Doubling time of an investment from its annual rate.
Rule-of-72 approximation. Exact: years = ln(2) / ln(1 + r). Diverges from 72/r above 15% APR.
The rule of 72 is the single most quoted shortcut in personal finance. It answers a question every saver eventually asks: at what annual rate, and over how many years, will my money double? The full compound-interest equation — years = ln(2) / ln(1 + r) — is correct but not memorisable; 72/r is correct enough across the rates that matter (3 % to 12 %) to be done in your head while reading a fund prospectus. A 6 % nominal return doubles your money in 12 years. An 8 % return doubles it in 9 years. A 4 % return takes 18. The rule short-circuits one of the most common failures of financial intuition: people massively underestimate how much a small rate difference compounds. A retirement portfolio earning 7 % beats one earning 4 % by a factor of two and a half over thirty years — and the rule of 72 makes that obvious in three seconds. This calculator runs the rule for any rate from 0.1 % to 30 %, shows the doubling, tripling, and quadrupling times, and projects an initial amount through each milestone so the abstract years map to concrete euros.
Two formulas live side by side here. The shortcut: years to double = 72 / r, where r is the annual rate as a percentage (so 6 % becomes 6, not 0.06). The exact value: years = ln(2) / ln(1 + r/100). For a 6 % rate the shortcut gives 12.0 years; the exact value is 11.90 years — accurate to within 1 % across the 3–12 % range. Above 15 % the divergence widens (at 30 % the shortcut says 2.4 years, exact is 2.64), which is why the calculator exposes the exact logarithmic computation and reports it under the headline 72/r number. Tripling time = ln(3) / ln(1 + r/100), and likewise quadrupling time = 2 × doubling time (because the second double doubles the first). The calculator multiplies the entered initial amount by 2, 3, and 4 to show the milestone balances alongside the years to reach them — the kind of side-by-side that makes "earn 7 % for 30 years" feel less abstract.
Two inputs: the annual rate (slider plus number input, 0.1 % to 30 %) and an initial amount (default €10 000). The result panel shows the doubling time as the headline metric, the tripling and quadrupling times alongside, and the corresponding balance multipliers. Slide the rate up and down to feel the convexity: moving from 5 % to 6 % saves more than two years; moving from 12 % to 13 % saves about half a year. That asymmetry is itself the lesson — every basis point matters more at low rates than at high ones. The defaults — 7 % and €10 000 — represent a long-term diversified equity portfolio under historical-average assumptions; the result of just over 10 years to double matches what most retirement planners assume.
A 30-year-old contributes €5 000 to an index fund averaging 7 % per year. Using the rule of 72: 72 / 7 = 10.29 years to double. By age 40 the balance reaches €10 000. By age 50 it doubles again to €20 000 (two doublings = quadruple of the original). By age 60 it doubles a third time to €40 000. By age 70 a fourth doubling brings it to €80 000. Same numbers in the calculator under the rule output: 2 × = €10 000 in 10.3 years; 3 × = €15 000 in 16.3 years (because tripling is more than 1.5 doubles — it follows ln(3)/ln(1.07) = 16.24); 4 × = €20 000 in 20.6 years. Now compare a 4 % rate over the same horizon: doubling in 18 years means by age 78 the contributor still has only €40 000 — half the 7 % outcome. The 3-percentage-point difference does not feel like much; over a 40-year horizon, it doubles the terminal balance.
First, applying the rule to nominal rates and treating the result as real (inflation-adjusted) growth. A 7 % nominal return at 3 % inflation is a 4 % real return — the doubling time for purchasing power is 18 years, not 10. Second, mistaking the rate for a decimal. The shortcut takes the percentage as-written (6 for 6 %), not the decimal (0.06). Plug 0.06 into 72/r and you get 1 200 years; the calculator handles this by reading the input as a percentage. Third, forgetting that the rule assumes compound interest. Simple interest at 6 % doubles in roughly 17 years, not 12 — and many fixed-income products (especially in the US) advertise nominal annual rates that compound monthly, which slightly accelerates the effective rate. Fourth, ignoring fees. A 7 % gross return with a 1.5 % expense ratio is a 5.5 % net return, which doubles in 13 years instead of 10 — a structural cost that the rule makes painfully visible. Fifth, expecting smooth doublings. Real markets don't compound steadily; they swing 30 % up one year and 20 % down the next. The rule of 72 describes the geometric mean over a long horizon, not the path; the path is choppy enough to feel nothing like the smooth exponential the calculator draws.
The rule of 72 has several refinements. Rule of 70 is more accurate for low rates (under 6 %) — 70/r aligns better with the natural log of 2 (which is 0.693). Rule of 69.3 is the exact constant: 69.3 / r matches the continuous-compounding formula. Rule of 114 for tripling: 114/r approximates ln(3) × 100. Rule of 144 for quadrupling. The calculator uses 72 because it has the most useful divisor properties (12 integer factors) and is the cultural standard. Inverted use: if you know your time horizon, divide 72 by the years to find the required rate. "I need to double my money in 6 years" → 72/6 = 12 % per year required. Real-rate use: subtract inflation from the nominal rate before applying the rule, to get the doubling time in purchasing power. Inflation context: at 5 % annual inflation, the rule of 72 says prices double every 14 years — which is why central banks target 2 %, where the doubling time stretches to 36 years and feels slow enough to ignore.