Financial
Compare compounding periods: Annual, Quarterly, Monthly, Daily, and Continuous
See exact dollar differences across compounding frequencies m = 1, 4, 12, 365 using P*(1 + r/m)^(m*t) alongside the continuous limit P*EXP(r*t).
Diminishing marginal gains as compounding approaches continuous EXP(r*t)
For the same positive nominal rate, more frequent compounding increases modeled terminal value toward the continuous limit.
The table illustrates $10,000 at 8% over five years; the daily-to-continuous difference is about $0.66 in this scenario.
| Compounding Frequency | Periods/Yr (m) | 5-Yr Ending Balance ($10k @ 8%) |
|---|---|---|
| Annual | 1 | $14,693.28 |
| Quarterly | 4 | $14,859.47 |
| Monthly | 12 | $14,898.46 |
| Daily | 365 | $14,917.59 |
| Continuous Limit | P * EXP(r * t) | $14,918.25 |
Discrete m-period vs Continuous compounding formulas
Pass any frequency m (1, 2, 4, 12, 52, 365) in D2 to evaluate ending value.
=ROUND(A2 * (1 + B2 / D2) ^ (D2 * C2), 2)Continuous compounding upper bound (P * e^(r*t))
Under the positive-principal, nonnegative-time assumptions, continuous compounding is the limit of increasingly frequent discrete compounding.
=ROUND(A2 * EXP(B2 * C2), 2)Inputs and interpretation
Assume a constant nominal annual rate, positive integer compounding frequency, nonnegative time and principal, and 1+r/m > 0. There are no fees, taxes, contributions or withdrawals. Fractional compounding periods are a mathematical interpolation, not necessarily an account-crediting policy. The dollar gap to continuous compounding scales with principal, rate and duration.
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Use the original values or change them to test your understanding.