Compound interest is the only financial calculation most people will ever need and the one most often entered wrong. The formula itself is short. The mistakes are not in the arithmetic — they are in the rate, the frequency, the timing of contributions, and the four drags that quietly reduce a headline return to something much smaller.
Compound interest grows a balance by applying interest to both the original principal and the interest already earned. The formula is A = P(1 + r/n)nt, where P is the principal, r is the nominal annual rate as a decimal, n is the number of compounding periods per year, and t is the number of years. With continuous compounding, the formula becomes A = Pert.
The effective annual rate — the number that lets you compare accounts with different frequencies — is EAR = (1 + r/n)n − 1, or er − 1 for continuous compounding. A quoted 6% becomes 6.00% compounded annually, 6.17% compounded monthly, and 6.18% compounded daily or continuously.
The formula and the variable trap
Every compounding calculation on every calculator reduces to one expression. The difficulty is not the algebra — it is that three of the four variables are easy to enter in the wrong form.
Three entries have to agree with each other, and this is where results go wrong:
- r is a decimal, not a percentage. A 7% rate is entered as 0.07. Entering 7 multiplies the balance by 8 each period.
- r/n is the periodic rate, and nt is the total period count. The two must be built from the same n. A 7% rate compounded monthly uses 0.07/12 = 0.0058333 per period, over 12t periods. Pairing a monthly periodic rate with annual periods produces a result that is neither.
- t is in years, and nt is periods. If you know the term in months, convert to years before entering t, or enter the month count directly as nt and divide the rate by 12.
The reason the exponent matters so much is that growth is multiplicative, not additive. Each period's interest joins the base for the next period, so the balance follows a curve rather than a line. The diagram below shows where the extra money comes from.
The shaded band between the straight simple-interest line and the curved compound line is interest earned on interest. It is zero in year one, small in year five, and dominant by year thirty. That is why the term length matters more than the rate for most realistic savings horizons — and why a calculator that only shows the final balance hides the most important part of the story.
Compound interest: A = P(1 + r/n)^(nt). Continuous compounding: A = Pe^(rt). Effective annual rate: EAR = (1 + r/n)^n − 1, or e^r − 1. Simple interest grows linearly; compound interest grows exponentially, with the difference equal to interest earned on previously credited interest.
A rate without a frequency is not a rate
This is the single largest gap in most online calculators. They ask for an “interest rate” and a “compounding frequency” as if they were independent inputs. They are not. The rate alone is ambiguous until you know whether it is a nominal rate or an effective one.
- Nominal rate (APR, or quoted rate): the annual rate before compounding is applied. It is the input to the formula above.
- Effective annual rate (EAR, APY, AER): the actual annual growth after compounding. It is the output you compare between products.
Two accounts can both advertise “6%” and grow at different speeds. The one compounding monthly grows 6.17% a year; the one compounding annually grows 6.00%. The difference is 17 basis points — small in one year, substantial over a decade.
| Compounding | n | Periodic rate | Effective annual rate | Gain vs annual |
|---|---|---|---|---|
| Annual | 1 | 6.0000% | 6.0000% | — |
| Semi-annual | 2 | 3.0000% | 6.0900% | +9.0 bp |
| Quarterly | 4 | 1.5000% | 6.1364% | +13.6 bp |
| Monthly | 12 | 0.5000% | 6.1678% | +16.8 bp |
| Daily | 365 | 0.01644% | 6.1831% | +18.3 bp |
| Continuous | ∞ | — | 6.1837% | +18.4 bp |
One basis point (bp) is 0.01 percentage point. The full span from annual to continuous at a 6% nominal rate is 18.4 bp — real, but far smaller than the difference between a 6% and a 7% product.
If a savings account advertises 4.50% APY, that figure already includes compounding. Entering 4.50% as the nominal rate with monthly compounding overstates the effective yield by about 9 basis points a year, because the calculator compounds a number that was already compounded. The correct nominal input that produces 4.50% APY at monthly compounding is 4.4098%. The conversion is n × ((1 + APY)1/n − 1).
The practical rule: when comparing products, compare effective rates; when entering a calculator, confirm which kind of rate the tool expects. Most calculators label the field “interest rate” and silently assume nominal. That assumption is correct for a mortgage APR and wrong for a savings APY.
Frequency, quantified — and why it stops mattering
Compounding frequency sounds like it should be a major lever. In practice it is a large lever between annual and monthly, and almost no lever between monthly and daily. The table below is the one most calculator pages omit: it shows not just the ending balance at each frequency, but the marginal gain from adding more periods.
| Frequency | Periods per year | Balance after 1 year on $10,000 at 6% | Marginal gain | Marginal gain as % |
|---|---|---|---|---|
| Annual | 1 | $10,600.00 | — | — |
| Semi-annual | 2 | $10,609.00 | +$9.00 | 0.085% |
| Quarterly | 4 | $10,613.64 | +$4.64 | 0.044% |
| Monthly | 12 | $10,616.78 | +$3.14 | 0.030% |
| Daily | 365 | $10,618.31 | +$1.53 | 0.014% |
| Continuous | ∞ | $10,618.37 | +$0.06 | 0.001% |
The marginal column is the key. Going from annual to monthly captures $16.78 of the available $18.37. Going from monthly all the way to continuous captures the remaining $1.59 — less than a tenth of the total.
This is a geometric series converging on a limit. Each additional compounding period adds less than the one before, because the amount of interest being reinvested within the year gets smaller. The limit is the continuous case, where the balance is Pert. That limit is why continuous compounding is used in derivatives pricing and quantitative finance: it makes the algebra additive across time, because er·t₁ × er·t₂ = er(t₁+t₂).
The practical takeaway for a saver: frequency is a tiebreaker, not a driver. A 6.18% rate compounded annually beats a 6.00% rate compounded daily, because the rate gap (18 bp) is almost exactly the entire frequency effect (18.4 bp). When comparing two products, compare the effective annual rate and ignore the frequency label — it is already baked in.
At a 6% nominal rate, the effective annual rate rises from 6.000% (annual) to 6.168% (monthly) to 6.183% (daily) to 6.184% (continuous). The entire frequency effect is 18.4 basis points. Increasing frequency from monthly to continuous adds only 1.6 basis points.
Contributions change the shape, not just the size
A lump sum grows as a single exponential curve. A series of contributions grows as a sum of overlapping curves, and the result is governed by a different formula — the future value of an annuity.
Two details are routinely ignored by calculators, and both change the answer:
- Ordinary annuity vs annuity due. If contributions arrive at the end of each period, the formula above applies. If they arrive at the beginning, every contribution earns one extra period, so the result is multiplied by (1 + i).
- Contribution frequency must match the compounding frequency. Monthly contributions to a monthly-compounding account use i = r/12 and n = 12t. Annual contributions to a monthly-compounding account require either converting the contribution stream to an equivalent monthly amount or using an effective annual rate for i.
| Scenario | Contribution | Rate | Years | Timing | Ending balance |
|---|---|---|---|---|---|
| A | $500 / month | 7% | 30 | End of month | $609,990 |
| B | $500 / month | 7% | 30 | Beginning of month | $613,548 |
| Difference from timing alone | +$3,558 | ||||
Seven-tenths of one percent of the ending balance comes from the timing of contributions rather than their size. Most calculators do not expose this switch, which means they silently assume one convention and present it as the answer.
Time in the market versus amount contributed
The contribution formula produces a counter-intuitive result that is worth internalising: starting early beats contributing more, because the earliest contributions have the longest to compound. The comparison below uses $2,400 a year at 7% with end-of-year contributions.
| Strategy | Contributing years | Total contributed | Balance at 65 | Interest earned |
|---|---|---|---|---|
| Start at 25, stop at 35 | 10 | $24,000 | $252,395 | $228,395 |
| Start at 35, continue to 65 | 30 | $72,000 | $226,706 | $154,706 |
| Early strategy advantage | −$48,000 | +$25,689 | +$73,689 | |
Contributing one-third as much, ten years earlier, produces a larger balance. The first strategy's final balance is 71% interest; the second's is 68% — and the first gets there with $48,000 less capital at risk.
Five worked examples, one for each question type
Each example shows the substituted values and the rounding decision. All use full precision internally and round only the final figure.
$25,000 at 6.5% for 25 years, compounded monthly
P = 25,000 · r = 0.065 · n = 12 · t = 25
Periodic rate = 0.065/12 = 0.00541667. Total periods = 12 × 25 = 300. Growth factor = (1.00541667)300 = 5.0558. A = 25,000 × 5.0558 = 126,395.
Answer: $126,395 — of which $101,395 is interest.
$25,000 at 6.5% for 25 years, compounded continuously
P = 25,000 · r = 0.065 · t = 25
A = 25,000 × e0.065 × 25 = 25,000 × e1.625 = 25,000 × 5.0784 = 126,960.
Answer: $126,960 — $565 more than monthly compounding, or 0.45%.
$500 per month at 7% for 30 years, end of month
PMT = 500 · i = 0.07/12 = 0.0058333 · n = 360
Growth factor = [(1.0058333)360 − 1] / 0.0058333 = 1,219.97. FV = 500 × 1,219.97 = 609,990.
Answer: $609,990 — $180,000 contributed, $429,990 interest (70.5%).
How long to reach $500,000 at $500/month and 6%?
PMT = 500 · FV = 500,000 · i = 0.005
Set 500,000 = 500 × [(1.005)n − 1]/0.005. This gives (1.005)n = 6. Solving with logarithms: n = ln(6)/ln(1.005) = 359.25 periods.
Answer: 359 months ≈ 29.9 years — just under 30 years.
What monthly deposit reaches $1,000,000 in 30 years at 7%?
FV = 1,000,000 · i = 0.0058333 · n = 360
PMT = FV / growth factor = 1,000,000 / 1,219.97 = 819.69.
Answer: $819.69 per month — about $820.
The reverse cases are where most free calculators stop. Investor.gov, NerdWallet and Calculator.net all run forward; only a few support “solve for time” or “solve for contribution,” and those that do usually hide it behind a separate tab. The algebra is the same formula rearranged, so a calculator that accepts only forward inputs is choosing to do less work, not doing something harder.
The four drags on nominal return
A projected balance is a nominal figure before anything is subtracted. Four forces reduce it in sequence, and they compound — which means their combined effect is larger than the sum of their individual percentages.
The ordering matters. Fees are charged on assets, so they reduce the return before tax. Tax is charged on the nominal return, not the real return, which is why inflation makes the effective tax rate higher than the statutory one. Inflation applies last, to whatever survives.
| Layer | Rate | Running return | Balance after 30 years on $10,000 | Lost vs nominal |
|---|---|---|---|---|
| Nominal return | 7.00% | 7.00% | $76,123 | — |
| less 1% annual fee | −1.00% | 6.00% | $57,435 | −$18,688 |
| less 22% tax on return | −1.32% | 5.46% | $49,274 | −$26,849 |
| less 3% inflation | −1.56% | 3.12% | $25,132 | −$50,991 |
The final figure is in today's purchasing power. A nominal projection of $76,000 is worth about $25,000 of current spending — a 67% reduction. The simple subtraction 7 − 1 − 1.32 − 3 = 1.68% is far off the true 3.12%, because the drags interact multiplicatively.
Subtracting percentages assumes each drag applies to the original amount. In reality each applies to the balance that survived the previous drag. A 7% return reduced by a 1% fee is 6.00%, not 6.00 percentage points off 7; then 22% tax on that 6.00% costs 1.32 points, not 1.54; then 3% inflation divides by 1.03 rather than subtracting 3 points. At small rates the difference is a few basis points. At larger rates or longer horizons it becomes thousands of dollars.
Fee drag in isolation
| Gross return | Annual fee | Net return | $10,000 after 30 years | Cost of the fee | Fee as % of gross result |
|---|---|---|---|---|---|
| 7% | 0.00% | 7.00% | $76,123 | — | — |
| 7% | 0.25% | 6.75% | $70,775 | $5,348 | 7.0% |
| 7% | 0.50% | 6.50% | $65,828 | $10,295 | 13.5% |
| 7% | 1.00% | 6.00% | $57,435 | $18,688 | 24.5% |
| 7% | 2.00% | 5.00% | $43,219 | $32,904 | 43.2% |
A 1% annual fee consumes nearly a quarter of the ending balance over 30 years — not 1% of it. The reason is that the fee is charged on a growing asset base while its opportunity cost compounds.
This is the number that makes fee disclosure worth reading. It is also the number most calculator pages omit, because their output is a gross projection with no expense line.
Doubling time and the rule of 72
The rule of 72 estimates how long an investment takes to double by dividing 72 by the annual rate. It is a mental shortcut, not a formula, and its accuracy varies predictably with the rate. The exact doubling time is derived from the compound formula itself.
| Annual rate | Rule of 72 | Rule of 69.3 | Exact (years) | 72's error | Error as % |
|---|---|---|---|---|---|
| 1% | 72.00 | 69.30 | 69.66 | +2.34 | +3.4% |
| 3% | 24.00 | 23.10 | 23.45 | +0.55 | +2.4% |
| 5% | 14.40 | 13.86 | 14.21 | +0.19 | +1.4% |
| 7% | 10.29 | 9.90 | 10.24 | +0.05 | +0.5% |
| 8% | 9.00 | 8.66 | 9.01 | −0.01 | −0.1% |
| 10% | 7.20 | 6.93 | 7.27 | −0.07 | −1.0% |
| 15% | 4.80 | 4.62 | 4.96 | −0.16 | −3.2% |
| 20% | 3.60 | 3.47 | 3.80 | −0.20 | −5.3% |
The rule of 72 is most accurate in the 7–8% band, which is why it is so often quoted — that is also the band of long-run equity returns. It understates doubling time at low rates and overstates it at high rates.
Two extensions are worth knowing:
- Halving time for inflation. The same rule applied to the inflation rate gives the years until purchasing power halves. At 3.5% inflation, 70/3.5 = 20 years — the reason a 20-year retirement projection in nominal dollars is misleading.
- Tripling and other multiples. Replace the 2 in the exact formula with the target multiple. Tripling time is ln(3)/ln(1 + r); a 50% gain is ln(1.5)/ln(1 + r).
Exact doubling time for periodic compounding is t = ln(2)/ln(1 + r). For continuous compounding it is t = ln(2)/r ≈ 0.693/r. The rule of 72 is an approximation that is most accurate between 7% and 8%; at 20% it overstates doubling time by 5.3%, and at 1% it understates it by 3.4%.
Anatomy of one year-by-year run
A final balance is a single number. The interesting part is how the composition of growth changes over time. This run uses $10,000 initial capital, $6,000 added at the end of each year, and 7% annual compounding.
| Year | Opening balance | Interest earned | Contribution | Closing balance | Interest share of growth |
|---|---|---|---|---|---|
| 1 | $10,000 | $700 | $6,000 | $16,700 | 10.4% |
| 2 | $16,700 | $1,169 | $6,000 | $23,869 | 16.3% |
| 3 | $23,869 | $1,671 | $6,000 | $31,540 | 21.8% |
| 4 | $31,540 | $2,208 | $6,000 | $39,748 | 26.9% |
| 5 | $39,748 | $2,782 | $6,000 | $48,530 | 31.7% |
| 6 | $48,530 | $3,397 | $6,000 | $57,927 | 36.1% |
| 7 | $57,927 | $4,055 | $6,000 | $67,982 | 40.3% |
| 8 | $67,982 | $4,759 | $6,000 | $78,741 | 44.2% |
| 9 | $78,741 | $5,512 | $6,000 | $90,253 | 47.9% |
| 10 | $90,253 | $6,318 | $6,000 | $102,570 | 51.3% |
Total contributed: $70,000. Total interest: $32,570. By year ten, interest exceeds the annual contribution for the first time — the point at which the portfolio begins to grow more from its own earnings than from new money.
This is the crossover every long-horizon saver is waiting for, and it happens earlier than most people expect. The interest share roughly follows 1 − (contribution ÷ total growth), so it rises steadily and then accelerates once the opening balance is large relative to the annual addition. A calculator that returns only the final number hides this entirely.
Common mistakes and what each one costs
Each error below produces a predictable direction and size of error. Knowing the signature makes it easy to spot in a result that looks wrong.
| Mistake | What it does | Direction of error | Size on a $50,000 / 20-year / 6% case |
|---|---|---|---|
| Entering 6 instead of 0.06 | Multiplies by 7 each period | Absurdly high | Quadrillions |
| Using an APY as a nominal rate | Compounds an already-compounded figure | Too high | ≈ +2% on balance |
| Mismatching r/n with nt | Monthly rate with annual periods | Far too low | ≈ −80% |
| Ignoring contribution timing | Assumes the wrong end of period | Too low (usually) | ≈ −0.6% |
| Adding contributions without growth | Treats deposits as non-compounding | Too low | −15% to −30% |
| Using simple interest | Interest on principal only | Too low | −27% |
| Ignoring tax and fees | Reports gross, not net | Too high | +30% to +60% |
| Ignoring inflation | Reports nominal, not real | Too high | +45% at 3% |
The two mismatched-unit errors (first and third rows) are the only ones that produce results off by an order of magnitude. The rest are accuracy problems rather than arithmetic failures, and they are the ones that matter for real planning.
It is treating a nominal projection as a spending plan. A calculator that reports $500,000 in 30 years is answering a question about nominal balance, not about lifestyle. After 3% inflation, $500,000 in 30 years buys what roughly $206,000 buys today. Any plan built on the nominal figure will overstate available spending by about 60%.
Compounding in reverse: decumulation
Every calculator on the market models accumulation. Almost none models the phase where the balance is being drawn down, and that phase has a property accumulation does not: the order of returns matters.
During accumulation, a −20% year followed by a +25% year leaves you exactly where you started before contributions. During decumulation, the same two years in the reverse order produce different ending balances, because the withdrawal in the bad year removes shares that would have participated in the recovery.
| Year | Return | Balance start | Withdrawal | Balance after withdrawal | Balance end |
|---|---|---|---|---|---|
| 1 | −20% | $1,000,000 | $40,000 | $960,000 | $768,000 |
| 2 | +25% | $768,000 | $40,000 | $728,000 | $910,000 |
| Same two years, reversed order | — | ||||
| 1 | +25% | $1,000,000 | $40,000 | $960,000 | $1,200,000 |
| 2 | −20% | $1,200,000 | $40,000 | $1,160,000 | $928,000 |
A 0% average return over two years produces $910,000 in the bad-first sequence and $928,000 in the good-first sequence — an $18,000 gap from ordering alone, on the same returns and the same withdrawals.
This is sequence-of-returns risk, and it is the reason a fixed-percentage withdrawal rule is safer than a fixed-dollar rule in the early years of retirement. It is also the reason a compound interest calculator, however precise, cannot answer “will my money last” — that question needs a distribution model with volatility, not a single deterministic rate.
Use a compound interest calculator for the accumulation phase, where a single average rate is a reasonable model. For the drawdown phase, model at least three return sequences (bad-first, good-first, and average-every-year) and size the withdrawal to survive the worst of them.
What the number cannot tell you
A compound interest projection is a deterministic answer to a probabilistic question. It is useful, but it is worth being explicit about the four things it does not model.
1. Volatility drag
A portfolio that returns +50% then −50% has an arithmetic average return of 0%, but a geometric return of −25%. The ending balance is 75% of the start. A calculator using a constant 0% rate would report no change. The gap between arithmetic and geometric return grows with volatility — roughly σ²/2 — which is why a volatile 7% average underperforms a smooth 7%.
2. Return sequencing
As §10 showed, the order of returns matters during withdrawals. It also matters during accumulation if contributions are large relative to the balance, because a bad early year reduces the base on which later contributions compound.
3. Taxes on accrual versus realisation
A tax-deferred account (401(k), IRA) compounds pre-tax and is taxed on withdrawal. A taxable account is taxed annually on dividends and interest, and on gains at sale. These two paths produce different ending balances for the same gross return, and only one of them can be modelled by a single tax rate applied at the end.
4. Liquidity and access
A projection assumes the money stays invested for the full term. Early withdrawal penalties, lock-up periods, and the need to sell at an inopportune moment are not in the formula. For a 30-year projection that is usually fine; for a 3-year one it is not.
A compound interest calculator answers: “If I earn exactly r every year for t years, with these contributions, what is the balance?” That is a useful hypothetical. It is not a forecast. Treat the output as a centre point around which real outcomes will scatter, not as the outcome itself.
A four-check verification routine
Any compound interest result can be sanity-checked in under a minute without redoing the full calculation.
Check 1 — the doubling bound
Divide 72 by the annual rate to estimate the doubling time, then confirm the balance roughly doubles that many times. At 7% over 30 years, the doubling time is about 10.3 years, so the balance should be about 2³ = 8 times the initial amount, before contributions. $10,000 at 7% for 30 years is $76,123, which is 7.6× — consistent with eight doublings less the continuous approximation error.
Check 2 — effective rate consistency
Compute the effective annual rate from the inputs, then confirm the balance after one year equals P × (1 + EAR). If the two disagree, the periodic rate and the period count were not built from the same n.
Check 3 — the contribution proportion
Contributions should be a plausible share of the ending balance. If you contributed $180,000 over 30 years and the ending balance is $610,000, contributions are 29.5% of the total and interest is 70.5%. If a calculator reports interest as 95% of the balance on those inputs, the rate or the term is wrong.
Check 4 — the real-value haircut
Divide the nominal balance by (1 + inflation)t. At 3% over 30 years the divisor is 2.427, so $610,000 becomes $251,000 in today's money. If the real figure seems implausible relative to the nominal one, the term or the inflation assumption is off.
Multiply the principal by the growth factor you can estimate from the rule of 72, then add contributions times roughly the same factor. If the calculator's answer is more than about 15% away from that estimate, one of the four inputs is wrong — most often the rate format.
Growth multiplier reference
The factor by which a single lump sum grows, for common rate and horizon combinations. Multiply the principal by the factor to get the future value with no contributions.
| Rate | 5 years | 10 years | 20 years | 30 years | 40 years |
|---|---|---|---|---|---|
| 3% | 1.159 | 1.344 | 1.806 | 2.427 | 3.262 |
| 4% | 1.217 | 1.480 | 2.191 | 3.243 | 4.801 |
| 5% | 1.276 | 1.629 | 2.653 | 4.322 | 7.040 |
| 6% | 1.338 | 1.791 | 3.207 | 5.743 | 10.286 |
| 7% | 1.403 | 1.967 | 3.870 | 7.612 | 14.974 |
| 8% | 1.469 | 2.159 | 4.661 | 10.063 | 21.725 |
| 10% | 1.611 | 2.594 | 6.727 | 17.449 | 45.259 |
The non-linearity is the whole story: moving from 30 to 40 years at 7% multiplies the ending balance by 1.97, while moving from 5 to 15 years multiplies it by only 1.83. The last decade contributes more than the first two combined.
Questions people actually search for
Twelve answers, each self-contained. Tap to expand.
What is the compound interest formula?
A = P(1 + r/n)nt, where P is principal, r is the nominal annual rate as a decimal, n is compounding periods per year, and t is years. For continuous compounding, A = Pert. The effective annual rate is (1 + r/n)n − 1 for periodic compounding and er − 1 for continuous.
What is the difference between compound and simple interest?
Simple interest is calculated only on the original principal, so it grows linearly. Compound interest is calculated on principal plus previously earned interest, so it grows exponentially. On $10,000 at 6% for 20 years, simple interest gives $22,000 and compound interest (annual) gives $32,071 — a 46% difference. The gap widens with every additional year.
How much difference does compounding frequency really make?
Less than most people assume. At a 6% nominal rate, the effective annual rate rises from 6.000% (annual) to 6.168% (monthly) to 6.183% (daily) and 6.184% (continuous). The entire span is 18.4 basis points. Going from monthly to continuous adds only 1.6 bp — under a tenth of the total frequency effect.
What is the difference between APR and APY?
APR is a nominal annual rate that does not include compounding. APY (also called EAR or effective annual rate) does include it. A 6% APR compounded monthly is a 6.168% APY. When comparing savings products, compare APY. When entering a calculator, confirm whether it wants the nominal rate or the effective one — entering an APY as if it were nominal double-counts compounding by a few basis points a year.
Can I enter a monthly contribution into an annually compounding calculator?
Not directly. The contribution frequency and the compounding frequency must be reconciled. Either convert the monthly deposits into an equivalent annual amount and use annual periods, or convert the annual rate into a monthly periodic rate (r/12) and use 12t periods throughout. Mixing a monthly deposit with an annual period count understates growth, because the deposits do not compound as often as the formula assumes.
Does it matter whether contributions are at the beginning or end of the period?
Yes, by one period's worth of growth per contribution. Contributions at the beginning of each period (an annuity due) are worth (1 + i) times more than the same contributions at the end (an ordinary annuity). On $500/month at 7% for 30 years, the difference is about $3,558 — roughly 0.6% of the ending balance. Most calculators silently assume one convention without saying which.
How long does it take to double my money?
Divide 72 by the annual percentage rate for a quick estimate. At 7%, 72/7 ≈ 10.3 years. The exact answer is t = ln(2)/ln(1 + r), which gives 10.24 years. For continuous compounding the exact form is ln(2)/r ≈ 0.693/r. The rule of 72 is most accurate in the 7–8% range and drifts by 3–5% at very low or very high rates.
Why is my calculator's answer different from another calculator's?
Three likely causes, in order of frequency: (1) one tool treats the rate as nominal and the other as effective; (2) contribution timing differs between beginning and end of period; (3) one tool compounds the contribution at the contribution frequency and the other at the account frequency. A fourth possibility is rounding of the periodic rate — a 7% monthly rate is 0.58333%, and truncating it to 0.58% shifts the ending balance by several percent over 30 years.
What is continuous compounding and when is it used?
Continuous compounding is the limit as the number of compounding periods goes to infinity: A = Pert. It is used in derivatives pricing (Black–Scholes), in quantitative finance for its additive time property, and in academic finance. It is not used by retail savings accounts — no bank compounds continuously. For a saver, continuous compounding is the theoretical ceiling of the frequency effect, worth about 0.18 percentage points over annual compounding at a 6% rate.
How do taxes and inflation change the result?
They compound in sequence and reduce the result more than adding their percentages suggests. A 7% nominal return with a 1% fee, 22% tax and 3% inflation nets about 3.12% in real after-tax terms — not the 1.68% that simple subtraction (7 − 1 − 1.32 − 3) would suggest. On $10,000 over 30 years, that is the difference between $76,123 nominal and about $25,132 in today's purchasing power.
What does a 1% fee actually cost over 30 years?
On a 7% gross return, a 1% annual fee reduces the net return to 6% and the 30-year balance from $76,123 to $57,435 on a $10,000 start. That is $18,688 — 24.5% of the ending balance, not 1% of it. The fee is charged on a growing asset base, so its opportunity cost compounds alongside the portfolio.
Can compound interest work against me?
Yes — on any balance you owe. Credit card debt at 22% APR compounded daily grows faster than almost any investment can. A $5,000 balance left unpaid grows to about $6,235 in one year and $13,700 in five. The same mathematics that builds a retirement account dismantles a household budget, which is why paying down high-rate debt is usually the highest-return financial action available.
Run the numbers with the rate and frequency separated
The calculator takes a nominal rate and a compounding frequency as separate inputs, shows the resulting effective annual rate, and returns the year-by-year split between contributions and interest — so you can see whether the growth is coming from your deposits or from the compounding itself.
Open the compound interest calculator