# Optimal Cash Models

A number of mathematical models have been developed to assist the financial manager
in distributing a company's funds so that they provide a maximum return to the company.

**William Baumol Model**

The model developed by William Baumol can determine the optimum amount of cash for a company to hold under conditions of certainty. The objective is to minimize the sum of the fixed costs of transactions and the opportunity cost of holding cash balances that do not yield a return. This is similar to the EOQ model used in inventory management. The costs can be expressed as follows, according to his model:

F (T/C) + I (C/2)

Where: F = Fixed costs of a transaction T = Total cash required for the specified time period I = Interest rate on marketable securities C = Cash balance The optimal level of cash is determined using the following formula: Optimal level of cash = √(2FT / I)

*Example:*

A Company estimates a cash requirement of $2,000,000 for a 1 month period. The opportunity interest rate is 6% per annum, which works out to 0.5 percent per month. The transaction cost for borrowing or withdrawing funds is $150.

Optimal level of cash = √(2 x $150 x $2,000,000) / 0.005 => $346,410.16

With the above optimal transaction size, we can now find the number of transactions required.

Number of transactions required = $2,000,000 / $346,410.16 => 5.77 or 6 transactions during the month.**Miller-Orr Model**

When the cash payments are uncertain, Miller-Orr model can be used. This model places upper and lower limits on cash balances. When the upper limit is reached, a transfer of cash to marketable securities is made; when the lower limit is reached, a transfer from securities to cash is made. As long as the cash balance stays within the limits, no transaction occurs. The various factors in this model are fixed costs of a securities transaction (F) which is assumed to be the same for buying and selling, the daily interest rate on marketable securities (I) and variance of the daily net cash flows, represented by σ2. This model assumes that the cash flows are random. The control limits in this model are d dollars as an upper limit and zero dollars at the lower limit. When the cash balance reaches the upper level, d less z dollars of securities are bought, and the new balance becomes z dollars. When the cash balance equals zero, z dollars of securities are sold and the new balance again reaches z. According to this model, the optimal cash balance z is computed as follows:

Z = 3√(3F σ2)/ 4I

The optimal value for d is computed as 3z.

Average cash balance (approx.) = (z + d)/3

Given are the following:*Example:*

Fixed cost of a securities transaction = $5

Variance of daily net cash flows = $25

Daily interest rate on securities = 0.0003 (10% per annum, so 10%/360 days = 0.0003 daily)

Optimal cash balance = 3√(3 x $5 x $25) / (4 x 0.0003) => $67.86

(or) $68 rounded off.

Upper limit, d = 3z = 3 x $68 = $204

Average cash balance = ($68 + $204)/3 => $90.67

So, when the upper limit of $204 is reached, $136 ($204 - $68) will be purchased. When the lower limit of zero dollars is reached, $68 of securities will be sold to again bring it to the optimal balance of cash calculated as $68 approximately.**Orgler's Model**

According to this model, the optimal cash management strategy can be determined through the use of a multiple linear programming model. It is a model that provides for integration of cash management with production and other aspects of the firm. The construction of this model comprises three sections namely:

- Selection of the appropriate planning horizon
- Selection of the appropriate decision variables and
- Formulation of the cash management strategy.

This model uses one year planning horizon with twelve monthly periods because of its simplicity. It has four basic sets of decision variables which influence cash management of a firm and which must be incorporated into the linear programming model of the firm.

These are

- Payment schedule
- Short-term financing and
- Purchase and sale of marketable securities and
- Cash balance.

The familiarity of all the above models provides the financial managers an insight into the normative framework as to how cash management should be conducted.

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