# American Institute of Mathematical Sciences

January  2018, 14(1): 1-17. doi: 10.3934/jimo.2017034

## The optimal cash holding models for stochastic cash management of continuous time

 1 Dept. of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China 2 School of Economics and Management, Yancheng Institute of Technology, Yancheng 224056, China 3 Dept. of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China 4 Southampton Statistical Sciences Research Institute and School of Mathematical Sciences, University of Southampton, SO17 1BJ, UK

Received  March 2015 Revised  August 2016 Published  April 2017

In business, enterprises need to maintain stable cash flows to meet the demands for payments in order to reduce the probability of possible bankruptcy. In this paper, we propose the optimal cash holding models in terms of continuous time and managers' risk preference in the framework of stochastic control theory in the setting of cash balance accounting with the interval of a safe area for cash holdings. Formulas for the optimal cash holdings are analytically derived with a widely used family of power utility functions. Our models can be seen as an extension of Miller-Orr model to solve the cash holding problem of continuous time from the accounting perspective. Numerical examples are also provided to illustrate the feasibility of the developed optimal cashing holding models of continuous time.

Citation: Zhengyan Wang, Guanghua Xu, Peibiao Zhao, Zudi Lu. The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial & Management Optimization, 2018, 14 (1) : 1-17. doi: 10.3934/jimo.2017034
##### References:
 [1] S. Baccarin, Optimal impulse control for cash management with quadratic holding-penalty costs, Decis. Econ. Finance, 25 (2002), 19-32. doi: 10.1007/s102030200001. Google Scholar [2] A. Bar-Ilan, Overdraft and the demand for money, Am. Econ. Rev., 80 (1990), 1201-1216. Google Scholar [3] A. Bar-Ilan, D. Perry and W. Stadje, A generalized impulse control model of cash management, J. Econ. Dyn. Control, 28 (2004), 1013-1033. doi: 10.1016/S0165-1889(03)00064-2. Google Scholar [4] A. Bar-Ilan and D. Lederman, International reserves and monetary policy, Econ. Lett., 97 (2007), 170-178. doi: 10.1016/j.econlet.2007.03.001. Google Scholar [5] W. Baumol, The transaction demand for cash -an inventory theoretic approach, Q. J. Econ., 66 (1952), 545-546. Google Scholar [6] A. Ben-Bassat and D. Gottlieb, Optimal international reserves and sovereign risk, J. Int. Econ., 33 (1992), 345-362. Google Scholar [7] A. Bensoussan, A. Chutani and S. P. Sethi, Optimal cash managementunder uncertainty, Operations Research Letters, 37 (2009), 425-429. doi: 10.1016/j.orl.2009.08.002. Google Scholar [8] F. Chang, Homogeneity and the transactions demand for money, J. Money Credit Bank., 31 (1999), 720-730. Google Scholar [9] H. G. Daellenbach, A stochastic cash balance model with two sources of short-term funds, Int. Econ. Rev., 12 (1971), 250-256. Google Scholar [10] H. G. Daellenbach, Daellenbach, H.G. Are cash management optimization models worthwhile?, J. Financ. Quant. Anal., 9 (1974), 607-626. Google Scholar [11] A. Dixit, A simplied exposition of the theory of optimal control of Brownian motion, J. Econ. Dyn. Control, 15 (1991), 657-673. doi: 10.1016/0165-1889(91)90037-2. Google Scholar [12] G. D. Eppen and E. Fama, Cash balance and simple dynamic portfolio problems with proportional costs, Int. Econ. Rev., 10 (1969), 119-133. Google Scholar [13] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Berlin, New York, 2006. doi: 10. 1007/978-0-387-26045-7. Google Scholar [14] J. Frenkel and B. Jovanovic, On transactions and precautionary demand for money, Q. J. Econ., 94 (1980), 24-43. Google Scholar [15] J. Frenkel and B. Jovanovic, Optimal international reserves: A stochastic framework, Econ. J., 91 (1981), 507-514. Google Scholar [16] N. Girgis, Optimal cash balance levels, Manage. Sci., 15 (1968), 130-140. Google Scholar [17] W. H. Hausman and A. Sanchez-Bell, The stochastic cash balance problem with average compensating-balance requirements, Manage. Sci., 21 (1975), 849-857. Google Scholar [18] I. Karatzas, J. P. Lehoczky, S.E. Shreve and G.L. Xu, Martingale and duality for utility maximization in an incomplete market, SIAM J. Control and Optimization, 29 (1991), 702-730. doi: 10.1137/0329039. Google Scholar [19] M. A. S. Melo and F. Bilich, Expectancy balance model for cash flow, Journal of Economics and Finance, 37 (2013), 240-252. Google Scholar [20] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous time case, Review of Economics and Statistics, 51 (1969), 247-257. Google Scholar [21] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. Google Scholar [22] R. Milbourne, Optimal money holding under uncertainty, Int. Econ. Rev., 24 (1983), 685-698. Google Scholar [23] M. Miller and D. Orr, A model of the demand for money by firms, Q. J. Econ., 81 (1966), 413-435. Google Scholar [24] M. B. C. Moraes and M. S. Nagano, Cash management policies by evolutionary models: A comparison using the Miller-Orr model, JISTEM, 10 (2013), 561-576. Google Scholar [25] D. Perry and W. Stadje, Risk analysis for a stochastic cash management model with two types of customers, Insur. Math. Econ., 26 (2000), 25-36. doi: 10.1016/S0167-6687(99)00037-2. Google Scholar [26] G. W. Smith, Transactions demand for money with a stochastic, time-varying interest rate, Rev. Econ. Stud., 56 (1989), 623-633. Google Scholar [27] N. Song, W. K. Ching, T. K. Siu and K. F. Yiu, On optimal cash management under a stochastic volatility model, East Asian Journal on Applied Mathematics, 3 (2013), 81-92. doi: 10.4208/eajam.070313.220413a. Google Scholar [28] J. Tobin, The interest elasticity of the transaction demand for cash, Rev. Econ. Stat., 38 (1956), 241-247. Google Scholar [29] R. G. Vickson, Simple optimal policy for cash management: The average balance requirement case, J. Financ. Quant. Anal., 20 (1985), 353-369. Google Scholar

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##### References:
 [1] S. Baccarin, Optimal impulse control for cash management with quadratic holding-penalty costs, Decis. Econ. Finance, 25 (2002), 19-32. doi: 10.1007/s102030200001. Google Scholar [2] A. Bar-Ilan, Overdraft and the demand for money, Am. Econ. Rev., 80 (1990), 1201-1216. Google Scholar [3] A. Bar-Ilan, D. Perry and W. Stadje, A generalized impulse control model of cash management, J. Econ. Dyn. Control, 28 (2004), 1013-1033. doi: 10.1016/S0165-1889(03)00064-2. Google Scholar [4] A. Bar-Ilan and D. Lederman, International reserves and monetary policy, Econ. Lett., 97 (2007), 170-178. doi: 10.1016/j.econlet.2007.03.001. Google Scholar [5] W. Baumol, The transaction demand for cash -an inventory theoretic approach, Q. J. Econ., 66 (1952), 545-546. Google Scholar [6] A. Ben-Bassat and D. Gottlieb, Optimal international reserves and sovereign risk, J. Int. Econ., 33 (1992), 345-362. Google Scholar [7] A. Bensoussan, A. Chutani and S. P. Sethi, Optimal cash managementunder uncertainty, Operations Research Letters, 37 (2009), 425-429. doi: 10.1016/j.orl.2009.08.002. Google Scholar [8] F. Chang, Homogeneity and the transactions demand for money, J. Money Credit Bank., 31 (1999), 720-730. Google Scholar [9] H. G. Daellenbach, A stochastic cash balance model with two sources of short-term funds, Int. Econ. Rev., 12 (1971), 250-256. Google Scholar [10] H. G. Daellenbach, Daellenbach, H.G. Are cash management optimization models worthwhile?, J. Financ. Quant. Anal., 9 (1974), 607-626. Google Scholar [11] A. Dixit, A simplied exposition of the theory of optimal control of Brownian motion, J. Econ. Dyn. Control, 15 (1991), 657-673. doi: 10.1016/0165-1889(91)90037-2. Google Scholar [12] G. D. Eppen and E. Fama, Cash balance and simple dynamic portfolio problems with proportional costs, Int. Econ. Rev., 10 (1969), 119-133. Google Scholar [13] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Berlin, New York, 2006. doi: 10. 1007/978-0-387-26045-7. Google Scholar [14] J. Frenkel and B. Jovanovic, On transactions and precautionary demand for money, Q. J. Econ., 94 (1980), 24-43. Google Scholar [15] J. Frenkel and B. Jovanovic, Optimal international reserves: A stochastic framework, Econ. J., 91 (1981), 507-514. Google Scholar [16] N. Girgis, Optimal cash balance levels, Manage. Sci., 15 (1968), 130-140. Google Scholar [17] W. H. Hausman and A. Sanchez-Bell, The stochastic cash balance problem with average compensating-balance requirements, Manage. Sci., 21 (1975), 849-857. Google Scholar [18] I. Karatzas, J. P. Lehoczky, S.E. Shreve and G.L. Xu, Martingale and duality for utility maximization in an incomplete market, SIAM J. Control and Optimization, 29 (1991), 702-730. doi: 10.1137/0329039. Google Scholar [19] M. A. S. Melo and F. Bilich, Expectancy balance model for cash flow, Journal of Economics and Finance, 37 (2013), 240-252. Google Scholar [20] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous time case, Review of Economics and Statistics, 51 (1969), 247-257. Google Scholar [21] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. Google Scholar [22] R. Milbourne, Optimal money holding under uncertainty, Int. Econ. Rev., 24 (1983), 685-698. Google Scholar [23] M. Miller and D. Orr, A model of the demand for money by firms, Q. J. Econ., 81 (1966), 413-435. Google Scholar [24] M. B. C. Moraes and M. S. Nagano, Cash management policies by evolutionary models: A comparison using the Miller-Orr model, JISTEM, 10 (2013), 561-576. Google Scholar [25] D. Perry and W. Stadje, Risk analysis for a stochastic cash management model with two types of customers, Insur. Math. Econ., 26 (2000), 25-36. doi: 10.1016/S0167-6687(99)00037-2. Google Scholar [26] G. W. Smith, Transactions demand for money with a stochastic, time-varying interest rate, Rev. Econ. Stud., 56 (1989), 623-633. Google Scholar [27] N. Song, W. K. Ching, T. K. Siu and K. F. Yiu, On optimal cash management under a stochastic volatility model, East Asian Journal on Applied Mathematics, 3 (2013), 81-92. doi: 10.4208/eajam.070313.220413a. Google Scholar [28] J. Tobin, The interest elasticity of the transaction demand for cash, Rev. Econ. Stat., 38 (1956), 241-247. Google Scholar [29] R. G. Vickson, Simple optimal policy for cash management: The average balance requirement case, J. Financ. Quant. Anal., 20 (1985), 353-369. Google Scholar
optimal cash holding model in case 1
optimal cash holding model in case 2
optimal allocation to cash for different p values
The daily average yield and variance
 stock number daily average yield variance A 0.000399 0.000437 B 0.000433 0.000687 C 0.000443 0.0019 D 0.000463 0.00055
 stock number daily average yield variance A 0.000399 0.000437 B 0.000433 0.000687 C 0.000443 0.0019 D 0.000463 0.00055
The optimal cash holdings in case 1: $X_t>H$
 stock number invest ratio $\lambda_{t}^{*}$ in (17) $(1-\lambda_{t}^{*})X_{t}$ optimal cash holdings A 0.756891 1944.871 1944.871 B 0.542302 3661.586 3661.586 C 0.04379 7649.684 7000 D 0.815414 1476.685 1476.685
 stock number invest ratio $\lambda_{t}^{*}$ in (17) $(1-\lambda_{t}^{*})X_{t}$ optimal cash holdings A 0.756891 1944.871 1944.871 B 0.542302 3661.586 3661.586 C 0.04379 7649.684 7000 D 0.815414 1476.685 1476.685
The optimal cash holdings in case 2: $X_t<L$
 stock number conversion ratio $\mu_{t}^{*}$ in (23) $\mu_{t}^{*}R_{t}+X_{t}$ optimal cash holdings A 0.124443 1944.8711 1944.8711 B 0.383288 4326.2458 4326.2458 C 0.770653 7890.0104 7000 D 0.163794 2306.9047 2306.9047
 stock number conversion ratio $\mu_{t}^{*}$ in (23) $\mu_{t}^{*}R_{t}+X_{t}$ optimal cash holdings A 0.124443 1944.8711 1944.8711 B 0.383288 4326.2458 4326.2458 C 0.770653 7890.0104 7000 D 0.163794 2306.9047 2306.9047
The optimal cash holdings with different p values
 p invest ratio $\lambda_{t}^{*}$ in (17) $(1- \lambda_{t}^{*})X_{t}$ optimal cash holdings 0.1 0.04379 7649.684 7000 0.2 0.04672 7626.262 7000 0.3 0.08911 7287.156 7000 0.4 0.14562 6835.016 6835 0.5 0.22475 6202.019 6202 0.6 0.34344 5252.523 5252 0.7 0.54125 3670.031 3670 0.8 0.93687 505.047 1000 0.9 2.12374 -8989.91 1000
 p invest ratio $\lambda_{t}^{*}$ in (17) $(1- \lambda_{t}^{*})X_{t}$ optimal cash holdings 0.1 0.04379 7649.684 7000 0.2 0.04672 7626.262 7000 0.3 0.08911 7287.156 7000 0.4 0.14562 6835.016 6835 0.5 0.22475 6202.019 6202 0.6 0.34344 5252.523 5252 0.7 0.54125 3670.031 3670 0.8 0.93687 505.047 1000 0.9 2.12374 -8989.91 1000
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