# 1.1 Compound Interest Terminology

## Learning Objectives

• Differentiate between the concept of compound interest and simple interest
• Define terms related to compound interest

## Formula & Symbol Hub

#### Symbols Used

• $f$ or $EFF$ = Effective interest rate
• $i$ = Periodic interest rate
• $j$ or $I/Y$ = Nominal interest rate per year
• $m$ or $C/Y$ = Number of compounds per year or compounding frequency
• $n$ or $N$ = Total number of compound periods for the term

#### Formulas Used

• ##### Formula 1.1 – Total Number of Compounds

$n=m \times \mbox{time in years}$

• ##### Formula 1.2 – Periodic Interest Rate

$i=\frac{j}{m}$

## Introduction

Compound interest involves interest being periodically converted to principal throughout a transaction, with the result that the interest itself also accumulates interest. This means that interest earned in the previous compounding period will earn interest in all subsequent compounding periods.

But how does compound interest compare to simple interest? The critical difference is the placement of interest into the account. Under simple interest, you convert the interest to principal at the end of the transaction’s time frame. For example, in a six-month simple interest GIC the balance in its account at any point before the maturity date is the original principal and nothing more. Only upon maturity does the interest appear. In contrast, a five-year compound interest GIC receives an interest deposit into the account at periodic times every year. Once this interest is deposited into the account that interest will begin to earn interest for the remainder of the term of the GIC.

Suppose you invested $\1,000$ into two different savings accounts: one account earns $10\%$ simple interest and the other account earns $10\%$ compounded annually. How much is in each account at the end of five years? Which account earns more interest?

From the previous chapter on simple interest, the simple interest savings account will earn $\500$ in interest and have a maturity value of $\1,500$ at the end of five years. Because this is simple interest, only the original $\1,000$ principal earns interest over the five years. The table below shows the balance in the account at the end of each year.

 Year Principal at Start of Year Accrued Interest Balance at End of Year $1$ $\1,000$ $\100$ $\1,000$ $2$ $\1,000$ $\200$ $\1,000$ $3$ $\1,000$ $\300$ $\1,000$ $4$ $\1,000$ $\400$ $\1,000$ $5$ $\1,000$ $\500$ $\1,000$

For the compound interest savings account, the $10\%$ interest is compounded annually, which means at the end of every year $10\%$ of the current principal is added to the account. This balance at the end of each year becomes the principal for the next year and is the amount of money that will earn interest in the next year. The table below shows the balance in the account at the end of each year when the interest is calculated and added to the principal.

 Year Principal at Start of Year Accrued Interest Balance at End of Year $1$ $\1,000$ $\100$ $\1,100$ $2$ $\1,100$ $\110$ $\1,210$ $3$ $\1,210$ $\121$ $\1,331$ $4$ $\1,331$ $\133.10$ $\1,464.10$ $5$ $\1,461.10$ $\146.41$ $\1,610.51$

The simple interest account earns a total of $\500$ in interest. But the compound interest account earns $\610.51$ in interest (the sum of the accrued interest column), which is $\110.51$ more interest than the simple interest account. This is the power of compounding interest. As an investor, you earn more interest on an investment when the interest is compounded than you would with simple interest. But this also means that as a borrower, you will pay more interest on a loan when the interest is compounded than with simple interest.

## Compound Interest Terms

The compounding period is the interval of time between two consecutive interest calculations. That is, the compounding period is the period of time between the dates of successive conversions of interest to principal. For example, an interest rate that compounds monthly means that the compounding period is monthly, and so the interest is calculated and added to the principal every month.

The compounding frequency ($m$)is the number of times the interest is compounded every year. That is, the compounding frequency is the number of compounding periods in one year. For example, an interest rate that compounds quarterly means that the interest is calculated and added to the principal four times every year, so $m=4$.

 Compounding Period Number of Times per Year Interest is Compounded Compounding Frequency Annually Every year/Once a year $1$ Semi-annually Every six months/Twice a year $2$ Quarterly Every three months/Four times a year $4$ Monthly Every month/Twelve times a year $12$ Daily Every day/$365$ times a year $365$

The term or time period ($t$)is the length of time for the investment or loan. That is, the term is the period of time when the interest is calculated. Because the interest rate is given as an annual interest rate and the compounding frequency is the number of compoundings per year, the term must be in years in order to use the term in compound interest calculations. If the term is not in years, the term must be converted to years. If the term is given in months, divide by $12$ to convert the term to years. If the term is given in days, divide by $365$ to convert the term to years.

Most compound interest calculations require the total number of compounding periods during the term ($n$).

## $\boxed{1.1}$ Total Number of Compounding Periods

\Large\begin{align*}\color{red}{n}&=\color{blue}{m}\times\color{green}{t}\end{align*}

${\color{red}{n}}\;\text{is the Total Number of Compoundings:}$ Most compound interest calculations require this value.

${\color{blue}{m}}\;\text{is the Compounding Frequency:}$ is the number of times the interest is compounded every year.

${\color{green}{t}}\;\text{is Time in Years:}$ is the length of time for the investment or loan.

### Example 1.1.1

For the following interest rates and terms, identify the compounding frequency and the term in years. Then calculate the total number of compounding periods during the term.

1. $3.5\%$ compounded monthly for five years.
2. $7\%$ compounded semi-annually for $18$ months.
3. $4.9\%$ compounded quarterly for three years and nine months.
Solution
1. $\mbox{Compounding frequency}=12$
$\mbox{Term}=5 \mbox{ years}$
$n=5 \times 12=60$
2. $\mbox{Compounding frequency}=2$
$\mbox{Term}=\frac{18}{12}=1.5 \mbox{ years}$
$n=1.5 \times 2=3$
3. $\mbox{Compounding frequency}=4$
$\mbox{Term}=3+\frac{9}{12}=3.75 \mbox{ years}$
$n=3.75 \times 4=15$

### Try It

1) Complete the following table.

 Interest Rate and Time Period Compounding Frequency ($m$) Term (in years) ($t$) Number of Compounding Periods for the Term ($n$) $7.2\%$ compounded quarterly for $27$ months $3.7\%$ compounded semi-annually for $6.5$ years $1.9\%$ compounded daily for $657$ days
Solution
 Interest Rate and Time Period Compounding Frequency ($m$) Term (in years) ($t$) Number of Compounding Periods for the Term ($n$) $7.2\%$ compounded quarterly for $27$ months $4$ $\frac{27}{12}=2.25$ $2.25 \times 4=9$ $3.7\%$ compounded semi-annually for $6.5$ years $2$ $6.5$ $6.5 \times 2=13$ $1.9\%$ compounded daily for $657$ days $365$ $\frac{657}{365}=1.8$ $1.8 \times 365=657$

## The Nominal and Periodic Interest Rates

The nominal interest rate ($j$) is the quoted or stated interest rate annually. It is the rate, expressed as a percent, that precedes the word “compounded”. For example, if the interest rate is $5\%$ compounded quarterly, the nominal interest rate is $5\%$.

The periodic interest rate ($i$) is interest rate earned or charged for a given compounding period.

### $\boxed{1.2}$ Periodic Interest Rate

\begin{align*}\Large\color{red}{i}=\frac{\color{blue}{j}}{\color{green}{m}}\end{align*}

${\color{red}{i}}\;\text{is the Periodic Interest Rate:}$ is interest rate earned or charged for a given compounding period.

${\color{blue}{j}}\;\text{is the Nominal Interest Rate:}$ is the quoted or stated interest rate annually.

${\color{green}{m}}\;\text{is the Compounding Frequency:}$ is the number of times the interest is compounded every year.

### Example 1.1.2

Calculate the periodic interest rate for the following nominal interest rates.

1. $9\%$ compounded monthly.
2. $6\%$ compounded quarterly.
Solution

1. For $9\%$ compounded monthly.

$\begin{eqnarray*} j & = & 9\% \\ m & = & 12 \\ \\ i & = & \frac{j}{m} \\ & = & \frac{9\%}{12} \\ & = & 0.75\% \mbox{ per month} \end{eqnarray*}$

This means that $9\%$ compounded monthly is equal to a periodic interest rate of $0.75\%$ per month. Interest is converted to principal $12$ times throughout the year at the rate of $0.75\%$ each time.

2. For $6\%$ compounded quarterly.

$\begin{eqnarray*} j & = & 6\% \\ m & = & 4 \\ \\ i & = & \frac{j}{m} \\ & = & \frac{6\%}{4} \\ & = & 1.5\% \mbox{ per quarter} \end{eqnarray*}$

This means that $6\%$ compounded quarterly is equal to a periodic interest rate of $1.5\%$ per quarter. Interest is converted to principal $4$ times throughout the year at the rate of $1.5\%$ each time.

### Example 1.1.3

Calculate the nominal interest rate for the following periodic interest rates.

1. $0.7\%$ per month.
2. $0.05\%$ per day.
Solution

1. For $0.7\%$ per month.

$\begin{eqnarray*} i & = & 0.7\% \\ m & = & 12 \\ \\ j & = & i \times m \\ & = & 0.7 \% \times 12 \\ & = & 8.4\% \mbox{ compounded monthly} \end{eqnarray*}$

This means that $0.7\%$ per month is equal to $8.4\%$ compounded monthly.

2. For $0.05\%$ per day.

$\begin{eqnarray*} i & = & 0.05\% \\ m & = & 365 \\ \\ j & = & i \times m \\ & = & 0.05 \times 365 \\ & = & 18.25\% \mbox{ compounded daily} \end{eqnarray*}$

This means that $0.05\%$ per day is equal to $18.25\%$ compounded daily.

### Example 1.1.4

Calculate the compounding frequency for the following nominal and periodic interest rates.

1. Nominal rate of $6\%$, periodic rate of $3\%$.
2. Nominal rate of $9\%$, periodic rate of $2.25\%$.
Solution

1. For $6\%$ nominal and $3\%$ periodic.

$\begin{eqnarray*} j & = & 6\% \\ i & = & 3\% \\ \\ m & = & \frac{j}{i} \\ & = & \frac{6\%}{3\%} \\ & = & 2 \end{eqnarray*}$

For the nominal interest rate of $6\%$ to be equal to a periodic interest rate of $3\%$, the compounding frequency is $2$, which means a compounding period of every six months, or semi-annually.

2. For $9\%$ nominal and $2.25\%$ periodic.

$\begin{eqnarray*} j & = & 9\% \\ i & = & 2.25\% \\ \\ m & = & \frac{j}{i} \\ & = & \frac{9\%}{2.25\%} \\ & = & 4 \end{eqnarray*}$

For the nominal interest rate of $9\%$ to be equal to a periodic interest rate of $2.25\%$, the compounding frequency is $4$, which means a compounded period of every three months, or quarterly.

### Try It

2) Calculate the periodic interest rate if the nominal interest rate is $7.75\%$ compounded monthly.

Solution

$\begin{eqnarray*} i & = & \frac{j}{m} \\ & = & \frac{7.75\%}{12} \\ & = & 0.65\% \mbox{ per month} \end{eqnarray*}$

### Try It

3) Calculate the compounding frequency for a nominal interest rate of $9.6\%$ if the periodic interest rate is $0.8\%$.

Solution

$\begin{eqnarray*} m & = & \frac{j}{i} \\ & = & \frac{9.6\%}{0.8} \\ & = & 12 \end{eqnarray*}$

### Try It

4) Calculate the nominal interest rate if the periodic interest rate is $2.0875\%$ per quarter.

Solution

$\begin{eqnarray*} j & = & i \times m \\ & = & 2.0875\% \times 4\\ & = & 8.35\% \mbox{ compounded quarterly} \end{eqnarray*}$

### Section 1.1 Exercises

1. Calculate the periodic interest rate if the nominal interest rate is $7.5\%$ compounded monthly.
Solution

$0.625\%$

2. Calculate the compounding frequency for a nominal interest rate of $9.6\%$ if the periodic interest rate is $0.8\%$.
Solution

$12$

3. Calculate the nominal interest rate if the periodic interest rate is $2.0875\%$ per quarter.
Solution

$8.35\%$

4. Lori hears her banker state, “We will nominally charge you $10.68\%$ on your loan, which works out to $0.89\%$ of your principal every time we charge you interest.” What is her compounding frequency?
Solution

$12$

5. You just received your monthly MasterCard statement and note at the bottom of the form that interest is charged at $11.9355\%$ compounded daily. What interest rate is charged to your credit card each day?
Solution

$0.0327\%$

6. Calculate the total number of compoundings for an investment at $3.5\%$ compounded monthly for four years and seven months.
Solution

$55$

7. Calculate the total number of compoundings for an investment at $4.2\%$ compounded quarterly for $39$ months.
Solution

$13$