Compounding frequency — how often interest is calculated and added to your balance — significantly affects how fast your money grows. Here's the exact math.
The Compound Interest Formula
A = P × (1 + r/n)^(n×t)
Where:
- A = final amount
- P = principal
- r = annual interest rate (as decimal)
- n = compounding periods per year
- t = time in years
Compounding Frequency Values
| Frequency | n |
|---|---|
| Annually | 1 |
| Semi-annually | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
| Continuously | e^(rt) |
Real Example: $10,000 at 8% for 10 Years
| Compounding | Final Amount | Interest Earned |
|---|---|---|
| Annual | $21,589.25 | $11,589.25 |
| Semi-annual | $21,911.23 | $11,911.23 |
| Quarterly | $22,080.40 | $12,080.40 |
| Monthly | $22,196.40 | $12,196.40 |
| Daily | $22,253.46 | $12,253.46 |
| Continuous | $22,255.41 | $12,255.41 |
Daily compounding earns $664 more than annual compounding over 10 years.
Continuous Compounding
The mathematical limit as n approaches infinity:
A = P × e^(r×t)
Example: $10,000 at 8% for 10 years:
A = 10,000 × e^(0.08 × 10) = 10,000 × e^0.8 = 10,000 × 2.2255 = $22,255
In practice, no bank offers true continuous compounding, but it approximates daily compounding closely.
The Effective Annual Rate (EAR)
To compare accounts with different compounding frequencies, convert to EAR:
EAR = (1 + r/n)^n - 1
Example: 8% compounded daily vs. 8.1% compounded annually
- Daily: EAR = (1 + 0.08/365)^365 - 1 = 8.328%
- Annual: EAR = 8.1%
The 8% daily account actually earns more than the 8.1% annual account.
What This Means for Loans
Compounding works against you in debt. Credit cards compound daily — a 20% stated APR becomes an effective rate of 22.13%. Always check whether rates are nominal or effective when comparing loan offers.
Use our Compound Interest Calculator to calculate any compounding scenario with a full year-by-year growth chart.