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Compound Interest Explained: Formula, Examples and the Rule of 72

Compound interest means you earn interest on your interest, so growth accelerates over time. See the formula, worked examples, APR vs APY and the Rule of 72.

By TradingDots Editorial 8 min read

Glass jar filled with coins with a small green plant growing from the top, symbolizing growing savings
In this guide
  1. What Is Compound Interest?
  2. Compound Interest vs Simple Interest
  3. The Compound Interest Formula
  4. How Compounding Frequency Changes the Result
  5. Compound Interest vs APY (and APR)
  6. The Rule of 72: Estimating Doubling Time
  7. Compound Interest on Debt: Why Credit Card Balances Grow
  8. Inflation-Adjusted Returns: What Your Money Is Really Worth
  9. Practical Ways to Put Compound Interest to Work
  10. Bottom Line
  11. Sources

Compound interest explained simply: you earn interest not only on the money you put in but also on the interest that money has already earned, so growth speeds up over time. Leave $1,000 at 5% interest compounded once a year and you have $1,050 after year one, $1,102.50 after year two, and $4,321.94 after 30 years, compared with $2,500 under simple interest.

The same math works against you when you borrow. Credit card balances compound too, which is why a balance that seems manageable can take years to clear.

This guide walks through the compound interest formula with step-by-step examples, how compounding frequency changes the result, the difference between APR and APY, the Rule of 72 and how accurate it is, what compounding does to debt, and how to adjust returns for inflation. All examples use hypothetical round numbers and assume a constant rate, which real investments do not deliver.

What Is Compound Interest?

Interest is the price of using money: a bank pays you for holding your deposits, and you pay a lender for borrowing. Compound interest means interest is added to the balance at regular intervals, and each new calculation uses that larger balance. The interval is the compounding period: daily, monthly, quarterly or annually.

Three things drive how much compounding does for you (or against you):

  • The rate. Higher rates compound faster.
  • Time. The longer money compounds, the more of the final balance comes from interest rather than deposits.
  • Frequency. More frequent compounding adds a little more, though far less than rate and time.

Strictly speaking, savings accounts and CDs pay compound interest. Stock and fund investments do not pay interest, but reinvested dividends and gains grow in the same compounding way, so the same math is used to model them.

Compound Interest vs Simple Interest

Simple interest is calculated only on the original amount, the principal. The formula is Interest = P × r × t. Many auto and personal loans, and a bond’s fixed coupon payments, work this way. Compound interest grows the base every period.

Years Simple interest, $1,000 at 5% Compound interest, $1,000 at 5% (annual) Difference
1 $1,050.00 $1,050.00 $0.00
5 $1,250.00 $1,276.28 $26.28
10 $1,500.00 $1,628.89 $128.89
20 $2,000.00 $2,653.30 $653.30
30 $2,500.00 $4,321.94 $1,821.94

The gap is tiny at first and large later. That is the core lesson of compounding: most of the benefit arrives in the final years, so time in the account matters more than almost anything else.

The Compound Interest Formula

The standard compound interest formula is:

A = P(1 + r/n)^(nt)

  • A = the final amount (principal plus interest)
  • P = the principal, or starting amount
  • r = the annual interest rate as a decimal (6% = 0.06)
  • n = the number of compounding periods per year (12 for monthly)
  • t = the number of years

Interest earned is simply A − P.

Example: a lump sum. Suppose you deposit $10,000 at 6% interest, compounded monthly, for 10 years, and add nothing else.

  1. Rate per period: r/n = 0.06 ÷ 12 = 0.005
  2. Number of periods: nt = 12 × 10 = 120
  3. Growth factor: (1.005)^120 ≈ 1.81940
  4. Final amount: A = $10,000 × 1.81940 ≈ $18,193.97

You earn about $8,193.97 in interest. With simple interest at the same 6%, you would earn $10,000 × 0.06 × 10 = $6,000, ending with $16,000. Compounding added roughly $2,194.

Adding monthly contributions

Most people save a little at a time rather than investing one lump sum. For regular deposits at the end of each period, the future value is:

FV = PMT × [((1 + r/n)^(nt) − 1) ÷ (r/n)]

where PMT is the amount you add each period. If you also start with a balance, add the lump-sum result from the first formula.

Example: monthly contributions. Suppose you invest $200 a month and earn a hypothetical 7% a year, compounded monthly, for 30 years.

  1. Rate per period: 0.07 ÷ 12 ≈ 0.0058333
  2. Periods: 12 × 30 = 360
  3. (1.0058333)^360 ≈ 8.1165, so the bracket is (8.1165 − 1) ÷ 0.0058333 ≈ 1,219.97
  4. FV ≈ $200 × 1,219.97 ≈ $243,994

You contributed $200 × 360 = $72,000. The other $171,994, about 70% of the total, is growth.

Now see what time does. The same $200 a month for 40 years grows to about $524,963 from $96,000 of contributions. Ten extra years of $200 deposits, just $24,000 more, roughly doubles the result. Starting with $5,000 already invested and adding $200 a month for 30 years gives about $284,577, because the $5,000 alone grows to about $40,582.

You can check any scenario with the SEC’s free compound interest calculator on Investor.gov. Remember that stock market returns vary year to year and can be negative; a constant 7% is a modeling assumption, not a promise.

How Compounding Frequency Changes the Result

The more often interest compounds, the sooner each period’s interest starts earning its own interest. Here is $10,000 at a 5% nominal annual rate for 10 years:

Compounding n per year Balance after 10 years Effective annual yield (APY)
Annually 1 $16,288.95 5.000%
Semiannually 2 $16,386.16 5.063%
Quarterly 4 $16,436.19 5.095%
Monthly 12 $16,470.09 5.116%
Daily 365 $16,486.65 5.127%
Continuously infinite $16,487.21 5.127%

Moving from annual to daily compounding adds about $198 over 10 years. Useful, but small next to the rate itself: the same $10,000 at 6% compounded annually reaches $17,908.48. When comparing accounts, focus on the APY, which already builds in the compounding frequency.

Continuous compounding uses a different formula, A = Pe^(rt), where e ≈ 2.71828. It is the mathematical limit of compounding more and more often and appears mostly in finance theory.

Compound Interest vs APY (and APR)

Two acronyms describe interest rates, and mixing them up causes real mistakes.

  • APR (annual percentage rate) is the yearly rate on a loan or credit card. For loans it can include certain fees, and it is usually quoted without the effect of compounding. In the US, lenders must disclose it under the Truth in Lending Act (Regulation Z).
  • APY (annual percentage yield) is the rate you actually earn on a deposit in one year with compounding included. In the US, banks and credit unions must show it on savings accounts and CDs under the Truth in Savings Act (Regulation DD).

To convert a nominal rate to APY: APY = (1 + r/n)^n − 1.

Example: A savings account advertises a 5% rate compounded monthly. APY = (1 + 0.05/12)^12 − 1 ≈ 0.05116, or 5.12%. A credit card with a 24% APR that compounds daily has an effective annual rate of (1 + 0.24/365)^365 − 1 ≈ 27.1%, so carrying a balance for a full year costs more than the headline APR suggests.

The rule of thumb: compare savings products by APY, compare loans by APR, and remember that compounding makes the true cost of revolving debt higher than its APR. The table below sums up the differences.

APR APY
Used for Loans, mortgages, credit cards Savings accounts, CDs, money market accounts
Includes compounding? Usually no Yes
You want it Lower Higher
US disclosure rule Truth in Lending Act / Regulation Z Truth in Savings Act / Regulation DD

For where to find competitive yields, see our comparison of high-yield vs regular savings accounts.

The Rule of 72: Estimating Doubling Time

The Rule of 72 is a mental shortcut: divide 72 by the annual rate (as a whole number) to estimate how many years it takes money to double.

  • At 6%: 72 ÷ 6 = 12 years to double.
  • At 8%: 72 ÷ 8 = 9 years. $10,000 becomes about $20,000 in 9 years, $40,000 in 18 and $80,000 in 27.
  • At 24% (a credit card APR): 72 ÷ 24 = 3 years for an unpaid balance to double, if no payments are made and no fees are added.

It also works in reverse. To double money in 10 years, you need roughly 72 ÷ 10 = 7.2% a year. And it applies to inflation: at 3% inflation, prices double in about 72 ÷ 3 = 24 years, meaning your money’s purchasing power halves.

How accurate is the Rule of 72?

The exact doubling time with annual compounding is ln(2) ÷ ln(1 + r). Here is how the shortcut compares:

Annual rate Rule of 72 estimate Exact doubling time Error
2% 36.0 years 35.0 years +1.0
4% 18.0 years 17.7 years +0.3
6% 12.0 years 11.9 years +0.1
8% 9.0 years 9.0 years 0.0
10% 7.2 years 7.3 years −0.1
12% 6.0 years 6.1 years −0.1
24% 3.0 years 3.2 years −0.2

The rule is most accurate around 8% and drifts at very low or very high rates. Some people use 70 or 69.3 for continuous or daily compounding, where they fit slightly better. For quick decisions, 72 is close enough, and it divides neatly by 2, 3, 4, 6, 8, 9 and 12.

Compound Interest on Debt: Why Credit Card Balances Grow

Credit cards usually charge interest daily. The Consumer Financial Protection Bureau (CFPB) explains that the daily periodic rate is the APR divided by 365, applied to your balance each day. Unpaid interest joins the balance, so tomorrow’s interest is charged on today’s interest. The CFPB reported that the average APR on cards that were assessed interest reached 22.8% in 2023; check the Federal Reserve’s G.19 consumer credit release for the latest figures.

Example: Suppose you owe $5,000 on a card with a 24% APR and stop using it. A 30-day billing cycle adds about $5,000 × (0.24 ÷ 365) × 30 ≈ $99 in interest. Using a simplified 2% monthly rate, here is how long payoff takes at different fixed payments:

Monthly payment Months to pay off Total interest paid
$100 Never (payment only covers interest) Unlimited
$150 56 About $3,322
$200 36 About $2,001
$250 26 About $1,449
$500 12 About $635

Going from $150 to $250 a month saves about $1,873 in interest and 30 months. Real results vary with daily compounding, fees and the issuer’s minimum payment formula.

Key point: Paying off a card charging 24% is the equivalent of earning a guaranteed 24% return on that money, before taxes. Few investments reliably beat that, which is why clearing high-interest debt usually comes before investing beyond any employer retirement match. Most cards also give a grace period: if you pay the full statement balance by the due date each month, you generally pay no interest on purchases.

For a structured plan, see how to pay off debt fast with the snowball or avalanche method.

Inflation-Adjusted Returns: What Your Money Is Really Worth

A balance that grows on paper may not grow in buying power. Inflation is the rise in the general price level, measured in the US by the Consumer Price Index (CPI) from the Bureau of Labor Statistics. The real return is what you earn after inflation:

Real return = (1 + nominal return) ÷ (1 + inflation rate) − 1

Example: Suppose an investment earns 7% a year while inflation runs at 3%. Real return = 1.07 ÷ 1.03 − 1 ≈ 3.88% (the quick approximation, 7% − 3% = 4%, is close). Over 30 years, $10,000 grows to $10,000 × 1.07^30 ≈ $76,123 in nominal dollars, but that is worth only about $31,361 in today’s purchasing power. Meanwhile, $100 of cash kept under a mattress for those 30 years would buy only about $41 of today’s goods.

Taxes and fees reduce returns further. A savings account paying less than the inflation rate is losing real value even though its balance rises. Learn more in our guide to how inflation affects your savings.

Practical Ways to Put Compound Interest to Work

  • Start early, even small. In the example above, 10 extra years turned $200 a month into about $525,000 instead of $244,000. If you are working with a tight budget, see how to start investing with little money.
  • Automate contributions on payday so they happen every month.
  • Reinvest dividends and interest instead of spending them.
  • Cut fees. Fees compound against you. $10,000 growing at 7% for 30 years reaches about $76,123; at 6% (the same return minus a 1% annual fee) it reaches about $57,435, roughly $18,688 less.
  • Use tax-advantaged accounts. Taxes paid each year interrupt compounding. In the US, accounts such as 401(k)s and IRAs let investments grow tax-deferred or tax-free; compare options in Roth IRA vs Traditional IRA. Other countries have their own versions, such as ISAs in the UK.
  • Kill high-interest debt first. Compounding at 20%+ on a card balance will outrun almost any savings rate.

Bottom Line

Compound interest rewards three things: a decent rate, regular contributions and, above all, time. Use A = P(1 + r/n)^(nt) or a free calculator to test your own numbers, compare savings by APY and loans by APR, and use the Rule of 72 for fast estimates. Treat high-interest debt as compounding in reverse and clear it quickly, and always judge growth after inflation and fees. More guides on saving and investing are in our personal finance hub.

Sources

Frequently asked questions

What is the compound interest formula?

The formula is A = P(1 + r/n)^(nt), where P is the starting amount, r the annual rate as a decimal, n the number of compounding periods per year and t the number of years. A is the ending balance, and A − P is the interest earned. For regular deposits, add PMT × [((1 + r/n)^(nt) − 1) ÷ (r/n)].

What is the difference between compound interest vs simple interest?

Simple interest is paid only on the original principal, so it grows by the same dollar amount each year. Compound interest is paid on the principal plus accumulated interest, so it grows faster each year. Over 30 years at 5%, $1,000 becomes $2,500 with simple interest and $4,321.94 with annual compounding.

Is APY the same as compound interest?

Not quite. Compound interest is the process; APY is a standardized rate that shows what that process earns in one year, including the compounding frequency. A 5% rate compounded monthly has an APY of about 5.12%, so APY lets you compare accounts that compound on different schedules.

How accurate is the Rule of 72?

It is very close for rates between about 6% and 10%; at 8% it matches the exact doubling time of 9.0 years. At 2% it overstates by about a year (36 vs 35 years), and at 24% it understates slightly (3.0 vs 3.2 years). It is meant for quick estimates, not precise planning.

How often is interest compounded on a savings account?

Many US banks compound interest daily and credit it monthly, but terms vary by account. The disclosed APY already accounts for the compounding schedule, so compare APYs rather than trying to compare compounding frequencies.

Does compound interest apply to credit cards?

Yes. Most credit cards calculate interest daily using a daily periodic rate (APR ÷ 365), and unpaid interest is added to the balance. That means you pay interest on interest. Paying the full statement balance each month usually avoids interest on purchases entirely.

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