Question

A loan amount of L is amortized over six years with monthly payments (at the end of each month) at a nominal interest rate of i(12) compounded monthly. The first payment is 500 and is to be paid one month from the date of the loan. Each subsequent payment will be 1% larger than the prior payment.

(a) If i(12) = 9%, find the principal repaid in the 25th payment. (b) If i(12) = 12%, find the amount of loan L.

Answer 1

(a) Loan Amount = L , Interest Rate = i% compounded monthly, Applicable Monthly Rate = (i/12) %

Payment Frequency: Monthly and Loan Tenure = 6 years or (12 x 6) = 72 months, payment growth rate = 1 %

If i = 9 %, then Monthly Rate = (9/12) = 0.75 %

Therefore, Loan Amount Post 24th payment = L24 = Present Value of Payment no 25 to 72.

25th Payments = 500 x (1.01)^(24) = $ 634.8673

Therefore, PV of Remaining Payments at the end of the 24th Payment = 634.8673 x [1/(0.0075 - 0.01)] x [1-{(1.01)/(1.0075)}^(48)}] = $ 32079.57

Loan Amout Post 25th Payment = L25 = Present Value of Payment no.26 to 72

26th Payment = 634.8673 x 1.01 = $ 641.216

L25 = 641.216 x [1/(0.0075-0.01)] x [1-{(1.01)/(1.0075)}^(47)}] = $ 31685.3

Principal Repaid in 25th Payment = Reduction in Loan Amount = L24 - L25 = 32079.57 - 31685.3= $ 394.27

(b) If i = 12, then applicable monthly rate = (12/12) = 1 %

Therefore, Loan Amount = 500 x [1/(0.0075-0.01)] x [1-{(1.01)/(1.0075)}^(72)] = $ 39069.93