# Understanding Effective Annual Interest Rate concept

I am trying to understand the effective interest rate concept with an example. All I know about it is the following formula:

``````EAR = (1+r/n)^n -1 , where

n = number of periods
r = stated interest rate
``````

So in my example we have the following:

``````PV = 1000
Rate annually is 10%
``````

FV in two years would be

``````FV = PV(1+r)^n
= 1000(1+0.10)^2
= 1210
``````

Now if we change the problem to

``````What will be the FV if interest rate is compounded quarterly?
``````

If the rate is applied quarterly, then should I use the same formula but with n = 8 compounding periods for two years?

Given a nominal rate, `i = 10%` compounded quarterly, the quarterly rate can be found in two ways.

1. Directly by `q = i/4 = 2.5%`

or 2. via the effective annual rate* :-

``````ear = (1 + i/4)^4 - 1 = 0.103813 = 10.3813 %
q = (1 + ear)^(1/4) - 1 = 0.025 = 2.5 %
``````

The amount accumulated in two years can be calculated using the quarterly rate or the EAR :-

1. `1000 (1 + q)^8 = 1218.4`

2. `1000 (1 + ear)^2 = 1218.4`

If, on the other hand, the effective annual rate is 10%

``````ear = 0.1
fv = 1000 (1 + ear)^2 = 1210
``````

then the quarterly rate will be

``````q = (1 + ear)^(1/4) - 1 = 0.02411 = 2.411 %

fv = 1000 (1 + q)^8 = 1210
``````

and the nominal rate compounded quarterly `i = 4 * q = 0.09645 = 9.645 %`