There is a thought experiment I return to often. Suppose you have two people, identical in every respect, except that one begins saving ten years earlier than the other. Both save the same monthly amount. Both earn the same return. At retirement, the early saver has roughly twice the wealth of the late one — not because she saved more, but because she gave time more time to work.
This is compound growth, and it is one of the few genuine surprises in economics. Most of what economists study is, at some level, intuitive: prices respond to scarcity, incentives shape behaviour, trade creates surplus. But compound growth is not intuitive. The human mind was not built to feel it.
The Mechanics
The formula is simple enough to fit on a business card:
where $A_0$ is the principal, $r$ is the annual rate of return, and $t$ is time in years. In the continuous limit — as compounding intervals shrink toward zero — this becomes:
The exponential $e^{rt}$ grows without bound. Slowly at first — imperceptibly, even — and then, past some threshold, with startling speed. At $r = 7\%$, money doubles roughly every ten years (the Rule of 72: $72 / r \approx$ doubling time in years). At $r = 10\%$, it doubles every seven.
Capital as Stored Time
The classical economists thought of capital as past labour embodied in tools, machinery, and infrastructure. There is something to this. A factory represents thousands of hours of human effort: the iron smelted, the bricks laid, the blueprints drawn. In this sense, capital is time that has been crystallised — extracted from the past and made available to the present.
But compound growth adds a twist. Capital does not merely preserve past time; it generates future time. If the return on capital exceeds the growth rate of the economy as a whole — Piketty's condition $r > g$ — then wealth accumulates faster than income, and the share of output flowing to owners of capital rises at the expense of workers. This is not a moral judgement. It is arithmetic.
"When the rate of return on capital exceeds the rate of growth of output and income, capitalism automatically generates arbitrary and unsustainable inequalities." — Thomas Piketty, Capital in the Twenty-First Century
The Discount Rate and the Long Run
The same logic that makes wealth grow also makes the future shrink. The present value of a cash flow $C$ received in $t$ years, discounted at rate $r$, is:
At even a modest discount rate of 5%, a pound received in a hundred years is worth less than half a penny today. This has consequences that extend well beyond finance. Cost-benefit analyses of climate change, infrastructure investment, and pandemic preparedness all depend on what discount rate you choose. A high rate says: future people matter less. A low rate says: they matter nearly as much as we do. This is not a technical question. It is an ethical one wearing a mathematical costume.
A Closing Thought
Keynes famously wrote that in the long run we are all dead. He meant it as an argument for action now, against those who counselled patience and left everything to market adjustment. But there is another reading. In the long run, the compounding of small advantages into vast fortunes, the discounting of future lives into present negligibility, the exponential divergence between those who own and those who work — all of this is arithmetic. And arithmetic, unlike politics, does not negotiate.
Understanding compound growth will not tell you what policies to adopt. But it will stop you from being surprised when the numbers arrive.