Effective Annual Rate

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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.

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Formula

S	Final value of investment
P	Initial value of investment
ieff	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

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Example calculation

Which plan is the better investment plan?

 

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 = (1.006666666)¹² - 1

i_eff = (1.082999498) - 1

i_eff = 0.082999498

i_eff = 8.2999%

 

i_eff for plan 2:

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

i_eff = (1 + 0.079/365)³⁶⁵ - 1

i_eff = (1.000216438)³⁶⁵ - 1

i_eff = (1.082194930) - 1

i_eff = 0.082194930

i_eff = 8.2194%

 

Plan 1 is better since 8.2999% > 8.2194%

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Calculator for effective annual rate:

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

ieff	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.)

First Plan

r: k:

Second Plan

r: k:

After entering values into the above input areas, click the following 'Calculate' button to get i_eff, the effective annual rate, for both plans.

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