Compound interest is the most powerful force in math: money earning interest on its interest grows exponentially rather than linearly. At 5% compounded monthly, $10,000 becomes $16,471 after 10 years — not $15,000 as simple interest would predict. The difference is $1,471 in interest earned on interest. Real savings accounts and investments compound at rates that change over time; this calculator shows the math for a fixed rate.
The formula
For a lump sum with no additional contributions:
A = P(1 + r/n)^(nt)
Where:
- A = final amount
- P = principal (initial deposit)
- r = annual interest rate as a decimal (e.g., 5% = 0.05)
- n = compounding periods per year (monthly = 12, daily = 365)
- t = time in years
With periodic contributions (PMT per compounding period), the total is:
A = P(1 + r/n)^(nt) + PMT × [((1 + r/n)^(nt) − 1) / (r/n)]
For contributions at the start of each period, multiply the PMT term by (1 + r/n).
Source: Investor.gov (U.S. Securities and Exchange Commission).
Practical examples
Example 1: Lump sum, 10 years
$10,000 at 5% annual rate, compounded monthly, for 10 years:
A = 10000 × (1 + 0.05/12)^(12 × 10)
= 10000 × (1.004167)^120
= 10000 × 1.6471
= $16,471
Total interest earned: $6,471 — nearly two-thirds of the original investment.
Example 2: Monthly contributions
$10,000 initial deposit, $200/month contributions, 5% compounded monthly, 20 years:
- Without contributions: $27,126
- Contribution total over 20 years: $48,000
- Final balance: approximately $100,968
- Interest earned: $15,968 (on contributions alone, in addition to the lump-sum growth)
Example 3: Effect of compounding frequency
$10,000 at 6% for 10 years:
| Compounding | Periods/year (n) | Final balance |
|---|---|---|
| Annual | 1 | $17,908 |
| Quarterly | 4 | $18,061 |
| Monthly | 12 | $18,194 |
| Daily | 365 | $18,220 |
Common mistakes
Using the rate as a whole number rather than a decimal. The formula requires r as a decimal. Entering 5 instead of 0.05 produces a result millions of times too large.
Confusing APR and APY. A savings account advertised at 5% APR compounded monthly actually yields 5.116% APY. The calculator uses the rate you enter as APR; if your account states APY, use the APR/APY converter to find the equivalent APR first.
Expecting the formula to include taxes. Real returns on savings and investments are subject to income tax on interest earned each year. The formula shows gross growth before any taxes.
Assuming the rate stays fixed. Real interest rates on savings accounts, CDs, and investment accounts change. The formula is mathematically exact only for a constant rate over the full period.
International and regional variations
| Country / Region | Common compounding convention | Notes |
|---|---|---|
| United States | Daily or monthly (savings); monthly (mortgages) | APY disclosure required by Truth in Savings Act |
| United Kingdom | Annual (AER — Annual Equivalent Rate) | AER is the UK equivalent of APY; mandated by FCA |
| European Union | Annual (EAR — Effective Annual Rate) | EU Consumer Credit Directive requires EAR disclosure |
| Canada | Semi-annual (mortgages); monthly (deposits) | Canadian mortgages legally compound semi-annually |
| Australia | Monthly or daily | Comparison Rate required for loan advertising |
Quick reference: Rule of 72
| Annual rate | Years to double (Rule of 72) | Exact years (monthly compounding) |
|---|---|---|
| 3% | 24 years | 23.1 years |
| 5% | 14.4 years | 13.9 years |
| 7% | 10.3 years | 9.9 years |
| 10% | 7.2 years | 6.96 years |
| 12% | 6 years | 5.81 years |
The Rule of 72 is a rough approximation. The calculator gives exact results for any rate and compounding frequency.