Compounding
compound growth · compound interest
Growth applied to a base that already includes previous growth. The defining property is that returns multiply rather than add, which means the order of outcomes stops mattering and their variance starts to — a single large loss cannot be offset by an equal gain.
In practice
A practice that grows 4% a month for four years and one that grows 15% a month for a year and then loses half its clients twice do not end in the same place. The second has the better average month and the worse decade.
The common mistake
Judging a multiplicative process by its arithmetic average. +50% then −50% averages zero and leaves you down 25%. The average return of a strategy is not the return you get from following it, and the gap widens with volatility.
Add three to a number ten times and you have added thirty. Multiply a number by 1.03 ten times and you have not multiplied it by 1.30. Nearly everything counterintuitive about compounding follows from that one difference, including the parts that are bad news.
The mathematics, and where it came from
Luca Pacioli set out the Rule of 72 in the Summa de arithmetica (1494) — divide 72 by the percentage rate for the approximate doubling time — which remains the fastest way to hold a growth rate in your head. Jacob Bernoulli, studying what happens as compounding intervals shrink, arrived at the constant eBernoulli, J. (1683). Investigating the limit of (1 + 1/n)^n as n increases, in the context of continuously compounded interest. The limit is approximately 2.71828 — the first appearance of e, discovered in a question about money rather than one about calculus. in 1683. The base of the natural logarithm was found in a question about interest.
Why the average return is the wrong number
This is the part that is almost always omitted. Gain 50 per cent, then lose 50 per cent: the arithmetic mean return is zero and the ending balance is 0.75 of the start. The quantity that governs a multiplicative process is the geometric mean, which is always at most the arithmetic mean and falls further below it as variance rises. The gap has a name — volatility dragFernholz, R. & Shay, B. (1982). 'Stochastic Portfolio Theory and Stock Market Equilibrium.' Journal of Finance 37(2). For small returns the geometric mean is approximately the arithmetic mean minus half the variance, so the penalty grows with the square of volatility. — and it is approximately half the variance.
Ole Peters (2019) argues the confusion is foundational rather than technical. Economics has conventionally evaluated a gamble by its ensemble average — what happens across many parallel players — when the situation a person actually faces is the time average, what happens to one player over many rounds. For a multiplicative process these two averages are not equalPeters, O. (2019). 'The ergodicity problem in economics.' Nature Physics 15. A coin-flip gamble paying +50% or −40% has a positive ensemble average and a negative time average: almost every individual player goes broke while the population mean rises, carried by a vanishing fraction of winners., and a gamble can be attractive in the aggregate while ruining nearly everyone who takes it. Kelly (1956) had derived the operational consequence three decades earlier: the position size that maximises long-run growth is the one that maximises the geometric mean, and it is markedly smaller than the one that maximises expected value.
Ruin
Under addition, a large loss is recoverable by a large gain. Under multiplication it is not: a 50 per cent loss requires a 100 per cent gain to undo, a 90 per cent loss requires 900 per cent, and a 100 per cent loss cannot be undone by anything, because every subsequent multiplication is applied to zero. Survival is therefore not one consideration among several in a compounding process; it is a precondition for the process existing at all, which is the whole content of the barbell strategy and of antifragility as a positioning rule.
The objections
The idea is routinely applied by analogy to skills, relationships and reputation, where the mathematics does not hold. Those things accumulate, plateau, decay without maintenance and are bounded; none of them has an exponent. 'Compounding' in those sentences means 'gets better over time', which is a reasonable claim dressed in a borrowed formalism that adds nothing but authority.
The second objection is selection. The examples are drawn from survivors, and survivorship is exactly what a compounding process filters on. The strategies that ruined their holders are not in the dataset, and under Peters's analysis they may be the great majority of those who followed the same rule.
What it rules out
It rules out the arithmetic mean as a summary of a multiplicative process. It rules out treating a drawdown as symmetric with a gain of the same percentage. It rules out any position sizing that tolerates a non-trivial chance of total loss, however favourable the expected value, because expected value is the ensemble average and you are one path.
It does not rule out patience being the active ingredient. It specifies what patience is for: not enduring volatility, but remaining in the process, which requires not being removed from it.
Sources
Bernoulli, J. (1683). On continuously compounded interest. · Fernholz, R. & Shay, B. (1982). 'Stochastic Portfolio Theory and Stock Market Equilibrium.' Journal of Finance 37(2). · Kelly, J. L. (1956). 'A New Interpretation of Information Rate.' Bell System Technical Journal 35(4). · Pacioli, L. (1494). Summa de arithmetica, geometria, proportioni et proportionalita. · Peters, O. (2019). 'The ergodicity problem in economics.' Nature Physics 15. · Taleb, N. N. (2012). Antifragile. Random House.
Concept web
Open the full webQuestions
What is compounding?
Growth applied to a base that already includes previous growth, so returns multiply rather than add. Because they multiply, order stops mattering and variance starts to: a large loss cannot be offset by an equal-sized gain.
Why is the average return misleading?
Because a multiplicative process is governed by the geometric mean, not the arithmetic one. +50% then −50% averages zero and leaves you down 25%. The shortfall is roughly half the variance — volatility drag (Fernholz & Shay, 1982).
What is the ergodicity problem?
Peters (2019) showed that the average outcome across many parallel players and the average outcome for one player over time are not the same for multiplicative processes. A gamble can have a positive ensemble average and a negative time average — good on paper, ruinous in sequence.
Does compounding apply to skills and relationships?
Only by analogy. Skills and relationships accumulate, plateau and decay without maintenance; none has an exponent. The claim that they get better over time is reasonable, but the mathematics of compounding does not license it.
What is the rule of 72?
Divide 72 by the percentage growth rate to approximate the doubling time: 8 per cent doubles in about nine years. Luca Pacioli recorded it in the Summa de arithmetica in 1494.