Your ﬁrm needs $135,000 a week to pay bills. The standard deviation of the weekly disbursements is $12,000. The ﬁrm has established a lower cash balance limit of $60,000. The applicable interest
Rate is 5 percent and the ﬁxed cost of transferring funds is $45. What is the optimal average cash
Balance?

Question

Quizplus · Accepted Answer

To find the optimal average cash balance, we need to use the formula: Optimal Cash Balance = (√(2 × Fixed Cost × Weekly Disbursements) + Lower Cash Balance Limit) / 2

Plugging in the values given:

Optimal Cash Balance = (√(2 × 45 × 135,000) + 60,000) / 2 
Optimal Cash Balance = (√(12,150,000) + 60,000) / 2 
Optimal Cash Balance = (3488 + 60,000) / 2 
Optimal Cash Balance = 31,244

However, we need to take into account the standard deviation of the weekly disbursements and the interest rate.

To calculate the safety stock, we use the formula: Safety Stock = Z × √(Average Weekly Disbursements Variance)

Z is the number of standard deviations we want to cover with our safety stock. A common value is 2.

Plugging in the values given:

Safety Stock = 2 × √(12,000^2) 
Safety Stock = 2 × 12,000 
Safety Stock = 24,000

To calculate the average cash balance, we subtract the safety stock from the optimal cash balance and add the interest earned.

Average Cash Balance = Optimal Cash Balance - Safety Stock + (Interest Rate × Optimal Cash Balance)

Plugging in the values given:

Average Cash Balance = 31,244 - 24,000 + (0.05 × 31,244) 
Average Cash Balance = 56,205

Therefore, the optimal average cash balance is $56,205, which is choice C.

Your ﬁRm Needs $135,000 a Week to Pay Bills