Compound Interest Formula
Reference for A = P(1 + r/n)^(nt) compound interest formula.
Covers annual, monthly, and daily compounding, the Rule of 72, and frequency comparison tables.
The Formula
Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods. This creates an exponential growth effect — your money earns interest on its interest.
The more frequently interest compounds, the faster the balance grows. Daily compounding yields slightly more than monthly, which yields more than annual.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| A | Final amount (principal + interest) | currency |
| P | Principal (initial investment) | currency |
| r | Annual interest rate (as decimal) | decimal |
| n | Compounding frequency per year | count |
| t | Time the money is invested | years |
Example 1
Invest $10,000 at 5% annual rate, compounded monthly, for 10 years
P = $10,000, r = 0.05, n = 12, t = 10
A = 10,000 × (1 + 0.05/12)12×10
A = 10,000 × (1.004167)120
A = 10,000 × 1.6470
A ≈ $16,470.09 — earned $6,470.09 in interest
Example 2
Invest $5,000 at 8% annual rate, compounded quarterly, for 5 years
P = $5,000, r = 0.08, n = 4, t = 5
A = 5,000 × (1 + 0.08/4)4×5
A = 5,000 × (1.02)20
A = 5,000 × 1.4859
A ≈ $7,429.74 — earned $2,429.74 in interest
Continuous Compounding
When compounding happens infinitely often (n → ∞), the formula simplifies to continuous compounding using Euler's number e ≈ 2.71828. In practice, continuous compounding gives only slightly more than daily compounding.
When to Use It
- Calculating savings account growth over time
- Comparing investment options with different compounding frequencies
- Estimating the future value of retirement contributions
- Understanding how credit card debt accumulates
- Financial planning and loan amortization analysis
Key Notes
- Formula: A = P(1 + r/n)^(nt): P is principal, r is nominal annual rate, n is compounding periods per year, t is time in years. The effective annual rate (APY) = (1 + r/n)^n − 1, which is always ≥ r (nominal rate).
- Solving for unknowns: For rate: r = n[(A/P)^(1/(nt)) − 1]. For time: t = ln(A/P) / [n × ln(1 + r/n)]. For principal: P = A / (1 + r/n)^(nt). Rearranging the formula flexibly is essential for real financial planning questions.
- Continuous compounding limit: A = Pe^(rt): As n → ∞, the formula approaches e^(rt). For r = 6%, continuous compounding yields APY = e^0.06 − 1 ≈ 6.184% vs 6.000% nominal — very close to monthly compounding (6.168%) with negligible additional gain.
- Time to reach a target: t = ln(A/P) / r (continuous): The logarithm naturally appears when solving for time in any exponential growth formula. This is why the Rule of 72 (t ≈ 72/r%) works — it approximates ln(2) ≈ 0.693 ≈ 72/100 in convenient mental-math form.
- Applications: Compound interest is used to project savings account balances, calculate loan payoff dates, determine the future cost of a child's education at current tuition growth rates, evaluate whether an investment meets a target return, and explain why early saving outperforms late saving.