# 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
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##### References:
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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