You have a loan outstanding. It requires making fiveannual payments of $ 4 comma 000each at the end of the next fiveyears. Your bank has offered to restructure the loan so that instead of making the fivepayments as originally agreed, you will make only one final payment in fiveyears. If the interest rate on the loan is 4 % , what final payment will the bank require you to make so that it is indifferent to the two forms of payment?

For the final payment be indifferent to the first payment form, the amount of money we need to pay is $22,531.9.

Explanation:

Notice that to the loan be indifferent, the money we wont pay for each of the years in the first form id money we need to borrow, then the bank is offering loans over eachone five yaers at a 4% interest rate. That mean we are borrowing the following loans:

The first year, a loan to pay in five years: L1 = $4,000 * (1+0.04) ^5

The second year, a loan to pay in 4 years: L2 = $4,000 * (1+0.04) ^4

The third year, a loan to pay in 3 years: L3 = $4,000 * (1+0.04) ^3

The fourth year, a loan to pay in 2 years: L4 = $4,000 * (1+0.04) ^2

The fiveth year, a loan to pay in 1 year: L5 = $4,000 * (1+0.04) ^1

By calculating each loan we have:

L1 = $4,000 * (1+0.04) ^5 = $4,866.6

L2 = $4,000 * (1+0.04) ^4 = $4,679.4

L3 = $4,000 * (1+0.04) ^3 = $4,499.5

L4 = $4,000 * (1+0.04) ^2 = $4,326.4

L5 = $4,000 * (1+0.04) ^1 = $4,816.0

If we add each one of those loans we obtain the equivalent final payment:

L = L1 + L2 + L3 + L4 + L5

L = $4,866.6 + $4,679.4 + $4,499.5 + $4,326.4 + $4,816.0 = $22,531.9