Math • Finance
Compound Interest Calculator
Project future value with compounding (annually to daily or continuous), regular contributions, APY and a year-by-year breakdown. Ordinary annuity vs due.
Direct Answer
Compound interest generates exponential return by earning interest on both initial principal and accumulated past interest. The standard formula is A = P(1 + r/n)^(nt) plus regular annuity contributions, maximizing future value through frequent compounding and investment duration.
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Future value
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Total contributions
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Interest earned
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Future − contributions
| Year | Balance | Contributions | Interest |
|---|
A = P(1+r/n)nt + PMT·[((1+r/n)nt−1)/(r/n)] (× 1+r/n if due). Continuous: P·ert. APY = (1+r/n)n−1.
How to use
- 1Enter principal, annual rate % and years. Choose compounding frequency (or continuous).
- 2Add a periodic contribution, its frequency (monthly/yearly) and timing — end (ordinary) vs beginning (due).
- 3See future value, total contributions, interest earned, APY and the yearly table.
Formulas & Mathematical Definitions
Future Value (Compound Interest):
A = P × (1 + r/n)^(n×t)
With Regular Contributions (Ordinary Annuity - End of Period):
A = P × (1 + r/n)^(n×t) + PMT × [((1 + r/n)^(n×t) − 1) ÷ (r/n)]
Continuous Compounding:
A = P × e^(r×t)
Annual Percentage Yield (APY):
APY = (1 + r/n)^n − 1| Variable | Name | Meaning & Example |
|---|---|---|
| P | Principal | Starting initial balance or deposit ($10,000) |
| r | Annual Rate | Nominal interest rate expressed as a decimal (0.06 (6%)) |
| n | Frequency | Compounding periods per calendar year (12 (Monthly)) |
| t | Time | Investment period in years (10 Years) |
| PMT | Contribution | Regular addition deposited per period ($100 / month) |
Variable Definitions:
- A = Total future accumulated balance (principal + earned interest)
- P = Initial principal deposit ($)
- r = Nominal annual interest rate as a decimal (e.g., 6% = 0.06)
- n = Compounding frequency per year (Daily: 365, Monthly: 12, Quarterly: 4, Annually: 1)
- t = Investment duration in years
- PMT = Regular periodic deposit amount ($)
- e = Mathematical constant (~2.71828)
When deposits occur at the beginning of each compounding period (annuity due), the contribution portion earns an extra compounding cycle and is multiplied by (1 + r/n).
Examples
$10,000 Initial at 6% Compounded Monthly for 10 Years (No Deposits)
Step 1: Identify variables: P = $10,000, r = 0.06, n = 12, t = 10.
Step 2: Periodic interest rate = 0.06 ÷ 12 = 0.005.
Step 3: Total compound periods = 12 × 10 = 120.
Step 4: Growth factor = (1 + 0.005)^120 ≈ 1.819397.
Step 5: Future Value A = 10,000 × 1.819397 = $18,193.97.
Total Interest Earned = $18,193.97 − $10,000 = $8,193.97.
Future Value: $18,193.97 (Interest: $8,193.97)
$10,000 Initial + $100/Month Contribution at 6% for 10 Years
Step 1: Principal growth = $10,000 × (1 + 0.06/12)^120 = $18,193.97.
Step 2: Monthly contributions annuity = $100 × [((1 + 0.06/12)^120 − 1) ÷ (0.06/12)] = $16,387.90.
Step 3: Total Future Value = $18,193.97 + $16,387.90 = $34,581.87.
Total Contributions = $10,000 + ($100 × 120) = $22,000. Total Interest = $12,581.87.
Future Value: $34,581.87 (Interest: $12,581.87)
About
Frequently asked questions
What is the difference between APR and APY?
What is the difference between Ordinary Annuity and Annuity Due?
How does compounding frequency impact returns?
How does daily compounding compare to monthly or annual compounding?
What is the difference between APR and APY in compound interest?
What is the Rule of 72 in compound interest?
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