The Basic Formula for Compound Interest
Compound interest is the interest you earn on your original money plus the interest that money has already made. The formula is:
A = P(1 + r/n)^(nt)
In this formula, A is your final amount, P is the principal (the money you start with), r is the annual interest rate as a decimal, n is how many times per year the interest compounds, and t is the number of years. The caret symbol (^) means you raise the number to a power — multiply it by itself that many times.
For example, if you put $1,000 in a savings account that pays 5 percent annual interest compounded once per year, after one year you would have $1,050. After two years, you would have $1,102.50, because you earn 5 percent on the $1,050, not just the original $1,000.
Key Takeaways
- The compound interest formula is A = P(1 + r/n)^(nt), where A is your final amount, P is what you start with, r is the annual rate as a decimal, n is how often interest compounds per year, and t is years.
- Converting your interest rate to a decimal means dividing the percentage by 100, so 5 percent becomes 0.05.
- Compounding more often — daily instead of yearly — means you earn interest on your interest more frequently, so your money grows faster.
- A scientific calculator or spreadsheet saves time and reduces arithmetic errors when you are working with large numbers or many years.
Converting Your Interest Rate to Decimal Form
The formula requires your interest rate as a decimal, not a percentage. To convert, divide the percentage by 100. If your rate is 5 percent, divide 5 by 100 to get 0.05. If your rate is 2.5 percent, divide 2.5 by 100 to get 0.025.
This step trips up many people because they forget to do it, then get a wildly wrong answer. Write down the decimal version before you start the rest of the calculation. If your bank statement says "5% APY," write "r = 0.05" on your paper.
Understanding Compounding Frequency
The n in the formula is how many times per year your interest compounds. Common frequencies are:
- Annually: n = 1 (interest compounds once per year)
- Semi-annually: n = 2 (twice per year)
- Quarterly: n = 4 (four times per year)
- Monthly: n = 12 (twelve times per year)
- Daily: n = 365 (every day)
The more often interest compounds, the more you earn, because you earn interest on your interest more frequently. A savings account that compounds daily will grow faster than one that compounds annually, even if both have the same interest rate. Your bank or investment statement will tell you the compounding frequency.
Working Through a Complete Example by Hand
Suppose you invest $2,000 at 4 percent annual interest, compounded quarterly, for 3 years. Here is how to solve it step by step.
First, convert your values: P = 2000, r = 0.04 (because 4 ÷ 100 = 0.04), n = 4 (quarterly), t = 3.
Next, calculate r/n: 0.04 ÷ 4 = 0.01. Now add 1: 1 + 0.01 = 1.01.
Then calculate nt: 4 × 3 = 12. This is your exponent — you will raise 1.01 to the 12th power.
Now raise 1.01 to the 12th power: 1.01^12 = 1.1268 (rounded). Multiply by your principal: 2000 × 1.1268 = $2,253.65. That is your final amount after 3 years.
Using a Scientific Calculator
A scientific calculator makes this much faster. Enter 1.01, then press the button labeled x^y or ^ (the exponent button), then enter 12, then press equals. You will get 1.1268 when ready instead of multiplying 1.01 by itself twelve times.
Most phones have a scientific calculator app. On iPhone, open the Calculator app and turn your phone sideways to landscape mode — the scientific buttons appear. On Android, open the Calculator app and swipe left to reveal the scientific view. Enter your base number, tap the ^ button, enter your exponent, and tap equals.
If you are working with many calculations or large numbers, a spreadsheet is even faster. In Excel, Google Sheets, or LibreOffice Calc, you can type the formula directly: =P*(1+r/n)^(n*t), replacing P, r, n, and t with your actual numbers or cell references.
The Difference Between straightforward and Compound Interest
straightforward interest pays you the same amount every year based only on your original principal. The formula is A = P(1 + rt). With $2,000 at 4 percent straightforward interest for 3 years, you would earn $240 total (4 percent of $2,000 is $80 per year, times 3 years), giving you $2,240.
Compound interest, as shown above, gave you $2,253.65 — about $13 more. The difference grows larger over time and with higher interest rates. After 10 years at 4 percent, straightforward interest would give you $2,800, but compound interest (compounded quarterly) would give you about $2,953. Compound interest is why banks advertise it — it genuinely pays you more.
Frequently Asked Questions
What does APY mean, and how does it relate to compound interest?
APY stands for Annual Percentage Yield. It is the interest rate your bank advertises, and it already accounts for compounding. You use the APY number directly as your r value in the formula. If a savings account says "4.5% APY compounded daily," you use r = 0.045 in the formula.
Does compound interest work the same way for loans and credit cards?
Yes, the math is identical, but it works against you instead of for you. A credit card balance compounds interest on the amount you owe, so your debt grows faster the longer you carry a balance. The formula is the same; you are just solving for how much you owe instead of how much you have earned.
What happens if interest compounds continuously?
Continuous compounding uses a different formula: A = Pe^(rt), where e is a mathematical constant approximately equal to 2.71828. This is rare in everyday banking but common in advanced finance. Most savings accounts and loans use daily, monthly, or quarterly compounding instead.
Can I use this formula to figure out how long it takes to double my money?
Yes, but you have to rearrange the formula to solve for t instead of A. A quick shortcut called the Rule of 72 says: divide 72 by your interest rate (as a percentage). At 6 percent interest, your money doubles in roughly 72 ÷ 6 = 12 years. This is an approximation, but it is close enough for planning.