 # Baumol-Tobin Model By Yuriy Smirnov Ph.D.

## Definition

The Baumol-Tobin model is used in corporate finance as a cash management technique to help determine the cash balance that grants the minimum amount of transaction cost and opportunity cost (foregone interest on marketable securities).

## Assumptions of Baumol-Tobin model

If the Baumol-Tobin model is applied, the following assumptions should be met:

1. The spending rate of cash is constant, and the total cash need is known in advance.
2. Funds can be held in either money or short-term investments (e.g., marketable securities).
3. The minimum cash balance is zero.
4. The opportunity cost for holding cash is constant.
5. The excess cash is invested in marketable securities.
6. The transaction cost is the same each time when marketable securities are converted to cash.
7. The line of credit or overdraft is not available.

## Formula

The optimal cash balance (OCB) is derived from the Baumol-Tobin model total cost equation,

 TC = C × K + T × F 2 C

where C is the cash balance, K is opportunity cost (e.g., interest rate on marketable securities), T is annual total cash need, and F is transaction cost (e.g., brokerage fee).

To find the cash balance with minimum total cost, we should calculate the derivative of the total cost function with respect to variable “C” and set it equal to zero assuming that other variables are constant.

 0 = K - T × F 2 C2
 C2 = 2 × T × F K

Solving the equation with respect to variable “C” will give the optimal cash balance. ## Opportunity cost vs. transaction cost

Opportunity cost and transaction cost are inversely related. To decrease transaction cost, the number of transactions should be reduced, which means a higher cash balance to be maintained. Such a scenario leads to an increase in opportunity cost (the interest foregone on marketable securities). In turn, reducing opportunity cost requires a lower cash balance to be maintained, which leads to an increase in the number of transactions and therefore an increase in transaction cost. Moreover, a low cash balance could negatively affect the solvency of a company.

This relationship between opportunity cost and transaction cost is shown in the graph below. ## Optimal cash balance calculation example

XYZ Company has an annual total cash need of \$3,000,000. The company has an opportunity to buy a certificate of deposit with an annual fixed interest rate of 3,75%. The brokerage cost is \$750 per transaction.

Let’s put all the data available in the formula above. Thus, the optimal cash balance of \$346,410.16 meets the minimum sum of opportunity cost and transaction cost. To prove this, we compute the total cost for OCB and a cash balance of \$250,000 and \$450,000 using the total cost equation above.

 TC = \$346,410.16 × 0.0375 + \$3,000,000 × \$750 = \$12,990.38 2 \$346,410.16
 TC = \$250,000 × 0.0375 + \$3,000,000 × \$750 = \$13,687.50 2 \$250,000
 TC = \$450,000 × 0.0375 + \$3,000,000 × \$750 = \$13,437.50 2 \$450,000

As we can see, the Baumol-Tobin model is always the best solution.

## Graph

If the basic assumptions of the Baumol-Tobin model are met, the graph of cash spending and cash balance replenishment is as follows: The maximum cash balance equals the optimal cash balance and is being spent at a constant rate until reaching zero. At this time, marketable securities should be sold to replenish the cash balance to the maximum level.