The future of branch cash holdings management is here: New Markov chains

Accepted Manuscript The future of Branch Cash Holdings Management is here: New Markov chains Julia Garc´ ıa Cabello PII: S0377-2217(16)30918-3 DOI: 10.1016/j.ejor.2016.11.012 Reference: EOR 14091 To appear in: European Journal of Operational Research Received date: 10 February 2016 Revised date: 27 September 2016 Accepted date: 4 November 2016 Please cite this article as: Julia Garc´ ıa Cabello, The future of Branch Cash Holdings Management is here: New Markov chains, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.11.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Highlights •This paper provides a new methodology: Markov Chains by blocks. •This would achieve knowledge on the branch cash holdings. •We study conditions for optimal cash holdings and their steady-states using Ergodicy. •These findings will also let bank managers know the time validity of the cash holdings •This incipient mathematical framework may also apply to other contexts. 1 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT The future of Branch Cash Holdings Management is here: New Markov chains Julia García Cabelloa,∗ aDpto. de Matemática Aplicada, Facultad Ciencias Económicas y Empresariales. Universidad de Granada. Spain. Abstract Liquidity management is one of the main concerns of the banking sector since it provides control in key areas such as treasury management, working capital financing and business valuation. Under the assumption that branch efficiency makes a fundamental contribution towards the effective performance of the global banking institution, this paper provides a new methodology (Markov Chains by blocks) in order to achieve knowledge on the branch cash holdings: conditions which ensure optimal cash holdings, recurring properties which help to better predict cash holdings shifts and the study of the branch cash holdings steady-states using Ergodic Theory. These findings will let bank managers know the time validity of the current cash holdings. This is a crucial advantage to ensure efficient cash management: while helping keep banking institutions on sound financial footing by guaranteeing the compulsory-by-law safety cushion, it also allows bank managers to make sound decisions upon fund investments. This incipient mathematical framework, based on the re-definition of classical theory on Markov chains, provides an alternative standpoint which may also apply to those dynamical systems which can be categorized into groups of similar features. Keywords: (D) Economics; Markov chains by blocks; Optimal cash balance; Time validity of the cash holdings; Ergodic Theory JEL classification: C44; C58; G10; G21 1. Introduction and Literature Review Corporate/bank cash holdings have always played a crucial role in the development of firms and financial institutions: without cash, they could both become insolvent and at risk of bankruptcy. Thus, efficient cash administration has traditionally focused the attention of managers and shareholders, especially during periods of uncertain market and credit conditions. In this regard, an accurate cash balance forecast is critical for successful management while also serving other strategic purposes such as controlling subsidiary groups. The banking industry has been in search of managerial measures to improve the control of its liquid resources in order to increase efficiency. While efficiency on all fronts (including cash management) has become a primary objective for banking industry over the last decade, there is a body of research which argues that branches have a role to play in helping to improve global bank institution performance. This was firstly suggested in Berger (1997), whose authors stressed the importance of the efficiency of branches as making a fundamental contribution towards the effective performance of the global banking institution. Moreover, the authors of Berger (1997) called attention to the fact that branch efficiency literature is much less complete than banking efficiency literature. As a matter of fact, specific literature to design techniques to improve branching ∗Corresponding author. Address: Dr. Julia García Cabello. Dpto. de Matemática Aplicada, Facultad de C.C.E.E. y Empresariales, Campus de Cartuja, Granada 18071, Spain. URL: [email protected]. (Julia García Cabello) Preprint submitted to Elsevier November 15, 2016 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT performance as far as cash management is concerned is quite short1apart from those papers which focus on regulatory measures to control under-performing branches. The present paper attempts to help to fill this gap by proposing specific conditions to improve cash (forecasting) management at branch level. In the area of management of corporate cash holdings, there have been a long series of attempts to determine the optimal investments that organizations should make in cash. Models of cash management or money demand can be categorized into two types: those with demand by households, pioneered by the Baumol-Tobin model, Baumol (1952) and continued by Frenkel and Jovanovic, (1980), Bar-Illan, (1990) and Chang, (1999) and those which concern cash management by firms, pioneered by the paper of Miller and Orr, (1966). Firms differ from households in that firms have daily cash inflow as well as daily expenditures. Also the size of financial transactions differentiates firms from households, as large and instantaneous transactions are more likely. In terms of their mathematical structures, the first describes the money stock between controls by means of Brownian motion with drift whereas Miller and Orr formulated a model under which an organization’s cash flow evolves in terms of a stationary random walk. Later proposals feature unified analysis of cash management by combining Brownian motion and compound Poisson processes, as in Bar- Illan (2004). Other authors classify cash management models according to their mathematical fundamentals. Following Melo (2011), these models can be grouped into Inventory Theory models (now both Baumol-Tobin and Miller-Orr belong to the same category), those developed with Linear Programming and those which are based upon Dynamic Programming. Further papers incorporate stochastic techniques in their patterns of cash management: Baccarin (2009) considers the optimal control of a multidimensional cash management system where the cash balances fluctuate as a homogeneous diffusion process in Rn. Cyert (1962) pioneered research using Markov chains for estimating the allowance of doubtful accounts while Hinderer (2001) analyzed a cash management system in which the distribution of the cash flow depends on a randomly varying environment. Ferstl and Weissensteiner (2008) considers a cash management problem by using a multi-stage stochastic linear program (SLP). In Higson (2010), the authors model the evolution of cash in terms of a square root process modified with a Brownian motion in such a way that the statistical properties of the cash flow process depend on the cash holdings. This functional relationship between cash holdings and cash flow process by means of Hamilton-Jacobi-Bellman equations and other elements of optimal control theory is the core qualitative finding of Anderson (2012). In Bensoussan (2009) the author uses a stochastic maximum principle to obtain an optimal transaction policy. In Sato (2011), a cash management model is built around Brownian motions and Poissson processes while Song (2013) discuss a cash management model for firms based on a stochastic volatility (SV) model. And more recently, Tangsucheeva (2014), where a cash flow forecasting model is developed in terms of Markov chains and bayesian models. However, there is little current literature about this subject for the banking industry, whose specific characteristics differ from firms and other economic organizations. While credit lines are freely available for the banking industry, the private sector must apply for external financing within the framework of those credit channels that are accessible to it. Actually, the relationship between cash holdings and credit risk is present in most former models of cash management: see Acharya (2012), where a dynamic continuous-time model allows for a description of the correlation between credit spreads and cash reserves. Or Anderson (2012), where the authors develop a model of optimal policy toward holding the liquid assets of a firm which faces external financing. Banks differ from firms also in the peculiar dynamics of their cash flow processes. Assorted entries of cash are specific to banks: at the branch level, as daily expected and unexpected deposits and withdrawals, similar to ATM dynamics, while at the aggregate level, large/huge transactions take place as a consequence 1“Short” unless the strand of research focused on improving the performance of automatic teller machines, ATMs, would be considered as part of the literature to improve branch cash management. 3 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT of movements of money amongst bank entities. Distinct regulation packages are also applied in order to control banking industry versus private sector. The ultimate aim as far as liquidity management is concerned is to find the optimal level of cash since it would help banking institutions facing short-term obligations at the aggregate and branch level, while minimizing the risk of bankruptcy in long-term projections. This “optimal” level of cash may be also read as “enough” cash. However, finance literature has given very little precise guidance on this question: how much money is enough for a banking institution? Common knowledge suggests that banks that have larger liquid assets should be safer. However, when banking firms keep liquid resources in cash, they renounce a part of their profitability, incurring the opportunity costs of not investing in other alternatives which do generate profits. Thus, the intuition recommends that a balance between minimizing costs and maximizing profits should be kept in order to ensure high levels of efficiency. But, how do the banks identify the right proportion to be held? This paper attempts to fill this gap by providing answers to the above questions in the context of bank branches. Stated briefly, the contributions of this paper are twofold: a deep study of the branch cash holdings from a dynamic point of view (first contribution) through an approach based on new stochastic financial analytics that we have developed in this paper (second contribution). Actually, this paper provides a new methodology (Markov Chains by blocks) in order to achieve knowledge on the branch cash holdings with proposals to be approached from a variety of perspectives: i)conditions which ensure optimal cash holdings, ii)recurring properties of the branch cash holdings derived from the natural cyclicity exhibited in the branch cash management practice which help to better predict their shifts and iii)the study of the branch cash holdings steady-states using Ergodic Theory, Braido (2013), which let bank managers know the time validity of the current cash holdings. In general, we find policies on holding optimal levels of liquid assets aimed at being useful for both bank and branch managers. These conditions are crucial to ensure efficient cash management: while helping keep banking institutions on sound financial footing by guaranteeing the compulsory-by-law safety cushion, it also allows bank managers to make sound decisions upon fund investments. As far as the author knows, this is the first time in the literature that such an analysis on cash holdings at the branch level has been carried out from a dynamic point of view. This incipient mathematical framework, based on the re-definition of classical theory on Markov chains, provides an alternative standpoint which may also apply to those dynamical systems which can be categorized into groups of similar features. The remainder of the paper is organized as follows. In Section 2, an overview of clustering methods (identification of similarities) is presented since the cyclicity that exists in the branch cash management practices (which is at the heart of our study) relies on the idea of grouping the weeks into blocks of weeks with similar features. Section 3 presents the general framework of the liquids funds of a branch. Section 4 is aimed at setting conditions to ensure optimal cash holdings. Recurring properties on branch cash holdings are presented in section 5, while time validity on cash holdings is analyzed in depth in section 6. Section 7 contains a numerical example, based on real data for a branched-bank. Finally, Section 8 concludes the paper. 2. Clustering: related approaches As mentioned before, one of the key insights of the paper is the natural cyclicity exhibited in the branch cash management practice: in detail, that refers to the usual branch managers’ partition of the year into blocks of weeks with similar features in order to require the same amount of case for all weeks inside the same block2. Around this idea, a new methodology -called Markov Chains by blocksis developed in this paper: specifically, the whole temporal sequence of branch cash holdings, 2For instance, a block of weeks is “first weeks of each month”. 4 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT which is previously shown to be a Markov chain, is partitioned into blocks such that this partition is well correlated with the partition of the year by branch managers3. This new methodology is aimed at providing knowledge on branch cash holdings. The practice of grouping weeks is close to clustering. As a matter of fact, following Manning and Schütze (2000), clustering is the process of partitioning a set of objects into groups or clusters. In the financial sector, where data are generated on large scale, the clustering can be improved by pattern recognition and data mining techniques. Although this paper’ approach is heading in a different direction (Markov Chains by blocks and the application of Ergodic Theory), a review on distance based methods for clustering might be of interest. Since clustering is the grouping of similar objects, some kind of measure which may conclude whether two objects are similar or not, is required. In global terms, two kinds of measures may be used to estimate this relationship: distance measures (which result in distance-based methods) and similarity measures (giving rise to similarity functions like the well-known Pearson correlation measure). Distance-based methods run on the basis of the shorter the distance between objects, the more similar to one another they are. These methods may be categorized according to the distance taken, which would vary depending on the type of attributes of data: numeric attributes (the similarity between two data instances may be calculated using the Minkowski metric, with the well-known Euclidean distance as a particular case), binary attributes (the distance between objects may be calculated through contingency tables), or other types as nominal, ordinal or mixed-type attributes for which specific definitions of distance are required. Together with a vast literature on clustering methods, there are also many criteria upon which they could be categorized. Mainly, they may be divided into hierarchical and partitional clustering, based on the way they produce the results, see Fraley and Raftery (1998): specifically, hierarchical methods construct the clusters by recursively partitioning the instances while partitional ones relocate instances by moving them from one cluster to another, starting from an initial partitioning. A more comprehensive classification, see Han and Kamber (2011), categorizes them into hierarchical’, partitional’, density-based methods (which assume that the points that belong to each cluster are drawn from a specific probability distribution, see Banfield and Raftery (1993)), model-based methods (which attempt to optimize the fit between the given data and some mathematical models), grid-based methods (which partition the space into a finite number of cells that form a grid structure on which all of the operations for clustering are performed, see Han and Kamber, (2011)) and finally soft-computing methods (with fuzzy clustering as main exponent). One example of model-based methods are Markov Mixture Models (MMM), which was firstly analyzed by Chib (1996) as models of a class of mixture distributions in which the component populations, from one observation to the next, were selected according to an unobserved Markov process. In the context of clustering, this is an approach that uses a Markov model to represent the data in each of the clusters: e.g, if there are two clusters, they should be represented by two different Markov models. The approach suggested in this paper is different from its conception to its implementation, as proved in next sections: firstly, the whole sequence of branch cash holdings, denoted as {CHn}n∈N, is shown to constitute a discrete-time Markov chain (Theorem 4.4). While it is shown not to be irreducible (hence, Ergodic Theory does not apply), a suitable definition of a new equivalent relation over the set of states of the Markov chain {CHn}n∈Nis proposed (equivalent Definitions 5.1 and 5.2) aimed at achieving irreducible chains. Subsequently, Theorem 5.6 establishes the correlation between the partition on the whole sequence of branch cash holdings originated by the definitions cited above, and the partition of the year as a result of the branch managers’ practices. Now, Ergodic Theory applies to these equivalent classes of cash holdings (called blocks) providing knowledge on them. 3This new methodology also applies to any dynamical system {Xn}n∈Nwhich holds the Markov property, see Conclusion section for further details. 5 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT 3. Dynamics of the liquids funds of a branch This section is devoted to stating the dynamics of the liquids funds of any branch. To this regard, we start with a few words on the functioning of the branches. Every day, branches perform numerous transactions which cause cash inflows and outflows. Each branch should keep its total liquid assets (known as cash holdings) at an optimal level, without generating either a surplus or a shortage of money. It should not be too low, in order to refrain from insolvency and to adhere to minimum capital regulations. But it should not either be too high so the branch would not have security problems due to heavy cash load or the bank would not suffer from opportunity cost (i.e., the opportunity loss of not investing in other alternatives which do generate profit). Hence, periodically, the branch adjusts its cash levels to its necessities -deposits and withdrawalsavoiding generating dormant money. To accomplish this task, the branch requires help from its cash central. Thus, an armoured van either evacuates the surplus or provides the deficit of cash up to reach a confident level of case. As for this “confident level of cash”, it should be mentioned that every branch has a cash upper bound fixed by the bank company as an internal control mechanism. This cash upper bound is assigned according to the branch size and it will be denoted by Cmax (i.e., maximum cash allowed to be held by the branch). The cash entries are the following: the own branch requests of cash to the cash central and the deposits made by individual users/companies. As far as the branch cash expenses are concerned, these include withdrawals and other costs which mainly consist of logistic costs for transport and handling as well as opportunity costs4. The branch movements of cash are reflected in Figure 1. ’ ← ← Weekly request of cash to the cash central Remaining money from previous week Deposits made by bank branch users ↓↓ Weekly Cash Holdings Expected Expenses Unexpected Expenses ↓ ↓ Figure 1 The dynamic of the branch cash holdings Let Cstands for the total amount of money that the branch requests from its cash central. This quantity Cis adjusted weekly5to the cash necessities. One of the aims as far as liquidity management at the branch level is concerned is to find the optimal amount Cwhich covers all branch expenses without generating either surplus or shortage of money. Habitual bank branches cash management routines as far as this computation is concerned consist of historical data handling. That means that the branch registers the cash quantity on some particular day (workable, weekend, holidays, etc) and the result obtained at the end of the journey (exceed or shortage of case) and copies the successful amounts. In the process of the decision-making, often the staff in charge reaches a decision with only partial information. The author of the present paper developed in García Cabello (2013)6a mathematical procedure in order to compute Csuch that Ccovers the 4Some of these costs can be anticipated and others are of a random nature. The same classification can be established for deposits. 5This unit of time -one weekmay be changed without loss of generality. 6A patent has been requested for the paper García Cabello (2013) by the University of Granada, “Method for managing liquidity in bank branches”, number ES201431094, United States 6 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT branch demand of cash as accurate as possible -i.e., without generating either a surplus or a shortage of money. This procedure is summarized below. Let Nbe the number of branch costumers during the considered period of time. Such arrival processes are described by a Poisson process N(t)or Nt. The counter tells the number of arrivals in the interval (0, t). The withdrawals and deposits movements are described as Nt=Nw t+Nd t+Ot, where Nw tstands for the number of withdrawals in the interval (0, t),Nd tis the number of deposits made in the interval (0, t)and Otgathers the rest of operations. Let Wirepresent the withdrawn amount made by i-branch user. The withdrawal process, parameterized by certain rate λ, is defined as the compound Poisson process Xt:= PNw t i=1 Wi, as independent and identically distributed (i.i.d.) random variables. Thus, the total amount of money which has been withdrawn for t= 1 is X1=PNw i=1 Wi. Then, in García Cabello (2013), some formula to compute the branch total expected withdrawals for a unit of time, EX1, and the branch total expected deposits for a unit of time, EY1, were given. Moreover, the key result of García Cabello (2013) is the following: Theorem 3.1 (García Cabello (2013)). Let K7be both branch expected expenses/deposits for a unit of time. The total amount of cash Cwhich will cover the branch demand of cash, C, may be computed as C=EX1−EY1+K C≤Cmax,(1) since no amount of cash should exceed the branch cash upper bound Cmax. Let CH stands for the branch cash holdings at some moment, that is, the total liquid assets to be held by the branch at some moment. Hence CH includes C.The main objective of this paper is to study CH with proposals to be approached from a variety of perspectives. To carry out this analysis, we shall use superscripts to point out a certain period of time (discrete-time). Hence, CHn are the branch cash holdings at week n; similary, Cnstands for the total amount of money that the branch requests from its cash central at week n. To start this study, let us first formalize the concept of size of a branch. The notion of branch size8is intuitively identified with the volume of its turnover. However, there are many criteria which quantify the size of a branch amongst bank managers. The most accepted is to consider size of a branch as increasing in function with the total branch cash needs: the bigger branch sizes correspond to the bigger branch cash needs, related to mayor larger moves -entries and exits- of liquid resources. Thus, we will identify branch size with the maximum value of branch cash holdings during some period of time: Definition 3.2. BS =branch size =max{CHn/n ∈N}. Note that the foregoing Cmax always performs under the size of the branch: Cmax ≤BS. 4. Optimal branch cash holdings This section is devoted to determining conditions which ensure optimal cash holdings for each branch. This involves a suitable definition of optimal cash holdings in terms of technical efficiency. Firstly, we specifically define branch weekly cash holdings following former Figure 1. Definition 4.1. For any bank branch, we define its cash holdings at the n-th week as CHn=Cn+Rn−1(2) where Rn−1stands for the remaining money from previous week. We shall write CH =C+Rwhen the week no longer needs to be emphasized. 7Kmay be considered as part of the security cash level (settlement accounts) that the banking institutions holds for precautionary reasons. 8Bank managers apply this term as benchmark to position the branches with respect to each other for several different purposes. 7 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Remark 4.2. Since Cis computed taking into account all branch cash entries and expenditures (see García Cabello (2013) for further details), if the computation of Cnwould be completely accurate, the remaining money from previous week Rn−1would be 0. That is, Ris the error committed in the computation of the cash requested to the branch cash central C. In consequence, Rn= se[Cn] := √V arCncould be considered as function of C. Hence, equation (2) CHn=Cn+Rn−1 could perform like a dynamical system provided initial and/or boundary conditions are given. Furthermore, according to the notion of technical efficiency (the maximum output produced from the minimum quantity of inputs), the optimal cash holdings should be those which cover all branch expenses without (or producing the minimum of) remaining money, as follows: Definition 4.3. For any bank branch, we define its optimal cash holdings CH∗as that which holds CH =C+Rwith almost no committed error (negligible) in the calculation of C. That is to say, CH∗≈C. 4.1. Main features of the cash holdings CH Before studying the main features of the branch cash holdings {CHn}n∈N, let us point out a few properties of the branch cash requests {Cn}n∈N. Firstly, {Cn}n∈Nconstitute a random walk. Moreover, as the set of states of Cnrepresents the set of cash amounts which may be required by branch managers, while Cnwould in principle be allowed to take values in R, in practice the set of states of Cnmay be considered finite, for every n. Indeed, branch managers envision only a few quantities to be required to central hubs when weekly adjusting branch cash needs, which vary depending on the characteristics of each week. This is the result of branch managers’ practice of categorizing the weeks according to their specific features in order to simplify the cash requirements. Let {Cnk, k = 1,2...,m}n∈Nbe the set of all quantities of cash which could be required to the central hub at the nth-week. These are illustrated in Figure 2. We proceed now with the study of {CHn}n∈N. The main result is the following: Theorem 4.4. The branch cash holdings constitute a discrete-time Markov chain {CHn}n∈N. Proof. CHn=Cn+Rn−1, where Cnis computed by Theorem 3.1 and the committed error in the computation is given by Rn=se[Cn] := √V arCn. Substituting Rn−1=se[Cn−1]into equation CHn=Cn+Rn−1would give CHn=Cn+se[Cn−1]which shows that every outcome CHndepends on the outcome of previous step. In consequence, the Markov property holds. As for the state space, the Markov chain {CHn}n∈Ntakes values in R. Nevertheless, while the set of states of {Cn}n∈Ncould be considered finite for the branch managers’s practice, the set of states of the Markov chain {CHn}n∈Nis not finite. Let {CHnk, k ∈N}be the set of feasible states of the Markov chain {CHn}n∈Nat week n. This is shown in Figure 3:. BS Cmax C1mC2mC3m. . . Cnm ... . . .. . .. . .. . . C12 C22 C32 . . . Cn2... C11 C21 C31 . . . Cn1... k k k k C1C2C3. . . Cn... Figure 2 The sequential structure of requests of cash C BS . . .. . .. . . CH1mCH2m. . . CHnm . . .. . .. . . CH12 CH22 . . . CHn2 CH11 CH21 . . . CHn1 k k k CH1CH2. . . CHn Figure 3 The sequential structure of branch cash holdings CH 8 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT which the branch cash holdings remain constant. It should be applied to chains of cash holdings corresponding to weeks into the same block (irreducible). We adopt here the notation Vi(n)to represent the number of visits to state ibefore step n. This theorem states that the long-run value of the ratio Vi(n)/n, which is the proportion of time spent in state ibefore step n, equals to the inverse of the expected return time to state i. Theorem 6.5 (Ergodic Theorem). For any irreducible Markov chain {Sn}, then limn→∞ Vi(n) n=1 mi .(6) The dominant interpretation of the Ergodic Theorem is that 1/miis the average time of permanence at state i. In our context, Ergodic Theorem provides thus conditions under which the cash holdings remain constant although this should be applied to each block since the Markov chain CHnn∈Nis not irreducible under the classical definitions. Thereby, the average amount of time that the Markov chain stays on state iequals to the inverse of the expected number of steps in return process to state i. That is to say, Theorem 6.6 (Ergodic Theorem for branches). For those weeks into the same block, the average time validity of the current cash holdings CHiis equal to gcd{i, d(i)} d(i).(7) Proof. It is sufficies to apply previous Ergodic Theorem, 6.5, toghether with the result achieved at Theorem 6.4. Next result can also be demonstrated: Theorem 6.7 (The average time validity). The average time validity of the current cash holdings is a class property. Proof. Let ibe a week and consider any other jsuch that [i] = [j], i.e., j=i+α·d(i),for some α∈ Z. Besides, the period of a state i,d(i), is a class property: this implies in turn that d(i) = d(j). From basic Number Theory, gcd{b+α·a, a}=gcd{b, a}. Hence, gcd{j, d(i)}=gcd{i+α·d(i), d(i)}=gcd{i, d(i)} ⇒ gcd{j, d(i)} d(i)=gcd{i, d(i)} d(i). 7. A numerical example This section is aimed at developing a case in point -intended for illustrative purposeswhich should show how the previous theorems may be run. This numerical example, supported by real banking records based on data transactions, should highlight the real gain for entities who adopt the proposed method as a complement for IT technologies9since it may be easily (and at no cost) converted into an algorithm10. 9The main features of this proposal -it is precise and very simple to be implemented at daily branch practices, assuring costs reductionswould allow it to co-exist with IT technologies, providing extra-support for branch managers’ decisions. 10This result shall be addressed in a forthcoming paper. 15 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT In order to carry out this task, real banking information has been processed. The dataset is based upon excel files that contain all daily branch operations from June to December 2012 of some representative Spanish branch of a well known Spanish bank11. Despite our initial database was originally written using the entity’s specific code, significant external operations have been extracted/separated from those internal organizational orders (accounting entries) as part of the database processing. The general procedure is based on the following steps: 1. Branch managers select a criteria for grouping the weeks. Once again (see former example 5.4), we select “first weeks of each month”, since it is amongst the customary criteria applied by branch managers. 2. Once the criterion is picked up, the year becomes partitioned into blocks of weeks. According to the “first weeks of each month” criterion, the period of a state iis d(i) = 4 and the quotient set is Z4={[0],[1],[2],[3]}, where each block [i]is as follows (see former Figure 4): [0] ={0,4,8...}=last weeks of month [1] ={1,5,9...}=first weeks of month [2] ={2,6,10 ...} [3] ={3,7,11 ...} 3. For those weeks inside the block of “first weeks of each month”, we will compute the average time validity of the current cash holdings CHi, according to Theorem 6.6. Next table shows how time only moves along the block of “first weeks of each month”, with a week as time unit: 2012 Jan Feb March April May June July Aug Sep Oct Nov Dec First weeks ✓ ✓ ✓ ✓ ✓ ✓ ✓ Second weeks Third weeks Last weeks Table 1: Average time validity of the cash holdings. As the time validity should be the same for those weeks inside the same block, according to Theorem 6.7, we only compute this for June. This is as follows: June 2012, i=1: gcd{1, d(1)} d(1) =gcd{1,4} 4=1 4week (1,75 days). It also applies for the average time validity of the cash holdings at states 5,9,..., (first week of July is state 5 = 1 + 1 ·4, first week of August is state 9 = 1 + 2 ·4... ) 4. The Theorem 6.6 should be applied now. Recall that this result provides conditions under which the cash holdings CHiremain constant. This sentence may be interpreted as CHido not deviate from the expected value stated for i(standard). These standards for each state iare an inherent feature for each branch. They may be easily computed by bank entities through their own huge amounts of data. Furthermore, for each specific branch, the expertise’ eye of the office director is certainly familiar with them. 11In order to comply with legislation, the name of the bank must be kept private. 16 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Thus, from previous point, cash holdings remain constant 1 4week for those weeks inside the block [1] ={1,5,9...}(=first weeks of month). This can be seen from following figures, which display the comparison between current cash holdings and the corresponding standard for the first week of months from June to December 2012. Grey bars represent real cash holdings and black ones the standards. The xaxis shows days of the corresponding first week (blanks represent weekends or holidays) while cash amounts in Euros appear on the yaxis. Figure 8 First week of June Figure 9 First week of July Figure 10 First week of August 17 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Figure 11 First week of September Figure 12 First week of October Figure 13 First week of November Figure 14 First week of December 8. Conclusions and direction for future research This paper attempts to provide a new methodology by re-defining the classical fundamentals of Markov chains, thereby providing an alternative point of view to the classical. This new theoretical setting is applied for analyzing bank branching cash holdings: conditions which ensure optimality, recurring properties to better predict cash holdings shifts and knowledge about their steady-states in order to envision the time validity of the current cash holdings. The marginal benefit of using the blocks approach -instead of the classical approachis based on the fact that having blocks makes the problem more granular and therefore closer to the desired optimality. Once the real scenario has been mathematically modeled, this new framework offers many chances to continue exploring other possibilities, apart from those employed in the present paper. This incipient perspective may be as fruitful as the classical theory in applying a wide variety of properties to the cause of modeling banking branch cash holdings. The results that we achieved in this paper enjoyed such broad support since they attempt to be suitable for all kind of branches, regardless their size or geographic location. Moreover, this set of theorems may be easily (and at no cost) converted into an algorithm which would co-exist with IT technologies, providing extra-support for branch managers’ decisions: this is a future research project within a foreseeable period of time. These suitability criteria also allowed for its application for different contexts apart from the banking scenario. This is the case for currency exchange offices as well as other settings where liquid provisions have to be made by adjusting monetary exits and entries. The breadth of our mathematical groundwork is an advantage: this new methodology may also apply to contexts where cash is not the star product. Actually, any dynamical system with the Markov property which may be categorized into groups of similar features is a potential application of the defined framework. Specifically, let {Xn}be a dynamical system where the Markov property holds with time variable nto be incremented discretely corresponding to the integers {0,1,2,3,4, ...}(discrete dynamical system holding the Markov property). Besides by hypothesis, any of the states Xnand/or the corresponding time unit nmay be categorized into groups of similar features. Thus, we let definition 18 ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT 5.1 remain the same: two states Xni(shorter state i) and Xnj(state j), corresponding to time units niand njcommunicate, i↔j, if both time units niand njexhibit the same set of features. Next, we set definition 5.2 to be adequate to the required context: we say that Xniand Xnjcommunicate if we put here the desired definition. As a result, not every state Xncommunicates each other, only those which correspond to time units with similar characteristics. Therefore, the set of these constitues a block. Moreover, irreducible subchains in {Xn}n∈Nare those formed by states corresponding to time units inside the same block. Next step is to apply Ergodic Theory to the blocks in order to determine the dynamical system steady-states as well as exploring other possibilities in the desired context. 9. Acknowledgements Acknowledgments Financial support from the excellence project of the Spanish Ministry of Science and Innovation “Mecanismos de resolución de crisis: cambios en el sistema financiero y efectos en la economía real” (P12-SEJ-2463). The author also thanks financial support from the project of the Regional Government of Andalusia “GAMMA (Grupo de Análisis Microeconómico y Macroeconómico Aplicado)” (SEJ340) . References [1] Acharya, V., Davydenko, S.A., Strebulaev, I.A. Cash holdings and credit risk. Review of Financial Studies, 2012, 25-12: 3572-3609. Doi: 10.1093/rfs/hhs106. [2] Anderson, R.W., Caverhill, A. Corporate liquidity and capital structure. Review of Financial Studies, 2012, 25-3: 797-837. Doi: 10.1093/rfs/hhr103. [3] Baccarin, S. Optimal impulse control for a multidimensional cash management system with generalized cost functions. European Journal of Operational Research, 2009, 196: 198-206. Doi: 10.1016/j.ejor.2008.02.040. [4] Bar-Illan, A. 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