Percent per year
Imagine you have borrowed money and pay two percent for it every month. How much is that per year? Almost anyone will answer without thinking: twenty-four. But the interest you did not pay in January is added to the debt in February, and interest starts to accrue on it as well. By December the total will be not twenty-four but nearly twenty-seven percent. Neither number is false: the first answers the question of what the rate is, and the second how much you will actually pay.
Percent per year is the price of money over time, converted to one year, and the same deal can have several such prices. The nominal rate APR simply multiplies the monthly rate by the number of months. The effective rate APY takes interest on interest into account. The total cost of credit adds fees and insurance to the interest, and then the rate can no longer be computed by a formula, it can only be found by trying numbers until the money received balances all the future payments. The real rate subtracts the rise in prices as well and answers the most important question: have you become richer.
Conversion
A rate for any period, converted to a yearly rate
Enter the rate the way you were quoted it: per year, per month, per two weeks or per day. On the left you will see all its annual and periodic forms, and on the right how long it takes at this rate for money to double and to grow tenfold. Notice how quickly simple multiplication and honest conversion drift apart when the period gets short and the rate high.
Etymology
For a hundred, between, and on trust
Three words without which no contract can do came from Latin, but each took its own road. One names the count, another the reason for the payment, and the third reminds us that every debt begins with trust.
01 · Definition
One deal, four rates
Percent per year shows what share of the sum has to be paid for having used the money for a whole year. If interest is added once at the end of the year, everything is simple, and the rate in the contract matches the growth of the debt. But banks add it monthly or even daily, and every portion of accrued interest immediately becomes part of the sum. Then we have to tell apart the nominal rate, which is divided by the number of periods, and the effective one, which shows the result of the year with all the interest on interest.
The gap between them grows with the rate. At five percent per year monthly compounding adds only twelve hundredths of a percent, but at twenty-four it adds almost three full points. And when fees are added to the interest, and payments of different sizes fall on different days, no simple formula is left. The total cost of credit is found as the root of an equation: it is the one rate at which the money received exactly equals all the future payments, discounted to today. That is how a financial calculator works, and so does the regulator's program that checks the bank.
A hundred percent, cut into pieces
Put one rouble at a hundred percent per year and let us add interest more and more often: once a year, every half-year, monthly, every day, every hour. It seems you could get as rich as you like. Choose into how many parts to cut the year and see what the rouble turns into.
Deposit, borrow, check the advert
Enter your own amount and choose a currency, and the canvas lays it out in real bundles of banknotes in bank bands. In the Deposit experiment change the rate, the compounding frequency, the tax and the inflation: you will see which part of the interest grows on interest and which part is eaten by prices. In the Loan experiment add a fee and insurance and watch how the rate in the contract turns into the total cost. And in the Trap experiment take apart the eight tricks that are used all over the world today to hide the true price of money.
02 · Laws of different countries
One loan, eight numbers in a frame
In 1968 the American Truth in Lending law was the first to require that the cost of credit be printed as one large number, and since then almost every country has introduced its own. All of them look for the same rate at which the money received balances the future payments, but they differ in details: whether to multiply the monthly rate by twelve or raise it to a power, which fees to include, whether to count tax. Set a loan of a hundred thousand and see what is printed in the contract in different parts of the world.
The rules are shown here in simplified form. Every country has detailed lists of which payments go into the total cost and which do not, its own rate ceilings and its own rounding tolerances. The fairest way to compare offers from different countries is by the same formula, best of all by the effective rate of all your own payments.
03 · Rate scale
From below zero to thousands of percent
Rates around the world differ a millionfold: from fractions of a percent on deposits in countries where prices stand still for years, to thousands of percent on loans taken for two weeks. At the lower end of the scale APR and APY almost coincide, at the upper end they differ by tens of times, and the higher the rate, the more important it is to ask which of them you were quoted. The bar here is logarithmic, otherwise everything but the last rows would shrink to a point.
04 · Rule of 72
Doubling in your head
Back in 1494 the Italian monk Luca Pacioli, a friend of Leonardo da Vinci, wrote down in his “Summa de arithmetica” a clever rule: divide seventy-two by the rate in percent and you will learn in how many years money doubles. He did not explain where the number came from. The answer lies in the natural logarithm of two, which equals 0.693, and seventy-two was chosen because it is slightly larger and divides without remainder by 2, 3, 4, 6, 8, 9 and 12. Move the rate and check how long the rule stays accurate.
05 · Writing rules
06 · Neighbouring units
07 · Historical section
The percent per year has no standard kept under a glass dome, as the kilogram once was. But it does have a verification procedure. The regulator takes the payment schedule from the contract, solves the equation again, looking for the rate at which the money issued equals all the payments, and compares the result with the printed number. In the USA a discrepancy of more than one eighth of a percentage point is already considered an error, and the bank must return the overpayment to its customers.
The mistakes found in such a check are almost always the same. A fee withheld on the day of issue was forgotten. The annual rate was divided by 360 days instead of 365, which quietly added to it almost one and a half percent of its own size. The monthly rate was multiplied by twelve where the law required raising it to the twelfth power. Each such trifle costs borrowers millions, and that is why the laws on rate disclosure are so detailed.
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