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A probabilistic approach to the stochastic fluid cash management balance problem

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Abstract

We consider a stochastic cash balance problem in which cash receipts and cash disbursements change linearly and form a fluid process. The firm has to decide how much cash to hold in order to meet its obligations. The cash balance is monitored continuously and a four thresholds (0, S, M, m) policy is used to control the cash (\(0<S<M<m\)). That is, (a) If, at any time, the cash level drops to an order-level 0, a cash transfer of size S is requested such that the cash level is immediately raised to level S. (b) Whenever the cash level lies within the interval (0, m), no action is taken. (c) Every time the cash balance reaches the salvage-level m, it is restored to a level M and the excess amount \((m-M)\) is invested in other earning assets that provide a suitable return. We further study the extended (s, S, M) policy, in which s be an order level and each request for a cash transfer takes some random time (called the lead-time) until it is approved. Using fluid-flow results and multi-dimensional martingales, we construct a closed-form expression for the discounted total cost of running the cash balance. Numerical study provides several guidelines for the optimal control. For example, we show that even when the holding cost is high and receipts occur more frequently, it would be wise for the firm to hold on some extra cash.

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References

  • Ahn, S., & Ramaswami, V. (2005). Efficient algorithms for transient analysis of stochastic fluid flow models. Journal of Applied Probability, 42, 531–549.

    Article  Google Scholar 

  • Asmussen, S. (2003). Applied probability and queues (2nd ed.). New York: Springer.

    Google Scholar 

  • Asmussen, S., & Kella, O. (2000). A multi-dimensional martingale for Markov additive processes and its applications. Advances in Applied Probability, 32, 376–393.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Barron, Y. (2016a). Performance analysis of a reflected fluid production/inventory model. Mathematical Methods of Operations Research, 83(1), 1–31.

    Article  Google Scholar 

  • Barron, Y. (2016b). An \({(s, k, S)}\) fluid inventory model with exponential lead times and order cancellations. Stochastic Models, 32(2), 301–332.

    Article  Google Scholar 

  • Barron, Y. (2018). An order-revenue inventory model with returns and sudden obsolescence. Operations Research Letters, 46(1), 88–92.

    Article  Google Scholar 

  • Baumol, W. (1952). The transactions demand for cash: An inventory theoretic approach. Quarterly Journal of Economics, 66(1952), 545–546.

    Article  Google Scholar 

  • Bar-Ilan, A., Perry, D., & Stadje, W. (2004). A generalized impulse control model of cash management. Journal of Economic Dynamics and Control, 28(6), 1013–1033.

    Article  Google Scholar 

  • Bean, N., O’Reilly, M., & Taylor, P. (2005). Hitting probabilities and hitting times for stochastic fluid flows. Stochastic Processes and Their Applications, 115, 1530–1556.

  • Bensoussan, A., Chutani, A., & Sethi, S. P. (2009). Optimal cash management under uncertainty. Operations Research Letters, 37(6), 425–429.

    Article  Google Scholar 

  • Chaouch, B. A. (2018). Analysis of the stochastic cash balance problem using a level crossing technique. Annals of Operations Research, 271(2), 222–429.

    Article  Google Scholar 

  • Chen, X., & Simchi-Levi, D. (2004). Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The infinite horizon case. Mathematics of Operations Research, 52, 887–896.

    Article  Google Scholar 

  • Chen, X., & Simchi-Levi, D. (2009). A new approach for the stochastic cash balance problem with fixed costs. Probability in the Engineering and Informational Sciences, 23(4), 545–562.

    Article  Google Scholar 

  • Constantinides, G. M., & Richard, S. F. (1978). Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time. Operations Research, 26(4), 620–636.

    Article  Google Scholar 

  • da Costa Moraes, M. B., & Nagano, M. S. (2014). Evolutionary models in cash management policies with multiple assets. Economic Modelling, 39, 1–7.

    Article  Google Scholar 

  • Dendievel, S., & Latouche, G. (2017). Perturbation analysis of Markov modulated fluid models. Stochastic Models, 33(4), 473–494.

    Article  Google Scholar 

  • Elliott, R. J., & Swishchuk, A. V. (2007). Pricing options and variance swaps in Markov-modulated Brownian markets. In Hidden Markov models in finance (pp. 45–68). Springer.

  • Elton, E. J., & Gruber, M. J. (1974). On the cash balance problem. Operational Research Quarterly, 25(4), 553–572.

    Article  Google Scholar 

  • Eppen, G. D., & Fama, E. F. (1999). Cash balance and simple dynamic portfolio problems with proportional costs. International Economics Review, 10, 119–133.

    Article  Google Scholar 

  • Fei, W. (2013). Optimal consumption and portfolio under inflation and Markovian switching. An International Journal of Probability and Stochastic Processes, 85(2), 272–285.

    Article  Google Scholar 

  • Feinberg, E., & Lewis, M. (2007). Optimality inequalities for average cost Markov decision processes and the stochastic cash balance problem. Mathematics of Operations Research, 32(4), 769–783.

    Article  Google Scholar 

  • Feng, L., Skouri, K., Wang, W. C., & Teng, J. T. (2020). Optimal selling price, replenishment cycle and payment time among advance, cash, and credit payments from the seller’s perspective. Annals of Operations Research, 1–22.

  • Girgis, N. M. (1968). Optimal cash balance levels. Management Science, 15, 130–140.

    Article  Google Scholar 

  • Gormley, F. M., & Meade, N. (2007). The utility of cash flow forecasts in the management of corporate cash balances. European Journal of Operational Research, 182(2), 923–935.

    Article  Google Scholar 

  • Grossman, S. J., & Laroque, G. (1990). OAsset pricing and portfolio choice in the presence of illiquid durable consumption goods. Eco0 nometrica, 58, 25051.

  • Harrison, J. M., Sellke, T. M., & Taylor, A. J. (1983). Impulse control of Brownian motion. Mathematics of Operations Research, 8(3), 454–466.

    Article  Google Scholar 

  • Harrison, J. M., & Taylor, A. J. (1978). Optimal control of a Brownian storage system. Stochastic Processes and Their Applications, 6(2), 179–194.

    Article  Google Scholar 

  • Heyman, D. P. (1977). Optimal disposal policies for a single-item inventory system with returns. Naval Research Logistics, 24, 385–405.

    Article  Google Scholar 

  • Higson, A., Yoshikatsu, S., & Tippett, M. (2009). Organization size and the optimal investment in cash. IMA Journal of Management Mathematics, 21(1), 27–38.

    Article  Google Scholar 

  • Jaggi, C. K., Gupta, M., Kausar, A., & Tiwari, S. (2019). Inventory and credit decisions for deteriorating items with displayed stock dependent demand in two-echelon supply chain using Stackelberg and Nash equilibrium solution. Annals of Operations Research, 274(1), 309–329.

    Article  Google Scholar 

  • Krugman, P. R. (1991). Target zones and exchange rate dynamics. Quarterly Journal of Economics, 106(3), 669–682.

    Article  Google Scholar 

  • Kumar, A. (2018). Business process management. Routledge.

  • Lashgari, M., Taleizadeh, A. A., & Ahmadi, A. (2016). Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain. Annals of Operations Research, 238(1–2), 329–354.

  • Li, R., Teng, J. T., & Zheng, Y. (2019). Optimal credit term, order quantity and selling price for perishable products when demand depends on selling price, expiration date, and credit period. Annals of Operations Research, 280(1), 377–405.

    Article  Google Scholar 

  • Liu, B., & Xin, C. (2008). An online model for managing cash: An alternative approach to the Miller–Orr model. In 2013 International conference on computing, networking and communications (ICNC) (pp. 314–317).

  • Miller, M., & Orr, D. (1966). A model of the demand for money by firms. Quarterly Journal of Economics, 81, 413–435.

    Article  Google Scholar 

  • Neave, E. H. (1970). The stochastic cash balance problem with fixed costs for increases and decreases. Management Science, 16, 472–490.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ramaswami, V. (2006). Passage times in fluid models with application to risk processes. Methodology and Computations in Applied Probability, 8, 497–515.

    Article  Google Scholar 

  • Salas-Molina, F., Pla-Santamaria, D., & Rodriguez-Aguilar, J. A. (2018). A multi-objective approach to the cash management problem. Annals of Operations Research, 267(1–2), 515–529.

    Article  Google Scholar 

  • Sato, K., & Suzuki, A. (2011). Stochastic cash management problem with double exponential jump diffusion processes. The Tenth International Symposium on Operations Research and Its Applications, 28(31), 186–194.

    Google Scholar 

  • Shen, Y., & Siu, T. K. (2013). Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching. Operations Research Letters, 41(2), 180–187.

    Article  Google Scholar 

  • Song, N., Ching, W. K., Siu, 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.

    Article  Google Scholar 

  • Taleizadeh, A. A., Zarei, H. R., & Sarker, B. R. (2019). An optimal ordering and replenishment policy for a vendor-buyer system under varying replenishment intervals and delayed payment. European Journal of Industrial Engineering, 13(2), 264–298.

    Article  Google Scholar 

  • Taleizadeh, A. A., Tavassoli, S., & Bhattacharya, A. (2020). Inventory ordering policies for mixed sale of products under inspection policy, multiple prepayment, partial trade credit, payments linked to order quantity and full backordering. Annals of Operations Research, 287(1), 403–437.

    Article  Google Scholar 

  • Tangsucheeva, R., & Prabhu, V. (2014). Stochastic financial analytics for cash flow forecasting. International Journal of Production Economics, 158, 65–76.

    Article  Google Scholar 

  • Tapiero, C. S., & Zuckerman, D. (1980). A note on the optimal control of a cash balance problem. Journal of Banking and Finance, 4(4), 345–352.

    Article  Google Scholar 

  • Tobin, J. (1956). The interest-elasticity of transaction demand for cash. The Review of Economics and Statistics, 38, 241–247.

    Article  Google Scholar 

  • Wang, Z., Xu, G., Zhao, P., & Lu, Z. (2018). The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial and Management Optimization, 14(1), 1.

  • Wu, J., Teng, J. T., & Skouri, K. (2018). Optimal inventory policies for deteriorating items with trapezoidal-type demand patterns and maximum lifetimes under upstream and downstream trade credits. Annals of Operations Research, 264(1), 459–476.

  • Ye, Q., & Duenyas, I. (2007). Optimal capacity investment decisions with two-sided fixed-capacity adjustment costs. Operations Research, 55(2), 272–283.

    Article  Google Scholar 

Download references

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  1. Ariel University, 40700, Ariel, Israel

    Yonit Barron

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Barron, Y. A probabilistic approach to the stochastic fluid cash management balance problem. Ann Oper Res 312, 607–645 (2022). https://doi.org/10.1007/s10479-021-04500-7

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