# Interest Rate Theory - Am I missing something?

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?

• Is this simple interest or is interest compounded annually? If simple interest, then there is no need to say effective interest rate for that suggests some type of compounding. Jul 11, 2019 at 2:48
• Annual compounding, annual payments - which is why I'm using the immediate annuity. Jul 11, 2019 at 2:57

$$559.12 \times 1.08$$ will give you the size of the final payment. And since all the payments are equal, will give you the size of each payment.
$$B = \frac{559.12\times 1.08}{1.08^4} + \frac{559.12\times 1.08}{1.08^3}+\frac{559.12\times 1.08}{1.08^2}+\frac{559.12\times 1.08}{1.08^1}$$
The first months interest payment is $$B\times 0.08$$
Which means that the first months interest payment is $$559.12\times 1.08 - B\times 0.08$$