## The Miller Orr Model A More General Approach

We now describe a cash management system designed to deal with cash inflows and outflows that fluctuate randomly from day to day. With this model, we again concentrate on the cash balance, but, in contrast to the situation with the BAT model, we assume that this balance fluctuates up and down randomly and that the average change is zero.

The Basic Idea Figure 20A.3 shows how the system works. It operates in terms of an upper limit to the amount of cash (U*) and a lower limit (L), and a target cash balance (C*). The firm allows its cash balance to wander around between the lower and upper limits. As long as the cash balance is somewhere between U* and L, nothing happens.

When the cash balance reaches the upper limit (U*), such as it does at Point X, the firm moves U* - C* dollars out of the account and into marketable securities. This action moves the cash balance down to C*. In the same way, if the cash balance falls to the lower limit (L), as it does at Point Y, the firm will sell C* - L worth of securities and deposit the cash in the account. This action takes the cash balance up to C*.

Using the Model To get started, management sets the lower limit (L). This limit essentially defines a safety stock; so, where it is set depends on how much risk of a cash shortfall the firm is willing to tolerate. Alternatively, the minimum might just equal a required compensating balance.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

VII. Short-Term Financial Planning and Management

20. Cash and Liquidity Management

© The McGraw-Hill Companies, 2002

### CHAPTER 20 Cash and Liquidity Management

As with the BAT model, the optimal cash balance depends on trading costs and opportunity costs. Once again, the cost per transaction of buying and selling marketable securities, F is assumed to be fixed. Also, the opportunity cost of holding cash is R, the interest rate per period on marketable securities.

The only extra piece of information needed is ct2, the variance of the net cash flow per period. For our purposes, the period can be anything, a day or a week, for example, as long as the interest rate and the variance are based on the same length of time.

Given L, which is set by the firm, Miller and Orr show that the cash balance target, C*, and the upper limit, U*, that minimize the total costs of holding cash are:2

C* = L + (3/4 X F X ct2/R)(1/3) U* = 3 X C* - 2 X L

Also, the average cash balance in the Miller-Orr model is:

The derivation of these expressions is relatively complex, so we will not present it here. Fortunately, as we illustrate next, the results are not difficult to use.

For example, suppose F = $10, the interest rate is 1 percent per month, and the standard deviation of the monthly net cash flows is $200. The variance of the monthly net cash flows is:

We assume a minimum cash balance of L = $100. We can calculate the cash balance target, C*, as:

= $100 + (3/4 X 10 X 40,000/.01)(1/3) = $100 + 30,000,000(1/3) = $100 + 311 = $411

Finally, the average cash balance will be:

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