The compound interest formula, fully worked (with a frequency table)

Most pages state the compound interest formula and stop, as if the stating were the understanding. I am going to do the opposite: take one problem — $8,000 at 6% for eight years — and solve it five times, once for each compounding frequency, showing every substitution and every intermediate number. By the end you will be able to check any compound formula calculator against figures you produced yourself, and the formula will have stopped being notation and started being a machine you own.

The compounding formula, term by term

A = P(1 + r/n)^nt

• A — the final amount, the answer.
• P — the principal you start with: $8,000 in our problem.
• r — the annual rate as a decimal: 6% becomes 0.06.
• n — how many times a year interest is credited: 1, 4, 12 or 365.
• t — the years: 8.

Two ideas do all the work inside that line. The rate per period is r/n — interest is credited in slices, never all at once. And the exponent nt is simply the total count of those slices. Interest alone is A − P. Whenever you meet the compounding formula wearing other clothes — a bank’s worked example, a textbook’s — it is this line, with n doing the only interesting decision.

The problem, solved five ways

Annually (n = 1). Rate per period 6%, 8 periods. A = 8000 × (1.06)⁸ = 8000 × 1.593848 = $12,750.78. Interest: $4,750.78.

Quarterly (n = 4). Rate per period 1.5%, 32 periods. A = 8000 × (1.015)³² = 8000 × 1.610324 = $12,882.59. Interest: $4,882.59.

Monthly (n = 12). The monthly compound interest formula is the same line with n = 12: rate per period 0.5%, 96 periods. A = 8000 × (1.005)⁹⁶ = 8000 × 1.614143 = $12,913.14. Interest: $4,913.14. If you want to feel it working, trace the first months: after month 1 the balance is $8,040.00 (interest $40.00), after month 2 $8,080.20, after month 3 $8,120.60 — and by month 12, when the balance reaches $8,493.42, that month’s interest alone is $42.26, because eleven months of interest are now earning interest too.

Daily (n = 365). Rate per period 0.06/365 ≈ 0.0164%, 2,920 periods. A = 8000 × (1 + 0.06/365)²⁹²⁰ = 8000 × 1.616011 = $12,928.09. Interest: $4,928.09.

Continuously (n → ∞). Compound ever more often and the formula converges on A = Pe^rt: A = 8000 × e^0.48 = 8000 × 1.616074 = $12,928.60. Interest: $4,928.60 — the ceiling the other four are climbing toward. Euler’s number is doing real work in that line: e^rt is the limit of (1 + r/n)^nt as n grows without bound, which is why continuous compounding is not a fifth method but the destination the other four are travelling toward.

One problem — $8,000, 6%, eight years — at every standard frequency. The full spread from annual to continuous is $177.81.

Adding deposits: the formula grows a second term

Real savers and investors rarely deposit once. Add a level monthly deposit and the compounding formula grows a twin term — the deposit stream, compounded the same way:

FV = P(1 + i)^N + PMT × [((1 + i)^N − 1) / i], with i the monthly rate and N the months.

This two-term version is the investment growth formula behind every retirement projection you have ever been shown. Run our problem through it — the same $8,000 at 6% compounded monthly for eight years, now with $150 added each month: the principal term is the $12,913.14 we already own, and the deposit term adds $18,424.28, for a final $31,337.42. Of that, $14,400 is deposits out of your pocket and $8,937.42 is interest — more interest than the original lump sum earned, generated by money that arrived gradually. That is the whole secret of the second term.

Spreadsheet-ready versions

Every result above, as formulas you can paste into a spreadsheet cell:

• Annual: =8000*(1+0.06)^8
• Monthly: =8000*(1+0.06/12)^(12*8)
• Daily: =8000*(1+0.06/365)^(365*8)
• Continuous: =8000*EXP(0.06*8)
• With a $150 monthly deposit: =FV(0.06/12, 96, -150, -8000) — the minus signs are the spreadsheet’s convention for money you pay in; the answer comes back positive: $31,337.42.

The Rule of 72, audited against the formula

You will hear that money at 6% doubles in 72 ÷ 6 = 12 years. The formula lets us grade that claim exactly: doubling time is ln 2 ÷ ln(1.06) = 11.90 years, and sure enough, 8000 × (1.06)^11.90 = $16,000.00. The rule overshoots by about a month at this rate — close enough for conversation, too loose for a contract. My standing advice: estimate with the rule, decide with the formula.

Three mistakes I catch in every spreadsheet

• A percent where a decimal belongs. Type 6 instead of 0.06 and the formula dutifully reports that $8,000 becomes more money than exists on Earth. The rate always enters as a decimal.
• A mismatched n. Pairing the monthly rate with the annual exponent — (1.005)⁸ instead of (1.005)⁹⁶ — turns eight years of monthly compounding into $8,325.66 and quietly deletes most of the interest. Rate per period and period count must always describe the same schedule.
• Deposits on the wrong side. The FV function wants money you pay in entered as negatives; flip the signs and it returns a negative balance — the spreadsheet politely telling you the cash is flowing backwards.

One last habit is worth copying: whatever a calculator or a spreadsheet tells me, I re-run the annual case by hand — it is the only frequency I can verify on a pocket calculator in under a minute — and if that one matches, I trust the machinery on the other four.

Sources: formula per standard college algebra references, with plain-language framing cross-checked against the SEC’s Investor.gov compound interest explainer. The 6% rate is illustrative; every figure was recomputed for this article.