Effective Annual Rate

Explanation

The effective annual rate is a value used to compare different interest plans. If two plans were being compared, the interest plan with the higher effective annual rate would be considered the better plan. The interest plan with the higher effective annual rate would be the better earning plan.

For every compounding interest plan there is an effective annual rate. This effective annual rate is an imagined rate of simple interest that would yield the same final value as the compounding plan over one year.

Formula symbols:

S	Final value of investment
P	Initial value of investment
i_eff	Effective annual rate
r	Annual percentage rate (APR)
k	Interest periods per year

After a term of one year the final value, S, of a compounded interest investment with an initial value P compounded k times per year at an annual percentage rate of r is given by:

S = P(1 + r/k)^k

After a term of one year the final value, S, of a simple interest investment with an initial value P at an annual percentage rate of i_eff is given by:

S = P(1 + i_eff)

Setting these two equal we get:

P(1 + i_eff) = P(1 + r/k)^k

Dividing each side by P we get:

(1 + i_eff) = (1 + r/k)^k

And solving for i_eff we get:

i_eff = (1 + r/k)^k - 1

Example calculation

Which of the following plans is the better investment plan? Note that the answer is not immediately obvious. Plan 1 has a higher interest rate than Plan 2, which causes it to earn more money. However, Plan 1 is compounded less frequently than Plan 2, and this would cause it to earn less money. It is exactly in these types of situations that calculating the effective annual rate can show you which plan will earn more.

Plan 1:

8.0% annual percentage rate, compounded monthly

Plan 2:

7.9% annual percentage rate, compounded daily

i_eff for plan 1:

i_eff = (1 + r/k)^k - 1

i_eff = (1 + 0.08/12)¹² - 1

i_eff = (1.0066...)¹² - 1

i_eff = (1.08299...) - 1

i_eff = 0.08299...

i_eff = 8.299...%

i_eff for plan 2:

i_eff = (1 + r/k)^k - 1

i_eff = (1 + 0.079/365)³⁶⁵ - 1

i_eff = (1.000216...)³⁶⁵ - 1

i_eff = (1.08219...) - 1

i_eff = 0.08219...

i_eff = 8.219...%

Plan 1is better since 8.299...% > 8.219...%

Here is a calculator for effective annual rate.

i_eff = (1 + r/k)^k - 1

i_eff	Effective annual rate
r	APR, Annual percentage rate
k	Times per year compounded

Enter values for the above formula:

(Example: For r enter 5.0% as 0.05, etc.)

Click the following 'Code' button to see the actual JavaScript code that executes in the above calculator. Use this button to see how your device calculates the result for the calculator.

Here's the code that runs when you click the above 'Calculate' button. It is shown here as an aid for your study and understanding.

You can change the code, if you like, and then click the following 'Reevaluate code' button. The program will then work as per your changes when you click the 'Calculate' button on the above calculator. Of course, your changes, especially random changes, can introduce errors, miscalculations, and browser crashes. If you need to get things back to their original condition, just reload this page using your browser's reload button.

Again, the intention here is to conveniently show you the inner workings of this program so that you understand how the program arrives at the result. Click the 'Code' button again to close this section.

window.calculate = function()
{
	var r1    = 0.0;
	var k1    = 0.0;
	var iEff1 = 0.0;

	var r2    = 0.0;
	var k2    = 0.0;
	var iEff2 = 0.0;

	r1 = parseFloat(document.firstPlanForm.firstPlanR.value);
	k1 = parseFloat(document.firstPlanForm.firstPlanK.value);

	r2 = parseFloat(document.secondPlanForm.secondPlanR.value);
	k2 = parseFloat(document.secondPlanForm.secondPlanK.value);
	
	iEff1 = Math.pow((1 + (r1 / k1)), k1) - 1;
	iEff2 = Math.pow((1 + (r2 / k2)), k2) - 1;

	alert("i eff, first plan: " + iEff1 + "\ni eff, second plan: " + iEff2);
};