Question

Find the EAR in each of the following cases. APR 9% quarterly, 16% monthly, 12% daily, and 11% infinite

Answer 1

• 9.31%
• 17.23%
• 12.75%
• 11.63%

Step-by-step explanation:

EAR is the effective annual rate, while APR is the annual percentage rate.

When the interests are compounded, the periodic interest rate is calculated dividing the APR by the number of periods.

The Effective Annual Rate (EAR) is the interest rate taking into consideration that the interest is compounded and is calculated as:

`EAR=\bigg(1+(APR)/(n)\bigg)^n}-1`

For APR = 9% quarterly

• n = 12 / 3 = 4

• EAR = (1 + 9%/4)⁴ - 1 = (1 + 0.09/4)⁴ - 1 = 1.09308 - 1 = 0.09308 = 9.31%

For APR = 16% monthly

• n = 12

• EAR = (1 + 16%/12)¹² - 1 = (1 + 0.16/12)¹² - 1 = (1 + 0.0133)¹² - 1 = 0.1723 =17.23%

For APR = 12% daily

• n = 365

• EAR = (1 + 12%/365)³⁶⁵ - 1 = 0.12747 = 12.75%

For APR = 11% infinetly

You might insert a very big number, like 1,000,000 in the formula and will obtain a correct value:

• n = 1,000,000

• EAR = (1 + 11%/1,000,000)¹⁰⁰⁰⁰⁰⁰ - 1 = 0.11628 = 11.63%

But that is just an approximation.

There is a formula for when the interest is compounded continuoslly (it is derived using limits):

`EAR=e^(APR)-1=e^((11\%))-1=e^(0.11)-1=0.11628=11.63\%`

That is the very same value obtained with the approximation; thus, when you do not remember the last formula you can use that trick.