Compound Interest
Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle and the interest that has accumulated in the past. The compounding period is the interval of time between the occasions when interest is added to the account.
If you invest $1000 at a simple interest rate of 5% annually, you will receive $50 dollars for every year your money remains invested. At the end of 1 year you will earn $50, after 2 yrs you’ll earn $100, after 3 years you’ll earn $150, etc… At the end of 10 years you will have earned $500 and you would have $1500.
Now, if you invest the same amount of money with compound interest, you will earn interest on the original principle plus any investment income that is reinvested.
Example: Invest $1000 at a rate of 5% interest compounded annually (once a year). The following table shows how your investment will grow.
Principle Interest Paid (principle * 5%) Annual Running Total
YR 1 1000 50 1050
YR 2 1050 52.5 1102.5
YR 3 1102.5 55.125 1157.625
YR 4 1157.625 57.88 1215.51
YR 5 1215.51 60.77 1276.29
YR 6 1276.29 63.81 1340.10
YR 7 1340.10 67.00 1407.11
YR 8 1407.11 70.36 1477.46
YR 9 1477.46 73.87 1551.33
YR 10 1551.33 77.57 1628.89
With compound interest you earn an additional $128.89 ($1628.89 - $1500).
Rather than using a table, there is a formula for calculating simple compound interest.
A = P (1 + r)t
P = the principle invested
r = annual interest rate as a percentage
t = the length of the term (investment or loan) A = the amount accumulated after n periods
In the above example, using the formula we get:
A = P (1 + r)t
A = 1000 (1 + 0.05)10 A = 1000 (1.628894627)
Try another example: Suppose you invest $1800 for 5 years at 6% interest that compounds annually. How much will you have at the end of 5 years?
A = P (1 + r)t
A = 1800 (1 + 0.06)5 A = 1800 (1.338225578) A = 2408.81
There are instances where interest compounds more than once a year. The basis of the formula remains the same but you must adjust the annual interest rate to the rate per period and be careful to include the correct number of periods (n). Now the formula looks like this:
A = P (1 + r/n)nt
P = the principle invested
r = annual interest rate as a percentage
n = the number of times per year interest is compounded t = the length of the term (investment or loan)
A = the amount accumulated after n periods
Example: How much money will you have at the end of 5 years if you invest $5000 at 8% annual interest compounded quarterly?
A = P (1 + r/n)nt where
r/n=0.08/4 = 0.02
nt= 5 years * 4 periods per year = 20 periods
A = 5000 (1 + 0.02)20 A = 5000 (1.485947396) A = 7429.74
Notice how the investment really adds up when you are earning interest on interest four times per year. The table below shows the different total amounts for the same
investment above with different compounding schedules.
Year 1 Year 5 Year 10
Simple 5400 7000 9000
Annual Compound 5400 7346.64 10794.62 Semiannually 5408 7401.22 10955.62 Quarterly 5412.16 7429.74 11010.20
Monthly 5415 7449.23 11098.20
Weekly 5416.10 7456.83 11120.87
Daily 5416.39 7458.80 11126.73
The difference in the compounding method over 1 year is miniscule but over 10 years the more compounding periods, the greater the increase. In this example the difference between annual and daily compounding interest was $332, or 6.6%.
This is the magic of compound interest, but remember, bank loans and mortgages also use compound interest so borrowing money has a much higher fee than a simple interest calculation.
Example: Calculate how much a 5-year loan of $20,000 with annual interest of 5%
compounded semiannually will cost you.
A = P (1 + r/n)nt
A = 20000 (1 + 0.05/2)10 A = 20000 (1.025)10 A = 20000 (1.280084544) A = 25601.69
Cost to you = A-P
25601.69 – 20000 = $5601.69
You can of course solve for any of the variables in the compound interest formula by algebraically rearranging the equation.
Example: What interest rate was paid on a 5-year loan with a principle amount of
$10,000, a cost to borrow of 3439, and a semiannual compound period.
A = P (1 + r/n)nt
10000+3439 = 10000 (1 + x/2)10 13439 = 10000 (1+ x/2)10
13439/10000 = (1 + x/2)10
101.3439 = 10(1+ x/2)∧10 1.2999 ≅ 1+ x/2
0.2999 ≅ x/2 0.5998 ≅= x
The interest rate is 6%
Compound interest is a way of life in our society. Understanding how it works and how it can be used effectively to grow your investments is a critical lesson. Investing early and accumulating compound interest over the long term is an excellent and safe way to make your money work for you. Conversely, when borrowing money try to limit the length of the borrowing term and pay as little interest overall as possible.
Student Worksheet – Compound Interest
1. How many compounding periods are there in a 12-year investment that compounds quarterly?
a. 6 b. 12 c. 24 d. 48
2. Compound interest is calculated a. on a semiannual basis
b. more than once a year
c. on the principle and interest earned d. all of the above
3. The formula for Compound Interest is:
a. A = P (1 + r/n)nt b. A = P (1 + r)t c. A = P (1 + r/t)nt d. A = P (1 + n/r)nt
4. An investment of $1425 earns 6.75% and compounds annually. What is the total amount after 8 years?
5. If you earn $3500 over 10 years on an investment that pays 5.3% compounded annually, what was the principle amount you started with?
6. Fill in the following table for an investment earning 4.5% annually and compounded semiannually.
Principle Interest Paid Per Year Annual Running Total YR 1 16,250
YR 2 YR 3 YR 4 YR 5
7. How much interest does a $10,000 investment earn at 5.6% over 18 years compounded quarterly?
8. How much does the same investment as above earn if the interest is compounded semiannually? Annually?
9. Calculate the amount of interest paid on a 7-year loan of $13,450 at 4.8%
compounded semiannually.
10.A $175,000 mortgage at 3.2% compounded monthly was paid off in 19 years. What
ANSWERS - Student Worksheet – Compound Interest
1. How many compounding periods are there in a 12-year investment that compounds quarterly?
a. 6 b. 12 c. 24 d. 48
2. Compound interest is calculated a. on a semiannual basis
b. more than once a year
c. on the principle and interest earned d. all of the above
3. The formula for Compound Interest is:
a. A = P (1 + r/n)nt b. A = P (1 + r)t c. A = P (1 + r/t)nt d. A = P (1 + n/r)nt
4. An investment of $1425 earns 6.75% and compounds annually. What is the total amount after 8 years? $2403.02
5. If you earn $3500 over 10 years on an investment that pays 5.3% compounded annually, what was the principle amount you started with? ~ $5,177
6. Fill in the following table for an investment earning 4.5% annually and compounded semiannually.
Principle Interest Paid Per Year Annual Running Total YR 1 16,250
YR 2 YR 3 YR 4 YR 5 Answer:
Principle Interest Paid Per Year Annual Running Total
YR 1 16,250 739.47 16989.47
YR 2 16989.47 773.12 17762.59
YR 3 17762.59 808.31 18570.90
YR 4 18570.90 845.09 19415.99
YR 5 19415.99 883.56 20299.55
7. How much interest does a $10,000 investment earn at 5.6% over 18 years compounded quarterly? $17,210.26?
8. How much does the same investment as above earn if the interest is compounded semiannually? 17,024.15 Annually? 16,665.55
9. Calculate the amount of interest paid on a 7-year loan of $13,450 at 4.8%
compounded semiannually. $5296.56
10.A $175,000 mortgage at 3.2% compounded monthly was paid off in 19 years. What was the amount paid in interest? 146,171.98
