What doubling time tells you that a growth rate does not
Doubling time converts an abstract percentage into a span of your own life. A 7% return means very little on its own; “this money doubles roughly every decade” immediately tells a 35-year-old that a balance today becomes roughly eight times as much by 65, because three decades hold three doublings.
The number depends on one thing only: the effective rate at which the balance grows. It does not depend on how much you start with. Ten dollars and ten million dollars both double in the same 9.0 years at 8% compounded annually, which is why the figure travels so well between conversations that involve wildly different sums.
The Rule of 72 exists because the exact answer needs a logarithm and nobody carries one. Divide 72 by the rate written as a whole number of percent and you get an estimate accurate to a fraction of a percent across the entire range of returns a diversified portfolio actually produces. The calculator above shows the shortcut and the exact figure side by side so you can see precisely where the approximation is safe and where it stops being safe.
Where 72 comes from, and why it is not 69.3
Start from compound growth. A balance multiplied by (1 + i) each year reaches twice its starting value when (1 + i)t = 2. Take logarithms of both sides and you get the exact expression the calculator uses:
t = ln 2 ÷ ln(1 + i), where ln 2 = 0.693147.
For small i, ln(1 + i) is very close to i itself, so t is close to 0.693147 ÷ i. Written in whole percent that is 69.3 ÷ r%. So the mathematically pure numerator is 69.3, and it is exactly right in the limit of continuous compounding, where t = ln 2 ÷ r with no approximation at all.
The reason practitioners use 72 rather than 69.3 is two-fold. First, ln(1 + i) is smaller than i, which pushes the true doubling time up above 69.3 ÷ r% for any discretely compounded rate, so a slightly larger numerator compensates. Second, 72 divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, which is most of the rates people quote. The compensation happens to be near-perfect around 7.8%, and that is the reason the rule earned its reputation: it is tuned to the exact region where long-run equity returns sit.
Reverse the question and you get the second formula. To double in T years with m compounding periods a year you need a nominal rate of m(21/(mT) − 1). The shortcut version is simply K ÷ T, so “double in 10 years” becomes 7.2% by the shortcut and 7.177% exactly. If you are solving the more general version of this problem — a target that is not exactly double, and contributions along the way — use the investment goal timeline calculator instead.
Worked example: 6% nominal, compounded monthly
Take a bond fund quoting 6% nominal with monthly compounding. Work it through by hand.
- Find the periodic rate. 6% ÷ 12 = 0.5% per month, or 0.005 as a decimal.
- Find the effective annual rate. (1.005)12 − 1 = 1.0616778 − 1 = 0.0616778, so 6.16778% a year. Monthly compounding buys you 0.16778 percentage points over the nominal quote.
- Apply the exact formula. ln 2 = 0.693147 and ln(1.0616778) = 0.0598505, so t = 0.693147 ÷ 0.0598505 = 11.5813 years.
- Cross-check in months. ln(1.005) = 0.00498754, so the balance doubles in 0.693147 ÷ 0.00498754 = 138.976 months. Divide by 12: 138.976 ÷ 12 = 11.5813 years. The two routes agree, as they must.
- Apply the shortcut. 72 ÷ 6 = 12.000 years.
- Measure the gap. (12.000 − 11.5813) ÷ 11.5813 = 0.03615, so the shortcut asks for 3.62% more time than the math requires. Five months of a twelve-year wait.
That 3.62% error is bigger than most people expect, and almost all of it comes from the compounding frequency rather than from the rule itself. Set the same 6% to annual compounding and the exact answer becomes 11.8957 years, cutting the shortcut error to 0.88%. The Rule of 72 is calibrated against annual compounding; every extra compounding period widens the gap.
How to read the error figure
The sign of the error tells you which way the shortcut is lying to you. Under annual compounding the two curves cross at approximately 7.85%: below that rate the shortcut quotes a longer wait than the exact calculation, and above it a shorter one. At 4% the shortcut says 18.00 years against an exact 17.673, overstating the wait by 1.85%. At 15% it says 4.800 years against an exact 4.9595, understating it by 3.22%.
Practically, that means the Rule of 72 is conservative exactly where you want conservatism — on bond-like returns of 3% to 7%, it tells you the money takes slightly longer than it really will. It becomes optimistic on high double-digit returns, which is also where a single-point return assumption is least trustworthy in the first place.
Inside the band from 5% to 10%, which covers almost every realistic long-run portfolio assumption, the shortcut is within 1.4% of exact. On a 14-year horizon that is about ten weeks. No planning decision turns on ten weeks, so for that band you can stop reading the exact column entirely.
One thing doubling time cannot do is tell you about purchasing power. A balance that doubles in nominal dollars over 12 years has not doubled in what it buys, because prices rise too. Feed the same nominal return through the inflation-adjusted return calculator and re-run the doubling time on the real rate: at 8% nominal with 3% inflation the real rate is 4.854%, and the real doubling time stretches from 9.0 years to 14.6 years.
Doubling time by annual rate: shortcut against exact
| Rate | 72 ÷ r | Exact years | Error |
|---|---|---|---|
| 1% | 72.000 | 69.661 | +3.36% |
| 2% | 36.000 | 35.003 | +2.85% |
| 3% | 24.000 | 23.450 | +2.35% |
| 4% | 18.000 | 17.673 | +1.85% |
| 5% | 14.400 | 14.207 | +1.36% |
| 6% | 12.000 | 11.896 | +0.88% |
| 7% | 10.286 | 10.245 | +0.40% |
| 8% | 9.000 | 9.006 | −0.07% |
| 9% | 8.000 | 8.043 | −0.54% |
| 10% | 7.200 | 7.273 | −1.00% |
| 12% | 6.000 | 6.116 | −1.90% |
| 15% | 4.800 | 4.959 | −3.22% |
| 20% | 3.600 | 3.802 | −5.31% |
Values generated by the calculator above with compounding set to annual. Switch the compounding selector and the on-page table regenerates for that frequency.
Mistakes that make a doubling time wrong
- Feeding in the rate as a decimal. The shortcut is 72 divided by the rate as a whole number of percent. 72 ÷ 0.08 is 900, not 9. The exact formula wants the decimal; the shortcut wants the percent.
- Using a nominal rate where the compounding is not annual. A 6% rate compounded monthly is a 6.16778% effective rate, and the exact doubling time shortens accordingly. Convert to an effective annual rate first, or let the compounding selector do it.
- Treating a nominal doubling as a doubling of purchasing power. Inflation runs the whole time. The real doubling time is always longer than the nominal one whenever inflation is positive.
- Applying it to a volatile return path. The formula assumes a constant rate. A portfolio averaging 8% with large swings compounds at less than 8% because of variance drag, so use a realised CAGR rather than an arithmetic average return.
- Forgetting fees and tax. A 0.75% expense ratio on an 8% gross return leaves 7.25%, which pushes the doubling time from 9.0 years out to 9.9 years. Over 40 years that is 4.44 doublings against 4.04 — the difference between multiplying your money by roughly 22 times and by roughly 16 times.
- Assuming it also works for halving. It does, but the sign matters: at a 6% annual loss the balance halves in ln 0.5 ÷ ln 0.94 = 11.2 years, not 12.
Rule of 70, rule of 69.3, and when to use each
Three numerators are in circulation and each is right in a different setting. Use 69.3 when growth is continuous — population models, bacterial growth, radioactive decay in reverse, anything modelled as ert. There the answer is not an approximation at all: ln 2 ÷ r is exact, so 69.3 ÷ r% is exact too.
Use 70 for inflation and GDP growth. Economists favour it because it is arithmetically friendly with the low single-digit rates macro data produces and because national accounts growth is closer to continuous than to annually compounded. At 3% inflation, 70 ÷ 3 = 23.3 years for prices to double, against an exact 23.45 years.
Use 72 for investment returns quoted as annual percentages, which is what this calculator defaults to. It is the only one of the three tuned for discrete annual compounding, and its accuracy peak sits inside the range where equity return assumptions live.
Doubling time is a communication tool, not a planning tool. Once you need to include contributions, a specific target that is not double, or a withdrawal phase, move to a full time-value model: the compound interest calculator for a single lump sum, the future value calculator when regular deposits are involved, and the Coast FIRE calculator when the question is whether existing savings will grow into a retirement number on their own.
Key terms
- Nominal rate
- The headline annual rate quoted before accounting for how often it compounds. A 6% nominal rate compounded monthly credits 0.5% twelve times.
- Effective annual rate
- The single annual rate that produces the same year-end balance as the stated nominal rate compounding at its stated frequency: (1 + r/m)m − 1.
- Continuous compounding
- The limit as compounding frequency goes to infinity, giving a growth factor of er. Doubling time is exactly ln 2 ÷ r.
- Variance drag
- The gap between the arithmetic average of a series of returns and the compound rate actually realised. It grows with volatility, so a bumpy 8% doubles slower than a smooth 8%.