# The insidious maths of payday loans (with added logarithms)

Compared to regular small print, it’s pretty big, but compared to the four-panel, faux-war-era comic story (disaster strikes, protagonist calls Rent-A-Loan*, cash magically appears and they all live happily ever after), you could easily overlook it. You might even look at the bottom of the poster and think “that can’t possibly be right.” But it is there, and it is true: **APR 1737% representative**.

Maybe they’re banking on you not knowing what APR is, or what percents are, or that ‘representative’ means there’s something different goes on with the maths, but no. Let’s put this to rest: it means, if you took out a £100 loan today, and paid it back this time next year, you’d have to pay back the original £100 plus an interest payment of £1737. Leave it another year, you’re in the hole for nearly £34,000. Three years, and you’re looking at well over half a million quid.

Let me say that again.

If you leave your £100 - the minimum possible amount you can borrow from Rent-A-Loan - for three years, **you end up owing very close to £620,000**. Of course, they’ll have sent the nice men with the big sticks around long before that.

And it’s up to you to decide whether that’s morally repugnant.

### Only following orders

They’ll argue, in their weaselly, ‘just providing a service’ way that’s so beloved of kebab shops and crack dealers, that that’s not what the loans are for, you pay them back in a week or two rather than three years, that the APR - the industry standard for measuring interest on a loan - is a flagrantly unfair way to measure, um, the interest on a loan - a ludicrous argument, to be sure, but let’s go with it for the sake of bringing logs into play. Let me suggest another way to compare the interest on loans: the time it takes for the size of your loan to double.

It turns out, if you do the sums, that if you compound interest continually (rather than every year, every microsecond or sooner), the size of your loan grows exponentially. I’m going to take that as read, but you’re welcome to verify it if you like!

That is to say: $L = L_0 e^{kt}$, where $L_0$ is the original loan and k is something related to the interest rate. I’m going to measure t in days, but you can pick whatever time unit you like and it’ll work just fine. But how?

### Your home may be at risk

So, let’s start by looking at a regular loan. In my bank - who are only slightly less horrible than Rent-A-Loan - they advertise an 8% interest rate. That means, a £100 loan will turn into a debt of £108 after a year. To turn that into a microsecond-by-microsecond sum, we can say:

$108 = 100 e^{365k}$, because there are 365 days in a year. Rearrange: $1.08 = e^{365k}$ and take logs:

$\ln(1.08) = 365k$, or $k = 2.1 \times 10^{-4}$.

How long would this take to double? Set $L$ to 200 and see what $t$ has to be.

$200 = 100 e^{kt}$

$2 = e^{kt}$

$\ln(2) = kt$

$t = \frac{\ln(2)}{k} \approx 3287.4$ days, or almost bang on nine years. Great! How about Rent-A-Loan? Their interest rate is different - 1737% - so they’ll have a different k.

$1837 = 100 e^{365k}$ - don’t forget to add the original 100%!

$18.37 = e^{365k}$

$k = \frac{\ln(18.37)}{365} \approx 0.008$, give or take.

So, to double your debt, you’d need:

$200 = 100 e^{kt}$

$2 = e^{kt}$

$\ln(2) = kt$

$t = \frac{\ln(2)}{k} \approx 86.9$ days, or less than three months.

So there you have it, a much fairer way of comparing loans! The loan from a respectable-ish bank takes about nine years to double in size; the payday loan firm doubles your debt every three months.

### Vandalism is bad, hmmkay?

Your homework is this: any time you see a poster for a payday loans company, work out how long it would take a loan to double. I’m not suggesting you should write your answer in sharpie on the poster, of course, that would be vandalism.

And it’s up to you to decide whether that would be morally repugnant.

* names have been changed to protect the guilty.