Question

Roberto borrows $10,000 and repays the loan with three equal annual payments. The interest rate for the first year of the loan is 4% compounded annually; for the second year of the loan the interest rate is 5% compounded annually; for the third year of the loan the interest rate is 6% compounded annually.

a. Determine the size of the equal annual payments.


b. Would interchanging the interest rates for first year and third year change the answer to part a?

Answer 1

(a)

Let the equal annual payment be P.

Hence Present Value of total payment must be equal to 10,000.

Present value of payment A after n years is given by;

PV = A/(1+i)ⁿ

Here i = interest rate, n = time

Hence, Total present values of a payment = P/(1+0.04)¹ + P/(1+0.05)² + P/(1+0.06)³

Hence this Value must be equal to 10,000

=> P/(1+0.04)¹ + P/(1+0.05)² + P/(1+0.06)³ = 10,000

=> P = 10,000/(1/(1+0.04)¹ + 1/(1+0.05)² + 1/(1+0.06)³) = 3692.51.

Hence, the size of the equal annual payments = $3692.51

(b)

Yes, changing the interest rate will definitely change our answer to first period. Now Lets calculate the change.

Now First period interest rate= 0.06 and Third period interest rate = 0.04

Let the equal annual payment be Q.

Hence Present Value of total payment must be equal to 10,000.

Present value of payment A after n years is given by;

PV = A/(1+i)ⁿ

Here i = interest rate, n = time

Hence, Total present values of a payment = Q/(1+0.06)¹ + Q/(1+0.05)² + Q/(1+0.04)³

Hence this Value must be equal to 10,000

=> Q/(1+0.06)¹ + Q/(1+0.05)² + Q/(1+0.04)³ = 10,000

=> Q = 10,000/(1/(1+0.06)¹ + 1/(1+0.05)² + 1/(1+0.04)³) = 3650.41.

Hence, New size of the equal annual payments = $3650.41.

Hence, Size of equal payment will decrease from to $3650.41 to $3692.51