SECTION 3.2: Compound Interest
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 [latex]\frac{1}{12}[/latex] of the 3% annual interest, or [latex]\frac{3\%}{12}[/latex] = 0.25% per month.
In the first month,
P = $1,000
r = 0.0025 (0.25%)
I = $1,000 (0.0025) = $2.50
A = $1,000 + $2.50 = $1,002.50
In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50.
In the second month,
P = $1,002.50
I = $1,002.50 (0.0025) = $2.51 (rounded)
A = $1,002.50 + $2.51 = $1,005.01
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 the ending balance for the first 12 months of the starting balance (rounding to the nearest cent) yields the table below.
| 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 |
This process can be generalized with the formula below.
Compound Interest over Time
A = P(1 + [latex]\frac{r}{n}[/latex])nt
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 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.
Compounding Terminology
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.
The most important thing to remember about using this formula is that it assumes that we put money in the account once and let it sit there earning interest.
Example 3
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, P0 = 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, A, when t = 10.
A = 3,000(1 + [latex]\frac{0.03}{4}[/latex])4(10) = $4,045.05
Answer: The account will be worth $4,045.05 in 10 years.
Example 4
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,
P = $3,000, the initial deposit
r = 0.06, 6% annual rate
n = 12, 12 months in 1 year
t = 20, after 20 years
A = 3,000(1 + [latex]\frac{0.06}{12}[/latex])12(20) = $9,930.61* (round to the nearest cent)
Answer: You will have $9930.61 in the account after 20 years.
Let us compare the amount of money earned from compounding against the amount you would earn from simple interest of 6% per year versus 6% compounded monthly.
| Years | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
| Simple | $3,900.00 | $4,800.00 | $5,700.00 | $6,600.00 | $7,500.00 | $8,400.00 | $9,300.00 |
| Compound | $4,046.55 | $5,458.19 | $7,362.28 | $9,930.61 | $13,394.91 | $18,067.73 | $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.
Example 5
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 P0, 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 P0 so that A = 40,000.
40,000 = P(1 + [latex]\frac{0.06}{2}[/latex])2(18)
40,000 = P(2.898278328)
P = [latex]\frac{40,000}{2.898278328}[/latex] ≈ $13,801.30
Answer: Lily will need to invest $13,801.30 to have $40,000 in 18 years.
A Note on Rounding: 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 5 using 2.898 instead of 2.898278328 and compare your answer with the example:
Section 3.2 You Try ProblemS:
A): If you invest $3,000 in an investment account paying 3% interest compounded weekly, how much will the account be worth in 10 years?
B) 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?
3.2 – Answers to You Try Problems
a) The account will be worth $4,049.23.
b) Lily will need to invest $13,585.03 to have $40,000 in 18 years.