## Frequency of Compounding

### Introduction Your grandmother sends you \$100 for your birthday, and you decide to invest it.

You visit three different borrowers and each offers you the same interest rate. You discover, however, that one borrower computes interest monthly, the other semiannually and the third annually.

Which borrower should you pick? Which one will result in more money for you?

As mentioned earlier, TVOM assumes a compound interest rate. This section highlights:

• The effects of frequency of compounding
• Differences between quoted rate, EAR and APR

### Objectives

Upon completing this section you will be able to:

• Calculate the amount of interest earned from borrowers with annual, semi annual and monthly compounding.
• Calculate periodic rate.
• Define quoted rate.
• Calculate EAR.
• Define APR.

### Frequency of Compounding

Frequency of compounding concerns the number of times that interest is computed per year. It can best be explained through an example.

Assume Maria wants to invest \$100 that she received from her grandmother for her birthday. She finds three borrowers that will pay her the same 12% interest rate. However, the interest for each borrower will be computed (and credited to Maria's account) differently as follows:

• Borrower#1 computes interest once a year (called annual compounding).
• Borrower#2 computes interest twice a year (called semiannual compounding).
• Borrower#3 computes interest twelve times a year (called monthly compounding).

What is the amount of interest that Maria earns from each borrower for one year?

To answer this question, you need to calculate future value at the end of one year for each borrower and compare it with the investment of \$100.

First, let's look at the time line and interest computation for Borrower#1 (annual compounding): From Section 1 of this tutorial, you know the formula for calculating a compound interest rate for the first year, X1, is:
X1 = X1+(X1×r)

Using this same equation, but substituting FV1 for X1:
FV1 = 100+(100×.12) = 100(1.12) = \$112

Since this timeline has only two values, PV and FV, you should also recognize it as a lump sum case. Thus you could also use the lump sum formula from Section 2-A to solve for FV1, which will give you the same answer as above:
FV1 = PV(1×r)t
FV1 = 100(1.12)1 = \$112

Given the investment of \$100, we can determine the amount of interest for the year by subtracting the FV from the PV:
\$112-100 = \$12

That is, Maria will earn a 12% return.

Next, let's look at the time line and interest computation for Borrower#2 (semiannual compounding): Because Borrower#2's interest rate is stated for the entire year, the interest per half a year equals to 12% divided by 2, which equals 6%. Thus, you can calculate the compound interest of the future value after the first six months (FV1) and the future value at the end of the year (FV2):
FV1 = 100+(100×.06) = \$106
FV2 = 106+(106×.06) = \$112.36

Alternatively, using the future value of lump sum equation, you get the same answer for future value at the end of the year:
FV2 = 100(1.06)2 = \$112.36

Given the investment of \$100, we can determine the amount of interest for the year by subtracting the FV from the PV:
\$112.36-100 = \$12.36

That is, Maria will earn a 12.36% return.

The time line and interest computation for Borrower#3 (monthly compounding) is: Because interest rate is stated for the entire year, interest per month equals to 12% divided by 12 or 1%. Calculating for compound interest, the future values at the end of each month are:
FV1 = 100+(100×1%) = \$101
FV2 = 101+(101×1%) = \$102.01
FV3 = 102.01+(102.01×1%) = \$103.03
FV4 = \$103.03+(\$103.03×1%) = \$104.06

FV11 = 110.46+(110.46×1%) = \$111.57
FV12 = 111.57+(111.57×1%) = \$112.68

Alternatively, using the future value of lump sum equation, the future value at the end of the year is:
FV12 = 100(1.01)12 = \$112.68

Given the investment of \$100, we can determine the amount of interest for the year by subtracting the FV from the PV:
\$112.68-100 = \$12.68

That is, Maria will earn a 12.68% return.

In sum, interest amount and return that Maria earns from each borrower are:

Borrower#1 Borrower#2 Borrower#3 Annual Semiannual Monthly \$12.00 \$12.36 \$12.68 12% 12.36% 12.68%

Although each borrower pays the same interest rate of 12%, Maria will earn different amount of interest and hence return because of different frequency of compounding. The interest amount and return are lowest for Borrower#1 and highest for Borrower#3.

This shows that the higher the frequency of compounding, the higher the future value and hence the return.

### Quoted Rate, EAR, and APR

The 12% interest rate quoted by each borrower in the example is called quoted rate, stated rate or nominal rate. The words 'rate' or 'interest rate' often means quoted rate.

The interest rate earned per period is called periodic rate, and can be calculated using this formula: where m = frequency of compounding.

From our example, the periodic rates are as follows:

• For Borrower#1, the quoted rate of 12% is divided by 1 (compounded annually) giving a periodic rate of 12%.
• For Borrower#2, the quoted rate of 12% is divided by 2 (compounded semiannually or every six months), giving the periodic rate of 6%.
• For Borrower#3, the quoted rate of 12% is divided by 12 (compounded monthly), giving the periodic rate of 1%.

The 12%, 12.36% and 12.68% return from each borrower are called effective annual rate (EAR). EAR is interest rate expressed as if it were compounded annually. That is, Maria will earn the same interest from either 12% semiannual compounding or 12.36% annual compounding. Similarly, the interest for Maria is the same for either 12% monthly compounding or 12.68% annual compounding.

The equation for converting from quoted rate to EAR is: For example, 12.36%EAR of 12% semiannual compounding can be obtained from the equation as follows: What is APR? APR stands for annual percentage rate. By law, lenders are required to disclose APR, which is interest rate charged per period multiplied by the number of periods per year.

In most types of loans such as auto loan and credit card, APR is the same as quoted rate. For example, if a bank charges 0.5% per month on a car loan, the bank must report 6% APR on the loan. The 6% rate is quoted rate, and 0.5% is periodic rate or interest rate charged per month.

However, in case of mortgage loan, APR is not the same as quoted rate because APR for mortgage loan is calculated by including not only interest but also some other fees such as discount points and loan-processing fees charged by lenders. APR is not used to calculate mortgage payments (quoted rate is), but is used to provide an estimate of the borrowing cost.