Compounding, properly

Seventy-two. Divide it by an annual rate and you get, near enough, the number of years until the thing doubles. Nine percent doubles in eight years. Six percent in twelve. Three percent in twenty-four.

It works on savings, on debts, on prices and on the number of users of anything. It is the only piece of financial arithmetic worth carrying around in your head, and the whole of this lesson is what it is hiding.

Doublings multiply

Here is where the rule stops being a party trick and starts being useful. Once you know one doubling time, you know all of them, because doublings do not add up — they stack.

Predict

A balance doubles every twelve years. How long until it is eight times its starting size?

Choose one answer

Twenty-four yearsThirty-six yearsForty-eight yearsNinety-six years

Lock in answer

Check the rule against the arithmetic

The model below runs the formula directly. Twelve years are fixed. Two controls move: what goes in each year, and the rate.

Compound growth

≈ Hypothetical model, not a forecast

Grows a starting balance plus a fixed yearly contribution at a constant rate.

It assumes

• The return is identical in every one of the twelve years.
• Any contribution lands at the end of its year and earns nothing during it.
• Growth is credited once a year rather than continuously.
• There is no tax, no fee and no year in which the return is negative.

It ignores

• Tax on growth, income or withdrawal
• Platform fees, spreads and trading costs
• Inflation — every figure shown here is nominal
• Any year in which the return happens to be negative

Turn the yearly contribution down to nothing, so the starting balance is on its own. Then find the lowest annual return that doubles it inside the twelve years — you want a final balance above 20,000 and below 21,000.

Added each year2,000

Paid in at the end of each year, so it earns nothing in that year.

Annual return300%

Final balance

42,642

Total paid in

34,000

Growth

8,642

Growth as a share of the final balance

0.203

• Balance
• Total paid in

◷ Not there yet — keep moving the controls.

Six percent gets there, and a tenth of a point less does not. The rule was not close — it was exact to the precision the control allows, which is the sort of thing that makes an approximation worth memorising.

Check

Two balances start together. One grows at 7% for thirty years. The other grows at 9% for the same thirty years. How much bigger is the second at the end?

Choose one answer

About 2% bigger, since the rates differ by two pointsAbout 30% biggerRoughly three quarters biggerMore than five times bigger

Check answer

›Why seventy-two and not seventyOptional

The exact answer uses the natural logarithm of two, which is about 0.693 — so seventy is the more accurate divisor for very small rates and for anything compounding continuously.

Seventy-two wins on two counts anyway. It is more accurate in the six to ten percent band where most of these questions live, because annual compounding is slightly slower than continuous. And it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is the difference between a rule you use in your head and one you write down.

Worth remembering

• A balance growing at a rate is multiplied by one plus that rate, raised to the number of years.
• Roughly: seventy-two divided by the annual percentage rate gives the years until doubling, and it is most accurate between about four and twelve percent.
• Doublings stack rather than add — three doublings is eight times the start, not six.
• Every doubling adds more than the entire history of the balance up to that point.
• The same rule applies to debts, prices and fees, because anything growing by a percentage of itself runs the same formula.