rule-of-72

Compounding is exponential, and the cleanest way to feel an exponential is to count its doublings. The Rule of 72 is the back-of-envelope shortcut for that: years-to-double ≈ 72 ÷ return rate. It works because the exact doubling time, ln2 ÷ ln(1 + rate), happens to track 72 ÷ rate almost perfectly across the range of ordinary returns — which is also why the number is 72 and not 69 or 75: 72 divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and it's most accurate right around the 8% where typical long-run stock returns live. This lesson plots years-to-double against the rate so you can see the hyperbola — a couple of extra points of return buys whole years off the clock — watch the rule track the truth through the middle and drift at the extremes, and count how many doublings fit in your horizon. Two doublings is 4×; three is 8×; ten is over a thousand times. Once you can do this in your head you can sanity-check any 'this will grow your money' pitch in about three seconds.