The Rule of 72, honestly bounded

What the rule answers, and what it does not

Divide 72 by the annual rate of growth and you get, near enough, the number of years the money takes to double. At 9% that is about eight years; at 6%, about twelve; at 12%, about six.

It works in your head, which is the entire point. Nobody carries a calculator into a conversation about a fixed deposit, and a shortcut you will actually use beats a formula you will not.

But it answers one narrow question, and the narrowness is where people come unstuck. The rule describes a single sum, left alone, growing at a constant rate. It says nothing about money you keep adding to, and nothing about a rate that moves — which between them describe almost every real portfolio. The time value of money is the idea underneath; this is one shortcut sitting on top of it.

Read in the other direction, the same arithmetic tells you how fast something shrinks in real terms — which is the use most articles skip, and the one worth the trouble.

Why 72, and not 70 or 75

Because the exact constant is 69.3, and 72 is 69.3 rounded up — up, because the approximation behind 69.3 runs short; and up to 72 rather than 70 or 75, because 72 divides cleanly by almost everything you would want to divide it by. Here is where both halves of that come from.

Doubling means the growth factor has reached 2. Written out, that is (1 + r) raised to the power t equals 2, and solving for t is what logarithms are for: t = ln 2 ÷ ln(1 + r).

The natural log of 2 is 0.6931. And for small rates, ln(1 + r) is very nearly r itself — at 5%, ln(1.05) is 0.0488 against r of 0.05. Substitute that approximation and the whole thing collapses to 0.6931 ÷ r, which in whole percentage points is 69.3 divided by the rate.

So the mathematically pure constant is 69.3, not 72. Two things push the working number up from there.

First, the approximation is biased. ln(1 + r) is always a little smaller than r, so the true doubling time is always a little longer than 69.3 ÷ rate suggests. Raising the numerator compensates.

Second — and this is the unglamorous reason — 72 divides cleanly by 2, 3, 4, 6, 8, 9 and 12. 69.3 divides cleanly by nothing. A shortcut whose arithmetic you cannot do in your head is not a shortcut.

Note which reason is doing what. The bias says the numerator should go up; only arithmetic convenience says it should go up to 72 specifically.

At 72, the approximation lands exactly right at 7.85%. Nobody tuned it to that rate — the divisors were the target and 7.85% is what fell out. Solve 72 ÷ r = ln 2 ÷ ln(1 + r) and you will find it yourself.

Why that constant outlived the more accurate 69.3 is a reading rather than a documented history, but it is not a mysterious one: the band the rule is accurate in is the band people actually assume when they talk about a deposit, a home loan or a long-run equity return. A shortcut that is near-exact where it gets used, and only drifts where it does not, has very little pressure on it to improve.

Where it drifts, and in which direction

The rule does not fail gradually in both directions. It has a single crossover point and errs one way below it and the other way above.

Every figure in the “exact” column is ln 2 ÷ ln(1 + r), so you can check any row yourself.

Between 6% and 10% the error never exceeds about a tenth of a year. On a twelve-year estimate, a month. That is well inside the precision anyone is entitled to from an assumed rate of return, so within that band the rule is not an approximation in any way that matters.

Outside it, the drift is one-directional and worth naming. Below the crossover the rule says doubling takes longer than it does; above the crossover it says doubling comes sooner than it does. And the error grows the further you go, in both directions.

Which produces an asymmetry nobody points out. Applied to a high return, the rule is flattering — it promises a double in four years at 18% when the honest answer is four years and two months. Applied to a high-rate debt, the same error is alarming in your favour: it says a balance at 36% doubles in two years when it actually takes about two years and three months. Same arithmetic, same sign of error, opposite emotional effect — and only one of the two is a mistake you would want to make.

The correction that repairs both tails

The rule has an extension that almost nobody teaches, and it costs one extra step.

Move the numerator by 1 for every 3 percentage points the rate sits away from 8% — up when above, down when below — then divide as usual.

At 36%, that is 8 subtracted from 36, divided by 3, which is 9.3; add it to 72 to get 81.3; divide by 36 and you have 2.26 years. The exact answer is 2.25. At 3%: 5 percentage points below 8, so subtract 1.7 from 72 to get 70.3; divided by 3 that is 23.4 years, against an exact 23.45.

Across the whole span from 1% to 48% the corrected version stays within about a hundredth of a year of the true answer — look down the last column of the table and compare it with the one before. Its worst miss anywhere in that range is under four days. A mental shortcut that accurate is a peculiar thing to leave out of every article on the subject.

One further variant, for a specific case. If a rate is compounded continuously rather than annually — the convention in some quoted yields, and the limiting case that daily compounding approaches — the exact constant is 69.3 with no approximation at all, because ln 2 is 0.6931 and the r-for-ln(1+r) substitution never has to be made. Rule of 69.3 for continuous, 72 for annual.

The cost of all this: three constants to remember instead of one. Whether that is worth carrying depends on how often you work outside the 6%–10% band — rarely, if the question is a deposit or a long-run return assumption; constantly, if it is a borrowing rate, because those are the ones that sit high enough for the plain rule to flatter them.

Point it at the thing that erodes you

The second use is the same arithmetic aimed in the opposite direction, and it is the one that changes how a number feels.

Run 72 over the inflation rate instead of the return, and you get the years in which prices double. Feed it whatever rate you want to test rather than a rate anyone has told you to expect: at 6% the answer is twelve years, at 4% it is eighteen, and the useful move is to run both and see how much the answer moves for two percentage points of assumption.

Prices doubling and your money halving are the same event described from two ends. A rupee that buys a litre of milk today buys half a litre after one doubling of prices. Which means the retirement figure someone quotes you in today's money is not a target — it is a target that keeps moving away at a rate you can now compute in your head. If a plan is twenty-four years out and prices are doubling every twelve, the number roughly quadruples before you get there. Two doublings, not twice as much.

Here is the trap in that, and it catches careful people. Doubling times do not subtract. Rates do. Take an assumed 11% nominal return, which doubles money in about 6.5 years, against an assumed 6% inflation, which doubles prices in 12. The real doubling of purchasing power is not 12 minus 6.5. You subtract the rates first, then apply the rule: roughly 5% real, so about fourteen years. The gap between those two answers is nearly eight years, and the reasoning that produces the wrong one looks entirely sensible on the page. The mechanics of why real returns are the ones that matter sit in time value of money.

Four ways the rule gets misused

Each of these produces an answer that looks reasonable, which is why they survive.

1. Feeding it an average instead of a compound rate. This is the expensive one. Take a year of +50% followed by a year of −33.3%: you are exactly back where you started, having grown at 0%. The arithmetic average of those two years is +8.3%, and 72 divided by 8.3 says the money doubles in about 8.6 years. The rule was not wrong; it was fed a number that describes nothing. It needs the compound annual rate — the single steady rate that would have produced the same end value — and an average of yearly returns is always higher than that whenever the returns vary at all. The difference between the two measures is set out in XIRR versus CAGR.

2. Applying it to money you are still adding to. The rule assumes one sum, deposited once. A SIP corpus crossing twice its starting value tells you nothing, because most of the increase is instalments you paid in, not growth. Doubling time is a meaningful idea only for capital that is sitting still.

3. Ignoring how often the rate compounds. A quoted 8% compounded quarterly is not 8% a year; it is an effective 8.24%, and the money doubles in about 8.75 years rather than 9. On a deposit that gap is a rounding error.

On a rate quoted per month it is not. Take 3% a month — a round number to work the arithmetic with, not a claim about what any particular card or lender charges. That compounds to about 42.6% a year, not 36%, and doubles what you owe in roughly 23 and a half months. Feed the rule the effective annual rate rather than the headline one; the difference between the two is set out in interest rates explained, and it is the single most common reason a borrowing cost is understated.

4. Using a pre-tax rate for a post-tax question. If the return is taxed as it accrues, the rate that compounds is what is left after tax, and that is the rate the rule needs. The stretch is proportional — knock any rate down to two-thirds of itself and the doubling time rises by half.

The delay in years is not proportional, and this is the part worth carrying. At 4%, that same haircut turns 18 years into 27. At 12% it turns six into nine — the identical 50% stretch, nine years of delay against three. Tax drag costs the most time exactly where the rate is lowest — which is the corner of the market usually described as the cautious one.

What the shortcut cannot be asked to do

Three honest boundaries, because a tool used past its range is worse than no tool.

It assumes a constant rate, and nothing has one. Feeding the rule a long-run average return gives you the doubling time of a smooth line that no investment has ever followed. The answer is a description of an assumption, not a projection — and the actual path, with its drawdowns, is what decides whether anyone stays invested long enough for the arithmetic to apply.

It is an estimate of time, not of certainty. The rule tells you what a rate implies. It has nothing to say about whether that rate will materialise, and no arrangement of the arithmetic can make it say so.

It gets unusable at very high rates. Past roughly 50% a year the plain rule is out by more than 15% and the correction, while still close, is being asked to work well outside where it was fitted. At those rates the honest move is the logarithm itself, which any phone will compute.

Working it against a real series

The gap between the rule's answer and a real outcome is not the rounding error in the table. It is the distance between a constant rate and a rate that arrived unevenly — and the only way to see that distance is to run a real series.

FNOTrader's Mutual Funds app runs lumpsum and instalment schedules against the full AMFI NAV history — around 34 million NAV rows — and reports the compound annual rate, the invested amount against the value, and the maximum drawdown along the way, plus the distribution of outcomes across every available start date rather than one. The rule of 72 will tell you what a rate implies. The drawdown column tells you what holding it would have felt like, which is the part no shortcut can compress.

Past performance is not indicative of future results, and no historical rate should be read as a rate that will repeat.

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