Rule of 72 Calculator

Rule of 72: years to double ≈ 72 ÷ annual rate; exact: years = ln(2) ÷ ln(1 + annual rate ÷ 100)

Annual rate

The yearly rate as a percentage — 8 means eight percent — which must be above zero because both expressions divide by something that vanishes there

72

The numerator of the shortcut, chosen so that the division is easy to do in your head at the rates people meet most often

ln(2)

The natural logarithm of two, about 0.6931, which appears because doubling means solving (1 + r) raised to the power n equals 2

Years to double

How long the money takes to double at a rate that is compounded; the shortcut and the exact answer are quoted side by side

Error

The shortcut's answer minus the exact one, which is positive at low rates and negative at high ones

Multiple at the rule's answer

What your money actually becomes if you wait exactly as long as the shortcut told you to, which is slightly under two at almost every rate

Use the shortcut when you need an answer in your head during a conversation, and use the exact figure when the answer is going to be written down. Because the two agree closely in the range where most ordinary rates live, the shortcut is at its best for rough planning at five to ten percent, and it is at its worst at both extremes: below about three percent it overstates the wait badly, and above about fifteen percent it understates it. The multiple output answers the question the error figure raises but does not settle, which is whether being off by a fifth of a year actually costs you anything. At twenty percent, waiting the rule's 3.6 years instead of the exact 3.802 leaves you with 1.93 times your money instead of 2 — a shortfall of about seven percent of your gain, which is worth knowing before you use the shortcut on a high rate.

1. Eight percent: the rate where the rule is nearly exact

   1. The rule: 72 ÷ 8 = 9.00 years
   2. The exact answer: ln(2) ÷ ln(1.08) = 0.693147 ÷ 0.076961 = 9.006 years
   3. Error: 9.00 − 9.006 = −0.006 years, about two days
   4. Growing at 8% for exactly 9 years: 1.08⁹ = 1.9990

   This is the page's default because it is the rule's best case, and it is worth seeing how good the best case is: the shortcut is off by about two days out of nine years. The multiple output is the tell that even here it is not exact — nine years at eight percent compounded gives you 1.9990 times your money, not 2, and it takes another couple of days to close that last thousandth.

2. One percent: the rule overstates the wait by more than two years

   1. The rule: 72 ÷ 1 = 72.00 years
   2. The exact answer: ln(2) ÷ ln(1.01) = 0.693147 ÷ 0.009950 = 69.661 years
   3. Error: 72.00 − 69.661 = +2.339 years
   4. Growing at 1% for 72 years: 1.01⁷² = 2.0471

   At a savings-account rate the shortcut is generous about how long you wait, and the direction is the opposite of what most people assume. The multiple makes it concrete: waiting the rule's 72 years at one percent leaves you with 2.0471 times your money rather than 2, so the extra two and a bit years are not wasted, they just overshoot. This is the largest error in the table.

3. Twenty percent: the rule now understates the wait

   1. The rule: 72 ÷ 20 = 3.60 years
   2. The exact answer: ln(2) ÷ ln(1.2) = 0.693147 ÷ 0.182322 = 3.802 years
   3. Error: 3.60 − 3.802 = −0.202 years
   4. Growing at 20% for 3.6 years: 1.2^3.6 = 1.9278

   Compare this with the one percent case and the point of the page is visible: the error has changed sign. The rule now promises a doubling a fifth of a year too early, and if you rely on it you get 1.9278 times your money instead of 2 — about seven percent of the gain you were expecting. The two cases are deliberately far apart in rate and close together in the size of the error in years, which is why the error in years is a poor guide on its own and the multiple is printed beside it.

The rule of 72 is an approximation and this page does not pretend otherwise; what it does is tell you how large the approximation is at your rate. It assumes the rate is compounded, and that it stays the same for the whole period — a rate that changes partway through has no single doubling time, and the rule has no way to express one. It works on a rate above zero only, because both the division and the logarithm fail at zero, which is why the field will not accept a rate below a tenth of a percent. It says nothing about tax or inflation, so the doubling time it reports is in nominal money: at three percent inflation a balance doubling in twenty-four years has not doubled in what it can buy. And it describes a single lump sum that is left alone, so it does not apply to a series of deposits, where the arithmetic is a different one and the answer is shorter. Finally, the shortcut is a rule about the number 72, not about your investment — the exact answer is the one to use when the figure matters, and the shortcut is the one to use when you are talking.

How does the rule of 72 work?

Divide 72 by the annual rate and the result is roughly how many years it takes the money to double. At eight percent that is 9.00 years and the exact answer is 9.006 years, so the shortcut is nearly perfect; at one percent it says 72 years when the exact answer is 69.661, and at twenty percent it says 3.60 when the exact answer is 3.802. The shortcut is a mental-arithmetic tool and this page shows the exact doubling time next to it.

Why is the rule of 72 only approximate?

Because it replaces a logarithm with a division. The exact doubling time solves the compound interest equation for the number of periods, which gives the natural logarithm of two divided by the logarithm of one plus the rate. The shortcut uses a fixed numerator instead, and the fixed numerator is a good match only over a range of rates: the two answers cross somewhere between seven and eight percent and drift apart in opposite directions on either side of that crossing.

Which way does the rule of 72 err?

Both ways, depending on the rate, and that is the part most people do not expect. Below the crossing point the shortcut gives an answer that is too long: at one percent it says 72 years against an exact 69.661, overstating the wait by more than two years. Above it the shortcut gives an answer that is too short: at twenty percent it says 3.60 against an exact 3.802. The sign of the error changes between seven percent, where it is still positive, and eight percent, where it has turned negative.

What does the multiple at the rule's answer mean?

It is what your money actually becomes if you wait exactly as long as the shortcut told you to. Growing a balance at eight percent for 9 years gives 1.9990 times your money rather than the 2 the rule implies, and at twenty percent, waiting the rule's 3.60 years gives only 1.9278 times. It converts the error from years into money, which is the form in which the error actually costs you something.

Can I use the rule of 72 with a low rate?

You can, but the error grows quickly as the rate falls, and it grows in the direction of overstating the wait. At one percent the shortcut says 72 years while the exact doubling time is 69.661, a gap of more than two years, and a balance grown at one percent for 72 years ends up at 2.0471 times its starting value. The field will not accept a rate at or below zero at all, because both the division and the logarithm break there.

Does the rule of 72 account for inflation or tax?

No, and neither does the exact figure on this page. Both work on the rate you type, so if that rate is a nominal return the doubling time is in nominal money, and a balance that doubles in nominal terms over twenty-four years at three percent inflation has not doubled in purchasing power. The same applies to tax: a return taxed along the way compounds at a lower effective rate, and the doubling time for the after-tax rate is longer than the one shown here.

• 31 CFR Part 356, Appendix B — Formulas and Tables: the present value factors and the price formula for Treasury notes and bonds — U.S. Department of the Treasury rule, via the Electronic Code of Federal Regulations (United States)
• Compound Interest Calculator — the forward form of the arithmetic, for checking a rate by growing a balance at it — U.S. Securities and Exchange Commission, Investor.gov (United States)
• H.15 Selected Interest Rates — the published benchmark rates a computed rate is compared against — Board of Governors of the Federal Reserve System (United States)