Optimum Cash Balance under Uncertainty: The Miller–Orr Model
One of the major limitations of the Baumol model is that it does not allow cash flows to fluctuate; it assumes a constant and predictable pattern of cash usage. In reality, however, firms experience uncertain and irregular cash inflows and outflows. The Miller–Orr (MO) model overcomes this limitation by allowing for random daily variations in cash flows.
The Miller–Orr model establishes two control limits an upper control limit and a lower control limit along with a return point (or target cash balance). When the firm’s cash balance fluctuates randomly and reaches the upper control limit, the firm invests the excess cash in marketable securities to bring the balance back to the return point. Conversely, when the cash balance falls to the lower control limit, the firm liquidates marketable securities to restore the cash balance to the return point.
Thus, the Miller–Orr model provides a practical framework for managing cash balances under conditions of uncertainty.
Assumptions and Working of the Miller–Orr Model
The Miller–Orr model assumes that net cash flows are normally distributed with a mean value of zero and a given standard deviation. This implies that daily cash inflows and outflows are uncertain and fluctuate randomly around a stable average.
The model provides for two control limits an upper control limit and a lower control limit along with a return point (target cash balance). When the firm’s cash balance fluctuates randomly and reaches the upper control limit, the firm purchases sufficient marketable securities to bring the cash balance back to the return point. Similarly, when the cash balance falls to the lower control limit, the firm sells adequate marketable securities to restore the cash balance to the return point.
Determination of Control Limits in the Miller–Orr Model
The difference between the upper control limit and the lower control limit in the Miller–Orr model depends on the following factors:
- Transaction cost
- Interest rate
- Standard deviation (variance) of net cash flows
The distance between the upper and lower control limits is known as the Z-spread.
The value of Z is determined by the following formula:
Where:
- C= transaction cost per transaction
- σ2= variance of net cash flows
- i= interest rate
It is observed that the upper control limit is three times the Z-spread above the lower limit, while the return point lies between the two limits. Accordingly:
- Upper control limit = Lower limit + 3Z
- Return point (Target cash balance) = Lower limit + Z
The net effect of the model is that firms maintain an average cash balance given by:
The Miller–Orr model is more realistic than the Baumol model because it allows cash balances to fluctuate within the upper and lower control limits. Moreover, the financial manager has the flexibility to set the lower limit according to the firm’s liquidity requirements.
Illustration: Miller–Orr Model
PKG Company follows a policy of maintaining a minimum cash balance (lower limit) of ₹5,00,000. The standard deviation of daily net cash flows is ₹2,00,000. The annual interest rate is 14%, and the transaction cost for buying or selling marketable securities is ₹150 per transaction.
You are required to determine the upper control limit, return point, and average cash balance as per the Miller–Orr model.
Step 1: Calculation of Z (Spread)
Since the standard deviation of cash flows is given on a daily basis, the annual interest rate is converted into a daily rate:
The formula for Z is:
Substituting the given values:
Step 2: Determination of Control Limits
Lower Control Limit (given) = ₹5,00,000
Upper Control Limit
Return Point (Target Cash Balance)
Step 3: Average Cash Balance
Interpretation
PKG Company will not allow its cash balance to fall below the lower limit of ₹5,00,000. If the cash balance reaches this level, the firm will sell marketable securities worth ₹2,27,226 (Z) to restore the cash balance to the return point of ₹7,27,226.
Conversely, if the cash balance rises to the upper limit of ₹11,81,678, the firm will purchase marketable securities worth ₹4,54,452 (2Z), thereby bringing the cash balance back to the return point:
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