Written by Colin+ in logarithms, ranting.

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.

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: [pmath]L = L_0 e^kt[/pmath], 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?

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:

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

[pmath]\ln(1.08) = 365k[/pmath], or [pmath]k = 2.1 * 10^{-4}[/pmath].

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

[pmath]200 = 100 e^{kt}[/pmath]

[pmath]2 = e^{kt}[/pmath]

[pmath]\ln(2) = kt[/pmath]

[pmath] t = {\ln(2)} / k[/pmath] = 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.

[pmath]1837 = 100 e^{365k}[/pmath] - don't forget to add the original 100%!

[pmath]18.37 = e^{365k}[/pmath]

[pmath] k = {ln(18.37)} / 365 = 0.008[/pmath], give or take.

So, to double your debt, you'd need:

[pmath]200 = 100 e^{kt}[/pmath]

[pmath] 2 = e^{kt}[/pmath]

[pmath]\ln(2) = kt[/pmath]

[pmath]t = {\ln(2)} / k[/pmath] = 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.

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.

## MathWiz

Maths and morality, eh?

Let\’s take a look at how you would run a loans business.

Suppose you are asked to loan an amount L for a time t (measured in years). What interest rate should you charge?

Suppose we have overheads ht, to pay for the men with sticks. (And also the receptionists, accountants, advertisers, office space, and all the other things a business needs.)

Suppose also we have a probability of default p. That\’s the probability that you loan the money out but don\’t get it back. You can send the men with sticks round, but satisfying as that may be it doesn\’t pay the bills.

So at the end of the loan period we get back L Exp(kt) with probability (1-p), so on average that\’s L(1-p)(Exp(kt)-1) profit. That has to pay for overheads ht and defaults Lp.

L(1-p)(Exp(kt)-1) = ht+Lp

Exp(kt) = 1+ (ht/L + p)/(1-p) = (ht/L + 1)/(1-p)

Exp(k) = APR = ((ht/L + 1)/(1-p))^(1/t)

That\’s just to break even. If you want to make as much as you would putting it in the bank, you need to add a little more.

So, we have got a formula, how do we interpret it?

We have got overheads – the bigger they are, the more we have to charge.

We have got the loan amount – the smaller it is compared to our overheads, the more we have to charge.

We have got the length of the loan – it\’s a bit harder to see what\’s going on here, because it appears twice. Well, it turns out if you check a few example numbers, the shorter the loan, the more you have to charge.

And the default probability p – as p gets closer and closer to 1, the amount you have to charge shoots up rapidly to some huge numbers.

And that\’s the real problem. Most banks will not loan money to people they\’re not sure are going to be able to pay it back – who are exactly the people who most need it: the poor. People who want small amounts for short periods of time, and who can\’t be sure they\’ll be able to pay it back.

Let\’s ignore the overheads for the moment – assume they\’re small compared to the loan. We have

APR = (1/(1-p))^(1/t)

If t is 1/50 years (about 1 week) and p is 10%, then you need to charge at least (10/9)^50 = 19,403.3% APR. That\’s working for free, for no profit. And *that* is why nobody moves in to take the trade away from the loan sharks and put them all out of business, by simply offering the poor a better deal.

—

Having a numerate society is such an important thing. People need to understand the reasons for the way things are, in business, finance, economics, as well as science and technology, if they are to make wise choices. I applaud your sterling efforts to bring that about.

## Colin

Thank you for your comment – always good to see things from the other side as well!

I quite agree – obviously – that a numerate society is vital; understanding the maths behind things certainly tickles me.

## Anthony

I agree that payday loan companies tend to take unfair advantage of the people who can afford it the least (you could argue that the problem originates in our lack of good financial education, and we should know better than to use them, and maybe that’s the price you pay for a free market, etc, but still I feel this is the kind of business that should be more carefully regulated – maybe cap the total repayable amount or something), but I think there is some validity to the “APR isn’t a fair measure for this kind of loan” argument. We accept up to 30% for a month-by-month credit card – you’re paying for a fundamentally different service to a planned long-term bank loan. One example I’ll use in compound interest lessons to lend an interesting perspective: You borrow £20 from a friend, and pay him back after a couple of weeks. Is it unreasonable for him to expect you to stand him a pint? But at £2, that’s 10% in a fortnight – the equivalent of around 1200% APR.

## Julia

I thought I should let you know that I’ve taught this blog post for a few years in my algebra 2 class. I’ve probably guided close to 400 students to understanding it. Then I have them write a letter to a state lawmaker to urge them to regulate the payday loan industry. Thank you!

## Colin

Tremendous work! Thanks for letting me know 🙂