# Interest Amount Calculator written 12 years ago

— Skip the intro, and go straight to the calculator —

## Background

Recently, as I’ve joined the real world, where rates of interest for loans aren’t the same as inflation. I’ve been wondering about how much interest rates differ when held for less time than a year. For example a bank overdraft is often quoted APR (Annual Percentage Rate), or even as a Monthly Percentage Rate, but what if I’m only overdrawn for 15 days of a month? How much interest will I be charged?

My university degree taught me the future value of money, which goes some way to explain the delicacies of compound interest. But this is how I understand it:

APR is great, at the end of each year the bank simply adds on say, 15% of your loan amount to the balance remaining. It doesn’t work like this though, as what If I keep the loan for only 6 months?

So the unit of time in which the bank adds on interest must be smaller than a year, but how much smaller? 1 day? 1 minute? 1 second? Well, no, it is in fact infinitesimally small, and has no duration whatsoever.

Wow, sounds magic. Well it’s actually a nice mathematical formula which works out the interest over a given period:

*V*=*A*e^{rt}Where

*A*- Initial amount of money
*r*- Interest rate. (note, this is NOT the APR)
*t*- Period of time.

Handily we can use the APR to work out the value of *r*.

Let *R* be the APR, *t* = 1, as the period is 1 year.

$$*A* + *AR* = *A*e^{r}

*1 + R* = e^{r}

*r* = ln(1 + *R*)

So now we can work out *r*, we can work out the amount of interest accrued for any period of time.

Secondly, there are lots of offers out there that say "get this car for ?7,523 today, and pay ?99 per month at %10.9 APR". How much does that cost to actually buy? (turns out it's over 12k). We can work this out by dividing the period into equal chunks of one month, and calculating each on individually and summing them.

## Instructions

Insert the Amount and APR.

If you are working out how much it will cost you to be overdrawn for 1 month, or how big your savings will be in 12 months, leave the "repayments" checkbox clear, and enter the appropriate length of time in the "period" box.

If you are working out the total cost of a loan with a fixed monhtly repayment amount, check the "repayments" checkbox, and then enter the

*monthly*repayment amount.Click Calculate.

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