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Online Calculators

The Compound Interest Formula Explained

Understand the compound interest formula, how to calculate it, and see real-world examples with step-by-step breakdowns.

The compound interest formula

The formula used for calculating compound interest is:

A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}

Where:

  • A = the future value of the investment
  • P = the principal balance (initial investment)
  • r = the annual interest rate (as a decimal)
  • n = number of times interest is compounded per year
  • t = the time in years

Step-by-step calculation example

Let's say you invest $10,000 at a 5% annual interest rate, compounded monthly, for 20 years.

  1. P = $10,000
  2. r = 0.05 (5% as a decimal)
  3. n = 12 (compounded monthly)
  4. t = 20 years

Calculation:

A=10,000×(1+0.0512)12×20A = 10,000 \times \left(1 + \frac{0.05}{12}\right)^{12 \times 20} A=10,000×(1.004167)240A = 10,000 \times (1.004167)^{240} A=10,000×2.71264A = 10,000 \times 2.71264 A=$27,126.40A = \$27,126.40

Total interest earned: $17,126.40

Yearly breakdown

Here's how your investment grows year by year:

Year Interest Calculation Interest Earned End Balance
1 $10,000 × 5% $500 $10,500
2 $10,500 × 5% $525 $11,025
3 $11,025 × 5% $551.25 $11,576.25
5 — — $12,762.82
10 — — $16,288.95
15 — — $20,789.28
20 — — $26,532.98

10,000investedat510,000 invested at 5% yearly, compounded annually, grows to 26,532.98 after 20 years — a total return of 165%.

Compounding frequency comparison

The frequency of compounding affects your final balance. Here's how $10,000 at 5% for 10 years compares:

Frequency n Final Balance Interest Earned
Annually 1 $16,288.95 $6,288.95
Quarterly 4 $16,436.19 $6,436.19
Monthly 12 $16,470.09 $6,470.09
Daily 365 $16,486.65 $6,486.65

The more frequently interest compounds, the more you earn — though the difference diminishes at lower rates.

Effective Annual Rate (APY)

The effective annual rate (also known as APY — Annual Percentage Yield) is the rate you actually earn after compounding has been factored in.

When interest compounding takes place, the effective annual rate becomes higher than the nominal annual interest rate. The more times interest is compounded within the year, the greater the difference.

Formula: Effective Rate=(1+rn)n−1\text{Effective Rate} = \left(1 + \frac{r}{n}\right)^n - 1

For example, 5% compounded monthly:

  • Effective Rate=(1+0.0512)12−1=5.116%\text{Effective Rate} = \left(1 + \frac{0.05}{12}\right)^{12} - 1 = 5.116\%

This means you earn slightly more than the stated 5% rate.

The Rule of 72

A quick mental shortcut to estimate how long it takes to double your money:

Years to double≈72Annual Interest Rate\text{Years to double} \approx \frac{72}{\text{Annual Interest Rate}}

Examples:

  • At 6%: 726=12 years\frac{72}{6} = 12 \text{ years}
  • At 8%: 728=9 years\frac{72}{8} = 9 \text{ years}
  • At 10%: 7210=7.2 years\frac{72}{10} = 7.2 \text{ years}
  • At 12%: 7212=6 years\frac{72}{12} = 6 \text{ years}