Use the Compounding Frequency Comparison Calculator to compare annual, monthly, weekly, daily, and continuous balances and effective yields over time.
Compounding Frequency Formulas
For a nominal annual rate compounded a fixed number of times per year, the ending balance is:
A_n = P(1 + r/n)^{nt}
For continuous compounding, use:
A_{continuous} = Pe^{rt}
The effective annual yield converts each frequency into the actual one-year percentage growth:
EAR_n = (1 + r/n)^n - 1
Variables:
- A is the ending balance
- P is the starting principal
- r is the nominal annual rate as a decimal
- n is the number of compounding periods per year
- t is time in years
- e is Euler’s number, approximately 2.71828
Comparison mode calculates annual, semiannual, quarterly, monthly, biweekly, weekly, daily, and continuous compounding from the same nominal rate. Required-rate mode works backward to find the nominal rate needed at one selected frequency to reach a target balance.
Frequency comparisons should use the same nominal annual rate. When accounts advertise an APY or effective annual rate instead, the frequency effect is already incorporated, so compare the APYs directly rather than compounding them again.
Compounding Frequency Comparison
This table shows the result of compounding $10,000 at a 6% nominal annual rate for 10 years.
| Frequency | Effective annual yield | Ending balance | Gain |
|---|---|---|---|
| Annual | 6.000% | $17,908.48 | $7,908.48 |
| Semiannual | 6.090% | $18,061.11 | $8,061.11 |
| Quarterly | 6.136% | $18,140.18 | $8,140.18 |
| Monthly | 6.168% | $18,193.97 | $8,193.97 |
| Weekly | 6.180% | $18,214.89 | $8,214.89 |
| Daily | 6.183% | $18,220.29 | $8,220.29 |
| Continuous | 6.184% | $18,221.19 | $8,221.19 |
Example Problems
Example 1: Compare annual and monthly compounding.
With $10,000 at a 6% nominal rate for 10 years, annual compounding produces $17,908.48. Monthly compounding produces $18,193.97, a difference of $285.49.
Example 2: Find a required nominal rate.
You want $15,000 to reach $25,000 in 8 years with monthly compounding. Rearranging the formula gives r = 12[(25,000/15,000)1/(12*8) – 1], or about 6.41% nominal annually.
Frequently Asked Questions
Which compounding frequency earns the most?
At the same positive nominal annual rate, more frequent compounding produces a larger balance. Continuous compounding is the mathematical upper limit, although its advantage over daily compounding is usually very small.
Is APY the same as the nominal interest rate?
No. The nominal rate states the annual rate before the effect of within-year compounding. APY, also called effective annual yield, includes the compounding frequency and is usually the better figure for comparing deposit accounts.
How much does daily compounding add over monthly compounding?
The answer depends on the rate, time, and principal. At a 6% nominal rate for 10 years, daily instead of monthly compounding adds about $26.32 per $10,000 of principal. The difference grows with higher rates and longer periods.
