Compound growth

Put 1,000 pounds in an account that pays 7% a year. After one year you have 1,070. The 70 pounds is interest on your money. In the second year you earn 7% on 1,070, not on 1,000. You get 74.90. The extra 4.90 is interest on your interest. That is the whole trick, and it is the reason the end looks nothing like the start.

Leave it alone for ten years and you have about 1,967. For twenty years, 3,870. For thirty, 7,612. The money has not grown three times faster in the third decade. It has grown by the same 7% each year. But 7% of a big number is a big number.

Here is a quick way to feel it. Divide 72 by the yearly rate and you get the years it takes to double. At 7%, that is about ten years. So thirty years is three doublings: 1,000 becomes 2,000, then 4,000, then 8,000. The last doubling adds more than the first two put together.

The same shape shows up far from money. A skill you practise a little each day builds on what you already know. A reader who tells two friends, who each tell two friends, reaches a town in a few weeks. A crack in a wall that grows by a share of its length each winter is slow for years and then sudden.

It also runs backwards. A fund that charges 1.5% a year does not take 1.5% of your money. It takes 1.5% of what the money would have become, every year, and the losses compound too. Over thirty years that fee can eat a third of the pot.

The honest warning is that nothing compounds for ever. Savings are taxed, companies stop growing, and a crack in a wall meets the end of the wall. The maths is not a law of nature. It is what happens while the rate holds, and the rate always stops holding at some point.