Complete Guide to Compound Interest

How compound interest works, the math behind it, and how compounding frequency, time, and rate affect your money. Includes worked examples and common scenarios.

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The compound interest formula

Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods. The formula is:

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

Where:

• A = the estimated future value of the investment
• P = the principal (starting amount)
• r = the annual interest rate (as a decimal)
• n = the number of times interest compounds per year
• t = the number of years

Simple interest pays only on the original principal. Compound interest pays interest on interest. That difference grows dramatically over time.

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Worked example: $10,000 at 5% for 20 years

Start with $10,000, an annual rate of 5%, compounded monthly, and no additional contributions.

• P = 10,000
• r = 0.05
• n = 12
• t = 20

A = 10,000 x (1 + 0.05/12)^(12 x 20)

A = 10,000 x (1.004167)^240

A = 10,000 x 2.7126

Estimated result: approximately $27,126.

The original $10,000 earned approximately $17,126 in interest over 20 years. Compare that to simple interest at 5%, which would produce only $10,000 in interest ($20,000 total). Compounding added roughly $7,126 on top of what simple interest would have returned.

How compounding frequency changes results

The more frequently interest compounds, the more you earn. Here is $10,000 at 5% for 20 years under different compounding schedules:

Compounding frequency	n value	Estimated future value
Annually	1	$26,533
Quarterly	4	$26,851
Monthly	12	$27,126
Daily	365	$27,181

The difference between annual and daily compounding on this amount is approximately $648. The gap is real but modest at 5%. At higher rates or larger principals, it becomes more meaningful. Most savings accounts and investment returns compound daily or monthly.

The Rule of 72

The Rule of 72 is a shortcut for estimating how long it takes money to double. Divide 72 by the annual interest rate:

Years to double = 72 / interest rate

At 6%, money doubles in approximately 12 years. At 8%, approximately 9 years. At 3%, approximately 24 years.

This is an approximation, not an exact calculation. It works best for rates between 2% and 12%. For a 5% return, the Rule of 72 predicts doubling in 14.4 years. The actual answer (compounded annually) is about 14.2 years. Close enough for quick mental math.

How regular contributions change the picture

Most people do not invest a lump sum and walk away. They add money regularly. The formula for the future value of regular contributions (an ordinary annuity) is:

FV = PMT x [((1 + r/n)^(nt) - 1) / (r/n)]

Where PMT is the regular contribution amount.

Consider $10,000 starting balance, $200/month contributions, 5% annual return, compounded monthly, for 20 years:

• The original $10,000 grows to approximately $27,126 (as calculated above)
• The $200/month contributions grow to approximately $82,207
• Total estimated value: approximately $109,333

You contributed $58,000 total ($10,000 initial + $200 x 240 months). The remaining $51,333 came from compounding. Regular contributions give compound interest far more to work with.

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Starting early: a 10-year head start

Time is the most powerful variable in the formula. Consider two scenarios, both contributing $200/month at 5%:

Person A starts at age 25 and invests for 35 years (until age 60):

• Total contributions: $84,000
• Estimated ending balance: approximately $227,000

Person B starts at age 35 and invests for 25 years (until age 60):

• Total contributions: $60,000
• Estimated ending balance: approximately $119,000

Person A contributed $24,000 more but ended up with approximately $108,000 more. That extra 10 years of compounding nearly doubled the final balance. Every year of delay costs more than the last because it removes the year with the largest potential growth (the final year, when the balance is highest).

Common compounding scenarios

Scenario	Principal	Monthly addition	Rate	Years	Estimated result
Emergency fund	$1,000	$100	4%	5	$7,700
College savings	$5,000	$250	6%	18	$111,600
Retirement (early start)	$0	$500	7%	40	$1,197,000
Retirement (late start)	$0	$500	7%	20	$260,500

All figures are estimates. Actual returns vary. Investment returns are not guaranteed, and past performance does not predict future results.

Key takeaways

• Compound interest earns interest on previously earned interest, which is what separates it from simple interest
• The formula is A = P(1 + r/n)^(nt), where n is compounding frequency and t is time in years
• $10,000 at 5% compounded monthly grows to approximately $27,126 in 20 years
• Compounding frequency matters, but time and rate matter more
• The Rule of 72 gives a quick estimate: divide 72 by the rate to find years to double
• Regular contributions have a massive effect; $200/month at 5% for 20 years adds approximately $82,000 from contributions alone
• Starting 10 years earlier can nearly double the final balance, even with similar contributions

Related Calculators

Compound Interest Calculator

Estimate how your investments may grow over time with compound interest and regular contributions.

Savings Goal Calculator

Calculate how long to reach a savings goal, the monthly contribution needed, or the projected final balance.

Rule of 72 Calculator

Quickly estimate how long it takes for an investment to double, or what rate is needed to double in a given time.