# 2. Calculate the Effective Annual Rate

## c. calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;

What are the three ways to quote interest rates for investments paying interest more than once per year?

1. Periodic interest rate - rate earned over a single compounding period

2. Stated annual interest rate (quoted interest rate) - the annual rate w/o taking compounding (in the case of multiple payouts per year) into account

3. Effective Annual Rate (EAR) - annual rate including compounding of multiple payouts per year

What is a periodic interest rate?

Periodic interest rate is the rate of interest earned over a single compounding period. For example, a bank may state that a particular CD pays a periodic quarterly interest rate of 3% that compounds 4 times a year.

What is a Stated annual interest rate? (or quoted interest rate?)

The Stated annual interest rate is the annual rate of interest that does not take multiple compounding periods into account. E.g. A CD that pays 3% per quarter would have a stated annual interest rate of 12%.

What is the Effective annual rate (EAR)?

The effective annual rate is the yearly rate of interest that takes compounding into account. The calculation is `EAR = (1 + periodic interest rate/m)^m - 1`

. E.g. A CD that pays 3% per quarter would have an effective annual rate of `(1 + .03/4)^4 - 1`

or 12.55%

How do you compute Effective Annual Rate (EAR) on the BA II Plus calculator?

You can either use the formula `(1 + period rate/m)^m - 1`

or you can use the 2ND ICONV (interest conversion) function.

When you see "continuous compounding" what does that mean?

"Continuous compounding" is compounding over a theoretically infinite number of periods. To calculate the effective annual rate for continuous compounding use `e^stated_rate`

. E.g. for an annual rate of 7.95%, do `e^0.0795`

= 8.2746%

What is the formula to get the continuously compounded EAR, given the stated rate of 12%?

$$EAR = (1 + \frac{.12}{365})^{365} - 1$$

If given the effective rate and the compounding term, how can you back into the stated rate (the rate that does not take compounding into account)?

Simplest way to get the stated rate is on the calculator: `ln(1 + EAR)`

How do you calculate the value of an investment that compounds annually at a given interest rate? E.g. 100,000 at 12% compounded annually for 5 years

Use the formula `(1 + interest rate))^compounding_period * initial amount`

. In the example given: `(1*.12)^5 * 100000`

= 176,234

##### Source:

- CFA