Understanding APR, APY, and EAR

Problem:

Jake goes to the bank and borrows $1000. The loan is at an APR of 5% and compounding is done on a monthly basis. The loan term is one year and he makes monthly loan payments.  What is the Effective Annual Rate (EAR) of the loan?

  • (A) 5.0000%
  • (B) 5.1932%
  • (C) 5.1162%
  • (D) 5.0999%

Answer:

Correct answer is Option C, EAR = 5.1162%

Explanation:

APR = Annual Percentage Rate = this is the interest rate; however, it does not provide how often compounding is done per year.

EAR = Effective Annual Rate = this is the real rate of interest Jake pays on the loan.

In the above loan, here are the facts:

  • Compounding is done on a monthly basis.
  • Interest rate per month = 0.05/12
  • Term = 1 year = 12 months
  • PV of loan = $1000

Now remember the PV and FV lesson – Time Value of Money – we will apply this lesson here.  Here we need to solve for FV. Now let us enter the following in an Excel cell: “=FV(0.05/12,12,0,1000)”. 

If you type “=FV” in Excel, it will show you what each field is.  In the above case, we don’t know what the PMT is, so I entered 0. We know the value for all the other fields.

You will get -$1051.162. The sign is negative because that is what Jake will end up paying the bank when he pays off the loan.

Now the interest amount = $1051.162 – $1000 = $51.162

Effective Annual Rate = (51.162/1000) * 100 = 5.1162%.

I took the long road to explain how it works.

Here is the formula to calculate EAR from APR:

EAR_x1

k = number of times loan is compounded per year.

In the example above,

EAR_x2

If you calculate, EAR = 5.1162.

If compounding was done only once per year, then EAR = APR.

Key Take Away’s:

  • When you are shopping for CD’s, banks will always quote the Annual Percentage Yield (APY).
    • If you are depositing money into a savings account or CD and the bank provides you the APR and how often compounding will be done per year, then you can calculate the APY using the same formula above (k is the number of times compounding is done per year):

APY_x1

  • When you are applying for a loan, banks will quote you the APR. You need to ask and understand what the EAR is. Always review and understand the EAR before you sign up for the loan.

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