Last week, I got an email from my dad. As usual, he made a weird loan system for the cooperative – well, not so weird actually.
Suppose that you borrow A $ from a bank with k% interest. Generally the bank take k% of A $ then divided it into N times of payment. For example, you borrow 10000 $ with 12% per year interest. You will pay back every month in two years. Therefore, you have to pay the bank 933.34 $ each month.
The cooperative has the main purpose to help members in finances. Hence, the cooperative needs less money than banks do.
Here is his loan system.
The same system, but this time, the interest won’t be calculated first. Each time you pay, the interest is decreased due to how much the capital is left. You will pay back in the same amount of money each time, however. Therefore, each time you pay, the capital part is getting larger and the interest part is getting smaller.
Confuse? Well, I’m confuse. Let’s see the example.
Suppose that you borrow 399000 $ from me. (I don’t have that much money actually. haha) We agree that you will pay me back in 84 months with 5% interest per year (or 0.4167 % per month) with my dad’s system. From the calculation, you have to pay me 5639.43 $ per month.
First month, I take 0.4167 % of 399000 $ which is 1662.5 $. You pay me 5639.43 $ which is 1662.5$ interest and (5639.43-1662.5) = 3976.93 $ capital.
Next month, I take 0.4167% of (399000 – 3976.93) = 395023.07 $ which is 1645.93 $. You pay me 5639.43$ which is 1645.93$ and 3993.50 $ capital.
…
and so on
in last month, you will pay 5639.43$ as usual, but it will be about 23.4$ interest and 5639.40$ capital.
I think we get the system now.
The problem is how can we calculate ‘how much you have to pay me each month in order to pay me back all the money included interest by the last payment – exactly.’
So my dad asked me this problem. He asked for a formula to calculate ‘the amount of money the borrower needs to pay each month’ when the amount of capital money, the interest rate, and the number of payments are given.
It took me such a long time for this problem because I tried to figure out how he came up with 5639.43$ at first. (Later on I found out that he adjusted it by hand.) First, I tried to find a recursive relation between each payment. I, however, could not see an easy pattern from its recursion.
I then assumed the amount of each payment to be X. Then I wrote out everything that would happen each payment in variables. Bang! I saw a pattern. I then summed up and solved the equation. I came out with a formula in the end.
I think this might me an interesting problem for somebody who is finding something to think in his/her leisure.
Here is the answer I got.
Let;
borrowed money = A
interest per payment = k
the number of payments = N
an amount of money would be paid each time = XDefine
then, X=AG
the sum of interest (I) ; I=(NG-1)A
The result comes out pretty cute, isn’t it?
