Scope and assumptions
A = final amount, P = principal, r = annual rate, n = compounding/yr, t = years. Illustrative.
Two accounts both pay 5%. Three years later, one holds $1,157.63 and the other $1,161.47. Nothing was misquoted and nobody cheated — the difference is entirely about how often the interest was allowed to compound.
The formula carries that frequency explicitly, which is the part most flat "5% a year" reasoning silently drops:
A = P × (1 + r ÷ n)^(n × t)
P is the starting amount, r the annual rate, n the compounding periods per year, t the years.
The same 5%, three frequencies
$1,000 for three years:
| Compounding | Calculation | Final |
|---|---|---|
| Annual (n = 1) | 1,000 × 1.05³ | $1,157.63 |
| Monthly (n = 12) | 1,000 × (1 + 0.05/12)³⁶ | $1,161.47 |
| Daily (n = 365) | 1,000 × (1 + 0.05/365)¹⁰⁹⁵ | $1,161.83 |
Monthly beats annual by $3.84 — about a third of a percent of the principal. Modest here, but it grows with both the balance and the horizon.
The gains flatten quickly
Frequency helps, then stops helping. At 5% over three years:
- annual → monthly: +$3.84
- monthly → daily: +$0.36
- daily → continuous: +$0.00
Each step adds less than the one before, because the extra compounding periods are being applied to an ever-smaller slice of time. There's a hard ceiling, approached but never crossed.
The continuous ceiling
Push n toward infinity and the formula collapses into:
A = P × e^(r × t)
At 5% for three years: 1,000 × e^0.15 ≈ $1,161.83 — identical to daily to the cent. In practice, daily compounding is continuous compounding; the remaining difference is invisible at normal rates.
Units must match
The formula has three separate places where a unit slip can happen:
- r is annual, always.
- t is in years. Three months is 0.25, not 3.
- n is periods per year, matching whatever the product credits.
A three-month term at 5% compounded monthly is n = 12, t = 0.25 — not n = 3. Getting that wrong changes the answer by a factor of four.
Compounding Assumptions That Fool Investors
Using A = P(1 + r)^t and calling it general (that's annual-only); dividing r by n but forgetting to multiply t by n; entering months as years; rounding inside the exponent; expecting large gains from frequency at low rates; and confusing the final amount with the effective annual rate, which is a different output of the same family of formulas.
Related growth & decay guides
For the rest of the growth-and-decay cluster, start at the hub: Population Growth Percentage. Adjacent: Annual Percentage Rate Explained and Doubling Time Percentage.
FAQ
How do I find the implied rate if I know the start and end amounts? Rearrange: r = (A ÷ P)^(1 ÷ t) − 1. Growing from $1,000 to $1,500 in five years implies about 8.45% a year.
How long does money take to double at 7%? About ten years, by the rule of 70 (70 ÷ 7). Compounding at 7% gives 1.07^10 ≈ 1.97, so the approximation is close.
Does compounding work against me on debt? Yes, symmetrically. An unpaid balance at 24% APR compounds just as relentlessly as an investment at 24% grows — with the sign reversed.
How do I handle regular contributions rather than a lump sum? Use an annuity formula: the future value of a series of deposits is PMT × ((1 + r)^t − 1) ÷ r. The compounding logic is the same; the sum is over payments, not years alone.
Sources: Investopedia, Compound Interest; Khan Academy.
Disclaimer: Illustrative. Real products have fees and terms; consult a professional. Informational only.
Sources and references
- Investopedia — Compound Interesthttps://www.investopedia.com/terms/c/compoundinterest.asp
- Khan Academy — Compound interesthttps://www.khanacademy.org/math
Maintenance and corrections
The named author or publishing brand is responsible for this article. Worked examples are illustrative, not user testimonials, professional advice or approval by the institutions cited. Maintenance responsibility and review methods · Report an error