Compound Interest Formula Explained
The compound interest formula A = P(1 + r/n)^(nt) is one of the most important equations in personal finance. It shows how an investment grows when interest is periodically added to the principal and then earns interest itself — a process called compounding.
This guide explains every variable in the formula, derives it from first principles, shows how compounding frequency affects results, introduces continuous compounding (A = Pe^(rt)), and covers the Rule of 72 for quick mental estimates.
Formula
A = P(1 + r/n)^(nt)
Understanding Each Variable
P — Principal: The initial amount of money you deposit, invest, or borrow. Everything starts here. If you invest $5,000, P = 5000.
r — Annual interest rate: Always convert the percentage to a decimal before plugging in. 6% → r = 0.06. 0.5% → r = 0.005. The rate is the stated annual rate, not the effective rate.
n — Compounding frequency per year: Annual n=1. Semi-annual n=2. Quarterly n=4. Monthly n=12. Daily n=365. The more frequently interest compounds, the more you earn. Each period, interest is added to the account at rate r/n.
t — Time in years: Convert months to years (months/12). Convert days to years (days/365). 18 months = 1.5 years. 90 days ≈ 0.2466 years.
A — Final amount: The total value after t years, including both principal and accumulated interest. Interest earned = A − P.
Derivation from First Principles
Start with simple interest for one period. If you invest P at annual rate r for one year with annual compounding: A = P + P×r = P(1+r).
For two years: After year 1, you have P(1+r). In year 2, this entire amount earns interest: A = P(1+r) + P(1+r)×r = P(1+r)(1+r) = P(1+r)².
For t years (annual compounding, n=1): A = P(1+r)^t.
For n compounding periods per year: Each period, the rate is r/n (monthly rate = annual rate ÷ 12). After one year (n periods): A = P(1+r/n)^n. After t years (nt total periods): A = P(1+r/n)^(nt).
The key insight: each compounding period, the new principal is the old principal × (1 + period rate). This self-similar multiplication is what creates exponential growth.
Worked Examples
Example 1 — Annual compounding: Invest $10,000 at 6% for 5 years, compounded annually. A = 10000 × (1 + 0.06/1)^(1×5) = 10000 × (1.06)^5 = 10000 × 1.33823 = $13,382.30. Interest earned = $3,382.30.
Example 2 — Monthly compounding: Same but compounded monthly. A = 10000 × (1 + 0.06/12)^(12×5) = 10000 × (1.005)^60 = 10000 × 1.34885 = $13,488.50. Monthly earns $106.20 more than annual.
Example 3 — Find P (present value): How much to invest now at 4% monthly compounding to have $20,000 in 8 years? Rearrange: P = A / (1+r/n)^(nt) = 20000 / (1+0.04/12)^96 = 20000 / 1.37713 = $14,523.
Example 4 — Find the rate: P=$5,000, A=$7,500, t=10 years, annual compounding. (1+r)^10 = 7500/5000 = 1.5. r = 1.5^(1/10) − 1 = 1.04139 − 1 = 4.14% per year.
Effect of Compounding Frequency
Same investment: $10,000 at 5% annual rate for 10 years. The table shows how increasing compounding frequency raises the final amount, with diminishing returns beyond monthly.
| Compounding | n | Formula | Final Amount | Interest Earned |
|---|---|---|---|---|
| Simple interest | — | A = P(1+rt) | $15,000.00 | $5,000.00 |
| Annual | 1 | A = P(1.05)^10 | $16,288.95 | $6,288.95 |
| Semi-annual | 2 | A = P(1.025)^20 | $16,386.16 | $6,386.16 |
| Quarterly | 4 | A = P(1.0125)^40 | $16,436.19 | $6,436.19 |
| Monthly | 12 | A = P(1.004167)^120 | $16,470.09 | $6,470.09 |
| Daily | 365 | A = P(1+0.05/365)^3650 | $16,486.65 | $6,486.65 |
| Continuous | ∞ | A = P × e^(rt) | $16,487.21 | $6,487.21 |
Continuous Compounding — A = Pe^(rt)
Continuous compounding is the mathematical limit as n → ∞. Using the limit lim(n→∞) (1 + r/n)^n = e^r (where e ≈ 2.71828), the formula becomes: A = Pe^(rt).
Example: $10,000 at 5% for 10 years continuously compounded. A = 10000 × e^(0.05×10) = 10000 × e^0.5 = 10000 × 1.64872 = $16,487.21.
Continuous compounding is used in: advanced financial modelling, options pricing (Black-Scholes formula), actuarial science, and physics (radioactive decay, population growth). In practice, daily compounding ≈ continuous compounding (difference of $0.56 in the example above).
The natural logarithm ln is the inverse of e^x. To find time for continuous compounding: t = ln(A/P) / r. To double: t = ln(2)/r ≈ 0.693/r (the basis for the Rule of 72).
Rule of 72 — Quick Doubling Time Estimate
The Rule of 72 estimates how long it takes an investment to double: Years to double ≈ 72 / Annual Rate (%).
Examples: 4% rate → 72/4 = 18 years to double. 6% → 72/6 = 12 years. 8% → 72/8 = 9 years. 9% → 72/9 = 8 years. 12% → 72/12 = 6 years.
Mathematical basis: We want A = 2P. From A = Pe^(rt): 2 = e^(rt) → rt = ln(2) ≈ 0.6931. So t = 0.6931/r. In percentage terms: t ≈ 69.3 / Rate%. Using 72 gives a slightly conservative (longer) estimate because 72 divides evenly by many numbers (2,3,4,6,8,9,12), making mental math easier.
Rule of 72 for other multiples: To triple: use Rule of 115. To quadruple: use Rule of 144. In general: years to multiply by M ≈ ln(M)/r (in decimal) or equivalently 100×ln(M)/Rate%.
Applications: quickly compare investment options. At 6% return, doubling takes 12 years. At 3% inflation, purchasing power halves in 24 years. At 2% GDP growth, the economy doubles in 36 years.
Effective Annual Rate (EAR) and APY
When comparing accounts with different compounding frequencies, convert all to the Effective Annual Rate (EAR), also called Annual Percentage Yield (APY): EAR = (1 + r/n)^n − 1.
Example: Bank A offers 5% compounded monthly. Bank B offers 5.1% compounded annually. Which is better?
Bank A EAR: (1 + 0.05/12)^12 − 1 = (1.004167)^12 − 1 = 1.05116 − 1 = 5.116%.
Bank B EAR: 5.1% (annual compounding means EAR = nominal rate).
Bank A (5.116% EAR) > Bank B (5.10% EAR) → Bank A is slightly better even at a lower nominal rate.
US law requires savings accounts to advertise APY (which equals EAR), making comparison straightforward. For loans, APR (Annual Percentage Rate) is used, which may or may not reflect compounding — always check the note rate vs effective rate.