Interest Rate Theory - Am I missing something?

Question

Seth borrows $X$ for four years at an annual effective interest rate of 8%, to be repaid with equal payments at the end of each year. The outstanding loan balance at the end of the second year is 1076.82 and at the end of the third year is 559.12. Calculate the principal repaid in the first payment.

I was thinking that $559.12 \times 1.08$ would give me the level payment $r$, because the last payment should pay off the loan. With the payment $r$, the $i=0.08$, and the 4 years lifetime of the loan, I can calculate the original loan balance $b = r\; a_{\overline{4}|i}$. Once I have that, the principal payment on the first payment must be $r - b \times 0.08$. That gives me the correct answer, too, but nowhere did I use "1076.82."

Did I get it right accidentally, or is this indeed redundant information?

Answer 1

$559.12 \times 1.08$ will give you the size of the final payment. And since all the payments are equal, will give you the size of each payment.

The initial balance is the NPV of the future payments.

$B = \frac{559.12\times 1.08}{1.08^4} + \frac{559.12\times 1.08}{1.08^3}+\frac{559.12\times 1.08}{1.08^2}+\frac{559.12\times 1.08}{1.08^1}$

The first months interest payment is $B\times 0.08$

Which means that the first months interest payment is $559.12\times 1.08 - B\times 0.08$

And that means that we never used the information regarding the balance at the end of the second year. But, we can use it as a check for our work.