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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?

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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 %

* ref: http://en.wikipedia.org/wiki/Effective_interest_rate#Calculation

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 %

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