5.2: Compound Interest
- Page ID
- 139276
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With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding.
Suppose that we deposit $1,000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
The \(3 \%\) interest is an annual percentage rate (APR) - the total interest to be paid during the year. Since our interest is being paid monthly, each month we will earn \(\frac{1}{12}\) of the \(3 \%\) annual interest, or \(\frac{3 \%}{12}=0.25 \%\) per month.
In the first month,
\[
\begin{array}{l}
P_0=\$ 1,000 \\
r=0.0025(0.25 \%) \\
I=\$ 1,000(0.0025)=\$ 2.50 \\
A=\$ 1,000+\$ 2.50=\$ 1,002.50
\end{array}
\]
In the first month, we will earn \(\$ 2.50\) in interest, raising our account balance to \(\$ 1002.50\).
In the second month,
\[
\begin{array}{l}
P_0=\$ 1,002.50 \\
I=\$ 1,002.50(0.0025)=\$ 2.51 \text { (rounded) } \\
A=\$ 1,002.50+\$ 2.51=\$ 1,005.01
\end{array}
\]
Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original \(\$ 1000\) we deposited, but we also earned interest on the \(\$ 2.50\) of interest we earned the first month. This is the key advantage that compounding of interest gives us.
Calculating out a few more months (rounding to the nearest cent):
|
Month |
Starting balance |
Interest earned |
Ending Balance |
|---|---|---|---|
|
1 |
1000.00 |
2.50 |
1002.50 |
|
2 |
1002.50 |
2.51 |
1005.01 |
|
3 |
1005.01 |
2.51 |
1007.52 |
|
4 |
1007.52 |
2.52 |
1010.04 |
|
5 |
1010.04 |
2.53 |
1012.57 |
|
6 |
1012.57 |
2.53 |
1015.10 |
|
7 |
1015.10 |
2.54 |
1017.64 |
|
8 |
1017.64 |
2.54 |
1020.18 |
|
9 |
1020.18 |
2.55 |
1022.73 |
|
10 |
1022.73 |
2.56 |
1025.29 |
|
11 |
1025.29 |
2.56 |
1027.85 |
|
12 |
1027.85 |
2.57 |
1030.42 |
The standard formula for compound interest is as follows:
\[
A=P_0\left(1+\frac{r}{n}\right)^{n t}
\]
\(A\) is the balance in the account after \(t\) years
\(t\) is the number of years we plan to leave the money in the account
\(P_0\) is the balance in the account at the beginning (starting amount, or principal).
\(r\) is the annual interest rate (APR) in decimal form (Example: \(5 \%=0.05\) )
\(n\) is the number of compounding periods in one year.
- If the compounding is done annually (once a year), \(n=1\).
- If the compounding is done quarterly, \(n=4\).
- If the compounding is done monthly, \(n=12\).
- If the compounding is done daily, \(n=365\).
If you invest \(\$ 3,000\) in an investment account paying \(3 \%\) interest compounded quarterly, how much will the account be worth in 10 years?
Solution
Since we are starting with \(\$ 3,000, P_0=3,000\)
Our interest rate is \(3 \%\), so \(r=0.03\)
Since we are compounding quarterly, we are compounding 4 times per year, so \(n=4\) We want to know the value of the account in 10 years, so we are looking for the ending value, \(\mathrm{A}\), when \(t=10\).
\[
A=3,000\left(1+\frac{0.03}{4}\right)^{4(10)}=\$ 4,045.05
\]
The account will be worth \(\$ 4,045.05\) in 10 years.
If you invest $3,000 in an investment account paying 3% interest compounded weekly, how much will the account be worth in 10 years?
- Answer
-
Add texts here. Do not delete this text first.
A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3,000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years?
Solution
In this example,
\[
\begin{array}{ll}
P_0=\$ 3,000 & \text { the initial deposit } \\
r=0.06 & 6 \% \text { annual rate } \\
n=12 & 12 \text { months in } 1 \text { year } \\
t=20 & \text { since we're looking for how much we'll have after } 20 \; years
\end{array}
\]
years
So \(A=3,000\left(1+\frac{0.06}{12}\right)^{12^{* 20}}=\$ 9,930.61\) (round your answer to the nearest penny)
Let us compare the amount of money earned from compounding against the amount you would earn from simple interest
|
Years |
Simple Interest |
6% compounded |
|
5 |
$3,900 |
$4,046.55 |
|
10 |
$4,800 |
$5,458.19 |
|
15 |
$5,700 |
$7,362.28 |
|
20 |
$6,600 |
$9,930.61 |
|
25 |
$7,500 |
$13,394.91 |
|
30 |
$8,400 |
$18,067.73 |
|
35 |
$9,300 |
$24,370.65 |
As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes the account will earn 6% compounded semi-annually (twice a year), how much will Lily need to invest in the account now?
Solution
Since the account is earning \(6 \%, r=0.06\)
Since interest is compounded twice a year, \(n=2\)
In this problem, we don't know how much we are starting with, so we will be solving for \(P_0\), the initial amount needed. We do know we want the end amount, \(A\), to be \(\$ 40,000\), so we will be looking for the value of \(P_0\) so that \(A=40,000\).
\[
\begin{array}{l}
40,000=P_0\left(1+\frac{0.06}{2}\right)^{2(18)} \\
40,000=P_0(2.898278328) \\
P_0=\frac{40,000}{2.898278328} \approx \$ 13,801.30
\end{array}
\]
Lily will need to invest \(\$ 13,801.30\) to have \(\$ 40,000\) in 18 years.
It is important to be very careful about rounding when performing calculations. If possible, enter the entire calculation in one step into your calculator to avoid rounding error. If this is not possible, you want to keep as many decimals during calculations as you can. Try calculating example 2 using 2.898 instead of 2.898278328 , and compare your answer with the example.
Over several compounding periods rounding can be manipulated to favor bank or the customer depending on the conditions of the loan. Rounding down will favor the bank, rounding down to the dollar amount in the account also. Rounding compounds also.
If Lily wants to set up a 529 account for her new granddaughter, wants the account to
grow to $40,000 over 18 years, and she believes the account will earn 6% compounded
daily, how much will Lily need to invest in the account now?
- Answer
-
Lily will need to invest $13,693.20 to have $40,000 in 18 years.