# Calculate repayments amount on loan

I doing some work on a client website, however I'm finding it difficult to calculate the correct interest amount.

• I have the principle amount, for example £200
• The loan is repaid over 3 months in monthly instalments
• I have an annual interest rate 264% (fixed), so a monthly interest rate is 22%
• The loan is repayable in equal instalments

I want to calculate the monthly repayment amount

Interest is only payable on the outstanding balance, so on the first payment:

Balance = starting balance + interest - repayment

I can work out interest ok, but I'm unsure of the work out the repayments so the balance can be reduced accordingly.

Anyone help?

• Just as a first question - if your annual interest rate is 264%, then the monthly interest rate will be slightly less than 22%, to factor in compound interest. – Mathmo123 Oct 1 '14 at 13:58
• Mathmo123 - I'm well and truly out of my comfort zone here - I should have pointed out that the interest rate is fixed - I don't know if this makes a difference. – John Oct 1 '14 at 14:00
• Compound interest is when you are taking interest of money already given as interest. For example, if I have £100 and get paid 10% a month, after the first month I will have £110, but after the second I will get £11 interest instead of £10. If you want the annual interest rate to be 264%, then the monthly interest rate should be roughly 11.3% – Mathmo123 Oct 1 '14 at 14:16

## 1 Answer

Let $x$ be the starting loan amount, and $d$ be the monthly interest expressed as a decimal (i.e. $22$% interest would be $d = 1.22$ so that multiplying by $d$ corresponds to adding $22$% of interest).

We are looking to find $y$ where $y$ is a fixed payment per month subject to:

$$d\cdot\left(d\cdot\left((d\cdot x) -y\right)-y\right)-y = 0$$

Expressing this equation as an equation in terms of $y$ gives

$$d^3x-d^2y-dy-y = 0$$

So $$y = \frac{d^3x}{1+d+d^2}$$

With a starting value of $200$ and monthly interest of $22$%, this becomes

$$y=\frac{1.22^3 \times 200}{1+1.22+1.22^2}\approx 97.93$$