I'm always paying, never make it, but you can't look back

[venta]

So, sums. I can, broadly speaking, do sums. I have a degree in maths.

However, interest rate calculations have always baffled me. Sure, if you ask me to calculate 3 months compound interest at a monthly rate of x% I know what to do. However, when it comes to real examples of mortgages and credit cards, I can't work out what the sum I need to do is. I'm still slightly baffled about the interest charged me when I was a day late paying my credit card off in full in March.

Today, the BBC carried a story about a loanshark. It includes the following statement about someone who borrowed £1000:

"...to pay £49 a week over 60 weeks, making the total amount he had to
repay £2,940 at 917% APR."

Now, if we approximate 60 weeks to a year, then surely that's an annual interest rate of no more than 294%. The quoted APR isn't even in vaguely the right ballpark.

So... have I completely failed to understand APR ? (Wikipedia's page on the subject didn't really help with the definition.) Or is the BBC publishing unmitigated wank in the name of investigative journalism ?

Edit It turns out I'd failed to understand APR, and the BBC is cleared in this instance.

Comments

[hughe.livejournal.com]

i havnt read the article yet but yes it sounds like the bbc are talking shite.

again

[hughe.livejournal.com]

unless they are badly refering to the interest on late payments? which would be in addition to the 2 grand?

[ewx.livejournal.com]

Surely 194% interest [again using the dodgy 60 weeks=1 year approximation], since the £2,940 presumably includes repayment of the initial £1,000. I'm not clear where they get the 917% figure from either.

[wechsler.livejournal.com]

194% would be the correct figure only if the £1000 was paid back in bulk at the end of the year. As the victim holds much of the money for much less time than that the rate must be higher.

Which isn't to say I can actually do the right sum either.

[ewx.livejournal.com]

Yes, that makes sense.

[venta.livejournal.com]

Yes, 194% is what I meant. Can do sums, honest.

[ewx.livejournal.com]

(In fact having worked out the weekly rate the 917% is just that to the power of 52.)

[bateleur.livejournal.com]

The quoted APR isn't even in vaguely the right ballpark.

The correct answer would be (I think) 1203% APR.

Each week the interest is 4.9%, so the annual interest rate is 100*1.049^52. It only comes out as low as 294% if you're paying each week. (Which I appreciate they say is compulsory, but that doesn't affect the rate charged.)

However, I have no idea how they got 917%.

[bateleur.livejournal.com]

Amendment: having seen ewx's answer I expect that explains the discrepancy. I'm not sure quite how one ought to do the sums in the case of fixed payments with the loan considered repaid at the end, since the consumer credit act allows you to repay in full at any time. How much would be repayable in such circumstances is not obvious (if it was the full £2940 only one month into the loan that would give an APR of over 2259754091681536350275064%).

[chrisvenus]

I got something more in that ballpark but slightly less. If the interest each week was 4.9% then he wouldn't actually be paying off any capital so its actually slightly lower than that.

I used excel and applied weekly interest after each payment and found that an interest rate of 4.83% would have it paid off completely after 60 payments and then doing the ^52 calculation you did above I got ~1165% APR. I too am confused by the 917% interest rate... I may give it more thought though. :)

[mister-jack.livejournal.com]

The thing to remember is that the capital decreases each month, so the APR% can be much higher than the total % paid.

[mister-jack.livejournal.com]

This online APR Calculator agrees with the BBC.

[undyingking.livejournal.com]

They're exactly right, according to this calculator.

Either that; or they also used that calculator, and it and they are both exactly wrong.

I think the trouble is that APR can be a bit of a misleading figure.

[venta.livejournal.com]

I wish I knew what sum that calculator was doing. In line with other comments above (suggesting my approach was too naive), I can believe the APR is much higher than expected. I still have no idea how to work it out, though :(

[hughe.livejournal.com]

yup that is right. i stand corrected. the apr is more symbolic of the value of the loan than just a percentage of the interest to the innitial loan amount.

Though i'm still trying to work out how they are calculating it...

  x = 1.0001
  z = 0
  do {
    fx = ( 49 * (x61 - x) / (x - 1) ) - 1000;
    dx = 49 * (60x61 - 61x60 + 1) / (x - 1)2
    z = fx / dx
    x = x - z
  }
  while (ABS(z) > 0.000000001)

  rate = 100 * ( (1 / x)52 - 1)

ok yeh i kind of see what they are getting at now. so it is usefull for working out which is the best rate of equivilant loans, but useless at a metric for actually working out cashflow or anything.

as far as cashflow goes the amount payed back is about 3 times that leant, over 14 months (200% or so intrest) but like a mortgage, at the start your repayments are mainly paying interest, and as time goes on more of your repayment goes on paying back the loan instead of interest. so i supose its a way to takee this into account too. ?

[feanelwa.livejournal.com]

I'm so glad it's not just me who looked at that and said "that's not 917%".

I think they must be wrong. I can't see any way you'd get 917% out of those numbers.

[venta.livejournal.com]

I'm starting to think they might be right (see other comments on this thread).

I'm still struggling to find a formula for APR (so far I've found lots of online calculators, which approximately agree with the BBC), a forum post which claims there are 14 different methods of calculating APR, and a lot of sites claiming that it's a "very complex calculation".

I'm about to go and wrestle with the impenetrable Javascript above, though.

I've also learned that APR isn't always a sensible metric - particularly if (say) I've lent you a sum of money for one week, with a flat fee due, then the APR can come out as 7 digits.

I'm kind of alarmed that between so many mathematically-able people none of us seems to have much of a clue how these things are calculated; it makes me suspect that as a measure it's deliberately opaque.

[battyblingtrash.livejournal.com]

maths really makes me panic :/

[venta.livejournal.com]

I hope you don't tell the kids that ;)

[ulfilias.livejournal.com]

60 weeks....Odd time for a loan, 60 months would make more sense as it would be 5 years, which is quite common !

[drdoug.livejournal.com]

I can, broadly speaking, do sums. I have a degree in maths.

You'll be aware, of course, that those two statements are orthogonal, or weakly correlated at best :-)

The people I know with the hugest disparity between expected and actual arithmetical prowess are mathematicians. In both directions - people who struggle to work out change for a fiver, and people who can factorise your telephone number as soon as you tell it to them.

[venta.livejournal.com]

They were intended to be orthogonal :) My mental arithmetic is reasonable, and I claim to have enough mathematical ability to work out what to add up in the first place...

However, I'm considering rescinding the latter statement. In working through formulae above, I've been alarmed at how little I seem to remember about maipulating indices and logs these days. Hopefully I can claim that that's just knowledge, and easily look-upable, and therefore doesn't count.

[onebyone.livejournal.com]

Now I want to change my telephone number to be the product of two large primes.

[john-the-hat.livejournal.com]

Part of my job, occasionally, is certifying the bank's own APR calculations.

Normally its pretty easy (mortgages, general loans, etc) but especially when you get into things like payday loans you can get some very silly and meaningless figures.

[mrlloyd.livejournal.com]

I'm sure I have a text book somewhere which covers this. However it's in the loft, and I'm not.