Keywords

Definition

A firm needs to keep cash, either in the form of cash on hand or as a bank deposit, to meet its daily transaction requirements. Daily inflows and outflows of cash are random. There is a finite target for the cash balance, which could be zero in some cases. The firm wants to select a policy that minimizes the expected total discounted cost for being far away from the target during some time horizon. This time horizon is usually infinity. The firm has an incentive to keep the cash level low, because each unit of positive cash leads to a holding cost since cash has alternative uses like dividends or investments in earning assets. The firm has an incentive to keep the cash level high, because penalty costs are generated as a result of delays in meeting demands for cash. The firm can increase its cash balance by raising new capital or by selling some earnings assets, and it can reduce its cash balance by paying dividends or investing in earning assets. This control of the cash balance generates fixed and proportional transaction costs. Thus, there is a cost when the cash balance is different from its target, and there is also a cost for increasing or reducing the cash reserve. The objective of the manager is to minimize the expected total discounted cost.

Hasbrouck (2007), Madhavan and Smidt (1993), and Manaster and Mann (1996) study inventories of stocks that are similar to the cash management problem.

The Solution

The qualitative form of optimal policies of the cash management problem in discrete time was studied by Eppen and Fama (1968, 1969), Girgis (1968), and Neave (1970). However, their solutions were incomplete.

Many of the difficulties that they and other researchers encountered in a discrete-time framework disappeared when it was assumed that decisions were made continuously in time and that demand is generated by a Brownian motion with drift. Vial (1972) formulated the cash management problem in continuous time with fixed and proportional transaction costs, linear holding and penalty costs, and demand for cash generated by a Brownian motion with drift. Under very strong assumptions, Vial (1972) proved that if an optimal policy exists, then it is of a simple form (a, α, β, b).

This means that the cash balance should be increased to level α when it reaches level a and should be reduced to level β when it reaches level b. Constantinides (1976) assumed that an optimal policy exists and it is of a simple form, and determined the above levels and discussed the properties of the optimal solution. Constantinides and Richard (1978) proved the main assumptions of Vial (1972) and therefore obtained rigorously a solution for the cash management problem.

Constantinides and Richard (1978) applied the theory of stochastic impulse control developed by Bensoussan and Lions (1973, 1975, 1982). He used a Brownian motion W to model the uncertainty in the inventory. Formally, he considered a probability space \((\Omega,\,\mathcal{F},\,P)\) together with a filtration \((\mathcal{F}_{t})\) generated by a one-dimensional Brownian motion W. He considered X t : = inventory level at time t, and assumed that X is an adapted stochastic process given by

$$\displaystyle{X_{t} = x -\int _{0}^{t}\mu ds -\int _{ 0}^{t}\sigma dW_{ s} +\sum \limits _{ i=1}^{\infty }I_{\{\tau _{ i}<t\}}\xi _{i}.}$$

Here, μ > 0 is the drift of the demand and σ > 0 is the volatility of the demand. Furthermore, τ i is the time of the i-th intervention and ξ i is the intensity of the i-th intervention.

A stochastic impulse control is a pair

$$\displaystyle\begin{array}{rcl} & & ((\tau _{n});\,(\xi _{n})) {}\\ & & = (\tau _{0},\tau _{1},\tau _{2},\ldots,\tau _{n},\ldots ;\xi _{0},\,\xi _{1},\xi _{2},\ldots,\xi _{n},\ldots ), {}\\ \end{array}$$

where

$$\displaystyle{\tau _{0} = 0 <\tau _{1} <\tau _{2} < \cdots <\tau _{n} < \cdots }$$

is an increasing sequence of stopping times and (ξ n ) is a sequence of random variables such that each \(\xi _{n} : \Omega \mapsto \mathbf{R}\) is measurable with respect to \(\mathcal{F}_{\tau _{n}}\). We assume ξ0 = 0. The management (the controller) decides to act at time

$$\displaystyle{X_{\tau _{i}^{+}} = X_{\tau _{i}} +\xi _{i}.}$$

We note that ξ i and X can also take negative values. The management wants to select the pair

$$\displaystyle{((\tau _{n});\,(\xi _{n}))}$$

that minimizes the functional J defined by

$$\displaystyle\begin{array}{rcl} J(x;\,((\tau _{n});\,(\xi _{n}))) :\,=\, E\left [\int _{0}^{\infty }e^{-\lambda t}f(X_{ t})dt\right.& & {}\\ \left.+\sum \limits _{n=1}^{\infty }e^{-\lambda \tau _{n} }g(\xi _{n})I_{\{\tau _{n}<\infty \}}\right ],& & {}\\ \end{array}$$

where

$$\displaystyle{f(x) =\max \, (hx,\,-px)}$$

and

$$\displaystyle{g(\xi ) = \left \{\begin{array}{*{20}c} C + c\xi \,\,\,\,\,\,\,\,\, &\mbox{ if}\,\xi> 0 \\ \min \,(C,D)&\mbox{ if}\,\xi = 0 \\ D - d\xi \,\,\,\,\,\,\, &\mbox{ if}\,\xi < 0\\ \end{array} \right.}$$

Furthermore, λ > 0, C, c, D, d ∈ (0, ∞), and h, p ∈ (0, ∞). Here, f represents the running cost incurred by deviating from the aimed cash level 0, C represents the fixed cost per intervention when the management pushes the cash level upwards, D represents the fixed cost per intervention when the management pushes the cash level downwards, c represents the proportional cost per intervention when the management pushes the cash level upwards, d represents the proportional cost per intervention when the management pushes the cash level downwards, and λ is the discount rate.

The results of Constantinides were complemented, extended, or improved by Cadenillas et al. (2010), Cadenillas and Zapatero (1999), Feng and Muthuraman (2010), Harrison et al. (1983), and Ormeci et al. (2008).

Cross-References