# 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 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 ieff is given by:

S = P(1 + ieff)

Setting these two equal we get:

P(1 + ieff) = P(1 + r/k)k

Dividing each side by P we get:

(1 + ieff) = (1 + r/k)k

And solving for ieff we get:

ieff = (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

ieff for plan 1:

ieff = (1 + r/k)k - 1

ieff = (1 + 0.08/12)12 - 1

ieff = (1.0066...)12 - 1

ieff = (1.08299...) - 1

ieff = 0.08299...

ieff = 8.299...%

ieff for plan 2:

ieff = (1 + r/k)k - 1

ieff = (1 + 0.079/365)365 - 1

ieff = (1.000216...)365 - 1

ieff = (1.08219...) - 1

ieff = 0.08219...

ieff = 8.219...%

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

Here is a calculator for effective annual rate.

ieff = (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 ieff, the effective annual rate, for both plans.

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