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ventaSo, 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.
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.
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no subject
Date: 2008-07-29 10:23 am (UTC)I think they must be wrong. I can't see any way you'd get 917% out of those numbers.
no subject
Date: 2008-07-29 10:28 am (UTC)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.
no subject
Date: 2008-07-29 10:36 am (UTC)I got as far as seeing that it's an iterative approximation technique and then gave up in disgust, as the underlying formula seems unlikely to be discernable from that :-(
no subject
Date: 2008-07-29 10:45 am (UTC)Assume you have your loan and you pay an amount of interest on it every month (or week). Now assume you make no payments in that time, APR is the % increase that will produce over the course of a year.
As such, it doesn't really measure anything you'll actually pay, but as it is entirely consistent you can use it to compare different payment plans. But if you try and compare, say, a mortgage to a personal loan, the values you get back are basically bunkum.
no subject
Date: 2008-07-29 10:59 am (UTC)Go and edit the Wikipedia article, will you ? There's a love. That's a much more coherent description of what APR is for.
no subject
Date: 2008-07-29 12:10 pm (UTC)no subject
Date: 2008-07-29 12:52 pm (UTC)So when you say the comparison is OK for comparable loans, I think that more loans are comparable than you might think, or than lenders admit. They're even somewhat comparable if one of the types of loan isn't available to you, since at least you know what you could have won, and can consider what it would be worth to you to somehow gain access to that lender.
Fewer people would take long-term "single low payment" consolidation loans if they compared the APR with that of their current loans, which charge more per month total, but at a lower interest rate because the repayment term is shorter.
Mortgages are cheaper than personal loans, and for good reasons - they're secured rather than unsecured debt, and they're allowed to invoke penalties for early payment. It's cheaper to borrow on a mortgage than a personal loan, because you're not buying the flexibility to pay early, or default on the debt. APR comparison fails to put a value on those freedoms, because it can't. But it does at least leave you with less maths to do than if two lenders are quoting non-standard "annual rates" which look similar, but have different compounding periods and hence are very different.
Loan sharks offer the worst of all worlds, of course - the loan is secured on your kneecaps, and it's also extremely expensive.
no subject
Date: 2008-07-29 01:13 pm (UTC)If we compare a mortgage paid back over 25 years to a car loan paid back over 3 but with double the APR; you'll actually pay more interest on the amount borrowed under the mortgage over those 3 years.
I'm not saying that APR is meaningless by any means; but it's not necessarily the most indicative of rates, especially for short term loans.
no subject
Date: 2008-07-29 02:51 pm (UTC)If you use an offset mortgage and voluntarily pay the money back in 36 equal instalments, then you'd pay less than the car loan iff the APR is less than the car loan APR. Conversely if you take out multiple 3 year loans over the course of 25 years to take your car debt down gradually, then it costs more than the mortgage iff their combined APR is higher.
I think lenders do also have to offer breakdowns of the total amount paid, which should help people decide how long a term they want to repay over, if offered multiple loans with different repayment periods. But fundamentally, as long as you have something worthwhile you can do with money, it's better to have a loan over 25 years than one over 3 years, which is why it costs more in total. If you don't have something worthwhile you can do with money, you should aim to be debt-free. APR gives a good idea what "worthwhile" means - does the cost each year justify continuing the debt another year?
no subject
Date: 2008-07-29 10:47 am (UTC)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.
no subject
Date: 2008-07-29 10:49 am (UTC)no subject
Date: 2008-07-29 11:38 am (UTC)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.
no subject
Date: 2008-07-29 11:59 am (UTC)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. :)
no subject
Date: 2008-07-29 12:10 pm (UTC)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 :)
no subject
Date: 2008-07-29 12:12 pm (UTC)To be fair you seemed to understand it perfectly the first time I explained it without any mistakes in my explanation. ;-)