Question

Today is January 1st, 2019 (T=0). You take out a 6 year fully amortizing auto loan of $24,000. Payments are made at the end of each calendar month. The loan has a fixed annual rate of 4.0% (or 4.0%/12 per month).

Assume you decide to make monthly payments of $503.50 instead. Approximately how many months sooner would you pay off the auto loan?

1.) 18

2.) 20

3.) 22

4.) 24

5.) 48

Answer 1

Present value of Annuity = A*[(1-(1+r)^-n)/r]

Where

A - Annuity payment = 503.50

r - rate per period = .04/12

n - no. of periods = ?

24000 = 503.50*[(1-(1+(.04/12))^-n)/(.04/12)]

[(1-(1+(.04/12))^-n)/(.04/12)] = 24000/503.50

= 47.6663356504

(1-(1+(.04/12))^-n) = 47.6663356504 * (.04/12)

= 0.1588877855

(1+(.04/12))^-n = 1 - 0.1588877855

= 0.8411122145

(1+ 0.00333333333 )^{-n =} 0.8411122145

-n = log 0.8411122145 / log 1.00333333333

= -0.07514606022/ 0.00144524087

n = 51.9955266834

n = 52 months

So it takes 52 months to payoff the loan under new EMI

no. of months sooner you would pay off the auto loan = (6*12) - 52

= 72-52

= 20