# Miller-Orr Model

By Yuriy Smirnov Ph.D.

## Definition

The Miller-Orr model of cash management is developed for businesses with uncertain cash inflows and outflows. This approach allows lower and upper limits of cash balance to be set and determine the return point (target cash balance). This is different from the Baumol-Tobin model, which is based on the assumption that the cash spending rate is constant.

## Assumptions

The Miller-Orr model of cash management can be used if the following assumptions are met:

1. The cash inflows and cash outflows are stochastic. In other words, each day a business may have both different cash payments and different cash receipts.
2. The daily cash balance is normally distributed, i.e., it occurs randomly.
3. There is a possibility to invest idle cash in marketable securities.
4. There is a transaction fee when marketable securities are bought or sold.
5. A business maintains the minimum acceptable cash balance, which is called the lower limit.

## Formula

The return point for the cash balance under the Miller-Orr model can be calculated as follows:

Return Point = Lower Limit + 1/3 × Spread

The lower limit is set by management. It depends on the acceptable risk of cash flows gap, creditworthiness of a business, and expected needs in cash. However, the lower limit can be set as zero if a business has sufficient investments in marketable securities or perfect creditworthiness and can raise additional short-term debt at any time.

The equation to compute the spread is as follows:

where F is the transaction cost, K is the opportunity cost of holding cash, and σ2 is a variance of a daily cash balance.

To find the upper limit of the cash balance, the following formula should be used:

Upper Limit = Lower Limit + Spread

## Limitations

When the Miller-Orr model of cash management is applied, we should take into account the following limitations:

1. An increase in transaction cost results in an increase of spread and a higher return point.
2. The higher the standard deviation (σ) of daily cash balance, the wider the spread and higher return point. A higher volatility of the daily cash balance also means a higher probability of reaching the lower or upper limit.
3. By contrast, an increase in the return on investment in marketable securities will lead to a narrower spread and lower return point because the opportunity cost of holding cash is also growing, so a business will seek to decrease its cash holdings.

## Graph

As mentioned above, the lower limit or safety cash balance is set by a company’s management, but both the upper limit and return point depend on the spread. The return point under the Miller-Orr model is a cash balance that has to be restored if the actual cash balance reaches a lower or upper limit.

See the graph below to illustrate such situations.

1. A is when the actual cash balance drops to the lower limit. A company’s management should replenish it to reach the return point, which can be done by selling investments in marketable securities.
2. B is when the actual cash balance touches the upper limit. In such cases, it is necessary to buy marketable securities and restore the cash balance down to the return point. The amount to be invested is the difference between the upper limit and return point.

## Example

The management of Stilmill Inc. has set a safety cash balance of \$50,000. The standard deviation (σ) of the daily cash balance during the last year was \$37,500, and the transaction cost was \$75. The company also has the opportunity to invest idle cash in marketable securities at an annual interest rate of 8%.

 Daily interest rate = 8% = 0.022% 365

 Return point = \$50,000 + 1 × \$213,325 = \$121,108 3

Upper limit = \$50,000 + \$213,325 = \$263,325