Introduction to Compound Interest

Compound interest is earned when interest is re-invested. That is, interest is earned on both the principal (i.e., the amount that was originally invested) and any interest that has already accumulated.

This can be contrasted with simple interest, in which interest is earned on the principal only.

Explore This 1

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Explore This Summary

Here are some things you may have noticed while completing the Explore This activity:

After the first year (or compounding period), compound interest always produces more interest than simple interest (at the same interest rate for the same length of time). In the first year (or compounding period), the interest earned is the same for both simple and compound interest.
Under compound interest, the amount of interest added increases every year.
The longer the investment, the greater the difference between simple interest and compound interest at the same interest rate.

Try This Revisited

$$10~000$ is invested at a rate of $5\%$ per year. At the end of each year, the interest is added to the account and the total amount is re-invested for the next year (still at a rate of $5\%$). What is the value of the investment after four years?

Solution

Try This Revisited Continued

$$10~000$ is invested at a rate of $5\%$ per year. At the end of each year, the interest is added to the account and the total amount is re-invested for the next year (still at a rate of $5\%$). What is the value of the investment after four years?

Solution

Year ($n$)	Interest ($$$)	Accumulated Value ($$$)
$0$		$10~000$
$1$	$10~000(0.05)=500$	$10~500$
$2$	$10~500(0.05)=525$	$11~025$
$3$	$11~025(0.05)=551.25$	$11~576.25$
$4$	$11~576.25(0.05)\approx578.81$	$12~155.06$

Writing an Investment as a Sequence

Year ($n$)	Accumulated Value ($$$), $t_n$
$0$	$10~000$
$1$	$10~500$
$2$	$11~025$
$3$	$11~576.25$
$4$	$12~155.06$

Writing an Investment as a Sequence Continued

Year ($n$)	Accumulated Value ($$$), $t_n$
$0$	$10~000$
$1$	$10~500$
$2$	$11~025$
$3$	$11~576.25$
$4$	$12~155.06$

The sequence is geometric: $t_n=ar^{n-1}$, or $t_n=10~500(1.05)^{n-1}$.

Writing an Investment as a Sequence Continued

Year ($n$)	Accumulated Value ($$$), $t_n$
$0$	$10~000$
$1$	$10~500$
$2$	$11~025$
$3$	$11~576.25$
$4$	$12~155.06$

Another representation: $r=1.05$

$\begin{align*} t_1 &= 1.05t_0 \\ &\class{timed add1-cover remove2-cover}{\;= 1.05(10~000)} \\ & \end{align*}$$\begin{align*} t_2 &= 1.05t_1 \\ &\class{timed add3-cover remove4-cover}{\;= 1.05(10~000(1.05))} \\ &\class{timed add3-cover remove5-cover}{\;= 10~000(1.05)^2 } \end{align*}$$\begin{align*} t_3 &= 1.05t_2 \\ &\class{timed add6-cover remove7-cover}{\;= 1.05(10~000(1.05)^2) }\\ &\class{timed add6-cover remove7-cover}{\;= 10~000(1.05)^3} \end{align*}$$\begin{align*} t_4 &= 1.05t_3 \\ &= 1.05(10~000(1.05)^3) \\ &= 10~000(1.05)^4 \end{align*}$

Formula for Compound Interest

The formula $A=P(1+i)^n$ gives the accumulated value of an investment or loan, $A$, after compound interest is applied to the principal, $P$.

$i$ represents the interest rate per compounding period.
$n$ represents the number of compounding periods.

Formula for Compound Interest Continued

Compound Interest: $A=P(1+i)^n$

In our example,

$P=10~000$.
$n$ is measured in years.
Interest rate is $5\%$ per year, so $i=0.05$ (Note that $i \ne 5$).
$A=10~000(1.05)^n$ represents the accumulated value after $n$ years.

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Alternate Representations of Compound Interest

We saw earlier that the accumulated value under compound interest can be represented by a geometric sequence.

It can also be represented by an exponential function: $f(x)=P(1+i)^x$.

$P$ represents the principal invested, $i$ represents the interest rate per compounding period, and $x$ represents the number of compounding periods.

When compound interest is earned, $i \gt 0$. This means that $f(x)=P(1+i)^x$ is an exponential function with a base, $1+i$, that is greater than $1$.

Therefore, compound interest is an example of exponential growth.

Graphing an Investment

Consider an investment of $$10~000$ at an annual interest rate of $5\%$, compounded annually.

Table of $A=10~000(1.05)^n$:

Year, $n$	Accumulated Value ($$$), $A$
$0$	$10~000$
$1$	$10~500$
$2$	$11~025$
$3$	$11~576.25$
$4$	$12~155.06$
$\ldots$	$\ldots$

Compound Interest as an Exponential Function

Graph of $f(x)=10~000(1.05)^x$:

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Example 1

Identify the amount invested or borrowed, the interest rate per compounding period, and the number of compounding periods for:

$A=500(1.01)^{12}$, which represents the accumulated value of an investment for which interest is compounded monthly.
$f(x)=2000(1.035)^x$, which represents the accumulated value of a loan for which interest is compounded yearly.

Solution — Part A

From $A=500(1.01)^{12}$, we can identify that:

The principal invested is $P=500$.
$1+i=1.01$, so $i=0.01$
- The interest rate is $0.01=1\%$ per month.
The number of compounding periods is $n=12$.

Interest is compounded monthly. So, the expression $500(1.01)^{12}$ represents the accumulated value of an investment of $$500$ invested at an interest rate of $1\%$ per month for a total of $12$ months.

Solution — Part B

From $f(x)=2000(1.035)^x$, we can identify that:

The principal amount borrowed is $P=2000$.
$i=0.035$
- The interest rate is $0.035=3.5\%$ per year.
The number of compounding periods is not specified.

Interest is compounded yearly. So, this function represents the accumulated balance of a loan of $$2000$ borrowed at an interest rate of $3.5\%$ per year. $x$ represents the number of years since the loan was issued.

Check Your Understanding 1

Example 2

On Xia's fifth birthday, she is given $$150$. Her parents deposit this money in a bank account with an interest rate of $3\%$ per year, compounded annually. If Xia does not deposit to or withdraw any money from this account, how much money will there be in the account on her $25$^th birthday?

Solution

We are asked to find the accumulated value of an investment with compound interest. We are given the following:

The principal invested is $P=150$.
The interest rate is $i=0.03$ per year.
The number of compounding periods is $n=20$, since there are $20$ years between her fifth and $25$^thbirthdays.

This is enough information to calculate $A$.

$\begin{align*} A &= P(1+i)^n \\ &= 150(1.03)^{20} \\ &= 150(1.8061 \dots) \\ &= 270.916685 \dots \end{align*}$

Therefore, Xia will have $$270.92$ in her account on her $25$^thbirthday.

Example 3

Reggie's credit card charges an interest rate of $1.6\%$ monthly, compounded monthly, if he does not pay his bill. This month, the balance on Reggie's credit card is $$625$. After six months, if he has not paid any of the balance:

How much money will he owe?
How much interest will he be charged?

Solution — Part A

Here, we are asked to find the accumulated balance on a loan with compound interest. We are given the following information:

The principal borrowed is $P=625$.
The interest rate is $i=0.016$.
The number of compounding periods is $n=6$.

$\begin{align*} A &= P(1+i)^n \\ &= 625(1.016)^{6} \\ &= 625(1.0999 \dots) \\ &= 687.4518 \dots \end{align*}$

Therefore, after six months of not paying his credit card bill, Reggie will owe $$687.45$.

Solution — Part B

The amount of interest is the difference between the accumulated value of the loan and the principal amount borrowed.

$687.45-625=62.45$

So, Reggie will be charged a total of $$62.45$ in interest if he does not pay his credit card bill for six months.

Year (\(n\))	Interest (\($\))	Accumulated Value (\($\))
\(0\)		\(10~000\)
\(1\)	\(10~000(0.05)=500\)	\(10~500\)
\(2\)	\(10~500(0.05)=525\)	\(11~025\)
\(3\)	\(11~025(0.05)=551.25\)	\(11~576.25\)
\(4\)	\(11~576.25(0.05)\approx578.81\)	\(12~155.06\)

Year (\(n\))	Accumulated Value (\($\)), \(t_n\)
\(0\)	\(10~000\)
\(1\)	\(10~500\)
\(2\)	\(11~025\)
\(3\)	\(11~576.25\)
\(4\)	\(12~155.06\)

Year (\(n\))	Accumulated Value (\($\)), \(t_n\)
\(0\)	\(10~000\)
\(1\)	\(10~500\)
\(2\)	\(11~025\)
\(3\)	\(11~576.25\)
\(4\)	\(12~155.06\)

Year (\(n\))	Accumulated Value (\($\)), \(t_n\)
\(0\)	\(10~000\)
\(1\)	\(10~500\)
\(2\)	\(11~025\)
\(3\)	\(11~576.25\)
\(4\)	\(12~155.06\)

Year, \(n\)	Accumulated Value (\($\)), \(A\)
\(0\)	\(10~000\)
\(1\)	\(10~500\)
\(2\)	\(11~025\)
\(3\)	\(11~576.25\)
\(4\)	\(12~155.06\)
\(\ldots\)	\(\ldots\)

Introduction to Compound Interest

Explore This 1

Explore This Summary

Slide Notes

Glossary

All Slides

Navigation

Try This Revisited

Solution

Try This Revisited Continued

Solution

Writing an Investment as a Sequence

Writing an Investment as a Sequence Continued

Writing an Investment as a Sequence Continued

Formula for Compound Interest

Formula for Compound Interest Continued

Alternate Representations of Compound Interest

Slide Notes

Glossary

All Slides

Navigation

Graphing an Investment

Compound Interest as an Exponential Function

Example 1

Check Your Understanding 1

Example 2

Example 3

Check Your Understanding 2