Compounding frequency comparison
| Frequency | n | Future Value |
|---|---|---|
| Annually | 1 | $32,071 |
| Semi-annually | 2 | $32,620 |
| Quarterly | 4 | $32,907 |
| Monthly | 12 | $33,102 |
| Daily | 365 | $33,198 |
| Continuous | ∞ | $33,201 |
Take $5,000 at 8% for 10 years: annual compounding grows it to $10,794.62, quarterly to $11,040.20, monthly to $11,098.20, and continuous to $11,127.70. That's a $333 spread across every compounding frequency at the same nominal rate.
APR vs APY
Lenders advertise APR (nominal annual rate); APY shows what you actually earn or pay after compounding. The gap is small at low rates and significant at credit-card rates: 24% APR monthly compounded is 26.82% APY. A high-yield savings account usually advertises APY rather than APR precisely because the bank already compounds daily and wants to quote the figure savers actually receive: $2,000 at a 4.5% APY left untouched for 3 years grows to 2,000 × (1.045)^3 = $2,282.33.
Compounding versus inflation
These calculations ignore inflation, so a nominal 6% return is not a 6% real return once prices are rising too. $10,000 growing at 6% nominal for 20 years reaches $32,071.35 in future dollars, but at 3% average inflation its purchasing power is only 32,071.35 / (1.03)^20, about $17,757 in today's dollars. The number on a savings statement overstates real wealth growth whenever prices are also climbing.
Why extra years matter more than they look
Extending the holding period compounds gains on top of gains rather than adding them up. $10,000 at 7% compounded annually reaches $19,671.51 after 10 years and $38,696.84 after 20 years. By 30 years it reaches $76,122.55, nearly double the 20-year figure even though the rate never changed. Each extra decade adds a larger dollar gain than the one before it, $9,671.51 in the first decade versus $37,425.71 in the third, because compounding always grows a bigger base.
Starting early matters more than the amount contributed later, for the same reason. $5,000 invested at 7% and left for 40 years grows to 5,000 × (1.07)^40 ≈ $74,872. Waiting ten years to invest the same $5,000, leaving only 30 years to compound, grows to just 5,000 × (1.07)^30 ≈ $38,061, a difference of roughly $36,811 from a single ten-year delay in getting started.
Most mortgages compound monthly but amortize with fixed payments, so extra principal paid early in the loan saves more interest than the same extra payment made later, because interest is charged on a shrinking balance that then compounds for a shorter remaining time. It is the same reason paying more than the minimum on a high APR credit card, discussed above, saves disproportionately more interest the earlier it happens.
A useful rule of thumb: an extra $100 principal payment made in month 1 of a 6% APR, 30-year loan saves more total interest than a $150 extra payment made in month 200, because the earlier dollar avoids compounding for hundreds of additional months while the later dollar was only ever going to compound for a few years regardless.
How long until your money doubles
A quick way to estimate doubling time is the Rule of 72: divide 72 by the interest rate as a whole number percentage. At 6% APR that gives 12 years; at 9% it gives 8 years. The shortcut is accurate to within a few months for rates between about 2% and 15%. For a precise answer at any rate or compounding frequency, use the doubling time calculator, which solves (1 + r/n)^(nt) = 2 directly instead of relying on the approximation.
Worked example
$10,000 at 7% APR compounded monthly for 30 years: A = 10,000 × (1 + 0.07/12)^(12 × 30) = $81,164.97. Total interest = $71,164.97. If you add regular monthly deposits on top of that lump sum instead of leaving it untouched, switch to the investment growth calculator, which layers ongoing contributions onto this same compounding formula.
FAQ
What is the compound interest formula?
The compound interest formula is A = P × (1 + r/n)^(nt) for discrete compounding, where P is the principal, r the annual rate, n the number of compounding periods per year and t the number of years. When compounding happens continuously instead of at fixed intervals, the formula becomes A = P × e^(rt). Both describe the same idea: interest earning interest over time.
How often does interest compound?
Common compounding frequencies are annually (n = 1), semi-annually (n = 2), quarterly (n = 4), monthly (n = 12), daily (n = 365) and continuously. The frequency matters less than it seems: a 6% annual rate compounded monthly returns a 6.17% effective annual yield, while compounding it continuously returns only 6.18%, a difference of about one hundredth of a percentage point.
What is the difference between APR and APY?
APR is the nominal annual rate before accounting for compounding, while APY (also called effective annual rate) includes it: APY = (1 + r/n)^n − 1. The gap grows with both the rate and the compounding frequency. A 12% APR compounded monthly produces a 12.68% APY, so the amount you actually earn or owe is always a bit higher than the advertised rate.
Is daily or continuous compounding much better?
Barely. At 5% APR, daily compounding yields 5.1267% APY versus 5.1271% continuous, a difference of roughly four cents per $1,000 per year. Continuous compounding is the limit of daily, hourly and by-the-second compounding all at once, but for a real savings account the extra return over daily compounding is too small to matter.
How long to double my money?
The Rule of 72 gives a fast doubling-time estimate: divide 72 by the interest rate expressed as a whole number percentage. A 4% return doubles money in about 18 years, a 9% return takes about 8 years, and a slower 3% return needs roughly 24 years. This shortcut holds up well for rates between about 2% and 15%; beyond that range, solve (1 + r/n)^(nt) = 2 directly for an exact answer.
Does compound interest work against me on loans?
Yes. The same formula that grows savings also grows debt. At 24% APR compounded monthly, a $1,000 unpaid credit card balance grows to roughly $3,281 after 5 years with no payments, more than tripling. Paying down high APR revolving debt quickly matters more, as a result, than almost any other personal finance decision.