A bet with a positive expected value that almost everyone loses.

The expected value is computed exactly right. Half the time you multiply your money by 1.5, half the time by 0.6, so the average multiplier is (1.5 + 0.6) / 2 = 1.05 — a 5% gain every round, forever. By that arithmetic the coin is a fortune. And the arithmetic is not wrong: if a million people each played one round, their combined wealth really would grow 5%. The expected value describes what happens to the crowd.

But you are not the crowd. You are one person, playing round after round, and your wealth doesn’t add to the next round’s — it multiplies into it. A 50% gain followed by a 40% loss does not leave you up 10%; it leaves you at 1.5 × 0.6 = 0.9 of where you started. Down 10%. Do that repeatedly and you decay toward nothing, which is exactly what the median player does: after 40 rounds in the simulation above, the typical player is left with pennies while the mean sails serenely upward, propped up by a vanishingly rare lottery winner. The number that says “get rich” is an average taken across parallel universes. You only get to live in one of them.

The trap has a precise mechanism, and it lives in the gap between two ways of averaging. The arithmetic mean adds the returns and divides — +50% and −50% average to 0%. The geometric mean multiplies them and takes the root — and multiplication is what your money actually does as it compounds. A year of +50% followed by a year of −50% has an arithmetic mean of zero, but it leaves £100 at £75. You didn’t break even; you lost a quarter of your money.

The size of this gap is governed by volatility, and the relationship is almost embarrassingly simple: the return you actually compound is roughly the average return minus half the variance. That subtracted term is volatility drag, and it is not a fee anyone discloses. It is the silent cost of bounciness, and it grows with the square of the swings.

Underneath the two averages sits a deeper idea, named and championed by the physicist Ole Peters: most of finance quietly assumes a system is ergodic — that the average across many parallel players at one moment equals the average for one player across time. For things that compound, that assumption is simply false. The crowd’s average and your trajectory through time are two different quantities, and conflating them is the whole illusion.

The expected value answers “if I cloned myself into a thousand parallel universes and averaged us, how would the average clone do?” Your actual life answers a different question: “how does one clone do, living through the rounds one after another?” When wealth multiplies, those answers diverge violently — the first dominated by rare lucky universes, the second tracking the lonely, downward median.

Once you see it, a lot of finance rearranges itself. Diversification isn’t just “don’t put eggs in one basket”; it is a way of lowering volatility and therefore drag, raising the return you actually compound. Position sizing — the Kelly criterion and its cautious cousins — is the maths of how much to bet so the time-average growth is maximised rather than the seductive but lethal expected value. And insurance, which has a negative expected value by construction, can still be rational: you pay a little arithmetic mean to escape the catastrophic tails that would end your one and only trajectory.

Leveraged ETFs. A fund promising “2× the daily return” of an index is a volatility-drag machine. In a market that ends flat but bounces along the way, the doubled daily swings compound into a steady loss — the fund can fall while the index it tracks goes nowhere. The product does exactly what it says each day; the multiplicative arithmetic across days does the damage. Held for months, these are not what most buyers think they are.

“Average annual return.” When a fund advertises its average annual return, that is usually the arithmetic mean of yearly figures — the bigger, friendlier number. What you actually earned is the compound annual growth rate, the geometric mean, which is always lower for anything that fluctuates. The gap is volatility drag wearing a marketing suit, and it widens precisely for the volatile, exciting funds whose big averages are doing the advertising.

The lottery of the lone stock. Most individual stocks underperform cash over their lifetimes; a tiny minority deliver almost all of the market’s gains. The mean stock return is healthy, carried by the handful of giant winners — but the median stock is a slow loser. Picking one stock is playing the coin: you’re betting your single trajectory on landing in the lucky tail that holds the average up.

Trust the geometric mean. For anything that compounds — returns, growth rates, portfolio values — the arithmetic average overstates what you keep. Ask for the compound annual growth rate, or compute it yourself from the start and end values; if someone quotes an “average return” on a volatile thing without it, assume the friendlier number was chosen on purpose.

Treat volatility as a cost. Bounciness is not merely risk in the sense of “might lose,” it is a guaranteed subtraction from long-run growth, roughly half the variance every period. A calmer path to the same average return is genuinely richer, not just more comfortable. This is the mathematical case for diversifying and for not over-betting.

Don’t read expected value as destiny. A positive expected value is necessary but nowhere near sufficient. If a bet’s downside multiplies your wealth toward zero, the rare upside that keeps the average positive will almost certainly not be yours. Size every bet so that you survive the bad rounds, because you only get one sequence of them.

So the question to carry into any rosy projected return is the one that separates the crowd from the customer: is that an average over many people, or the likely path of my one account? If the process compounds and the number is a mean, quietly subtract the drag and look at the median instead. The rest of the library is full of averages that mislead about a population; this is the one where the average misleads you about yourself.