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

[chrisvenus]

No, it just means I did the maths wrong. ;-)

It really is pretty easy in most ways... Especially with a spreadsheet or something that can do "goal seek" such as excel.

The basic thing as I said in my JS analysis above (though trying to explain in terms of the JS it might have got confusing) is that you work out the total amount repaid (given in this case). You then assume that each payment you make starts getting interest on it equal to the interest rate of the loan from when you pay it. You then add these up and you get a power series.

1st repayment + virtual interest = 49*interest^59
2nd repayment + virtual interest = 49*interest^58
3rd repayment + virtual interest = 49*interest^57
...
59th repayment + virtual interest = 49*interest^1
60th repayment + virtual interest = 49

You add up all these payments and then solve for sum("repayments+virtual interest") = total loan repaid and you then have an equation with one variable (interest) to solve.

The main problem is that its a power series which is why the javascript then uses approximation methods to solve it because that's usually the only practical way. That or trial and error once you know the ballpark its likely to be in - you are fortunate that the curve you are trying to solve will I think, be monotonically increasing so trial and error should allow you to narrow in relatively easily.

[chrisvenus]

Oh, I should also add that you may need to do some kind of calculation to convert the interest rate you get to an annual one since the interest rate you'll get from my method will be the interest per repayment period. Use powers to convert it. :)

[venta.livejournal.com]

I think I've got the basic outline now, but there's still details I haven't quite figured out.

When you said: "You then assume that each payment you make starts getting interest on it equal to the interest rate of the loan from when you pay it." what did you mean ?

Surely when you make a payment, the payment shouldn't get interest on it, as that payment's been made, and returned to the loaner. The interest should be on the outstanding remainder.

[chrisvenus]

I've realised I am making an absolute pigs ear of explaining this due trying to put my understanding of things in terms of how other people are expressing things. :) I made a pretty fundamental mistake in the above in that what I meant was:

You add up all these payments and then solve for sum("repayments+virtual interest") = amount borrowed*interest^60.

Essentially the technique above was meant to simulate putting your repayments in separate savings so your loan is gaining interest on the full amount until you repay it in lump once your payment plan has matched that value.

Breaking it down to simple parts is probably the best way to understand it though. And if I try to do it from first principles then hopefully any mistakes I make will be obvious. :) To get the 917% all you need to do is say each time you make a repayment you apply weekly interest to the loan amount and subtract the repayment.

In mathsy terms if f(n) = amount owed after n weeks ( i = interest, r = repayment) then:

f(0) = 1000.
f(1) = 1000*i-r (applied interest and subtracted repayment)
and in general:
f(n) = f(n-1)*i-r (applied interest to previous owed amount and subtracted repayment)
you can substitute things to get:

f(n) = (f(n-2)*i-r)*i-r = f(n-2)*i^2-r*i-r
f(n) = (f(n-3)*i-r)*i^2-ri-r = f(n-3)*i^3-r*i^2-r*i-r

Since you have paid off the loan after 60 payments then f(60) = 0.

You can then combine the series expansion that you could see developing with the known endpoint to get something like:

f(60) = f(0)*i^60 - r*i^59 - r*i^58 - r*i^57 ... - r^2 - r

(this is where I occasionally get confused by how many terms I should be having here and so on)

f(60) of course in this case is 0 so we can solve

f(0)*i^60 = r*i^59 + r*i^58 + r*i^57 ... + r^2 + r

And this is the equation I was badly trying to replicate above.

Also the right hand side simplifies to r*((i^60-i)/(i-1).

This is where the javascript fx comes from. I always get confused as I say in expanding things and working out where my series stop which is probably why I have i^60 whereas they had i^61.

You get the general principle and if you bothered to work it out properly then you'd get right answers.

And now I am stopping. If I've not explained myself properly this time I'm just goign to give up my teaching career. :)

[venta.livejournal.com]

Wow! Thanks, that is actually very well explained.

Essentially the technique above was meant to simulate putting your repayments in separate savings so your loan is gaining interest on the full amount until you repay it in lump once your payment plan has matched that value.

That was a fairly useful starting point, and the numbers all seem sensible now. I can understand getting confused with endpoints, and I was always crap at summing series.

You can be Designated Hero of this Week, and you may now stop wasting your time explaining things to the stupid :)

[chrisvenus]

explaining things to the stupid

To be fair you seemed to understand it perfectly the first time I explained it without any mistakes in my explanation. ;-)