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I'm trying to work out interest on short term loans - these are loans that extend to months not years, and are typically repaid in monthly chunks, but I also know that some are repayable in weekly instalments. A 'month' can also have slightly different meanings - a calendar month (28,30 or 31 days), or it could be 1/12th of a year.

I'm looking to create a formula that will allow me to calculate interest (and total repayment) based on these variables - a typical example can be found at the link below. Assuming a £100 loan over 3 months @ 29% per calendar month.

https://sunny.co.uk/?sc=SunnyFlexipay

I found this question on the site which seems to be very similar to what i'm asking - monthly compounded interest on a variable interest period (28-31 days). However, I'm no maths expert and the formula in its given state is baffling.

formula to calculate the monthly repayments of this contract

I'd appreciate it if anyone could show working out alongside the formula to help me understand it better.

Thanks in advance.

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There's a limit to how much I can explain these formula to you, since they require quite a bit of build-up to understand, however in general the form of the equation you want is:

$$P=X\dfrac{1-(1+i)^{-n}}{i}$$

Where $P$ is the Principal (i.e. the quoted value of the loan, in this case £100), $X$ is the repayments per period (which we'll work out later), $i$ is the interest rate per period (e.g. 29% would be $i=0.29$), and $n$ is the number of periods (in this case 3). To use this equation, you first need to make sure that everything is in the same period - i.e. for your example, you're already given that the interest rate is 29% per month, and payments occur each month. You'll always be missing one piece of information, usually $X$, so you can just plug this all into wolfram like so and get it to solve it for you. In this particular case, you'll be paying £54.29 a month for 3 months, or £162.87 in total. Since £100 of that is the principal value of the loan, the other £62.87 is the interest that you'll pay.

The only trouble you should hopefully have with this method is if the interest rate runs under a different period than the payment period (e.g. monthly payments but you're given a yearly interest rate). In that case, you'll need to convert the interest rate. This is easy, by itself, however it requires you to understand what form of rate is being provided. In most cases, if you get a yearly interest rate, and it does not say it's an effective rate, then your monthly interest rate is just that yearly rate divided by 12. If, instead, the yearly rate, $i$, is an effective interest rate, then you can use the formula $i_m=(1+i)^{\frac{1}{12}}-1$, with $i_m$ being the monthly interest rate that you'll then substitute into the first formula.

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  • $\begingroup$ Thanks for your help its really appreciated. The link is awesome. I still don't understand how I can adapt the solution, say if its weekly, or if the first payment is made after say 20 days instead of the full calendar month. This is entirely possible due to the way this type of loan works. $\endgroup$
    – Johnnyboy
    Dec 8, 2014 at 15:42
  • $\begingroup$ @Johnnyboy the problem is, those things are very easy to deal with, but they require you to understand what the above formula is doing, and then to adapt it to your needs. No one formula will cover everything you want to do with this, because there are just too many ways in which loans can be structured. I suggest you read up on what an Annuity is here - en.wikipedia.org/wiki/Effective_interest_rate, and how to convert interest rates into different effective periods here - en.wikipedia.org/wiki/Effective_interest_rate. $\endgroup$
    – mardat
    Dec 8, 2014 at 21:53

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