Compound Interest and APR/APY: A Deeper Dive
- erin2678
- 6 days ago
- 7 min read
Updated: 1 day ago
Compound interest happens when timelines are longer. Interest is applied at predetermined periods. Compound Interest is simply a bunch of Simple Interest formulae lined up together one after the other.
What do I need to understand before reading this blog?
Simple Interest
Arithmetic: Adding and multiplying
Pre-Algebra: Applying Exponents
Algebra: Concept of "n" and "n-1"
Algebra: Applying values to variables, Factoring, and the Distributive Property over Addition
Click here if you need help with these prerequisites
Check out Simple Interest: A Deeper Dive to learn more about Simple Interest
Click here for help with Pre-Algebra and Algebra.
Check out moment.of.math for Algebra practice.
To learn more about this series, please see the first blog entry: Introduction to Basic Financial Concepts
See the below example showing how to use the moment.of.math scratchpad.
What is Compound Interest?
Compound interest is the next concept after Simple Interest to learn about financial calculations. Compound interest happens when timelines are longer, and this allows money to grow increasingly fast. When broken down, Compound Interest simply strings a bunch of Simple Interest formulae together one after the other.
Period | Present Value (PV) | Interest | Future Value (FV) | ||
Month 1: | Initial Money or Investment (PV) | + | Month 1 Interest | = | 1st Period's FV |
Month 2: | 1st Period's FV becomes 2nd Period's PV | + | Month 2 Interest | = | 2nd Period's FV |
Month 3: | 2nd Period's FV becomes 3rd Period's PV | + | Month 3 Interest | = | 3rd Period's FV |
Month "n": | "n-1" Period's FV becomes "n" Period's PV | + | Month "n" Interest | = | "n" Period's FV |
At the end of the first month, or period, apply Simple Interest to determine the 1st month's Future Value (FV or F). The total amount owed at the end of the first period (initial Present Value plus the interest incurred) becomes the 2nd period’s Present Value (PV or P) and so on.
This continues through the lifecycle of the transaction until the last month "n" FV, which is calculated by adding FV from the "n-1" period and the interest applied during the "n" period. Recall from Algebra that a series "1, 2, 3" that ends some time in the future is shown as "1, 2, 3...n-2, n-1, n".
Interest is compounded with savings accounts that build value over time and loans that have monthly payments.
What is APR and APY?
Because compound interest happens over many periods of time, Interest usually described using APR and APY
APR: The Annual Percentage Rate you pay, such as in the interest rate of a loan.
APY: The Annual Percentage Yield you gain, such as the interest rate in an interest bearing savings account.
APR and APY are the same, and the only difference is whether you are gaining money, or spending money. It is important to note that APR and APY are the annual rate.
To understand the actual Interest rate applied at each period, you divide the APR or the APY (r) by the number of payment periods per year (n). For example if your loan is compounded Monthly, divide the APR by 12 to understand the interest rate per month (r/12 in this example).
The formula for calculating Compound Interest is:
F=P*(1+r/n)^(n*t)
In this formula,
F = the Future Value of the loan or investment
P = the Present Value of the loan or investment
r = Annual Rate (APR or APY)
n = the Number of periods that Interest is applied in one year
t = the Time (number of years of the loan or investment at the time you want to know "F")
Click here to understand how APR and APY work using the moment.of.math scratchpad
APR and APY use the same very simple equations. To perform Compound interest, one must calculate the interest applied at each period. If monthly compounding, the monthly rate is the APY/12. For quarterly the rate is APY/4.
The example below shows how to calculate the monthly Interest Rate for a 3% APY savings account that compounds monthly. Note, in this case we are using the variable "y" as the monthly interest rate (yield) which is calculated from the APY (r=.03), and the number of periods in a year (n=12).
Note above how the green dots track your progress and shows you are on the right track. The double check mark confirms that your answer and all your math is correct.
*Note: You can also duplicate a line by holding down the “Shift” key and hit the “Enter” key.
**Note: You can also right-click on the “r” and select the value “.03” from the “Substitute” menu at the bottom of the list. Then right-click on the “n” and select the value “12” from the “Substitute” menu at the bottom of the list.
Click here to learn how to derive Compound Interest Formula “F=P*(1+r/n)^(n*t)” using the moment.of.math scratchpad
Problem: David has $1000 which he wants to let grow in a savings account at his local Credit Union. They are offering 3% APY on their new savings accounts for the first year. How much money would David have in his savings account after the 1 month? After 2 months? After 3 months?
See how the Compound Interest formula "F=P*(1+r/n)^(n*t)" is derived from the simple interest formula "F=P*(1+R)" using the moment.of.math scratchpad examples below. Remember, the monthly interest rate is the APR divided by the number of periods in a year (R=r/n)
Note above how the green dots track your progress and shows you are on the right track. The double check mark confirms that your answer and all your math is correct.
Now you will use the first month's Future Value ($1002.50) as the Present Value of the 2nd month's calculation.
Note above how the green dots track your progress and shows you are on the right track. The double check mark confirms that your answer and all your math is correct.
Now you will use the second month's Future Value ($1005.006) as the Present Value of the 3rd month's calculation.
Note above how the green dots track your progress and shows you are on the right track. The double check mark confirms that your answer and all your math is correct.
Do you notice a pattern in the compound interest? The Future Value is a function of the Present Value, the APR, the number of times the interest is compounded in a year, and how much time has progressed into the future in the form of number of periods, or nper.
F=P*(1+r/n)^(nper)
The number of periods (nper)can be calculated by multiplying the number of periods per year “n” by the amount of time in years “t”. In the case of 3 periods, t=3/12 and n=12. Therefore t*n=3. The final formula for compound interest therefore is:
F=P*(1+r/n)^(n*t)
*Note: You can also duplicate a line by holding down the “Shift” key and hit the “Enter” key.
**Note: You can also right-click on the "P" and select the value "1000" from the "Substitute menu, then right-click on the “r” and select the value “.03” from the “Substitute” menu at the bottom of the list, and finally, right-click on the “n” and select the value “12” from the “Substitute” menu at the bottom of the list.
Compound Interest: Worked Example Problem
Problem: Fred has $5,000 and he wants to invest it in a CD. His Bank is offering 3.5% APY on their new 5 year CD compounded monthly. Using the Compound Interest formula
"F=P*(1+r/n)^(n*t)", determine how much total Interest Fred would earn after 6 months.
Note above how the green dots track your progress and show you are on the right track and the the double check mark confirms the final solution matches all setup constraints.
*Note: You can also duplicate a line by holding down the “Shift” key and hit the “Enter” key.
**Note: You can also right-click on the “P” and select the value “5000” from the “Substitute” menu at the bottom of the list, then right-click on the “r” and select the value “.035” from the “Substitute” menu at the bottom of the list, then right-click on the “n” and select the value “12” from the “Substitute” menu at the bottom of the list, and finally, right-click on the “t” and select the value “.5” from the “Substitute” menu at the bottom of the list.
Try the below Compound Interest problem yourself!
Problem: Lisa has $8,000 cash saved up and wants to invest it in a Bond. She finds a Bond paying for bridge construction in California that is offering 3.8% APY for a term up to 5 years compounded quarterly. Using the Compound Interest formula F=P*(1+r/n)^(n*t), determine how much total Interest Lisa would earn after 2.5 years of the Bond.
Click here to see the step by step to the answer
Setup:
"Let" statement assigns 8000 to "P"
"Let" statement assigns .038 to "r" (because 3.8% = 3.8/100)
"Let" statement assigns 4 to "n" (Interest is applied 4 times a year)
"Let" statement assigns 2.5 to "t" (time will be 6 months or 0.5 years. Notice that the 5 year term of the Bond is irrelevant in this analysis)
"Let" statement defines the formula "F=P*(1+r/n)^(n*t)"
Begin Work:
Recreate the math equation by clicking on the 5th line then right-click and select “Start from numbered equation …” then select the line for “F=P*(1+r/n)^(n*t)". This will pop that equation at the bottom of your work.*
Substitute the variables with their values:
Replace the "P" with 8000, “r” with “.038”, “n” with “4”, and "t" with 2".5" in the last line.**
Calculate the answer:
Click anywhere into the last line that contains only numbers, right-click, then select “Calculate Line” and the answer ~8793.327 pops up in the final line. This shows that Lisa earned $793.33 in interest from 2.5 years of this investment.
Remember that the green dots track your progress and shows you are on the right track. The double check mark confirms that your answer and all your math is correct.
*Note: You can also duplicate a line by holding down the “Shift” key and hit the “Enter” key.
**Note: You can also right-click on the “P” and select the value “8000” from the “Substitute” menu at the bottom of the list, then right-click on the “r” and select the value “.038” from the “Substitute” menu at the bottom of the list, then right-click on the "n" and select "4" from the "Substitute" menu at the bottom of the list, and finally right-click on the "t" and select "2.5" from the "Substitute" menu at the bottom of the list.
Catch your algebra mistakes before they cost you
The scratchpad in this guide runs on moment.of.math — a free Chrome extension that checks your algebra line by line, on any problem. It can generate new variations problems, so you can get as much practice as you need. On many homework sites, it automatically detects the math on the page, so you don't have to copy anything over. Try it on tonight's homework.
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