As the number of compounding period increases annual percentage rate (APR) remains same

EAR = Effective Annual Rate, APR=Annual Percentage Rate
Look at this… 4% compounding annual. APR = 4%, EAR=4%
4% compounded semi-annual -> 1.02^2 = 1.0404 => EAR=4.04%, APR = 2% * 2 = 4%
4% compounded quarterly -> EAR => 1.01^4 => 1.040606 => EAR= 4.0604% But APR = 1% x 4 = 4%
APR is always 4% per year, but EAR changes - and inceases as the number of times the compounding occurs.
The APR is the way someone tries to tell you that the rate is lower – but what you actually end up paying is the EAR … which is higher.
So someone will tell you - you will be paying 10% APR - but if you are paying monthly -> you would be paying (1+.1/12) ^ 12 = 10.47%.
Quote:
An effective annual interest rate of 10% can also be expressed in several ways:
  • 0.7974% effective monthly interest rate, because 1.00797412=1.1
  • 9.569% annual interest rate compounded monthly, because 12×0.7974=9.569
  • 9.091% annual rate in advance, because (1.1-1)÷1.1=0.091
These rates are all equivalent, but to a consumer who is not trained in the mathematics of finance, this can be confusing. APR helps to standardize how interest rates are compared, so that a 10% loan is not made to look cheaper by calling it a loan at “9.1% annually in advance”.
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You should look at the difference betwen Effective Annual Rate, which is affected by the compounding period, and the Annual Percentage Rate, which isn’t. Particularly, look at pages 287 - 289 of the Quanatative Methods book.
Think of the APR as a pie that can be sliced into as many pieces as you want…. No matter how many slices (compounding periods) you divide it into, the size of the pie doesn’t change.
The EAR is different. That is equivilant annual rate is where you look at the effect of all of your compounding over the course of the year and describe that as if there was a single rate applied for the whole year. So, if you compound a bunch (e.g., daily) you will have more money at the end of the year than if you either don’t compound or compound just a little (e.g.,semi-annually) . So, the more slices, the bigger the pie gets.
Thus, your EAR will always be the same or a little higher than the APR.
Hope this helps.
 
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