Mathematical Modeling of a Cash Concentration and Disbursements System

[ES] Este trabajo presenta un modelo de simulación para un sistema de concentración de fondos y desembolsos (SCFD) visto como un sistema de gestión de inventario, basado en ecuaciones en diferencias y técnicas de ingeniería de sistemas. El modelo asume la existencia de retardos por trámite o traslado bancario y analiza la aplicación del concepto de operación con cuentas de saldo cero. Se plantea el caso de una empresa genérica cuyas agencias o distribuidores geográficamente están dispersos en diferentes regiones. El modelo supone la existencia de una cuenta principal operada centralizadamente y política de saldo mínimo. Esta cuenta recibe las transferencias de los ingresos depositados en las cuentas de ingresos de cada agencia y, también, desde la cuenta principal son transferidos los fondos para cubrir los sobregiros ocasionados en las cuentas de egresos de las agencias. Existe una cuenta de inversión a la cual se transfiere el superávit de efectivo en la cuenta principal y una línea de crédito que cubre los déficits de saldo en esa cuenta. Se definen las reglas de operación del SCFD y se consideran los ingresos y costos involucrados. El modelo representa el flujo del dinero entre los elementos identificados del sistema y el flujo de requerimientos u órdenes de transferencia. Se deriva un modelo equivalente representado por ecuaciones algebraicas utilizando la transformada z con el fin de abrir perspectivas al uso riguroso de técnicas de control en el campo de las finanzas.[EN] This paper presents a simulation model for a cash concentration and disbursements system (CCDS) seen as an inventory management system, based on difference equations and systems engineering techniques. The model assumes the existence of delays due to banking procedures and analyzes the application of the zero balance accounts concept. The case of a generic company whose agencies are geographically distributed in different regions is proposed. The model assumes the existence of a centrally operated main account and minimum balance policy. This account receives money transfers from the revenues accounts of each agency and, also from the main account, money is transferred to the agencies’ expense accounts in order to cover overdrafts. There exist an investment account into which any cash surpluses of the main account are deposited and a credit line in order to avoid the cash deficits. The operating rules for the CCDS are defined, and income and financial costs involved are considered. The model represents the flow of money between the identified elements of the system and the flow of money requirements or transfer orders. An equivalent model represented by algebraic equations through the z-transform is derived, which opens perspectives for using rigorous control techniques in the field of finance.Los autores desean agradecer al Ministerio de Economía y Competitivad su apoyo parcial a este trabajo a través del proyecto DPI2013-47825-C3-1-RHerrera Cáceres, CA.; Ibeas, A. (2016). Modelado Matemático de un Sistema de Concentración de Fondos y Desembolsos. Revista Iberoamericana de Automática e Informática industrial. 13(3):338-349. https://doi.org/10.1016/j.riai.2015.07.008OJS338349133Albanese, C., Jackson, K., Wiberg, P., 2004. A new Fourier transform algorithm for value-at-risk. Quantitative Finance 4(3), 328-338.Anvari, M., 1981. An application of inventory theoretical models to cash collection. Decision Sciences 12, 126-135. DOI: 10.1111/j.1540- 5915.1981.tb00067.x.Anvari, M., 1986. Efficient scheduling of cross-border cash transfers, Financial Management 15(2), 40-49.Anvari, M., 1987. Cash transfer scheduling for concentrating noncentral receipts. Management Science 33(1), 25-38.Anvari, M., Goyal, S. K., 1985. Optimization of the decentralized cash management problem. European Journal of Operational Research 20(2), 198-205.Anvari, M., Mohan, N., 1980. A computerized cash concentration system. Omega-International Journal of Management Science 8(4), 459-464.Baccarin, S., 2009. Optimal impulse control for a multidimensional cash management system with generalized cost functions. European Journal of Operational Research 196(1), 198-206.Bar-Ilan, A., Perry, D., Stadje, W., 2004. A generalized impulse control model of cash management. Journal of Economic Dynamics & Control 28(6), 1013-1033.Baumol, W. J., 1952. The transactions demand for cash: an inventory theoretic approach. Journal of Finance, LXVI.Beyer, D., Sethi, S. P., 1999. The Classical Average-Cost Inventory Models of Iglehart and Veinott-Wagner Revisited. Journal of Optimization Theory and Applications 101(3), 523-555.Boissard, J., 2012. Applications and uses of digital filters in finance. Master's thesis, Swiss Federal Institute of Technology (ETHZ).Buser, S. A., 1986. Laplace Transforms as Present Value Rules: A Note. The Journal of Finance 41(1), 243-247.Cagan, P., 1956. The monetary dynamics of hyperinflation. In M. Friedman (Ed.), Studies in the Quantity Theory of Money. Chicago: University of Chicago Press.Carr, P., Madan, D., 1999. Option valuation using the fast Fourier transform. The Journal of Computational Finance 2(4), 61-73.Carr, P., Wu, L. R., 2004. Time-changed Levy processes and option pricing. Journal of Financial Economics 71(1), 113-141.Cerny, A., 2004. The risk of optimal, continuously rebalanced hedging strategies and its efficient evaluation via Fourier transform. London: Tanaka Business School, Series: Tanaka Business School discussion papers, pp. 1-41, ISSN 1744-6783, http://ssrn.com/abstract=559417.Cerny, A., 2006. Introduction to Fast Fourier Transform in Finance. Cass Business School Research Paper, pp. 1-29.Cerny, A., 2009. Mathematical Techniques in Finance: Tools for Incomplete Markets (Second Edition). Princeton University Press, Princeton and Oxford, p.p. xxii + 390.Cerqueti, R., 2012. Financing policies via stochastic control: a dynamic programming approach. Journal of Global Optimization 53(3), 539-561.Chourdakis, K., 2005. Option pricing using the fractional FFT. The Journal of Computational Finance 8(2), 1-18.Christev, A., 2005. The hyperinflation model of money demand: Some new empirical evidence from the 1990s. Centre for Economic Reform and Transformation, pp. 1-34.Cui, X. Y., Gong, L. T., Zhao, X. J., Zou, H. F., 2013. The Z-transform method for multidimensional dynamic economic systems. Applied Economics Letters 20(11), 1081-1088.Dempster, M. A. H., Hong, S. S. G., 2002. Spread option valuation and the fast Fourier transform. Mathematical Finance - Bachelier Congress 2000, Springer Finance, pp. 203-220.Dennis, S. B., 2009. Matrix mathematics: Theory, facts and formulas (Second Edition). Princeton University Press, Princeton, NJ, p.p. xlii+1139.Duffy, D. J., 2006. Finite difference methods in financial engineering: A Partial differential equation approach. John Wiley & Sons, Ltd, 442 pages, ISBN: 978-0-470-85882-0.Eppen, G. D., Fama, E. F., 1968. Solutions for cash-balance and simple dynamic-portfolio problems. Journal of Business 41(1), 94-112.Eppen, G. D., Fama E. F., 1969. Cash balance and simple dynamic portfolio problems with proportional costs. International Economic Review 10(2), 119-133.Fusai, G., Roncoroni, A., 2008. Implementing models in quantitative finance: Methods and cases. Springer, Series: Springer Finance, p.p. xxiii + 607.Fusai, G., Marazzina, D., Marena, M., Ng, M., 2012. Z-transform and preconditioning techniques for option pricing. Quantitative Finance 12(9), 1381-1394.García, C. A., Ibeas, A., Herrera, J., Vilanova, R., 2012. Inventory control for the supply chain: An adaptive control approach based on the identification of the lead-time. Omega-International Journal of Management Science 40(3), 314-327.García Salcedo, C. A., Ibeas Hernández, A., Vilanova, R., Herrera Cuartas, J., 2013. Inventory control of supply chains: Mitigating the bullwhip effect by centralized and decentralized Internal Model Control approaches. European Journal of Operational Research 224(2), 261-272.Girgis, N. M., 1968. Optimal cash balance levels. Management Science 15(3), 130-140.Grubbstrom, R. W., 1998. A net present value approach to safety stocks in planned production. International Journal of Production Economics 56(7), 213-229.Grubbstrom, R. W., Tang, O., 1999. Further developments on safety stocks in an MRP system applying Laplace transforms and input-output analysis. International Journal of Production Economics 60(1), 381-387.Grubbstrom, R. W., Huynh, T. T. T., 2006. Multi-level, multi-stage capacityconstrained production-inventory systems in discrete time with non-zero lead times using MRP theory. International Journal of Production Economics 101(1), 53-62.Grubbstrom, R. W., 1999. A net present value approach to safety stocks in a multi-level MRP system. International Journal of Production Economics 59(1-3), 361-375.Gupta, S., Dutta, K., 2011. Modeling of financial supply chain. European Journal of Operational Research 211(1), 47-56.Hurd, T. R., Zhou, Z. W., 2010. A Fourier transform method for spread option pricing. SIAM Journal on Financial Mathematics 1(1), 142-157.Khan, M., 1975. The monetary dynamics of hyperinflation: A note. Journal of Monetary Economics 1(3), 355-362.Lee, R. W., 2004. Option pricing by transform methods: extensions, unification and error control. The Journal of Computational Finance 7(3), 51-86.Liang, Z. X., Sun, B., 2011. Optimal control of a big financial company with debt liability under bankrupt probability constraints. Frontiers of Mathematics in China 6(6), 1095-1130.Lin, P. H., Wong, D. S. H., Jang, S. S., Shieh, S. S., Chu, J. Z., 2004. Controller design and reduction of bullwhip for a model supply chain system using Z-transform analysis. Journal of Process Control 14(5), 487- 499.Marquis, M. H., Witte, W. E., 1989. Cash management and the demand for money by firms. Journal of Macroeconomics 11(3), 333-350.Miller, M. H., Orr, D., 1966. A Model of the demand for money by firms. The Quarterly Journal of Economics (Published by: Oxford University Press) 80(3), 413-435.Miller, M. H., Orr, D., 1968. Demand for money by firms - extensions of analytic results. Journal of Finance 23(5), 735-759.Naim, M. M., Wikner, J., Grubbstrom, R. W., 2007. A net present value assessment of make-to-order and make-to-stock manufacturing systems. Omega-International Journal of Management Science 35(5), 524-532.Obstfeld, M., Rogoff, K., 1995. Exchange-rate dynamics Redux. Journal of Political Economy 103(3), 624-660.Obstfeld, M., Rogoff, K., 1996. Foundations of international macroeconomics. The MIT Press, p.p. xxiii + 804.Ogata, K., 1996. Discrete time control systems. Prentice-Hall International.Premachandra, I. M., 2004. A diffusion approximation model for managing cash in firms: An alternative approach to the Miller-Orr model. European Journal of Operational Research 157(1), 218-226.Sethi, S. P., Thompson, G. L., 1970. Applications of mathematical control theory to finance - modeling simple dynamic cash balance problems. Journal of Financial and Quantitative Analysis 5(4-5), 381-394.Song, N., Ching, W. K., Su, T. K., Yiu, C. K. F., 2013. On optimal cash management under a Stochastic Volatility Model. East Asian Journal on Applied Mathematics 3(2), 81-92.Stone, B. K., Hill, N. C., 1980. Cash transfer scheduling for efficient cash concentration. Financial Management 9(3), 35-43.Stone, B. K., Hill, N. C., 1981. The design of a cash concentration system. Journal of Financial and Quantitative Analysis 16(3), 301-322.Tenorio, V. A. F., Martín C. A. M., Paralera, M. C., Contreras R. I., 2013. Ecuaciones diferenciales y en diferencias aplicadas a los conceptos económicos y financieros. Revista de Métodos Cuantitativos para la Economía y la Empresa (16), 165-199. ISSN: 1886-516X. D.L: SE-2927- 06. URL: http://www.upo.es/RevMetCuant/art.php?id=83.Tobin, J., 1956. The interest-elasticity of transactions demand for cash. Review of Economics and Statistics 38(3), 241-247.Vasconcellos, G. M., 1988. On the application of optimal-control theory to financial-planning and forecasting. Journal of Economics and Business 40(4), 309-318.Woodford, M., 1998. Control of the public debt: A requirement for price stability? Debt burden and its consequences for monetary policy. IEA Conference (118), 117-154