How this instrument works
Take the annual rate, split it evenly across the year's compounding periods, and apply it that many times. That is the whole instrument: r/n is one period's rate, n·t counts how many periods elapse, and the exponent does everything else. Splitting a rate evenly is a disclosure convention rather than a law of arithmetic, which is why a quoted 12% compounded monthly is really 1% applied twelve times — $1,000 finishes a year at $1,126.83, not $1,120. Those extra $6.83 are interest earning interest, and US banks must publish that effect as an annual percentage yield of 12.6825%.
Raising n does not raise the answer without limit. Jacob Bernoulli asked in 1683 what a 100% rate does when compounded ever more often, and found a ceiling. Annually it doubles; quarterly it reaches 2.4414 times the stake, monthly 2.6130, daily 2.7146, and no schedule ever passes e = 2.71828. The same flattening happens at ordinary rates, only far sooner: 5% nominal yields 5.1162% compounded monthly against 5.1267% compounded daily, roughly a penny per hundred dollars a year between them.
Two things this sheet deliberately leaves out. It models a single deposit left alone, so money added later belongs in a different calculation, and it holds the rate fixed for the entire term, which no savings account actually does. Tax on the interest, account fees and inflation all sit outside the arithmetic too. The formula is also indifferent to which side of a ledger you stand on: a $25,000 card balance at 22.99% compounded daily grows to $31,459.58 in a year by the identical route.
- Put the opening sum in Principal, $ — one deposit that sits untouched, with nothing added later.
- Type the quoted nominal figure in Annual rate, % as a percentage: 5 for five percent, not 0.05.
- Set Compounding to your bank's schedule, not your saving habits. Monthly, daily and quarterly return different answers from the same rate.
- Enter the term in Years. Halves are allowed, so eighteen months is 1.5.
- Read Final balance for the total and Interest earned for growth alone — the gap between them is always your principal.
Worked example — the doubling check
Set Principal, $ to 1000, Annual rate, % to 100, Compounding to Annually, and Years to 1. Exactly one period elapses, so the growth factor is 1 + 1.00 ⁄ 1 = 2 and A = 1000 × 2¹ = $2,000. Interest earned reads $2,000 − $1,000 = $1,000. Every digit is checkable in your head, which is precisely why it is worth running before you trust the same sheet with a thirty-year figure you cannot verify.
Now change nothing but Compounding. Quarterly returns $2,441.41, monthly $2,613.04, daily $2,714.57 — one rate, one year, one deposit, and $714.57 of spread from frequency alone. Reset to the defaults, $1,000 at 5% compounded monthly for 10 years, and the readout gives $1,647.01. Annual compounding on identical terms gives $1,628.89. That $18.12 is what frequency is worth at a realistic rate, and it is a useful corrective to how the 100% case looks.
Questions
Why does 12% compounded monthly earn more than 12%?
Because the 12% is nominal — twelve applications of 1%, not one of 12%. Each month's interest joins the balance that next month is computed on, so $1,000 ends the year at $1,126.83 rather than $1,120, an effective yield of 12.6825%. US institutions must disclose that second figure as APY under the Truth in Savings Act, which is why APY, and never the nominal rate on its own, is what compares two accounts running on different schedules.
Does this account for money I add every month?
No. This formula describes one deposit left alone for the whole term. Regular contributions require a future-value-of-a-series calculation, which grows each payment separately according to how long it personally has left to sit. Adding up your planned deposits and entering that total as Principal will overstate the result badly, because it credits every dollar with the full term instead of the months it was actually there.
Is there a limit to how often interest can compound?
Yes. Push n toward infinity and the expression converges on continuous compounding, A = Pe^(rt). At 100% for one year that ceiling is $2,718.28 per $1,000 against $2,714.57 for daily — under four dollars apart. At 5% the daily-to-continuous gap is about one cent per $1,000 per year. Frequency does real work between annual and monthly; beyond daily it is a rounding detail.
Why is my bank's balance a few dollars off this figure?
Day-count conventions, usually. This sheet cuts the year into n equal periods, while banks often accrue on actual days, so a 31-day month pays more than a 30-day one and February pays least. Some credit on the final business day; some divide by 360 while counting 365. Those differences run to cents or a few dollars on ordinary balances. A large discrepancy points instead at fees, withheld tax, or a rate that moved mid-term.
Does the Rule of 72 agree with this instrument?
At ordinary rates, closely. Dividing 72 by the rate estimates a doubling time: 72 ÷ 8 = 9 years against a true 9.006 under annual compounding. The shortcut comes from a logarithm and degrades as rates climb. At the 100% used in the worked example it predicts 0.72 years when the exact answer is 1. It stays reliable for single-digit rates; past those, the exponent here is the honest route.
Is Interest earned the amount I actually keep?
Rarely all of it. Interest on savings accounts, CDs and most bonds is generally taxable in the year it is credited, even when it stays in the account, and US payers report it on Form 1099-INT. Maintenance fees come off as well, and inflation eats into whatever survives. What this readout gives you is nominal, pre-tax growth — the number the contract produces, before your own tax position touches it.
References
- SEC Investor.gov — investor education and calculators
- IRS Topic no. 403 — interest income and Form 1099-INT
- FDIC — deposit insurance and consumer banking resources
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.