Question

Please post with mathematical formulas please, not an excel sheet!

1. Mr. X is repaying a loan by monthly payments of $146.75 at a nominal annual rate of 9% compounded monthly. Immediately after one of the pay- ments is made, when Mr. X has still 50 payments ahead of him, the lender lowers the interest rate to 7.8% nominal annual rate compounded monthly. Mr. X chooses to keep the same monthly payments, except the last payment that is larger than the regular ones. How many months from the time of the interest change will it take for Mr. X to repay the loan and what will be the total amount of the last payment?

Answer 1

Just when interest rate is lowered, the PV of the loan will be PV of all the balance 50 payments ahead of him at the old interest rate.

Hence the loan outstanding when the rate is lowered = PV = A / r x [1 - (1 + r)^-n] where A = payment per period = $ 146.75; r = old interest rate per month = 9% / 12 = 0.0075; n = number of balance payments = 50

Hence, PV of loan = 146.75 / 0.0075 x [1 - (1 + 0.0075)^-50] = $  6,099.88

Now this loan is serviced at interest rate per period of i = 7.8% / 12 = 0.0065 by paying the same old amount of = 146.75

Hence, let m be the number of such payments required to payoff the loan along with interest. Hence, PV of such payments = PV of loan = A / i x [1 - (1 + i)^-m]

Hence, 6,099.88 = 146.75 / 0.0065 x [1 - (1 + 0.0065)^-m]

Hence, 1.0065^-m = 1 - 6,099.88 x 0.0065 / 146.75 =  0.73

hence, m = - ln (0.73) / ln (1.0065) =  48.61

So, the loan can be repaid by 47 full payments of 146.75 and the 48 payment will be the entire balance amount.

So, let the 48th payment be P

Hence, PV of loan = 6,099.88 = PV of 47 annuity payments of 146.75 + PV of the last payment P = A / i x [1 - (1 + i)^-47] + P x (1 + i)^-48 = 146.75 / 0.0065 x [1 - (1 + 0.0065)^-47] + P x (1 + 0.0065)^-48 =  5,926.81 +  0.7327P

Hence, P = (6,099.88 - 5,926.81) / 0.7327 =  236.19

Hence, the loan will be repaid in 48 months after the interest rate change and the last payment will be of an amount = P = $ 236.19