What is the effective interest rate corresponding to a nominal rate of 10% compounded quarterly?

Equivalent interest rates are interest rates that produce the same future value after one year

For example, 10% compounded quarterly and 10.125% compounded semiannually are equivalent nominal interest rates. If you calculate the future value of $100 invested at either rate for one year, you will obtain $110.38.

You can see that equivalent interest rates have different numerical values but produce the same effect.

The term “equivalent rates” carries with it the same concept as “effective rates” but takes into account interest rates that are compounded more than once per year. Note that “effective rates” refer to interest rates that are compounded annually.

To use Excel to find equivalent interest rates we can use the EFFECT and NOMINAL functions

Steps:

• First, find the effective rate using the EFFECT function for the given nominal rate.
• Then, using the effective rate and the new compounding frequency to compute the new nominal rate using the NOMINAL function.

Example:

To be equivalent to 10% compounded quarterly, what must be the nominal rate with monthly compounding?

Given:

• nominal rate j1=10%
• compounding frequency m1=4

Step 1:

Find the Effective rate to 10% compounded quarterly.  We can use the EFFECT function.

We have now found the effective rate of 10.3813%.

However, we are looking for the equivalent rate to 10% compounded quarterly.

Step 2

Find the nominal rate that is compounded monthly that is equivalent to the effective rate of 10.3813%.

We will use the compounding frequency m2=12 and the NOMINAL function.

Hence, the 9.9178% compounded monthly is equivalent to 10% compounded quarterly.

To see the spreadsheet used for this example:equivalent rates – template

Interest rate adjusted for compounding over a given period

What is the Effective Annual Interest Rate?

The Effective Annual Interest Rate (EAR) is the interest rate that is adjusted for compounding over a given period. Simply put, the effective annual interest rate is the rate of interest that an investor can earn (or pay) in a year after taking into consideration compounding.

EAR can be used to evaluate interest payable on a loan or any debt or to assess earnings from an investment, such as a guaranteed investment certificate (GIC) or savings account.

The effective annual interest rate is also known as the effective interest rate (EIR), annual equivalent rate (AER), or effective rate. Compare it to the Annual Percentage Rate (APR) which is based on simple interest.

The EAR formula is given below:

Where:

• i = Stated annual interest rate
• n = Number of compounding periods

Effective Annual Rate Based on Compounding

The table below shows the difference in the effective annual rate when the compounding periods change.

Table: CFI’s Fixed Income Fundamentals Course

For example, the EAR of a 1% Stated Interest Rate compounded quarterly is 1.0038%.

Importance of Effective Annual Rate

The effective annual interest rate is an important tool that allows the evaluation of the true return on an investment or true interest rate on a loan.

The stated annual interest rate and the effective interest rate can be significantly different, due to compounding. The effective interest rate is important in figuring out the best loan or determining which investment offers the highest rate of return.

In the case of compounding, the EAR is always higher than the stated annual interest rate.

EAR Example

For example, assume the bank offers your deposit of $10,000 a 12% stated interest rate compounded monthly. The table below demonstrates the concept of the effective annual interest rate:

Month 1 Interest: Beginning Balance ($10,000) x Interest Rate (12%/12 = 1%) = $100

Month 2 Interest: Beginning Balance ($10,100) x Interest Rate (12%/12 = 1%) = $101

The change, in percentage, from the beginning balance ($10,000) to the ending balance ($11,268) is ($11,268 – $10,000)/$10,000 = .12683 or 12.683%, which is the effective annual interest rate. Even though the bank offered a 12% stated interest rate, your money grew by 12.683% due to monthly compounding.

The effective annual interest rate allows you to determine the true return on investment (ROI).

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How to Calculate the Effective Interest Rate?

To calculate the effective interest rate using the EAR formula, follow these steps:

1. Determine the stated interest rate

The stated interest rate (also called the annual percentage rate or nominal rate) is usually found in the headlines of the loan or deposit agreement. Example: “Annual rate 36%, interest charged monthly.”

2. Determine the number of compounding periods

The compounding periods are typically monthly or quarterly. The compounding periods may be 12 (12 months in a year) and 4 for quarterly (4 quarters in a year).

For your reference:

• Monthly = 12 compounding periods
• Quarterly = 4 compounding periods
• Bi-Weekly = 26 compounding periods
• Weekly = 52 compounding periods
• Daily = 365 compounding periods

3. Apply the EAR Formula: EAR = (1+ i/n)n – 1

Where:

• i = Stated interest rate
• n = Compounding periods

Example

To calculate the effective annual interest rate of a credit card with an annual rate of 36% and interest charged monthly:

1. Stated interest rate: 36%

2. Number of compounding periods: 12

Therefore, EAR = (1+0.36/12)^12 – 1 = 0.4257 or 42.57%.

Why Don’t Banks Use the Effective Annual Interest Rate?

When banks are charging interest, the stated interest rate is used instead of the effective annual interest rate. This is done to make consumers believe that they are paying a lower interest rate.

For example, for a loan at a stated interest rate of 30%, compounded monthly, the effective annual interest rate would be 34.48%. Banks will typically advertise the stated interest rate of 30% rather than the effective interest rate of 34.48%.

When banks are paying interest on your deposit account, the EAR is advertised to look more attractive than the stated interest rate.

For example, for a deposit at a stated rate of 10% compounded monthly, the effective annual interest rate would be 10.47%. Banks will advertise the effective annual interest rate of 10.47% rather than the stated interest rate of 10%.

Essentially, they show whichever rate appears more favorable.

Related Reading

Thank you for reading CFI’s guide on Effective Annual Interest Rate. To continue developing your career as a financial professional, check out the following additional CFI resources:

• Expected Return
• Basis Points
• Capital Gains Yield
• Weighted Average Cost of Capital (WACC)