Compound interest is used for most transactions lasting one year or more. In simple interest, interest is converted to principal at the end of the transaction. Therefore, all interest is based solely on the original principal amount of the transaction. Compound interest, by contrast, involves interest being periodically converted to principal throughout a transaction, with the result that the interest itself also accumulates interest.
Calculating the Periodic Interest Rate
The first step in learning about investing or borrowing under compound interest is to understand the interest rate used in converting interest to principal. You commonly need to convert the posted interest rate to find the exact rate of interest earned or charged in any given time period.
The Formula
[latex]\colorbox{LightGray}{Formula 9.1}\; \color{BlueViolet}{\text{Periodic Interest Rate:}\;i=\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}}[/latex]
Concept Check:
Example 9.1.1: The Periodic Interest rate (i)
Calculate the periodic interest rate, [latex]i[/latex], for the following nominal interest rates:
a) 9% compounded monthly
b) 6% compounded quarterly
Solution:
Step 1: Given information:
a) I/Y = 9%; C/Y = monthly = 12 times per year
b) I/Y = 6%; C/Y= quarterly = 4 times per year
Step 2: For each question apply the periodic interest formula.
a) [latex]i=\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}=\frac{9\%}{12}=0.75\%\;\text{per month}[/latex]
Nine percent compounded monthly is equal to a periodic interest rate of 0.75% per month. This means that interest is converted to principal 12 times throughout the year at the rate of 0.75% each time.
b) [latex]i=\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}=\frac{6\%}{4}=1.5\%\; \text{per quarter}[/latex]
Six percent compounded quarterly is equal to a periodic interest rate of 1.5% per quarter. This means that interest is converted to principal 4 times (every three months) throughout the year at the rate of 1.5% each time.
Example 9.1.2: The Nominal Interest Rate (I/Y)
Calculate the nominal interest rate, I/Y, for the following periodic interest rates:
a) [latex]0.58\overline{3}\%[/latex] per month
b) 0.05% per day
Solution:
Step 1: Given information:
a) [latex]i=0.58\overline{3}\%[/latex]; C/Y = monthly = 12 times per year
b) i = 0.05%; C/Y = daily = 365 times per year
Step 2:For each question, apply the periodic interest formula and rearrange for the nominal rate, I/Y.
a) [latex]I/Y=i\times C/Y=0.58\overline{3}\times12=7\%[/latex]
A periodic interest rate of [latex]0.58\overline{3}[/latex] per month is equal to a nominal interest rate of 7% compounded monthly.
b) [latex]I/Y=i\times C/Y=0.05\times365=18.25\%[/latex]
A periodic interest rate of 0.05% per day is equal to a nominal interest rate of 18.25% compounded daily.
Example 9.1.3: Compounds per Year (C/Y)
Calculate the compounding frequency (C/Y) for the following nominal and periodic interest rates:
a) nominal interest rate = 6%, periodic interest rate = 3%
b) nominal interest rate = 9%, periodic interest rate = 2.25%
Solution:
Step 1: Given information:
a) I/Y = 6%; i = 3%
b) I/Y = 9%; i = 2.25%
Step 2: For each question, apply the periodic interest formula and rearrange for the compounding frequency, C/Y.
a) [latex]C/Y=\frac{I/Y}{i}=\frac{6\%}{3\%}=2\;\text{compounds per year}=\text{semi-annually}[/latex]
For the nominal interest rate of 6% to be equal to a periodic interest rate of 3%, the compounding frequency must be twice per year, which means a compounding period of every six months, or semi-annually.
b) [latex]C/Y=\frac{I/Y}{i}=\frac{9\%}{2.25\%}=4\;\text{compounds per year}=\text{quarterly}[/latex]
For the nominal interest rate of 9% to be equal to a periodic interest rate of 2.25%, the compounding frequency must be four times per year, which means a compounded period of every three months, or quarterly.
Exercises
In each of the exercises that follow, try them on your own. Full solutions are available should you get stuck.
- Calculate the periodic interest rate if the nominal interest rate is 7.75% compounded monthly. (Answer: 0.65%)
- Calculate the compounding frequency for a nominal interest rate of 9.6% if the periodic interest rate is 0.8%.(Answer: 12)
- Calculate the nominal interest rate if the periodic interest rate is 2.0875% per quarter. (Answer: 8.35% compounded quarterly)
- After a period of three months, Alese saw one interest deposit of $176.40 for a principal of $9,800. What nominal rate of interest is Alese earning?
(Answer: 7.21 compounded quarterly)
