Question 5.3: Let’s say that you are now three years into your $30,000 car...

PLAN
We have already determined that the monthly payments on the loan are $590.50. The remaining balance on the loan is the present value of the remaining two years, or 24 months, of payments. Thus, we can just use the annuity formula with the monthly rate of 0.5625%, a monthly payment of $590.50, and 24 months remaining.

EXECUTE

Balance with 24 months remaining    =\$590.50\times \frac{1}{0.005625} (1-\frac{1}{1.005625^{24} } )=\$13.222.32 

Thus, after three years, you owe $13,222.32 on the loan.

Using a financial calculator or spreadsheet:

You could also compute this as the FV of the original loan amount after deducting payments:

The 24 cent difference is due to rounding on the payment amount.

EVALUATE

At any point in time, including when you first take out the loan, you can calculate the balance of the loan as the present value of your remaining payments. Recall that when the bank gave you the $30,000 in the first place, it was willing to take 60 monthly payments of $590.50 in return only because the present value of those payments was equivalent to the cash it was giving you. Any time that you want to end the loan, the bank will charge you a lump sum equal to the present value of what it would receive if you continued making your payments as planned. As the second approach shows, the amount you owe can also be thought of as the future value of the original amount borrowed after deducting payments made along the way.