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The formula given is the Baumol Cash Management Model. An increase in the cash requirements for the year would cause an increase, not a decrease, in the optimal cash balance according to the model. That is because an increase in the numerator of the fraction under the radical sign will increase the value of the fraction, thereby increasing the square root of the value and the optimal average cash balance. Using the Baumol Cash Management Model, this can be illustrated by assigning some values to it. Let’s say the cost per sale of a T-Bill is $20 (that is the commission paid, or the fixed cost per transaction to convert a T-Bill into cash). And let’s say the annual cash disbursement amount, or the total demand for cash for the period, is $50,000. And let’s say the interest rate on marketable securities is 2%, or .02. The optimal cash balance will be the square root of [(2 × 20 × 50,000) / .02]: (2 × 20 × 50,000) = 2,000,000. 2,000,000 / .02 = 100,000,000. The square root of 100,000,000 = $10,000. Now, let's increase the cash requirements for the year to $75,000. The optimal cash balance will be the square root of [(2 × 20 × 75,000) / .02]: (2 × 20 × 75,000) = 3,000,000. 3,000,000 / .02 = 150,000,000. The square root of 150,000,000 = $12,247. So because the cash requirements for the year have increased, the optimal cash balance has also increased from $10,000 to $12,247.
The formula given is the Baumol Cash Management Model. The uncertainty of cash outflows is a fact that limits the effectiveness of the Baumol Cash Management Model. It does not result in either an increase or a decrease in the optimal cash balance calculated using the model.
An increase in the return on marketable securities would increase the opportunity cost of holding cash instead of investing it in marketable securities. When the opportunity cost of holding cash increases, the amount of cash a business will want to hold will decrease. That is because an increase in the denominator of the fraction under the radical sign will decrease the value of the fraction, thereby decreasing the square root of the value and the optimal average cash balance. The formula given is the Baumol Cash Management Model. Using the Baumol Cash Management Model, this answer can be illustrated by assigning some values to it. Let’s say the cost per sale of a T-Bill is $20 (that is the commission paid, or the fixed cost per transaction to convert a T-Bill into cash). And let’s say the annual cash disbursement amount, or the total demand for cash for the period, is $50,000. And let’s say the interest rate on marketable securities is 2%, or .02. The optimal cash balance will be the square root of [(2 × 20 × 50,000) / .02]: (2 × 20 × 50,000) = 2,000,000. 2,000,000 / .02 = 100,000,000. The square root of 100,000,000 = $10,000. Now, let’s increase the interest rate on marketable securities to 3%, or .03: 2,000,000 / .03 = 66,666,666.67. The square root of 66,666,666.67 = $8,165. So because the interest rate has increased, the optimal cash balance has decreased from $10,000 to $8,165.
The formula given is the Baumol Cash Management Model. An increase in the cost of a security trade would cause an increase, not a decrease, in the optimal cash balance according to the model. That is because an increase in the numerator of the fraction under the radical sign will increase the value of the fraction, thereby increasing the square root of the value and the optimal average cash balance. Using the Baumol Cash Management Model, this can be illustrated by assigning some values to it. Let’s say the cost per sale of a T-Bill is $20 (that is the commission paid, or the fixed cost per transaction to convert a T-Bill into cash). And let’s say the annual cash disbursement amount, or the total demand for cash for the period, is $50,000. And let’s say the interest rate on marketable securities is 2%, or .02. The optimal cash balance will be the square root of [(2 × 20 × 50,000) / .02]: (2 × 20 × 50,000) = 2,000,000. 2,000,000 / .02 = 100,000,000. The square root of 100,000,000 = $10,000. Now, let's increase the cost per sale of a T-Bill to $30. The optimal cash balance will be the square root of [(2 × 30 × 50,000) / .02]: (2 × 30 × 50,000) = 3,000,000. 3,000,000 / .02 = 150,000,000. The square root of 150,000,000 = $12,247. So because the cost per sale of a T-Bill has increased, the optimal cash balance has also increased from $10,000 to $12,247.