The Rule of 72 and what it actually approximates
The Rule of 72 says: years to double = 72 ÷ annual rate (as a whole number). At 8% return, doubling takes 9 years; at 6%, 12 years. It is correct within about 1% for rates between 4% and 12%. Most people who quote it don’t know why 72, or where the approximation breaks. Here is the math, in plain language, and the rules-of-thumb that share its DNA.
The exact doubling formula
For annual compounding at rate r: doubling time = ln(2) / ln(1+r). For continuous compounding it simplifies to ln(2) / r ≈ 0.693 / r. At 8% annual compounding: ln(2) / ln(1.08) = 0.693 / 0.077 = 9.006 years. The Rule of 72 gives 72 / 8 = 9. Difference: 0.07%.
Where the number 72 comes from
For small rates, ln(1+r) ≈ r − r²/2 + r³/3 − ... (the Taylor series of the natural logarithm). The constant that best fits doubling time across the equity-return range (6–10%) happens to be 72. The choice of 72 versus 70 versus 69 depends on which rate range you optimise for: 69 for continuous compounding at any rate, 70 for low rates (2–3%, savings accounts), 72 for moderate rates (6–10%, historical equities), 76 for high rates (15%+, private equity). 72 also divides cleanly by many integers (1–9 except 5 and 7), which made it convenient before calculators were ubiquitous.
Where the Rule of 72 breaks
High rates (above 20%): error grows fast. At 20% annual, actual doubling is 3.80 years; Rule of 72 says 3.60 — error of 5%. Low rates (below 3%): error grows too, in the other direction. At 2%, actual is 35.00 years; Rule of 72 says 36.00 — error of 3%. Variable rates: the Rule of 72 assumes a constant return. Real investments don’t deliver that — the sequence of returns matters as much as the average, especially during withdrawal phases.
What compounding actually does
$10 000 at 7% real return: doubles to $20 000 in roughly 10 years (Rule of 72 says 10.3), reaches $40 000 at year 20, $80 000 at year 30, $160 000 at year 40. The last 10 years account for nearly half the total. The first 10 years contribute almost nothing in absolute terms but everything in time. This is the math behind “start early outranks save more” for most retirement scenarios — not a slogan, a consequence of exponential growth.
Related shortcuts: tripling and quadrupling
Rule of 114 for tripling: years to triple ≈ 114 / rate. (Exact: ln(3) / ln(1+r).) Rule of 144 for quadrupling: years to quadruple = 144 / rate, which follows mechanically from doubling twice (2 × 72 = 144). Useful for back-of-envelope retirement math without opening a spreadsheet.
Takeaway: The calculator on this page does the exact math, no approximation, including periodic contributions which the Rule of 72 cannot model at all. The Rule of 72 is a useful mental shortcut for the 4–12% rate range, accurate to about 1%. Outside that range, or with variable returns, or with contributions, use the calculator. Most importantly: the spectacular numbers in any compound-interest table come from the final decades, not the first. Time matters more than rate, within a reasonable range.
Sources: U.S. SEC Investor.gov compound interest education · Bogleheads: Rule of 72 derivation.