Compound Interest Formula
The compound interest formula looks intimidating but breaks down cleanly once you walk through it. This page covers the lump-sum version, the version with regular contributions, and three worked examples you can verify by hand.
The Lump-Sum Formula
- A = future value (the amount you end up with)
- P = principal (the amount you start with)
- r = annual interest rate, as a decimal (5% = 0.05)
- n = compounding periods per year (12 for monthly)
- t = number of years
Walking Through It
- r / n — split the annual rate across the compounding periods. At 6% compounded monthly, the per-period rate is 0.06 / 12 = 0.005, or 0.5% per month.
- 1 + r/n — adds the per-period growth factor. 0.5% growth in a month means your balance multiplies by 1.005.
- (1 + r/n)n × t — raises that growth factor to the total number of compounding periods. 10 years × 12 months = 120 periods. 1.005120 = 1.8194.
- P × (1 + r/n)n × t — multiplies by your starting principal. $10,000 × 1.8194 = $18,194.
Worked Example #1: Lump Sum
Question: How much will $5,000 grow to in 15 years at 7% compounded monthly?
Plug in: P = 5000, r = 0.07, n = 12, t = 15
A = 5000 × (1 + 0.07/12)12×15
A = 5000 × (1.005833)180
A = 5000 × 2.8489
Answer: $14,244.35
The Formula with Contributions
When you add money regularly (a monthly contribution to a 401k or savings account), the formula adds a second term:
- PMT = contribution per compounding period (e.g., monthly deposit)
- The first term grows your starting principal.
- The second term sums the future values of every contribution.
Worked Example #2: Monthly Contributions
Question: $10,000 starting balance, $500/month, 7% annual return, 20 years.
Plug in: P = 10000, r = 0.07, n = 12, t = 20, PMT = 500
A = 10000 × (1.005833)240 + 500 × [((1.005833)240 − 1) / 0.005833]
A = 10000 × 4.0387 + 500 × 520.93
A = 40,387.39 + 260,463.33
Answer: $300,850.72
Of that, $130,000 came from your pocket (10k start + 240 × $500). The other ~$170,000 is compound interest.
Worked Example #3: Find the Rate
You can also solve the formula for rate, time, or principal. To find the rate needed to grow $5,000 into $20,000 in 25 years:
20,000 = 5,000 × (1 + r)25
4 = (1 + r)25
1 + r = 41/25 = 1.0573
Answer: r ≈ 5.73% per year
Quadrupling in 25 years requires about 5.7% returns.
The Continuous-Compounding Variant
For continuous compounding (the theoretical limit as n → ∞), the formula simplifies dramatically:
Where e≈ 2.71828 is Euler's number. This is mostly used in academic finance and options pricing — retail accounts compound at discrete intervals (daily at finest), so the discrete formula above is what matters for personal finance.
Common Variables to Solve For
| Find | Rearranged Formula |
|---|---|
| Future value (A) | P × (1 + r/n)n×t |
| Principal (P) | A / (1 + r/n)n×t |
| Rate (r) | n × [(A/P)1/(n×t) − 1] |
| Time (t) | ln(A/P) / [n × ln(1 + r/n)] |
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Frequently Asked Questions
What is the compound interest formula?
For a lump sum: A = P × (1 + r/n)^(n×t), where A is the future value, P is the principal, r is the annual rate (as a decimal), n is compounding periods per year, and t is years. For continuous compounding, the formula simplifies to A = P × e^(r×t).
What is the formula for compound interest with monthly contributions?
A = P × (1 + r/n)^(n×t) + PMT × [((1 + r/n)^(n×t) − 1) / (r/n)]. The first term grows the initial principal. The second term sums the future value of a series of equal contributions made at the start of each period.
How do I calculate compound interest by hand?
Convert your rate to a decimal, divide by the number of compounding periods per year, add 1, raise that to the power of total periods, then multiply by your principal. Example: $1,000 at 5% monthly for 10 years = 1000 × (1 + 0.05/12)^(12×10) = 1000 × 1.6470 = $1,647.01.
What is the difference between the discrete and continuous compounding formulas?
Discrete compounding uses A = P × (1 + r/n)^(n×t) with a finite number of compounding periods. Continuous compounding uses A = P × e^(r×t), which is the mathematical limit as n approaches infinity. Discrete is what banks actually use; continuous is mainly used in finance theory.
What does 'compounding period' mean in the formula?
A compounding period is the interval at which interest is calculated and added to the balance. n in the formula equals the number of compounding periods per year: 1 for annual, 4 for quarterly, 12 for monthly, 365 for daily. The smaller the period, the more often interest gets re-invested.