What compound interest is
Compound interest is interest paid on both the principal and any interest already earned. Each compounding period, the previous balance becomes the new base — so growth accelerates over time. Albert Einstein is widely (though probably apocryphally) credited with calling it "the eighth wonder of the world." The intuition holds: even modest rates produce enormous balances over decades because the curve is exponential, not linear.
The compound interest formula
The general formula for compound interest with periodic compounding is:
A = P × (1 + r/n)^(n × t)where A is the final amount, P is the principal, r is the annual interest rate as a decimal, n is the number of compounding periods per year, and t is the time in years.
If you also make a regular contribution PMT at the end of each compounding period, the future value becomes:
A = P × (1 + r/n)^(n × t) + PMT × ((1 + r/n)^(n × t) − 1) / (r/n)Continuous compounding
As compounding becomes more frequent — daily, hourly, every second — the formula approaches a limit known as continuous compounding:
A = P × e^(r × t)For practical purposes, daily compounding is already very close to continuous. The difference between monthly and daily compounding on a 10-year, 7% account is only about 0.3% of the balance.
A worked example
Invest $10,000 at 7% annual return, compounded monthly, for 30 years:
A = 10000 × (1 + 0.07/12)^(12 × 30) ≈ $81,165The same investment with an additional $200 contributed at the end of each month:
A ≈ $81,165 + 200 × ((1.005833)³⁶⁰ − 1) / 0.005833 ≈ $325,486Of that final balance, roughly $10,000 came from the initial deposit, $72,000 from the monthly contributions, and the remaining $243,000 from compounding alone. After about year 20, you start earning more from interest each year than you contribute.
The Rule of 72
A useful mental shortcut for doubling time:
Years to double ≈ 72 / annual return %- At 6% growth, money doubles roughly every 12 years.
- At 9% growth, doubles every 8 years.
- At 12% growth, doubles every 6 years.
The Rule of 72 works well for rates between 4% and 15%. For continuous compounding, the Rule of 70 (use 70 instead of 72) is slightly more accurate.
Why starting early matters more than contributing more
Two savers, both earning 7%:
- Saver A invests $300/month from age 25 to 35, then stops. Total contributed: $36,000.
- Saver B invests $300/month from age 35 to 65. Total contributed: $108,000.
At age 65, Saver A has approximately $386,000 and Saver B has approximately $367,000. Saver A contributed one-third as much but ended up with more because the early money had thirty extra years to compound. This is the central argument for starting retirement contributions in your twenties — even small amounts.
Real return vs nominal return
Compound interest calculators show nominal returns. To understand actual purchasing power you need to subtract inflation. The approximate real return is:
real return ≈ nominal return − inflationA 7% nominal return at 3% inflation is roughly 4% real. Over 30 years, $325,000 with 3% average inflation has the buying power of about $134,000 in today's money — still a substantial gain, but less impressive than the headline number.