On adverts for payday loan deals (UK), the representative APR is always displayed in small print, by law.
I always feel as though these loans, with their x-thousand % interest rates are ripping off those most in need.
However, I have stumbled across this claim on Wonga.com;
We don't actually charge thousands of percent in interest
In fact we don't charge anything like the large Representative APR
Wonga charges the equivalent of just under 1% interest per day and that rate is
applied to the loan amount and transmission fee for the period of the
short term loan - usually between one and 31 days. The annual rate of
interest is 360%. We know this may be hard to believe, because of the
much larger Representative APR we are obliged to display. But that's
because the calculation required by law means that, where a loan is
not taken out for a year, the interest rate must be compounded the
same number of times the actual loan period would fit into a year.
Because an annualised measure is distorting over short periods of
time, we always show the total cost of repayment very clearly too.
The current Wonga representative APR is 4214%, so how can they claim it is in fact 360% ?
If you borrow $100, at 6% per annum simple interest for half of a year, you must pay back:
$100 * (6/100 * 1/2 year) = $3 interest, plus the original $100 principal.
But, if you borrow with compound interest, then you have to pay interest on the unpaid interest. The rate of compounding (how often you calculate the interest) makes a difference.
If it compounds every 6 months, it is still the same amount: $3.
If it compounds ever 3 months - i.e. twice - it is:
$100 * ((1 + (6/100 * 1/2 year)/2)2 - 1) = $3.022 interest, plus the original $100 principal.
If it compounds every day - i.e. 183 times - it is:
$100 * ((1 + (6/100 * 1/2)/183)183 - 1) = $3.045 interest, plus the original $100 principal.
So the frequency of the compounding is important, even though the nominal interest rate (6% per annum) didn't change. The nominal rate is also known as the Nominal Annual Percentage Rate (APR).
Governments (and most legitimate businesses) understand that the nominal APR is misleading, and they insist that people publish the Representative APR (a.k.a. Effective APR, EAR, Annualised APR, or AAPR) which is standardised as being the interest rate you'd pay if you borrowed for one year.
Now, lets use Wonga's figures, to see how the representative figures look:
If you borrow $100 for 1 year at a nominal rate of interest of 360% per annum compounding every day (non-leap year), you actually pay:
$100 * ((1 + (360/100)/365)365 - 1) = $3495.84 interest, plus the original $100.
This corresponds to an Representative APR of 3495%, but there may be other charges that make the real figure even higher.
Now, Wonga is right that, because the loan is much shorter than a year, your total bill will not come to that figure. However, it is a representative figure because it allows you to compare their interest rates to other lenders.
Wonga can honestly quote 360% (or 1% per day) as the Nominal APR, but (as they admit) the regulations require them to also state the Representative APR which is much higher.
It appears that they charge simple interest over the short period of the loan (plus a fee), which I assume means the Representative APR represents the worst case of a 1 day loan. If you borrow for 30 days, the annualised APR would be lower.
Here's one online tool to try this stuff out for yourself. It computes the last figure as $3522.43 - I am hoping the difference is taking into account leap years and rounding errors.