What's Annual Percentage Yield?

If you have a credit card or car loan, you probably are familiar with APR, or annual percentage rate. APR is the simple interest you pay to borrow money or earn on an investment. Annual percentage yield, or APY, is similar, except it includes the effects of intra-year compounding. That is, APY is sensitive to how often interest is applied - monthly, daily or continuously. If you’re a borrower, compounding is unwelcome, but investors love it. The faster the compounding rate, the larger the cost or return.

Compounding 101

APY depends on compounding, but what is that? Compounding is applying interest on previous interest and adding it to the principal amount of a loan or deposit. You’ll find compounding is ubiquitous within the financial realm, as investors seek to maximize return while borrowers want to minimize their interest charges. Simple interest differs from compounding interest in that it doesn’t pay interest on interest.

The APY Formula

The formula for annual percentage yield is:

APY = (1 + r)n -1

where r is the periodic interest rate and n is the number of compounding periods per year. For instance, for daily compounding, n = 365 and r is the daily interest rate (i.e., the annual rate divided by 365). For monthly compounding, n =12 and r = the annual rate/12. Credit cards use daily compounding, whereas many passbook savings accounts use monthly compounding.

This is the simple-interest APR formula:

APR = rn

Where r is the periodic rate and n is the number of periods in the year. The lack of an exponent in the APR formula is your tipoff that compounding is not occurring.

Example Calculations

Let’s assume you have a credit card that markets an 18% APR. What is the APY, assuming daily compounding?

APY = (1 + 0.18/365)365 -1= 19.7164%

The APR for this credit card is simply:

APR = (0.18/365 x 365) = 18%

Note that the APY is almost 2% points higher than the advertised APR.

Continuous Compounding

Continuous compounding is the most extreme version of compounding. For savers, continuous compounding represents the absolute best return you’ll get for a given APR. Clearly, the opposite is true for borrowers.

The continuous compounding formula is derived by performing a little calculus on numbers representing present value and future value. Luckily, the resulting formula is simplified because of the presence of a constant known as e, which is approximately 2.7183.

The formula for APY under continuous compounding is:

APYcontinuous =  2.7183r

where r is the annual interest rate.

Let’s say you deposit \$10,000 in a savings account that pays an annual interest rate of 3%, continuously compounded. Applying the formula, your money would have grown in one year to the following future value:

FV = \$10,000 x 2.7183 0.03 = \$10,304.55

In other words, you picked up an extra \$4.55 for the year because of the power of continuous compounding. While that might not seem like an overwhelming difference, just think of big banks transacting with each other, where the money is in the millions or billions.

Suppose you are seeking a private student loan of \$30,000 and are shopping around for the best deal. Private student loan providers will often quote their loan rates in APR terms but charge a higher rate by compounding interest on a daily or monthly basis. Over the course of a 20-year student loan, that little omission might cost you hundreds or thousands in extra interest. Be wise: Always compare APYs among competing lenders to get the true cost of a loan. For example, if a private lender advertises a 5% APR on a \$30,000 student loan that compounds interest daily, the true cost (APY) to you will be 5.1267%. That’s an extra \$1,538 over the life of the loan. Credit cards that fail to disclose the APY are pulling the same trick. You can fight back by paying your entire balance every month. That way, your APY is 0%.

Apples to Apples

One place you probably will see APY quoted is on savings accounts as banks and credit unions compete for your money. Always compare APY to APY or APR to APR when you are doing comparison shopping. The other factor to consider with a savings vehicle is whether the interest rate is fixed or variable. For example, a certificate of deposit usually offers a fixed interest rate, which typically is higher than the rate on a demand savings account. However, the savings account might offer variable-rate, which can outperform the CD if interest rates are trending strongly higher.

Usually, you’ll find the best savings APYs from online-only banks. These banks compete totally on value, because they don’t offer person-to-person contact you enjoy at a brick-and-mortar bank branch. While you don’t receive the personal touch from online banks, you also don’t pay for it. Their lower overhead costs allow online banks to offer a higher APY, as well as lower fees and smaller initial deposit requirements.

Reinvestment Risk

Bonds are an interesting case in which APY is inappropriate. Typically, bonds pay interest periodically — usually monthly, quarterly, or semi-annually. The bond interest payment, or coupon, is sent to the investor rather than plowed back into the investment principal. Now, suppose you purchase an 8%, 10-year semi-annual bond with a face value of \$10,000. That means you will receive 20 payments of \$400 each, spaced in six-month intervals. However, what happens if interest rates drop soon after you purchase the bond? On the one hand, you celebrate your timing in that you locked in a relatively high interest rate for 10 years (assuming the bond cannot be called back by the issuer). However, you now have the problem of reinvesting the coupon payments at a lower interest rate. This will reduce your overall yield on the bond compared to that of a CD with similar characteristics, since CDs offer interest reinvestment and compounding at a fixed rate.

In this example, the CD will reinvest and compound your interest at 8%, no matter what happens to prevailing interest rates. That means you earn fixed-rate compounding that you do not get from most bonds. The only exception is a zero-coupon bond, which pays all its interest at maturity based upon a fixed rate. The interest rate is imputed from the deeply discounted purchase price of the bond. That imputed interest is figured as though it was compounded, usually semiannually, which means these bonds do not have reinvestment risk.

BankingAndy Misek