Rule of 72 Calculator
See how long money takes to double at a given annual return, or the return needed to double in a set time — Rule of 72 estimate plus the exact math.
Divide 72 by an annual return to get the years it takes money to double — or divide 72 by a target number of years to get the return you’d need. The exact figure sits alongside so you can see the approximation gap.用 72 除以年化收益率，得到本金翻倍所需的年数；或用 72 除以目标年数，得到所需的收益率。旁边同时给出精确值，让你看清近似的误差。
Enter an annual return to see how long money takes to double.输入年化收益率，即可看到本金翻倍所需的时间。
- Rule-of-72 estimate72法则估算
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- Exact math精确值
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- Approximation gap近似误差
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To estimate how many years it takes money to double at a steady annual return, divide 72 by that return written as a whole number — at 8% a year, roughly 72 ÷ 8 = 9 years. Flip it to find the return a doubling in a set time would demand: divide 72 by the years instead. It is a mental-math shortcut for the exact compounding formula, and it lands remarkably close in the range investors actually use.要估算在稳定的年化收益率下本金翻倍需要多少年，就用 72 除以以整数表示的收益率——每年 8% 时，约为 72 ÷ 8 = 9 年。反过来，若想知道在给定年数内翻倍所需的收益率，则用 72 除以年数。它是精确复利公式的一个心算捷径，而在投资者实际使用的区间里，它的结果出奇地接近。
01The formula公式
Return needed ≈ 72 ÷ Years所需收益率 ≈ 72 ÷ 年数
Exact years = ln(2) ÷ ln(1 + Return% ÷ 100)精确年数 = ln(2) ÷ ln(1 + 收益率% ÷ 100)
Exact return = 100 × ( 2^(1 ÷ Years) − 1 )精确收益率 = 100 × ( 2^(1 ÷ 年数) − 1 )
The exact line comes from solving (1 + r)^t = 2 for the missing term. Seventy-two is chosen because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, making the mental arithmetic easy — and because its error is smallest right around the 6–10% band where the true multiplier of ln(2) ≈ 69.3 would otherwise undershoot.精确的那一行来自求解 (1 + r)^t = 2 中缺失的那一项。选用 72，是因为它能被 2、3、4、6、8、9、12 整除，心算方便——而且在 6–10% 这个区间里它的误差最小，此处真实系数 ln(2) ≈ 69.3 反而会偏低。
02A worked example一个算例
Suppose a fund compounds at 8% a year and you want to know how long it takes to double.假设一只基金以每年 8% 复利增长，你想知道它翻倍需要多久。
- Rule of 72 = 72 ÷ 8 = 9.00 years72法则 = 72 ÷ 8 = 9.00 年
- Exact = ln(2) ÷ ln(1.08) = 9.01 years精确值 = ln(2) ÷ ln(1.08) = 9.01 年
- The estimate is off by about four days — close enough for a decision made in your head.估算值仅相差约 四天——对于心算决策已足够接近。
Now run it the other way: to double in 10 years you would need roughly 72 ÷ 10 = 7.2% a year; the exact requirement is 7.18%. Both directions rest on the same shortcut, and both stay honest inside the ordinary range of returns.再反过来算：要在 10 年内翻倍，你大约需要 72 ÷ 10 = 7.2% 的年化收益；精确要求为 7.18%。两个方向依赖同一个捷径，在通常的收益区间内都保持诚实。
03The common trap常见陷阱
The rule assumes one fixed, positive return compounding without interruption — a straight line that real markets never draw. An average 8% return made of a −20% year and a +40% year does not double capital on the same schedule as a steady 8%; the swings, taxes, fees and any withdrawals all bend the path. The number is a back-of-envelope sketch, not a forecast, and it says nothing about the odds of earning that return in the first place.该法则假设一个固定的正收益率不间断地复利——一条真实市场从不画出的直线。由 −20% 与 +40% 拼成的平均 8% 收益，翻倍的节奏并不等同于稳定的 8%；波动、税费、成本以及任何提取都会改变这条路径。这个数字是信手勾勒的草图，而非预测，它也完全没有说明你能否首先赚到那样的收益率。
04Where this breaks它的局限
The shortcut is a linear approximation of a curve, so it drifts as you leave the middle. A few honest limits:这个捷径是对一条曲线的线性近似，因此一旦偏离中段就会失真。几点诚实的局限：
- High rates overshoot. At 20%+ a year, 72 exaggerates the doubling time — the true multiplier drifts toward 76 or higher. Some traders switch to 70 for low rates and 78 for high ones; the exact math above never has this problem.高收益率会高估。年化超过 20% 时，72 会夸大翻倍所需时间——真实系数会漂移到 76 甚至更高。有人对低收益率改用 70、对高收益率改用 78；上面的精确公式则从无此问题。
- It needs a positive return. A zero or negative return never doubles, so the rule has no answer there. This tool asks for a return above zero rather than printing a meaningless number.它需要正收益率。零或负的收益率永远无法翻倍，法则在此没有答案。本工具会要求输入大于零的收益率，而非印出一个无意义的数字。
- Compounding frequency matters. The formula assumes the return compounds once per year. Monthly or continuous compounding doubles slightly faster, so the same headline rate reaches 2× a touch sooner than the annual figure suggests.复利频率有影响。公式假设每年复利一次。按月或连续复利翻倍略快，因此同样的名义收益率达到 2× 的时间会比按年计算略早一些。
Seventy-two over the rate is your doubling time in years — a fast, honest sketch inside the 6–10% band, and a curve worth checking exactly once you stray far from it.72 除以收益率就是以年计的翻倍时间——在 6–10% 区间里是一张快速而诚实的草图，一旦远离这个区间，就值得用精确公式核对。
See the full curve behind the shortcut with the 用 compounding calculator复利计算器, or turn a growth rate into an annualized figure with the 查看捷径背后的完整曲线，或用 CAGR calculator年化增长率计算器.把增长率换算成年化数字。