The Core Compound Interest Formula
The basic formula: A = P(1 + r/n)^(nt) where A = future value, P = principal, r = annual interest rate (decimal), n = compounding periods per year, t = time in years. For accounts with regular contributions, an annuity formula is added.
A = P(1 + r/n)^(nt) = $5,000 × (1 + 0.07/12)^(12×20) = $5,000 × (1.005833)^240 = $5,000 × 4.0387 = $20,194. Your $5,000 grew to over $20,000 in 20 years — quadrupled — from compound interest alone, with no additional contributions.
How Each Variable Affects the Outcome
Sensitivity analysis of each formula variable on $10,000 over 20 years at 7%
| Variable | Increase By | Effect on $10,000 over 20 years at 7% |
|---|---|---|
| Principal (P) | +$5,000 (50% more) | +$10,097 (50% more outcome) |
| Rate (r) | +1% (to 8%) | +$7,200 more (from $38,697 to $45,882) |
| Time (t) | +5 years (25 years) | +$16,400 more (from $38,697 to $54,274) |
| Compounding (n) | Monthly vs. Annual | +$185 more (minimal effect) |
The Formula With Regular Contributions
When you add regular contributions (PMT), the formula adds a future value of annuity component: FV = P(1+r)^t + PMT × [((1+r)^t − 1) / r] where r is the period rate. This is what makes monthly investing so powerful — each contribution starts its own compounding clock.
Compound interest with contributions — impact of starting balance vs. contribution amount
| Starting Balance | Monthly Contribution | Years | Rate | Final Value | Interest Earned |
|---|---|---|---|---|---|
| $0 | $500 | 30 | 7% | $589,000 | $409,000 |
| $10,000 | $500 | 30 | 7% | $665,000 | $475,000 |
| $50,000 | $500 | 30 | 7% | $965,000 | $775,000 |
| $0 | $1,000 | 30 | 7% | $1,178,000 | $818,000 |
The e^rt Continuous Compounding Formula
Continuous compounding (theoretical maximum): A = Pe^(rt). At 7% for 20 years on $10,000: A = $10,000 × e^(1.4) = $40,552. Compare to monthly compounding: $40,388. The difference is just $164 — confirming that contribution amount and time matter exponentially more than compounding frequency.
The Rule of 72: Mental Math for Compounding
Divide 72 by the annual rate to find the doubling time. At 7%: 72÷7 = 10.3 years to double. At 10%: 7.2 years. At 4%: 18 years. This quick mental math shortcuts the formula and gives you an instant sense of any investment’s long-term potential.
Rule of 72 accuracy by interest rate
| Annual Rate | Doubling Time (Rule of 72) | Actual Doubling Time | Accuracy |
|---|---|---|---|
| 2% | 36 years | 35.0 years | 97% |
| 4% | 18 years | 17.7 years | 98% |
| 6% | 12 years | 11.9 years | 99% |
| 8% | 9 years | 9.0 years | 100% |
| 10% | 7.2 years | 7.3 years | 99% |
| 12% | 6 years | 6.1 years | 98% |
See the Formula in Action for Your Numbers
Enter your starting balance, contributions, and rate — watch the compound interest formula build your wealth.