The model: the shortcut that has held for centuries

The rule of 72 is finance's most celebrated shortcut: divide 72 by the annual rate, and there are the years in which capital doubles. This calculator takes it seriously and completes it — with the four numbers that count: the chosen rule (69, 70 or 72), the annual rate, the horizon and the expected inflation. Out come the years to the nominal doubling and the more honest ones to the real doubling — and the metric that orders everything: how many doublings truly mature in your horizon.

The count is universal and currency-free: only percentages, years and doublings — the same on every brokerage account in the world.

The four knobs: rule, rate, horizon and inflation

The rule has three legitimate values: the textbooks' 72 (divisible by almost everything), mental math's 70, the purists' 69 — closest to the exact value 100·ln 2 = 69.3. The rate goes from 1 to 20 percent: from the cautious deposit to the aggressive return. The horizon goes from 5 to 50 years: from the short project to a working life. The inflation goes from 0 to 8 percent — with the model's honest constraint: the real rate must be worth at least one point, or the real doubling never arrives.

The rate scale: from the deposit to the engine

Rule 72, a 30-year horizon, inflation at 2 percent; only the rate changes:

The second row is the factory recipe: at 5 percent, nominal capital doubles in 14.4 years — but the real one, with inflation at 2, asks for 24. And the scale tells the non-linear law of patience: from 4 to 20 percent the real doublings multiply by nine — because the rate works in the denominator, and every point wrestled at the bottom is worth more than one added at the top.

The years to doubling: the doubly immobile number

The first result row has an immobility that orders the model: the years to nominal doubling depend neither on the horizon nor on inflation — they are the pure ratio of rule to rate. The 14.4 years of 5 percent stay 14.4 for the five-year project and for the fifty-year plan. It is the doubling's separation of powers: rule and rate decide the speed; the horizon decides how far you go; inflation decides how much of the distance is real.

The three identities: the horizon that returns

The model rests on three exact identities, verified at every version by the test bench. The most elegant: doublings times years-to-doubling equals the horizon — the 2.08 nominal doublings times the 14.4 years give back exactly the 30 years, and the same holds for the real pair. The third is the definition: nominal years times rate equals the rule — 14.4 times 5 makes 72, always. Here the shortcut loses no pieces on the way.

The four readings: from the sleeping capital to the interest machine

The verdict reads the real doublings in the horizon. Below the half it is the sleeping capital: the horizon ends before compounding walks on stage. Between the half and one it is the half doubling: engines on, wings still folded. Between one and two it is the completed doubling: purchasing power has truly doubled. Beyond two it is the interest machine: real capital quadruples — and each next doubling is worth as much as all the previous ones together.

The inflation that eats half: the engine threshold

Above the readings, the model marks the erosion threshold: when inflation is worth at least half the rate — touched exactly at 4 percent with prices at 2. From there up, half the engine works just to stand still: the real doubling sits more than twice as far as the nominal one, and every promise made in nominal terms must be divided by honesty. It is not a defect of saving: it is a contract — whoever reasons in nominal also signs the illusion, and this count shows it before the signature.

The three rules: 72, 70 or 69

Why three rules? Because the shortcut approximates the exact value 100·ln 2 = 69.3 — and each constant strikes a different pact. The 69 is truest at low rates and continuous compounding; the 70 is the mental-math compromise; the 72 — divided exactly by 2, 3, 4, 6, 8, 9 and 12 — is the most convenient, and at middle rates of 6-10 percent it is also surprisingly accurate, because discrete compounding helps it. The rule knob is there for this: three different pacts with precision, all declared.

The limits of the model: what the shortcut does not cover

The perimeter is declared: the rule as an approximation, the real rate as a simple difference. Taxes on returns are not in the count, and in many countries they take a fifth or more of the growth; product fees neither — and one point of fees lengthens the doubling as much as one point of rate lost; the sequence of real returns, which is never a straight line, is a separate chapter. The number that comes out is the base of planning: the right order of magnitude, before the wrong promise.

The horizon scale: the speed still, the road that grows

Rule 72, rate at 5 percent, inflation at 2; only how long you let it run changes:

The columns that do not appear are made of stone: the years to doubling stay 14.4 nominal and 24.0 real across the whole scale — the horizon does not touch the speed of compounding, only how much road it is allowed. But the road is not all equal: the fifth doubling is worth sixteen times the first in absolute value. It is the table that explains the paradox of compound interest: it is an exponential that disguises itself as a straight line in the early years — and whoever judges by the first five years judges the disguise.

The inflation scale: the rate still, the truth that sinks

The same nominal 5 percent, the same 30 years; only the race of prices changes:

The nominal doubling stays immobile at 14.4 years on every row — and meanwhile the real one goes from 14.4 to 72: a factor of five, without the bank account showing any difference. The last row is the model's threshold: at 4 percent inflation against a 5 percent rate, prices run at more than half the engine — and thirty years of patience buy less than half a true doubling. Inflation does not show on the statement: it shows only here.

The rules scale: three constants, one choice

The third scale holds rate, horizon and inflation still (5 percent, 30 years, 2) and changes only the constant. The 69 gives 13.8 nominal years and 23.0 real — 1.30 doublings; the 70 gives 14.0 and 23.3 — 1.29; the 72 gives 14.4 and 24.0 — 1.25. The total spread between the sternest rule and the handiest is worth half a year out of fourteen: 3-4 percent, less than the uncertainty on the rate itself. It is the practical proof of why the shortcut has survived the centuries: the choice of the constant is a detail; the choice of rate and horizon is destiny.

The extremes of the grid: from a fifth to seven and a half doublings

The model's minimum is the shy project: five years at 5 percent with inflation at 2 — 0.21 real doublings: a fifth of the road, compounding that has not even tied its shoes. The maximum is the twenty-percent exception: thirty years at 20 percent — 7.50 real doublings, purchasing power multiplied by more than one hundred (and the word «exception» is in the name for honesty: no market gifts 20 percent for thirty years). Between the two extremes runs a factor of thirty-six: the whole grid fits inside four knobs — and that is why the same patience produces financial lives that do not resemble each other.

The doubling in reverse: the same rule, the dark side

The rule has a stern twin: 72 divided by inflation gives the years in which prices double — that is, cash halves. At 3 percent, 24 years; at 6, twelve. And it also holds for debt: revolving credit at 18 percent doubles in barely four years, faster than any reasonable portfolio. It is the symmetry that closes the circle: the same machine works for whoever owns capital and against whoever owns debt — and the rule tells both, with the same number, how fast.

The doubling strategy: the levers in order of honesty

When the verdict says fewer doublings than needed, the ranking of levers is sharp. First: the horizon — every added year is certain linear gain, without a gram of extra risk. Second: fees and taxes — a point of fees is a point of rate lost, and it is recovered with a signature. Third: personal inflation — cash beyond the safety buffer sleeps at rate zero, which is to say it retreats. Fourth and last: more return — which is always more risk, and deserves to be the last item, not the first. The doubling is not chased: it is waited for well.

Frequently asked questions

How does the rule of 72 work?

Divide 72 by the annual rate in percent: the result is the years to the doubling of capital. At 6 percent, 72/6 = 12 years; at 8, nine years. It is an approximation of exact compound interest — surprisingly accurate between 4 and 12 percent — and the calculator completes it with the horizon and inflation.

Why do the rules of 70 and 69 also exist?

Because the mathematically exact value is 100·ln 2 = 69.3. The 69 is faithful to it at low rates and continuous compounding; the 70 is handy in your head; the 72 has more integer divisors (2, 3, 4, 6, 8, 9, 12) and is the most accurate at middle rates with annual compounding. The calculator offers all three, declared.

What is the real doubling?

The doubling of purchasing power, not of the numbers on the account: it is computed with the real rate — rate minus inflation, as a declared simple difference. At 5 percent with inflation at 2, the nominal doubles in 14.4 years but the real one in 24: the difference is the money illusion the count unmasks.

Why does the model require a real rate of at least one point?

Because below that margin the real doubling passes seventy years — beyond any practical horizon — and at a zero or negative real rate it never arrives: nominal capital grows while the true one stands still or retreats. The constraint is the honest way of saying that without a real rate there is no doubling to compute.

How accurate is the rule against the exact calculation?

Between 4 and 12 percent, the 72's error on the years typically stays under 2-3 percent: small change, for a shortcut. At very low rates the 69 or 70 serve better; at high rates the rule slightly understates the times. For planning the order of magnitude, the shortcut is more than enough.

How many doublings to quadruple, or to make eight times?

Two doublings quadruple, three multiply by eight: each doubling starts from the previous finish line. That is why the calculator's metric is the doublings in the horizon: 3 real doublings mean purchasing power times eight — and the fourth alone would be worth as much as the first three together.

With typical rates, how fast does a portfolio really double?

A cautious balanced at 4-5 percent nominal with inflation at 2 doubles in real terms in 24-36 years: once per working life. A global equity at 7-8 nominal drops to 12-14 real years: two-three doublings in a career. It is the spread the page's table lines up.

Does the rule also work for inflation alone?

Yes, in reverse: 72 divided by inflation gives the years in which prices double — that is, purchasing power halves. At 3 percent, prices double in 24 years; at 6, in 12. It is the dark mirror of the same mathematics, and an excellent intuition test on idle cash.

Does the rule also work for debt?

Perfectly, and it is its sternest use: a revolving debt at 18 percent doubles in barely four years if untouched. The same machine that works for the saver works against the debtor — with the difference that on debt the rate is certain, on returns it is not.

Why does the horizon matter more than the rate, past a threshold?

Because doublings are horizon divided by doubling time: lengthening the horizon adds them linearly, but each added doubling doubles the balance. Ten extra years at 5 percent are worth almost three quarters of a real doubling — more than an extra rate point held for twenty years with higher costs and risks.

How much do taxes and fees lengthen the doubling?

A lot, and it is outside the declared perimeter: a 26 percent levy on gains and one point of fees can turn a nominal 7 into a real net 4 — and the doubling goes from 10 to 18 years. The rule stays the same: just enter the true net rate instead of the list one.

What do I do if the verdict says sleeping capital?

The levers in order of honesty: lengthen the horizon if the project allows it — the linear lever that asks for no risk; check fees and taxes, which erode the real rate like private inflation; and only then consider more return, knowing that more rate is always more risk. The count above, redone with the net numbers, is the compass.

Updated July 2026