# You have some extra cash this month and are considering putting it toward your car loan. Your...

## Question:

You have some extra cash this month and are considering putting it toward your car loan. Your interest rate 7%; your loan payments are $600 per month; and you have 36 months left on your loan. If you pay an additional$1000 with your next regular \$600 payment (due in one month), how much will it reduce the amount of time left to pay off your loan?

## Monthly Loan Payments:

Loans often require monthly payments. The monthly payments constitute an annuity, and the amount of each payment is calculated such that the present value of these payments is equal to the amount borrowed.

The time left will be reduced by roughly 24 months.

We first compute the outstanding balance on the loan, which is the present value of the remaining 36 monthly payments. We can use the following formula to compute the present value of an annuity with periodic payment {eq}M {/eq} for {eq}T{/eq} periods, given periodic return {eq}r{/eq}:

• {eq}\displaystyle \frac{M(1 - (1 + r)^{-T})}{r} {/eq}

Applying the formula, the remaining balance is:

• {eq}\displaystyle \frac{600(1 - (1 + 7\%/12)^{-36})}{7\%/12} = 19,431.88 {/eq}

Next, we apply the same formula to compute the number of months to pay off the loan if the monthly payment is instead 1600, which requires:

• {eq}\displaystyle \frac{1600(1 - (1 + 7\%/12)^{-T})}{7\%/12} = 19,431.88 {/eq}

and the solution is:

• {eq}T = 12.63{/eq}

That is, by paying an additional 1000 a month, the number of payments is reduced by 36 - 12.63 = 23.47