# How to determine annual payments on a partially repaid loan?

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 $$X$$.

My Attempt

$$\ 500$$ 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?

Another update!
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?

• I like that this question explains what they've tried and where they got stuck. Jul 24, 2010 at 2:00
• I just deleted some of my previous comments as they might not have been correct. You said that each of the first 10 payments must be 150% of interest due and used that to get that each payment must be \$75 and the calculation in my answer depends on that figure. However, the amount of interest due decreases as payments are made, so each subsequent repayment will be less than \$75. The correct way to deal with this is to just reduce the debt by 5% per year Jul 24, 2010 at 3:46

This problem is in two stages. For the first stage, notice that you are paying 150% interest, but ending up owing more. This is because you subtracted $\$ $750 from the future value, when in fact each$\75 amount was paid at a time in the past and needs be converted to a future value too. The payment in the nth year has a future value of $75\times(1.1)^{10-n}$. The total future value of the repayment is: $$75(1.1^9)+75(1.1^8)+\cdots+75(1.1^0).$$
Note that I have assumed that the interest is charged before the repayments are made. This sequence is a geometric progression. We consider it as a geometric sequence in reverse to make the maths easier. It has first term ($a$) 75, each term 1.1 times the previous ($r$) and 10 terms ($n$). The sum is given by the formula: $$\frac{a{r^{n-1}}}{r-1}=\frac{75{1.1^{10-1}}}{0.1}\approx\1195.31$$