The Interest Rates: APR, APY And EAR

By Greg McFarlane AAA

When most people shop for financial products, all they focus on is the listed interest rate. Human eyes instinctively dismiss the fine print, which usually includes the terms APR (annual percentage rate) and APY (annual percentage yield) – or to use a synonym for the latter, EAR (effective annual rate) – as just arcane trios of letters.

In fact, there’s plenty of difference between the similar but not identical APR and APY. Each expression sounds straightforward enough, but misidentifying one as the other can cost you plenty. Here’s an explanation of these oft-confused terms and what distinguishes them.

Basic Concept of Interest

Grasping the concept of interest should be easy enough. If you borrow $100,000, and pay $5,000 back in interest each year, that should be a 5% interest rate now and forever, right? Why complicate things?

Because there’s more to taking out a loan than just making simple interest payments. For instance, there are often origination fees and other one-time charges levied at the time of borrowing.

Say Alpha Mortgage offers you a 5% rate on the $100,000 loan. Meanwhile, Beta Mortgage offers to lend you the same amount of money, but at 4.75%. Over the course of a standard 30-year mortgage, the reduced interest rate could mean thousands of dollars in savings; and would, if all the other conditions of the two lenders’ loans were identical. However, Beta Mortgage requires $3,000 in upfront closing costs – a reasonable figure for a new home purchase. Which is the better deal? If only there were a single metric we could use to compare loans with such varying characteristics.

There is, and it’s the APR.

The APR

Annual Percentage Rate (APR) is a measure that attempts to calculate what percentage of the principal you’ll pay per time period (in this case a year), taking every charge – monthly payments over the course of the loan, upfront fees, etc. – into account.

By the way, the Alpha Mortgage loan in the example above carries the lower APR. With the Beta Mortgage loan, you’re essentially paying $3,000 for the privilege of borrowing $100,000, and thus effectively borrowing only $97,000. However, you’re still making interest payments that the lender is basing on a $100,000 loan, not a $97,000 one. A lower denominator has the same effect as a higher numerator. The APR on the Alpha Mortgage loan is 5.00%, but the APR on the Beta Mortgage loan is 5.02%.

To calculate the APR for a loan that incorporates costs beyond those of the principal borrowed, first determine how much the periodic payments are. For the Beta Mortgage loan, each monthly payment is:

The $100,000 is the gross principal borrowed, .0475 the interest rate, 12 the number of periods in a year, and 360 the number of periods over the course of the loan. Break out your calculator, and you’ll find that the monthly payment is $521.65.

Then, divide the monthly payment into the net amount you’re borrowing,


 

The APR is the unknown quantity that solves this equation:

Which, alas, you can’t figure out through any amount of algebraic manipulation. You need either a fondness for trial-and-error and an awful lot of patience, or a computer. [In Microsoft Excel, the formula is ‘RATE (nper, pmt, pv, fv, type, guess).' Use 360; -521.65 (rounded); and for the first three values, respectively]. Multiply by 12 to get the annual rate. The resulting rate is 5.02%.

The APY or EAR

APY differs from APR in that the latter takes only simple interest into account. APY incorporates the additional complication of compound interest: interest charged on the simple interest, which again distorts the numbers and increases a borrower’s obligations beyond the standard simple interest rate.

Note that APY and EAR are identical. They represent the same quantity, but are quoted by one name or the other depending on the circumstance. The expressions are two sides of the same coin, in much the same way that an accounts payable for one business is an accounts receivable for another. A credit card issuer, for example, would use the term EAR rather than APY, because it’s not good public relations to talk in terms of the “yield” that the cardholders’ payments are generating for the issuer.

Compound interest – interest on interest – is a subject that should warrant its own article, and does, but suffice it to say that knowing that compound interest differs from simple interest isn’t enough. When calculating APY/EAR, the compounding period is everything. Interest that compounds semiannually is far different from interest that compounds daily, as it does on most credit cards.

Difference Between APR and APY

To determine the APR and APY on accounts with compounding interest, start with the interest rate per compounding period – again, in this case that means per day. Target Corp. offers a credit card that levies interest of 0.06273% daily. Multiply that by 365, and that’s 22.9% per year, which is the advertised APR. If you were to charge a different $1,000 item to your card every day, and waited until the day after the due date (when the issuer started levying interest) to start making payments, you’d owe $1,000.6273 for each thing you bought (disregarding for a moment that the issuer probably won’t let you make daily payments on your card, let alone have them post immediately, and also disregarding that pennies don’t carry out to two decimal places).

To calculate the APY, instead of multiplying 0.06273% by the number of compounding periods in a year, add 1 (which represents the principal) and take that number to the power of the number of compounding periods in a year. Subtract 1 from the result to get the percentage form.

.0006273 x 365 = 22.9% APR

(1.0006273365) - 1 = 25.72072% APY

That’s pretty much it. The difference between APR and APY can be illustrated more forcefully in a couple of equations than in any amount of prose. The higher the interest rate, and to a lesser extent the smaller the compounding periods, the greater the difference between APR and APY. Understand that of the two, APY is the more universally applicable measure, the one that states how much you’ll be paying in interest charges (or receiving, in the case of deposit accounts) regardless of compounding frequency. That’s why the Truth in Savings Act of 1991 mandates that APY be disclosed with every deposit account offered by financial services firms.

Given that an APR and a different APY can be used to represent the same interest rate, it stands to reason that lenders and borrowers will pick the more flattering number to state their case. A bank will advertise a savings account’s APY in a large font and its corresponding APR in a smaller one, given that the former features a superficially larger number. The opposite happens when the bank acts as lender, rather than borrower, and thus tries to convince its own borrowers that it’s charging a rate as close to zero as possible.

The Bottom Line

So what can a borrower overwhelmed with data do? As always, caveat emptor. Look for a listed APY before paying attention to APR. If no APY is listed, calculate it from the listed periodic interest rate via the method shown here. And if you’re concerned about how much your credit card issuer is charging you in interest, one foolproof way around that is to pay your balance in full every month. That’s a nominal rate, an APR and an APY of 0.

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