Question : A $10$-year loan of $\$500$ is repaid with payments at the end of each year. The lender charges interest at an annual effective rate of $10\%$. Each of the first ten payments is $150\%$ of the amount of interest due. Each of the last ten payments is $X$. Calculate $X$.
$\$ 500$ will earn $\$50$ interest each year, so each of the first $10$ payments must be $\$75$.
Then after $10$ years, a total of $750$ has been repaid. In $10$ years, I can find the accumulated debt by saying
PV=500 I/Y=10 N=10
giving me FV=$\$1296.87$.
So the balance would be $\$ 1296.87-\$ 750=\$ 546.97$.
Now I am stuck! How do I find out what the last payments should be? I know I can't just divide $\$ 546.97/10=\$ 54.697$, because the lender is still charging interest while the borrower pays off this remaining debt, so there would still be the interest left over.
This situation isn't mentioned anywhere in my calculator manual! Can one of you give me some explanation about what is going on so that I can do it by hand?
I tried working on it some more, and came up with a really great idea! Since the payments are $1.5\times$ the $10\%$ interest, it's just like paying off $5\%$ of the principal each year! This saves me a lot of time, because I can just set $I/Y=-5$ and get $FV=598.74$ on my calculator. I did it the long way by calculating the future value of each interest payment (turns out they were not all $\$75$, because the outstanding principal got smaller), and they were the same. Is this always going to work, or did I just get lucky here?
I think I solved it. All I needed to do was to set
FV=0 I/Y=10 N=10 PV=598.74
and then I got $PMT=97.44$. I never used the
PMT button before, though, so is there some other way I can check the answer is right?