The question you opened
Fifteen dollars for two weeks
A storefront lends you 100 dollars and asks for 115 dollars back in two weeks.
What is that as a yearly rate, in percent?
390 percent
Fifteen percent for two weeks repeats 26 times in a year. Twenty six lots of 15 percent is 390 percent a year before any compounding.
Fifteen dollars on a hundred is 15 percent. The period is two weeks, and a year holds 26 of those, so the simple annual rate is 15 times 26, which is 390 percent. That is the number lenders in many countries are required to print, and it is the honest way to compare this loan with a credit card at 22 percent.
Compounding makes it worse. If the borrower rolls the loan over every fortnight rather than repaying, the debt is multiplied by 1.15 twenty six times. That is 1.15 to the power 26, about 37, so 100 dollars becomes roughly 3,700 dollars in a year. The gap between 390 percent and 3,700 percent is the difference between paying the fee each time and letting it ride.
The general move is to attach a period to every rate before comparing anything. A fee of 2 percent a month is 24 percent a year simple, and about 27 percent compounded. A 1 percent daily late charge is over 3,600 percent a year. Any rate quoted over a short window looks small, and short windows are exactly where fees live.
There is a fair counter argument that annualising a two week loan overstates the cost for someone who genuinely borrows once. That is true, and it is also true that most of this lending is repeat lending, which is why the annual figure is the one that regulators require. The habit that transfers is simply to convert everything to the same clock before judging it. Per month, per fortnight, per day, and per year are not comparable numbers, and the shorter the window, the friendlier the number looks.
Technique: Every rate needs a time frame