Question

You have a loan outstanding. It requires making five annual payments at the end of the next five years of $4000 each. Your bank has offered to restructure the loan so that instead of making five payments as originally agreed, you will make only one final payment at the end of the loan in five years. If the interest rate on the loan is 5.63%, what final payment will the bank require you to make so that it is indifferent between the two forms of payment?

Answer 1

the bank will require you to make a final payment of $22,004.52 at the end of the loan in five years.

Explanation:

To calculate the final payment that the bank requires you to make, we need to find the present value of the five annual payments of $4000 each, and then compound that present value to the end of the loan in five years at the interest rate of 5.63%.

Let's begin by calculating the present value of the five annual payments. We can use the formula for the present value of an annuity:

PV = C * [(1 - (1 + r)^-n) / r]

where:

PV = present value

C = annual payment amount

r = interest rate per period (annual rate divided by number of periods per year)

n = number of periods

Plugging in the given values, we get:

PV = $4000 * [(1 - (1 + 0.0563/1)^-5) / (0.0563/1)]

= $4000 * [(1 - (1.0563)^-5) / 0.0563]

= $4000 * 4.169942

= $16,679.77

So the present value of the five annual payments is $16,679.77.

Next, we need to compound this present value to the end of the loan in five years. We can use the formula for future value:

FV = PV * (1 + r)^n

where:

FV = future value

PV = present value

r = interest rate per period

n = number of periods

Plugging in the given values, we get:

FV = $16,679.77 * (1 + 0.0563/1)^5

= $16,679.77 * 1.319695

= $22,004.52

Therefore, the bank will require you to make a final payment of $22,004.52 at the end of the loan in five years.