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What is the difference between APY and APR? How is interest calculated?

Update: I’ve made an online calculator to convert between APR and APY: APR and APY converter.

Whether you are financing a loan or getting a savings account, you need to know about interest rates. Banks talk about interest rates using the acronyms APR and APY, but what exactly do they mean? I’ll explain why banks conveniently quote one figure or the other. But first, I will discuss simple interest and compound interest which are what APR and APY really boil down to.

Simple Interest (no compounding)

The nominal interest rate is an annual rate quoted in percentages. The simple interest method does not consider the effects of compounding. This method calculates interest as the product of the original balance, the nominal interest rate, and the time period (in years).

Here’s an example. Consider a $5,000 savings account balance with a 12% nominal interest rate, using the simple interest method. In one year, your account would equal the interest payment of $5,000 x 12% = $600 plus your original balance of $5,000, for a grand total of $5,600.

Let’s suppose you could request to be paid in half of a year. In this case, your balance would be $5,000 x (1 + 12% x 0.5) = $5,300. As you can see, when you use the simple interest method, the yearly interest of $600 is exactly double the semi-annual interest of $300.

To generalize, simple interest can be computed once you know:

  1. the balance of the loan or savings account (P )
  2. the annual nominal interest rate (r )
  3. the time in years (t )

The general formula is:

Compounding Interest

Compound interest is calculated very much like simple interest, but it takes into account that the balance changes after each time interest is paid out.

Consider a $5,000 savings account with a 12% simple interest rate, interest is paid semi-annually, but this time use the compound interest method.

What is the account balance in one year? For the first half of the year, no interest has been paid, so the compound method is the same as the simple method. The balance grows to $5,300 = $5,000 x (1+12%/2) in half of a year.

Now comes the interesting part. For the second half of the year, interest is computed on top of the interest that’s already accumulated. Interest is therefore computed using a $5,300 balance (instead of a $5,000 balance in the simple interest case). This is what differentiates compound interest from simple interest. After one year, the ending balance is $5,618 = $5,000 x (1+12%/2)^2, which is slightly higher than the $5,600 from the simple interest method.

Compound interest uses the same variables as simple interest, but we also need to know the frequency of compounding:

  1. the balance of the loan or savings account (P )
  2. the annual nominal interest rate (r )
  3. the time in

    years (t )

  4. the frequency of compounding in one year (m )

The compound interest formula is:

The formula differs from simple interest in a few ways: (1) the nominal interest rate is expressed as an interest rate per m periods (so a 12% nominal rate is a 6% semi-annual rate) (2) interest is compounded on top of what’s already paid: the exponent takes care of compounding m times each year.

Annual Percentage Yield (APY)

The annual percentage yield (APY) is the interest yield you would get on a balance held for one year in a financial product, taking compounding into account.

In the previous example, a $5,000 savings account balance becomes $5,618 in one-year, so the APY is equal to $618 / $5,000 = 12.36%. Another way of saying this, is that a 12% nominal interest rate, compounded semi-annually, has an APY of 12.36%.

As you can see, the APY is a simplification of compound interest formula, where we consider an investment of one year (t=1) and an investment of one dollar (P=1) .

Here is the formula for the APY of an investment where r is the annual interest rate and m is the number of compounding periods:

Example: ING Electric Orange Checking

ING advertises interest rates and APYs for its various checking accounts (see table below).

Let’s check that ING has made a correct conversion between APY and interest rates. Based on the August 16, 2007 rates, ING shows a 5.25% APY for a 5.13% interest rate (compounded monthly). Using the APY formula, we have:

Wonderful–ING did the math correctly.

Annual Percentage Rate (APR)

The annual percentage rate (APR) is the same as what I’ve been calling the nominal interest rate. It is the rate of interest in one year, without taking compounding into account. In the ING example, the 5.25% APY is equal to a 5.13% APR.

You can convert APR to APY once you know the frequency of compounding per year (m). Many credit cards, loans, and savings accounts are monthly (m=12) :

Because of how they are defined, the APY is higher than the APR. This mathematical property is why banks like to quote both figures:

  1. For savings accounts, companies advertise the APY to inflate how high their rates are.
  2. For loans and credit cards, companies advertise the APR to conceal the true cost of the loan.

The APY is more realistic because it takes compounding into effect. For this reason, I typically ignore APRs and only consider APYs. The law says that the APY must be quoted by banks. though it usually appears in small print. I find it is easier just to calculate it myself.

Update: I’ve made an online calculator to convert between APR and APY: APR and APY converter.

Category: Bank

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