MD Groceries currently sets its initial cash balance at $5,000, which is depleted every four days. The current market rate of return is 4.5% per year. The firm pays $2.50 each time it replenishes its cash account. Use a 360-day year. What is the optimal initial cash balance for MD using the BAT model?

A) $5,127
B) $5,449
C) $7,071
D) $7,968
E) $8,937

The optimal initial cash balance can be calculated using the Baumol Model (also known as the BAT model), which is given by the formula: $C^{\ast }=\sqrt{\frac{2bT}{i}}$ , where $C^{\ast }$ is the optimal cash balance, $b$ is the transaction cost per replenishment, $T$ is the total cash needed for the period, and $i$ is the interest rate per period.Given:- Initial cash balance depletes every 4 days, implying the firm needs to replenish its cash every 4 days in a 360-day year. This means the firm replenishes 360/4 = 90 times a year.- If the initial balance is $5,000, the total cash needed for the period (T) is 90 * $5,000 = $450,000.- The transaction cost ( $b$ ) is $2.50.- The annual interest rate ( $i$ ) is 4.5%, or 0.045 when expressed as a decimal. For the Baumol model, we need the interest rate per period, which in this case is per year, so we use 0.045.Plugging these values into the Baumol formula gives: $$C^{\ast }=\sqrt{\frac{2\ast 2.50\ast 450,000}{0.045}}=\sqrt{\frac{2,250,000}{0.045}}=\sqrt{50,000,000}=7,071$$ Therefore, the optimal initial cash balance for MD Groceries, according to the BAT model, is $7,071.