Question

Xavier Martinez took out a $5,000 loan at 7.5% interest for reconstructing his patio. His monthly payment on the 12-month loan was $437.26. After 6 payments, the balance was $2,556.03. He paid off the loan with the next payment. What did he save by paying off the loan early?

(Hint: First calculate the payoff: Find the interest on the balance due and add it to the balance due. Add the amount of the first six payments to the payoff in month 7. Compare this total to the monthly payment multiplied by 12.

Answer 1

You should follow those nice instructions.

i = 0.075
j = i/12 = 0.075/12 = 0.00625
v = 1/(1+j) = 1/(1 + 0.00625) = 0.99378882

\($5000 = Pmt\cdot(v + v^2 + ... + v^12) = Pmt\cdot\dfrac{v - v^{13}}{1-v}\)

This leads to a payment of $433.79. Well, that's a little annoying as the problem statement clearly states $437.26. Oh, well. Sometimes, other things must be considered.

\(437.26\cdot\dfrac{v-v^{13}}{1-v} = 5040.03\) Maybe there was a $40 service charge for loan origination, or something.

In any case:

Payoff at Loan Origination: \(437.26\cdot\dfrac{v-v^{13}}{1-v}\) (You may not get your service charge back!)

Payoff after the 1st payment: \(437.26\cdot\dfrac{v-v^{12}}{1-v}\)

Payoff after the 2nd payment: \(437.26\cdot\dfrac{v-v^{11}}{1-v}\)

Payoff after the 3rd payment: \(437.26\cdot\dfrac{v-v^{10}}{1-v}\)

Are you seeing a pattern?