Rule of 72 / doubling time

Rule of 72 — the mental math that teaches compound growth

The Rule of 72 is the simplest financial-math shortcut in existence: divide 72 by your annual rate of return, and you get the number of years for an investment to double. At 8% return, that's 9 years. At 6%, it's 12 years. At 10%, it's 7.2 years. The rule works because of the mathematical convenience of compound growth.

Where the 72 comes from

Exact doubling time for compound growth is t = ln(2) / ln(1 + r), where r is the rate expressed as a decimal. ln(2) is approximately 0.6931, and for small r, ln(1 + r) is approximately r. So t ≈ 0.6931 / r = 69.31 / r%.

That's the Rule of 69.3, which is mathematically cleaner but inconvenient for mental math. 72 is divisible by 2, 3, 4, 6, 8, 9, 12 — so 72/6 = 12, 72/8 = 9, 72/12 = 6, all clean. The cost is a slight under-estimation of doubling time, which is actually an advantage: the Rule of 72 is conservative for most rates people care about.

Why the rule matters

Most people don't have a feel for compound growth — it looks linear in the short term and then takes off. The Rule of 72 is the bridge between the first-year "$1,000 × 7% = $70" and the 20-year "that original $1,000 is now $3,870".

Key insight: the total number of doublings matters more than the rate for long-horizon compounding. A 30-year-old investor with 35 years to retirement at 8% gets ~4 doublings. At 10%, they get ~5 doublings. One extra doubling means 2× the money at retirement. Over 40 years at 10%, it's 5.5 doublings vs 3.6 at 6% — a 4× difference in ending portfolio value.

Applications in financial planning

• Retirement — how many doublings between now and 65? At 30, you get 4+ at historical market rates. At 50, you only get 1. This is why starting early matters so much.
• Inflation — at 3% inflation, $100k today is worth $50k in 24 years. At 4%, 18 years. A retiree's purchasing power gets halved during a typical 25-year retirement even at moderate inflation.
• Debt — a credit card at 24% APR doubles your balance in 3 years if you make no payments. A medical bill sent to collections with a 12% judgment rate doubles in 6 years.
• Business growth — a company growing earnings at 15%/year doubles them in under 5 years. This is why compound growth companies trade at premium multiples.

Rule of 72 vs the exact formula

The error is under 3% for all rates between 4% and 20%. The Rule of 72 is essentially exact for the range of returns most people encounter.

The Rule of 114 — tripling time

Similarly, 114 divided by rate gives approximate time to triple. At 8%, triple time is 14.25 years. The Rule of 144 gives quadruple time (= 2 doublings).

Why compound growth is so hard to intuit

Human intuition is linear. $100 grows at 10%/year, so in 10 years you think it's $200. The actual answer is $259 because each year's growth compounds. Over 30 years it's not $400, it's $1,744. The Rule of 72 is a shortcut that forces you to think in doublings, which is the language compound growth actually speaks.