How long until my money doubles?

How many years a balance takes to double at a given return, what the rule of 72 says instead, and how far apart the two figures are. It doubles the number, not what the number buys, and the two part fast where prices rise, and it is one balance left alone, never one being added to.

• The rule of 72 is an approximation rather than the arithmetic. It is closest at rates near 8% and drifts further either side of that.

At 6.0% a year 50,000 doubles in 11.9 years. The rule of 72 says 12.0, which is 0.1 too long, assuming the return holds every year.

A balance grows by the same proportion every year, so the year it reaches twice its size is the year the compounding factor reaches two. That gives ln 2 ÷ ln(1 + rate), where ln is the natural logarithm. It is exact, it depends on nothing but the rate, and the amount you start with does not enter it at all.

The rule of 72 is the version everyone carries instead: divide 72 by the rate as a percentage. It survives because it is arithmetic you can do while somebody is still talking, and it is close enough to be useful over the middle of the range. It is closest between about 8% and 10% — at half-point steps the two figures print the same one decimal at 7.5, 8, 8.5, 9, 9.5 and 10.5, and differ by a tenth of a year at 10, because both are rounded before they are compared. Below that band it runs long: at 1% it says 72 years against a true 69.7. Above it, it runs short: at 30% it says 2.4 against 2.6.

The gap on the card is the subtraction of the two figures printed above it, not of the numbers behind them. At 7% the underlying difference is four hundredths of a year, which would print as 0.0 beside a shortcut of 10.3 and a doubling time of 10.2 — a card contradicting its own reader's subtraction. Taking it from what is displayed is what makes the third figure checkable.

At a rate of zero or below there is no doubling time. Nothing multiplies by one repeatedly and arrives at two, and dividing 72 by zero is not a number, so both figures and the gap between them print nothing rather than an infinity.

years = ln 2 ÷ ln(1 + rate)

ln is the natural logarithm and the rate is the annual return as a decimal. The rule of 72 is 72 divided by the same rate written as a percentage.

A worked example

At 6% a year money doubles in 11.9 years, and the rule of 72 says 12 — a tenth of a year too long. At 8% the two land on the same figure, 9.0. At 1% the shortcut is 2.3 years long, and at 30% it is 0.2 short.

What it assumes

• The return is the same in every year and compounds once a year. Two years at 12% and two at nothing is not four years at 6%, and it takes longer to double.
• The amount does not enter the answer. The figure at the top is the same for 500 as for five million — it is here so the picture has a scale, and dragging it changes the height of the bars and nothing else.
• Tax and charges are not counted. Where either applies it comes off the return before this arithmetic starts, and the doubling takes longer than the figure here.