Compound Interest and APY
Learn How to Calculate Compounding Interest, and APY Rates.
For Calculators, see my Compound Interest Calculators Page

In order to calculate how much interest you will get from a bank, you first must know the formula used to calculate compounding interest.

F = D(1+r/n)nt

where:
F = Future Value
D = Initial Deposit
r = Interest Rate in Decimal Form
n = Number of Compounding Periods per Year
t = Number of Years Invested

For an example, let us say that a bank is offering a CD (Certificate of Deposit) that has a 6.45% interest rate and is compounded quarterly (4 times a year) for 3 years. You decide to invest \$5500 into this CD and want to know how much money you will have at the end of the 3 years?  First plug all the numbers into the formula, and then simplify it until you can solve for F.
Example 1

F = \$5500 (1 + 0.0645/4)4*3
F = \$5500 (1 + 0.016125)12
F = \$5500 (1.016125)12
F = \$5500 * 1.2111617783
F = \$6663.90

So once the CD matures (in 3 years), you will have \$6663.90.  To determine how much interest you will have made, take this future value and subtract the initial amount:
Example 2

Interest Made (IM) = F - D
IM = \$6663.90 - \$5500
IM = \$1163.90

So if you were to deposit \$5500 into this CD, after 3 years you will have earned \$1163.90 in interest.  At this point you may be saying, "Wait!, my bank doesn't tell me how many compounding periods there are each year, they only give the percentage as APY (Annual Percent Yield)".  New laws require banks to provide you with the APY figure because it makes it easier to shop around and find the best interest rate.  The APY is the interest rate stated as if there was only one compounding period per year.  In case you are curious, the formula to calculate APY from a normal interest rate and the compounding periods per year is:

APY = (1+r/n)n - 1

So in our example the bank stated that it has an interest rate of 6.45% compounded quarterly, and you want to calculate the APY (although they should give it to you), just plug the numbers into the formula and solve.
Example 3

APY = (1 + 0.0645 / 4)4 - 1
APY = (1 + 0.016125)4 - 1
APY = (1.016125)4 - 1
APY = 1.066076 - 1
APY = 0.066076 or 6.608%

When a bank gives you an APY, you can re-write the compounding interest formula to make it simpler to solve:

F = D(1+r)t

If we use this formula and the APY to calculate how much interest we will make using the same numbers as in example 1, we should come out with the same amount.
Example 4

F = \$5500 (1+0.06608)3
F = \$5500 (1.06608)3
F= \$5500 * 1.2116282
F= \$6663.96

Notice that this is not the same amount we calculated in the first example.  There is a difference of \$0.06 between the two examples, and this is due to small rounding differences between the problems.  Rounding starts to create a problem in these examples because the small amount of rounding gets amplified since we keep raising rounded numbers to powers.  Again, to determine how much interest we have earned using this APY rate, take the future value and subtract the initial deposit.
Example 5

IM = \$6663.96 - \$5500
IM = \$1163.96

I hope this unravels the mysteries of compounding interest and APY's.  Please email me if you have any questions, comments or if you found this explanation on compounding interest and APY helpful.

Email me at:  Romero@pharmacy.arizona.edu