## 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:

k = number of times loan is compounded per year.

In the example above,

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).

- 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.