What is the Future Value of a $\$5{,}000$ loan paid back at $\$1{,}000$ per month, with a $6\%$ nominal interest rate?

Question

I'm trying to figure out a problem with the formulae I have, but I'm having some difficulty. The problem is:

Susan borrows $5{,}000$ dollars from a finance company at a nominal interest rate of $6.6\%$ compounded monthly. If she makes payments of $\$1{,}000$ at the end of each year, how much does she owe five years after borrowing the money?

So basically, I know that I need to use the formula $A = P(1+i)^n$ where $n =$ (number of years) $\times$ (number of compounding periods), and $i = $(annual interest rate)$/$(number of compounding periods in a year). The issue but what I'm confused about is how to factor in the payments of $1000$ dollars at the end of each year. Where could that fit into the equation? I know I could make a chart and calculate if out per year, but my professor doesn't want that.

Any help would be greatly appreciated!

Answer 1

Hint:

This is a basic exercise of the concept future value(FV), so how much she owe after five years will be the future value of 5000 after five years, minus the future values of the amounts she paid each year.

Answer 2

You can think of each payment as drawing interest separately from the rest. She borrows 5000 at year zero, then borrows -1000 each of the next five years. Compound each of them up to the end of five years and you have your answer. The five payments make a geometric series with ratio $(1+i)^{12}$ that you can sum.

The way you write it, $n$ should be in months, not years. You say years in your text.

Answer 3

Hint

If you have not learnt a specific formula to cover such types of problems,

(a) Compute how much the loan will be worth after 5 years.

(b) Compute the worth after 5 years of the stream of payments, as a sum of a G.P.

(c) The difference between the two is the balance owed after 5 years.