# What is Effective Annual Rate?

## Compound Interest Terminologies

When thinking about compounding interests of your investment, you need to take into account compounding periods m, nominal rate r as well as the compounding frequency t.

• Nominal rate r is typically the quoted annual rate.
• Compounding periods m means how many times we are cutting up one year for compounding (a quarterly compounding will mean we cut one year into four compounding periods).
• Compounding frequency t is the number of times we are compounding our investment over the entire term of our investment (if our term of investment is 10 years, then we have to calculate how many times we need to do compounding = how many periods m there are in a year x 10)

## Example

Consider this scenario: An investment over one year at an annual interest rate of 10%, compounded semiannually. Typically the quoted rate is the nominal rate r, and “compounded semiannually” means the amount of money we invested will be compounded every half-yearly, so there are two compounding periods over our investment term of one year. Therefore:

• nominal rate r = 10%
• compounding periods m = 2
• compounding frequency t = 2

So for every \$1 we invested, we will get back:

\$1 x (1 + r/m)^t = \$1 x (1 + 10%/2)² = \$1 x (1.05)² = \$1.1025

And the interest we earned on this investment of \$1 is \$0.1025, which translates to 10.25%. Notice how the quoted rate, or nominal rate r = 10% but we are effectively earning 10.25%!

This interest rate that we are effectively earning on our investment is called the Effective Interest Rate, which in this case is the 10.25%. The 10% nominal rate is actually the interest rate we would earn in one year if we do not have the nasty compounding frequencies t and periods m. In that case:

• nominal rate r = 10%
• compounding periods m = 1
• compounding frequency t = 1

So for every \$1 we invested we will get back:

\$1 x (1 + r/m)^t = \$1 x (1 + 10%/1)¹ = \$1 x (1.1)¹ = \$1.10

## In short

We are actually using the same formula to calculate — effective interest rate r’ but substituting different m and t. The effective interest rate r’ will satisfy this equation:

1 + r’ = (1 + r/m)^t

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