strategies to retain depositors and to manage liquidity as efficiently as possible to avoid financial fragility that, overall, makes the whole financial system weaker.

This problem can be related with a classic issue: the transaction demand for the cash. The transaction demand for the cash consists of managing an inventory of cash holdings: the decisión maker holds two distinct types of assets, one asset which bears interest at a given rate, and a noninterest bearing asset where periodic receipts of deposit and expenditures are made. Transfers of funds between the two accounts are permissible but at a cost (transfer cost). There are also other costs of different nature involved: the opportunity costs derived from the fact that funds into the noninterest bearing asset are losing money while they are not into the interest bearing portfolio. Apart from the opportunity costs associated with cash inventories, there are many others: personnel costs, investments costs, material and clearing costs, insurance and logistic costs for transport and handling.

Current Approximations to the solution

Several studies have dealt with the implications of wholesale debt markets of retail deposit markets on liquidity tensions as well as on the possible policy actions to solve such market-related liquidity shortages (as shown, ínter alia, in [Fecht F, Nyborg KG, Rocholl J (2011) The price of liquidity: The effects of market conditions and bank characteristics. J Finan Econ 102: 344-362] or [Loutskina E (2011) The role of securitization in bank liquidity and funding management. J Financ Econ 100: 663-684]. Although there are significant potential efficiency improvements in liquidity from branch- level cash management, there are only a few studies dealing with these issues, as most of the banking models in the financial literature assume the standard allocation models of cash from central hubs to branches as given. A relevant exception is the work of Pokutta and Schmaltz [Pokutta S, Schmaltz C (2011) Managing liquidity: Optimal degree of centralization. J Bank Finance 35: 627-638] who study how to improve the optimal allocation of cash at large banking groups. They compare the alternatives of a central liquidity hub and the case of many decentralized branches. They provide an analytical solution for a 2-branch model and show that a liquidity center can be interpreted as an option on immediate liquidity where the valué can be interpreted as the price of information. Importantly, Pokutta and Schmaltz derive the threshold above which it is advantageous to open a liquidity center and show that it is a function of the volatility and the characteristic of the banking network.

Other studies have also dealt with liquidity management not just at the bank level but as a general feature of firm management. In this front, some papers have made use of stochastic and inventory theory to propose some models of firms' cash management. Among all these approaches, the closest to the invention described in this document is the model described by Ferstl and Weissensteiner [Ferstl R, Weissensteiner A (2008) Cash management using multi-stage stochastic programming. Quant Finance 10: 209- 219]. They consider a cash management problem where a company with a given financial endowment and given future cash flows minimizes the Conditional Value at Risk of final wealth using a lower bound for the expected terminal wealth. Ferstl and Weissensteiner use a multi-stage stochastic linear program (SLP) where a company can choose between a riskless asset (cash), several default- and option-free bonds, and an equity investment, and rebalances the portfolio at every stage. They explicitly estímate a function for the market price of risk and change the underlying probability measure and simúlate scenarios for equity returns with moment-matching by an extensión of the interest rate scenario tree. Also, Castro [Castro J. (2009). A Stochastic Programming approach to Cash Management in Banking. European Journal of Operation Research, 192: 963-974] describes a stochastic programming approach but it's only applicable in ATMs.

Qptimization based on histórica! data

Other implemented methods use previous data in similar circumstances (similar day, similar month, working day or saint day) as a process of trial and error for the present day, being this one the unique reference in the daily decision-making on cash flow. Moreover, this exercise of historical data implicitly carries a human error of valuation, since most of the process is based on the valuation of the person in charge of the branch, who decides, after consulting the previous information, which quantity it must demand for (if equal, higher or lower to that of the historical data). Moreover, it is a common practice to overload ATMs with case to avoid users' penalty actions Although the introduction of AMTs and other technological innovations has reduced cash management costs, there is still a pressing need to optimize resources, imposed by the high competition among bank institutions in the present scenario of economic and financial crisis.

BRIEF DESCRIPTION OF THE INVENTION Since the notion of liquidity management comprises all short- and medium-term cash flows no matter whether they are accounting-based or cash, in this invention we focus on improvements in optimization of cash inventories within banks and, in particular, on how to improve cash management inventories at branch level, as the scope of this paper deals essentially with the cash both for the cash desks and/or in ATM's.

The first aspect of the invention is a method to manage an inventory of bank cash holdings in a set of bank branches and/or Automatic Teller Machines using optimization theory as well as stochastic calculus.

This method provides benefits as compared to the known branch cash management models. In particular, the invention describes a program of optimization of the bank cash for all branches or ATMs. This program consists of some precise, simple formulae which could be implemented in branches at a set of instructions from the bank company. Among the major virtues of the program that we propose are that these formulae are very easy to implement and practically cost-free.

Specifically, we propose to substitute the oíd procedures the branches use, based on their historical data, by a method based on mathematical formulae. The mathematical tools that we have used, which combine optimization programs with stochastic calculus as well as some aspects of Inventory Theory, conform a rigorous and powerful weapon to reduce cash inventory.

Many banking systems are going through intense processes of consolidation which include not only mergers and acquisitions but also efficiency plans. In these plans, the number of bank branches, their size and different aspects of their management are a critical feature for determining the success of the restructuring process in terms of efficiency gains. The proposed method of the invention relies on a series of simplifying assumptions that, however, do not affect its practical implementation.

The proposed method focuses on optimization of cash inventories which has been always a critical feature of financial intermediation given the maturity transformation that banks face by converting short-term liquidity to long-term funding. The results shows this method also has implications for bank restructuring processes in what it provides a way to evalúate efficiency improvements from changing branch size (or simply closing branches) from the cash management perspective. The method as well suggests that the (significant) differences in the size of bank branches across banking sectors internationally may respond, ínter alia, to different ways of dealing with branch cash management.

Finally, it is important to point out that the simplicity of the method proposed makes it particularly feasible to implement. It is also worth noting that some of the simplifying assumptions of the model (such as making the máximum of cash available equal for each branches irrespective of their location) do not affect the overall validity of the results as banks tend to apply their branch cash management methods in a very standard way for all the branches within a bank. DETAILED DESCRIPTION OF THE INVENTION

In the context of the present invention, the following abbreviations are used:

i. i. i.: independent and identically distributed;

IT: Information technology;

SLP: Stochastic linear program

In this context, the first aspect of this invention is a method to optimize the cash resources of the bank companies across its branches that comprises the minimization of banks objective function, defined as a smart approximation of the sum of logistic cost for transport and handling and costs derived from the funds being into the non interest bearing, such as C ₀ must cover the whole branch expenses

Different approaches to the problem can be made in order to optimize the bank cash handling: in a global approach, modeling the situation of a bank's company viewed a solid piece of the puzzle. In this case, we have to consider the set of international rules as constrains to the optimization program. This course of action is very complicated because the cash movements are not known. In fact, the cash funds are strongly tied up as they are invested in many other bank producís. The method of the invention substitute the oíd mechanism the branches use until now, based only on their historical data. Problem's plot

As for the invention, we have chosen to consider any bank's company as an institution composed by several pieces in contrast with the unique and solid piece of the global point of view. The pieces which conform a bank's company are its branches: henee, we will carry out the optimization process across all the branches.

Branches' Operation In their labor of attention to the users, any branch must have a quantity of ready money, which comes in part from the cash central, as well as from the deposits made by the individual users or companies. Periodically, the branch adjusts its cash to its necessities -deposits and expenses- avoiding generating a quantity of dormant money. After this adjustment, both of the following possibilities may take place: the branch has generated a surplus of cash or, on the contrary, the branch needs an injection of money.

In both cases, the branch needs help from its cash central (the closest one): in the first case, the branch requires that an armoured van evacuates the surplus of money. In the second one, that of the branch needing more cash, the cash central should move money to the branch, by means of an armoured van as well. In any case, these cash movements from the cash central to the branches take place after a concrete demand of money from branches to the cash central.

Anticipated refunds

As for the demand of money from branches to the central office, this often oceurs with a forecast of funds according to a concrete demand. This is the case, for instance, of users who want to cancel a deposit and to recover their money. Henee, the branch can anticípate this expenditure, in time and in quantity. This argument of anticipation of the bank in some cases is based on the real normative of some bank companies, who require that clients announce the physical withdrawal one day before such that the bank has twelve hours to demand the funds from its headquarter.

The above example suggests that there are two types of expenditures: those which can be anticipated and those of a strictly random nature. The same classification can be established in the case of deposits. Thus, we refer to both the above types as expected and unexpected expenditures and deposits. Turning again into the movement of money from the cash central, this event implicitly carries a cost that we analyze deeply in later sections. Mainly, it consists of a logistic cost for transport and handling, due to the costs generated by renting a security company (which includes costs for security personnel and costs for transport in armoured vans). There is involved as well an opportunity cost, derived from the fact that funds into the non interest bearing asset are loosing money while they are not in the interest bearing portfolio. One of the targets of this paper is, then, to minimize the different costs implicated in the currency management while keeping the user satisfaction as high as usual.

Dynamics of the funds of a branch

For simplicity, we will assume that the bank capital can be separated into two producís: cash funds and the rest of bank goods, even those with high capacity of turning out to be in ready money without loss of their valué. This hypothesis can seem naive, but it allows us to be focused only in the liquid assets of the bank company.

As we mentioned before, each branch of any bank company has to assume several periodic expenses, some of them are predictable while many others are not. This implies that an entry of money must exist, available when necessary, from the closest cash central. This channel of entry of money is complemented with the daily deposits that the individual users or companies make. The cash central sends cash to the branch after a specific request from the branch. This request is done after adjusting both the programmed expenses and those which are of random nature, with the own expected deposits and the cash available in the branch at this moment. All this process is done with the target of not generating a surplus of money.

In the real world, the criterion with which such a request is done is purely based on the histórica! data of the branch. This expresses that the branch compares such a day as today with a similar day of the past. The concept of "similar day" means a day of chosen similar characteristics: working day versus holiday, day at the beginning of the week versus day at the weekend, etc. The branch, having identified such a similar day, examines the success or failure of the quantity required for such a day of the past and does its request after valuating if doing so in an equal, higher or lower quantity. As it is the banks company who makes this optimization process, it establishes some mechanism of control on the branches to restrict these movements of enfries and exits of money:

1. A first mechanism of control to optimize its resources is set by the bank's company by fixing a certain number of periodic stops: branches cali stop to each of the armoured van stops next to the branch. Each branch has a certain number of stops available which might or might not be required. Let us recall that stops occur after a request of the branch, and due to both possibilities: caused by the existence of a surplus of cash or, on the contrary, caused by the absence of cash to cover all the branches necessities.

2. The branch will know that there is not enough cash when they will not be able to cover all the programmed (or unexpected) expenses. However, the existence of a surplus of ready money can be established only with a comparative to a reference, which allows us to introduce the second mechanism of control on behalf of the bank's company.

Such reference to set how much money exceeds in this branch consists of an upper bound C _z (máximum of cash), which is fixed by the bank's company for each branch, attending some parameters, particularly, the volume of the branch turnover which, as we will see later, can be identified with the size of the branch. In comparison with C _z , if the branch liquid funds exceeds this margin or not, the branch will know if it must require or not one stop to evacúate the surplus.

Depending on the internal normative of each bank company, the bank can decide (or not) to add a bit more over C _z for precautionary motives (i.e., for prevention) in order to fulfill the clients demand up to a certain confidence level. Since such precautionary "bit more" will be fixed for each bank company in attendance of their internal rules of functioning, the method of the invention provides the optimum C _z for minimizing the bank costs assuming that this precautionay "bit more" is external to C _z and henee, it is not included in C _z . Then, once we have finished our task of determining the optimal amount C _z for minimizing bank costs for any bank company, each bank could (or not) add more or less precautionary amount, following its internal normative.

Summary of variables used in the model: A denotes the variance of changes in the cash balance in the considered period of time. A = μ ² ί

B is a constant related to Insurance Costs. BCz represents the Insurance Costs C ₀ is the money that the branch demands to its cash central

C _z is the máximum cash which the bank's company will allow in its branches

Di denotes the deposit that has been made by the / ^' th-user

γ represents the costs per stop (as defined later).

λ = E[N] is the average of branch consumers in the considered period of time

A ^d = E[Di] is the average of deposits in the considered period of time

2 ^W = E[W _t ] is the average of withdrawals in the considered period of time μ denotes the amount of euros that the branch cash balance increases or decreases in some small fraction of the period of time 1/t.

N is the number of customers which make use of the bank branch during the considered period of time. N = N _t .

N _t represents the number of bank branch users in the interval (0,t) N _t = N™ +

N™ represents the number of withdrawals in the interval (0,f),

N represents the number of deposits made in the interval (0,f)

O _t gathers the rest of operations and requests (apart from withdrawals and deposits) made by the branch users.

ϋ represents the rate of interest earned on the portfolio in the considered period of time.

Wi denoted the withdrawal made by the / ^' -th user

X _t is the total amount that has been taken off by the N™ withdrawals made in the interval (0,f). X _t =∑vf ₁ w _t

Y _t is the total amount that has been deposited by the N deposits made in the interval (0,t). Y _t =∑^ ₁ D _i

The final objective of the method of the invention is to provide to the branches with some precise instructions under the shape of formulae, which will be deduced in later sections. These instructions will constitute a set of simple formulae, with no cost in their implementation in practice. These features conform the more valuable advantages of our approach compared with other techniques for different bank branches.

The underlying optimization problem The method of the invention is a method to optimize the cash resources a bank's branch and/or ATM that comprises the minimization of a smart approximation of the sum of stop costs, opportunity costs and insurance costs as a function of the money that the branch demands to its cash central, such as the money that the branch demands to its cash central must cover the whole branch expenses.

Where

• "Stop Costs" are the sum of logistic cost for transport and handling

• "Opportunity costs" are those costs derived from the fact that funds into the noninterest bearing asset are losing money while they are not into the interest bearing portfolio; and

• "Insurance Costs" are the payment on the bank ^' s company to a theft insurance policy. This cost will be directly proportional to the máximum amount of money allowed by the bank ^' s company for this branch in particular.

In particular, the method comprises the periodical minimization of e(C ₀ , C _z ) as defined in Eq. 1 , f ^(C °' ^{Cz): = Y} ^ ^ ^{+ ü} HT + ^sc z (Eq. 1 )

Where

C _z ≥ EIXJ - £[¾ + K,

Being

C ₀ , the money that the branch demands to its cash central

C _z , the máximum point of the cash level which the bank's company will allow in this branch

K , the valué of both expected expenditures and expected deposits (known constant)

A, the variance of changes in the cash balance in the considered period of time

B, a constant related to Insurance Costs. BC _Z represents the Insurance Costs γ, the costs per stop, defined as total costs per contracting a security company (which are constant and in consequence, independent of the amount transferred) divided by the mean of total stops for this branch.

ϋ, the rate of interest earned on the portfolio in the considered period of time. Ep J , the expected valué of the no anticipated withdrawals in the considered period of time; and

E[Y _t ] , the expected valué of the no anticipated deposits in the considered period of time

In a particular realization, the consider period is one day.

In other particular realization, branches pay a constant amount per month for all services to the security company, independently of the number of stops they have to make. So, γ is calculated by distributes the total costs per contracting a security company (which are constant and in consequence, independent of the amount transferred) by the mean of total stops for this branch.

Regarding constrains of the optimization problem, C ₀ must cover the whole branch expenses, this can be written in mathematical terms as C ₀ = E[X ₁ ] - Ε^] + K, where K includes both expected expenditures and deposits. Note that both terms E[X _t ] and E[X _t ] are known.

C _z is the upper bound of cash that the bank's company allows to the branch. Henee, a natural second constrain will be C _z ≥ C ₀ . Note that both constrains together state that C _z ≥ E[X _t ] - E^] + K, which it is the logical rule to ensure a well-functioning of the branch.

In a preferred realization, the distribution of X and Y is a Compound Poisson:

Where W _t denoted the withdrawal made by the / ^' -th user, so X _t is the total amount of

1

the iV ^w = withdrawals done in the considered period of time, and W follows a

Poisson distribution with parameter 2 ^W ; and Where D _t denoted the deposit made by the /-th user, so Y _t is the total amount of the N ^d =— ¹ — _¿ deposits done in the considered period of time, and D follows a Poisson distribution with parameter ^d . In other particular realizations the distribution of X and Y is selected from the group consisting in Mixture Poisson-Lognormal, Poisson - Inverse Gaussian and Poisson- Triparametric Lognormal (with threshold parameter). X and Y can follow the same or different distributions. A further aspect of the invention is a computer program that implements the method of the invention.

A further aspect of the invention is related to means that support the computer program, in particular, an article of computer readable media bearing a plurality of computer executable instructions to cause a computer to carry out the method of the invention.

Development of the model: election of variables and function of costs Variables of the method

Let us start by choosing the variables of our model: the first one is the total amount of money that the branch demands from its cash central, C ₀ . The requirements for C ₀ are that it should

· be enough to cover both expected and unexpected expenditures of the branch and

• minimize the costs function of the bank's company.

A formula to calcúlate an accurate C ₀ for each monetary circumstances of each branch, will be described later on.

A negative C ₀ should mean that there is a surplus of cash in the branch. From the bank's company point of view, the second variable to be considered is the upper bound (the máximum) of cash resources that the bank's company fixes for each branch as a reference to determine if a surplus of cash exists in there. Let C _z be the máximum point of the cash level which the bank's company will allow in its branches. We assume that the cash balance in the bank branch is allowed to fluctuate until it reaches the upper bound, C _z . Once the branch exceeds the quantity C _z , it must demand from the cash central one stop to evacúate the surplus of money. After having selected the variables, let us continué with the banks objective function.

As for the first addend, this can be written as the product of the cost per transfer (say y) and the total expected number of transfers per period of time considered. While in the classical issue of transactions demand for cash, the costs due to cash flow are simply transfer costs (from one current account to another), in our particular context, these costs include personnel costs, investments costs, material and clearing costs, insurance and logistic costs for transport and handling.

Henee, this first addend of the function represents the costs due to cash flow,

y ' , (C0- ^A Cz)Cz

where A denotes the variance of changes in the cash balance in the specific period of time. Specifically, considering that the random behavior of the cash balance can be characterized as a sequence of independent Bernoulli triáis, if μ denotes the amount of euros that the branch cash balance increases or decreases in some small fraction of a working day 1/t, thus / = μ ² ί.

As for the costs per stop, γ, we simply distributes the total costs per contracting a security company (which are constant and in consequence, independent of the amount transferred) by the mean of total stops for this branch. Let us point out that in reality, branches (bank companies) pay a constant amount per month for all services to the security company, independently of the number of stops they have to make.

As for the second addend, this will be the product of the rate of interest earned on portfolio (e.g., other banks producís which yield higher benefits) in the considered period of time, say ϋ, and the average cash balance in the considered period of time. _β ^C ° ⁺ ° ^ζ

3

This steady-state distribution of cash holdings is of a discrete triangular form with base C _z and mode C ₀ . Henee the mean of such a distribution is ^ ⁵ - Then, summarizing all the above information, y represents the costs per ^sto P' - ' ^{s tne tota} ' ^numDer of stops in the period of time, while θ is the rate of interest earned on the portfolio in the considered period of time, and represents the cash balance on bank branch in the considered period of time. We must also consider other costs involved with a large quantity of stored money inside its branches: the payment on the bank's company to a theft insurance policy. This cost will be directly proportional to the máximum amount of money allowed by the bank's company for this branch in particular. So, the insurance costs are BC _Z where B is constant.

Henee, the objective function is f ^(C _° ' ^{Cz): = Y} ^ ^ ^{+ ü} HT + ^sc z (Eq. 1 ) Note that in the objective function (in the first addend) there is a parameter directly related to the branch's users behavior. This is μ, with A = μ ² ί.ννηίοη acts as an índex of how active the branch users are: the more times they make withdrawals or deposits - and the higher are the quantities they move- the bigger is μ and the bigger are the fluctuations of the cash balance. Henee, from now on, we consider μ as the indicator of the fluctuations of the branch cash flow. Let us remark that μ e [0, +∞) ¡f ¡t is considered as a symmetric function.

In practice, μ is strongly directly related with the geographic location of the branch, more than to its size: nearer the branch is situated to an industrial, commercial or financial center, higher will be the fluctuations of the branch cash flow, so higher will be μ. The random deposits and expenditures

As mentioned before, the quantity C ₀ of money that the branch demands to its cash central must fulfill some requirements, which can be condensed in only one: C ₀ must cover the expenses of the branch during the considered period of time.

As mentioned above, the branch expenses can be classified into expected and unexpected expenditures. While the first ones are easily modeled like a constant (which could be left aside of the model, as we will argüe in short), the second ones are much more difficult to be designed, due to their random nature. This section is, thus, devoted to capturing the randomness of our probiem by means of some stochastic processes.

Through the analysis of our probiem, some stochastic elements have been kept hidden: first of all, the number of bank branch users during the considered period of time. These consumers make deposits and withdrawals through the available ATM of the branch as well as from the cash desks. A second stochastic element involved is the quantities of money that these branch users take into and out. Note that the bank branch deals with both stochastic elements using its historical data.

Let N be the number of costumers which make use of the bank branch during the considered period of time. In a particular realization of the invention the arrival process of users, N, will be consider a "Poisson process" Mathematically the process is described by the so called counting process N(t) or N _t . The counter tells the number of arrivals that have occurred in selected period of time, the interval (0, t) . That is, Λ/^number of bank branch users in the interval (0, t).

From this definition and considering the period of time as the unit of time, note that N =

NI.

For that reason, if N _t is a Poisson counter process of parameter Á, some of its properties are the following:

•the number of arrivals to the bank branch in an interval of length t has a

Poisson distribution with parameter Á t; that is, P[N _t = n] =— measures the probability of n bank branch users in the time t.

· The mean and variance of N _t are E[Nt] = Át and var[Nt] = Át. Particularly, since λ = E[N], follows that the rate of the Poisson process N, λ is the average of branch consumers in the considered period of time. In order to properly structure the later withdrawals and deposits movements, note now that not all the users who come to a branch do so in order to make withdrawals or deposits. For this reason, the counting process N _t can be separated into many counting processes, which we summarize for simplicity like Nt = N™ + N + O _t , where N™ represents the number of withdrawals in the interval (0, t) , N represents the number of deposits made in the interval (0, t), and O _t gathers the rest of operations and requests (apart from withdrawals and deposits) made by the branch users.

The withdrawal process Apart from the number of branch consumers per period of time, another random element of our problem is the amount of each withdrawal made by each consumer. Let Wi the withdrawal that the / ^' -th user makes.

In a particular realization of the invention, we consider all these quantities V _¿ by means of a compound Poisson process.

In this document, a "Compound Poisson process" is a (random) stochastic process with jumps such as the jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution.

In this realization, the withdrawal process, parameterized by certain rate Á, will be the

N ^w

compound Poisson process given by X _t =∑ _¿= ^t ₁ V _¿ , viewed as independent and identically distributed (i.i.d.) random variables. Then, X _t is the total amount that has been taken off by the N™ withdrawals the interval (0, t) .

N ^w

Note that, particularly, X _t =∑ _i=1 V _¿ is the total amount of money that is taken from the bank accounts, either through the ATMs or through the cash desks, for the whole considered time period (t = 1). At this point, remember that the mean of a compound Poisson process can be calculated via the mean of one of the i.i.d. variables, as E[X _t ] = X ^w■ t ^■ E[Wi], where 2 ^w is the rate of the Poisson process N™. Consequently, for f = 1 , we have £¾] = X ^w · £ ^" [V _¿ ] = Average number of withdrawals per considered period of time by Average quantity taken off from bank branch per considered period of time.

The deposit process

Once we have modeled the withdrawal process, we can opérate in a similar manner for the deposit process. That is, in this particular realization, the deposit process parameterized by certain rate X ^d ·, as the compound Poisson process given by

Yt =∑¿ _i where D _¿ denotes the deposit that has been made by the / ^' th-user, viewed as independent and identically distributed (i.i.d.) random variables. In a parallel form to the withdrawal process, the mean of this compound Poisson process can be expressed as E[Y _t ] = X ^d■ t ^■ E[Di] where X ^d is the rate of the second Poisson process N ^d

For t = 1, we also have that E^] = X ^{d ■} E\D\ = Average number of deposits per considered period of time by Average quantity taken into the bank branch per considered period of time.

Considering the particular case in which the deposits and withdrawals are supposed to be compound Poisson process, £¾] = X _t and Ε^] = Y ₁ , so the needed cash is C ₀ = X ₁ - Y ₁ + K

EXAMPLES

In a particular implementation, we considered the users' arrival process is distributed as a Poisson process and the deposits and withdrawals can be considered compound Poisson processes and the considered period of time is a day.

In this scheme, the method of the invention can be implemented in a computer program to compute the total amount of money that the branch or ATM demands from its cash central, C ₀ , subject to the followings requirements: • C ₀ must be enough to cover both expected and unexpected expenditures of the branch and

• C ₀ should minimize the costs function of the bank ^' s company. Consisting on the following steps:

1 . Write down the constants which the fixed costs of the branch depends on.

These are numerical constants specific for each banking branch, and they are known by the branch staff. These are

^■ The cost per transfer (y) and the total expected number of transfers per day.

^■ The variance of daily changes in the cash balance (A).

^■ The daily rate of interest earned on portfolio, e.g. , other banks producís which yield higher benefits (υ).

^■ The insurance costs, e.g. , the payment on the bank ^' s company to a theft insurance policy. This cost is directly proportional to the máximum amount of money allowed by the bank ^' s company for any particular branch (B).

2. Compute the average of daily withdrawals.

3. Compute the average daily quantity taken off from bank branch.

4. Compute the product of both quantities 2 and 3, E[X _X ]

5. Compute the average of daily deposits

6. Compute the average daily quantity taken into the bank branch.

7. Compute the product of both quantities 5 and 6, E[Y _t ]

8. Write down the expected branch expenses, K.

9. Compute C ₀ = E[X _± ] - E[Y _± ] + K.

Instructions from the bank's company to its branches

As we mentioned before, the first aspect of the invention is a method to optimize the bank cash inventories across its branches. The method accurately adjusts the demand for money to the real needs of the branches, avoiding a surplus of money and, henee, minimizing the opportunity costs.

For this, the method objective is that the bank's company must impose some restrictions to its branches cash movements. These restrictions are imposed in the shape of some formulae which help the branches to calcúlate their needs of cash, before they make a cash requirement to the cash central.

This section is devoted to rewriting these restrictions, designed as formulae, which the bank's company imposes to its branches as the new mechanism to calcúlate the required cash.

The branches cash needs, C ₀ , are quite simple to define in view of the deposits and withdrawals of money: they are equal to the total amount of expenditures minus the total amount of deposits (expected and unexpected). Let us associate the expected quantities of money as well as the unexpected ones in separated addends, since both addends exhibit opposite qualities. Thus, C ₀ = unexpected expenses - unexpected deposits + expected expenses + expected deposits). C ₀ = £¾] - £Ί¾] + K, assuming that both expected expenses and expected deposits are constant since they could be anticipated because they do not exhibit any random features.

In the particular case in which the distribution of X and Y are a Compound Poisson: C _Q = Xi — Y- + K .

Another reason for leaving aside of the model the quantity K derived from expected movements of money, is that this may be considered as part of the security cash /eve/ (like settlement accounts) that each bank holds in order to fulfill the clients demand up to a certain confidence level for precautionary motives. As we mentioned in previous sections, the total amount of cash which the bank holds for security reasons (K is part of this) depends mainly on the internal normative (or rules of functioning) of each bank.

Error of the approximation Let us return then to the task of estimating the unexpected cash moves of the branch, X ₁ and Y _t . If we approximate each random variable X ₁ and Y ₁ by its mean, we have that C ₀ = E[X _± ] - E[Y _± ] + K.

Then, using the specifications made for f l and in the former section, the above formula turns out to be C ₀ = Average number of withdrawals per considered period of time by Average quantity taken off rom bank branch per considered period of time + Average number of deposits per considered period of time by Average quantity taken into the bank branch per considered period of time + K. Thus, this formula is the new mechanism of control that the bank's company must impose in its branches in substitution of their oíd methods, based on historical data. The introduction of this formula must be imposed as part of the program of minimizing costs for internal cash flow. In the light of this, we emphasize that this formula is very simple itself as well as very simple to manage with.

Let us make now an slight approach to the adequacy of the model for different branch sizes, by evaluating the error committed in the approximation used before on the random variables X ₁ and Y _t . The previous formula is constructed by means of the stochastic variables, X _t and Y _t (the so called withdrawal process and the deposit process, respectively) which captures both random processes of withdrawing and incoming money, for a random number of users, N(t). Moreover, in order to transform into determinist a random problem as far as possible, we have used the approximation of the stochastic variables X _t and Y _t by their means

E[X _± ] = X _t and E[Y _± ] = Y _t

As any approximation, this carries a little error of calculus. We are now going to examine the committed mistake of approximation as well as to interpret this for different branch sizes. Intuitively, the notion of branch size is identified with the volume of its turnover. Actually, there are many criteria to quantify the size of a branch: the volume of its credits, the number of its clients, the number of its staff or the volume of its deposits, among others. In practice (i.e., among bank managers), the most accepted criterion to measure the size of a branch is simply to quantify its total needs of cas/7 and then, to state that branch size as an increasing function of these needs. In short, bank managers assume that the bigger branch sizes correspond to the mayor needs of cash of the branch, related to mayor moves of money (enfries and exits).

Let's R defined as R = £¾] - £Ί¾] + K. R represents the total needs of cash of the branch. For the reasons exposed before, we will assume from now on that the parameter R quantifies specifically the size of the branch. As a parameter, R takes valúes on [0, +∞), although for simplicity, in practice ? is considered as belonging to one of the three categories: small, médium and big size.

Then, regarded simply as the branch size, the parameter R w \\ allow us to calíbrate for which sizes our model is more or less accurate. Recall that R is the second indicator we define since actually we introduced the parameter μ as an índex for measuring the cash flow fluctuations of the branch.

Let's m ^* be the optimal valué for e(C ₀ , C _z ) , x = C ₀ - C _z and y = C _z .

Let us observe that m is function of y, m = m(y), via the objective function

A x + 2y

Ύ—+ ϋ—— + By

xy 3

and with the change x = R - y, that is,

A v

m y) = Y r _R _ _} + 3 + y) + By,

so we may calcúlate ^ = ^ This partial derivative is always positive since this is the result of algebraic operations (sum and product) among positive quantities. This shows that the optimal valué m ^* is increasing on the optimal policy y ^* . Let us note that this fact, m ^* is increasing on my ^* is not related with the parameters of the model but it is simply a logical consequence of the model: mayor will be this upper bound (máximum of cash) C , higher will be the optimal valué for costs. This statement should be kept in mind for the bank companies when deciding (or not) to add the "bit more" over C _z for precautionary motives according with their internal rules of functioning, independently either of the size of the branches or the fluctuations of their cash balance. On the other hand, the function m() may also be regarded as a function of R:

dm dm

Henceforth we may calcúlate „

which could be positive or negative, depending on the valúes of both -^— _t and -.

x y 3

We note that we have the following chain of logical implications:

When x ^*2 y ^* → 0 = >∞ => ^ < 0

x* ² y* dR

And when x ^*2 y ^* →

^J ∞ = x* ² y* > o => ^ dR > 0

On the other hand, it is not difficult to see that ² y is increasing on y since - x ² > 0

Thus, combining both results, as y = C _z , we have that

When C _z → 0, then m ^* is decreasing on R.

When C _z → oo, then m ^* is increasing on R.

At this point, we return to the fact that C _z is fixed by the bank's company for each branch (or ATM), attending some parameters, particularly, the size of the branch. Although other considerations (like geographic location) may influence, in the reality C _z is fixed by the bank company in a directly proportional way to the branch size: higher is the branch size, bigger will be C _z . That is, if R / => C _z /, where the symbol / means the preceding variable increase.

So, this result states that for small branches (branches for which size R takes small valúes) the optimal valué for branch costs m ^* decreases in despite of i? increases. This may be interpreted in the sense that the bank company can expected to have the same optimal valué for costs m ^* for all branches of this category, even for branches with sizes in a neighborhood of small size, that is, for small/medium branches.

However, the opposite phenomena occurs for the category of big branches: dealing with them, the bank company may expect that the optimal costs will blow up as the size of the branches increase. Now we'll discuss the features of the Lagrange multiplier associated to this problem. This will allow us to state some complementan/ properties of the índex i?.

With the variable shift x = C ₀ - C _z and y = C _z , and substituting x = £¾] - £Ί¾] + K - y , the optimization problem can be written as

Minimize:

m(fi) = _γ — ₊ - (χ + 2y) + By

xy 3

such as

x + y = R and x≤ 0

This minimization problem can be resolved this by applying the Lagrangian method. In this context, let us see then which features the associated Lagrange multiplier exhibits and which economic implications could be derived from this. When the objective is to minimize the cost function subject to the "output constraint", it is well known that the Lagrange multiplier turns out to be the marginal cost of "production" that is, the increase in total costs when one more unit of output is permitted. Note that the "output" in our constrain corresponds to the branch needs of cash (or alternatively, the branch size), R.

If we define the Lagrangian as L(x, y; Ϊ) = y— + - (x + 2y) + By - l(x + y - R), the xy 3

equations given by this Lagrange multiplier method are

dL vA 1 v

ox y x ^¿ 3

dL vA 1 2v

oy x y ^¿ 3

dL

- = x + y = fi

where the multiplier l has an economic interpretation. As it is well known,

dm ^*

l =

dR

represents the rate of change of the optimal return point = £(C _Q , C , ) when R increases one unit.

From the first and second equations, we have that vA 2v

l = ^ +— + B

yx ^¿ 3 From algebraic computation, it is clear that l is increasing function both xy ² and yx ² , both of then increasing functions of R. So.

dm ^* dm ^*

R _/ => / _/ => -== _/ <=>—— _/

dR dR

As a consequence of this,

dm*

^1Ím ñ→co ~~dR ⁼ +°° . ^anC '

dm ^*

The economical interpretation of this result is the following: this outcome states that, as the branch size increases, the rate of changes in the optimum for branch costs increases as well. This means that the situation of branch costs deteriorates at major speed when dealing with branches of big size. All the contrary, the situation of possible changes on optimal costs for the branch of small size diminishes (improves) for small valúes of i?.

The fluctuations of the cash flow

Now, we analyze the influence of the branch or ATM users behavior as it indicates how active the branch users are in form of fluctuations of the branch cash balance. This behavior is represented by the páramete^, and we analyze its influence on the behavior of the optimal return point m ^* .

Let us start then with some remarks about the meaning of the parameter μ.

Recall that μ appears at the first addend of the objective function

^ , is the total number of stops in the period of time where A = μ ² ¿.denotes the variance of changes in the cash balance in a determined period of time and μ denotes the amount of euros that the branch cash balance increases or decreases in some small fraction of a working day 1/t.

Recall also that, in practice, μ is strongly directly related with the geographic location of the branch since the fluctuations of the cash balance -in the major part of the cases- are caused by somep/7ys/ ^' ca/ reason (in a similar way that a physical phenomena disturbs the state of rest of any object, changing this initial síafe of rest -with a graphical representation cióse to a straight line- into a motion sfate-represented by a periodic wave motion, bigger or smaller depending on the intensity of the disturbs). In our particular context, this physical phenomena which disturbs the branch cash flow may be identified with the proximity to a cash center, in the wide sense of this expression: that is, understanding that a cash center is an industrial, financial, shopping center or whatever other physical zone where the moves of money are higher than usual.

Thus, for our purposes of calibrating the model, will quantify the fluctuations in the branch cash flow and μ e [0, +∞). However, due to the direct relationship exposed before between the fluctuations of the cash flow and the physical reason that causes the fluctuations (cash center), μ could be interpreted as geographic location of the branch -in reference to a cash center- as well. As

™ ^* = Y _(R _ y* _)y * + ( ^β + y ^* + ^B y ^*> in a similar way, m ^* may be regarded as a function of μ:

m* = _(μ) = γ— — _μ 2 + 1 _{(R + yl + By}

From this expression it can be derived that

df 2vt

= -^—μ < 0

ομ x ^* y ^*

As a consequence of this, m ^* is decreasing on μ in all cases.

In contrast with the results based on the parameter R, where the result clearly depends on the valúes of R, the case of the parameter μ can be written in absolute terms: for all the valúes of μ the optimal valué for branch costs m ^* is always decreasing on μ. That may be interpreted as, if the fluctuations in the branch cash balance increases, the optimum for branch costs decreases: specifically, the situation of branch costs improves when dealing with branches of big fluctuations in the cash flow, that is, when dealing with branches closely located near to cash centers.