1. Position size matters more than getting the signal right
Risk 2: Position sizing
There is a question everyone asks when they start out: what to buy. And there is another that almost nobody asks until it is too late: how much to buy. This whole course exists to argue that the second matters more than the first, and this module is the arithmetic proof of that claim.
Two people, the same signal, opposite endings
Imagine a repeatable trade with a clear, known edge. Half the time you make 50% of what you put in; the other half you lose 40%. The expectation is unambiguously favorable: for every euro risked you expect to get back 1.05 euros. Nobody would dispute that the signal is a good one.
Ana runs it with her entire capital on every round. Bruno runs it with a quarter, and leaves the rest untouched. They have the same information, the same hit rate and the same market. Only one figure differs: the fraction of capital risked per trade.
Ana multiplies her wealth by 1.5 or by 0.6 on each round. Two rounds, one of each, and her capital sits at 1.5 times 0.6, that is, 0.9: she has lost 10% on two trades whose average is positive. Round after round, Ana loses around 5% compounded, and with enough rounds her capital tends to zero with near certainty.
Bruno, risking a quarter, multiplies by 1.125 or by 0.90. Two rounds, one of each: 1.0125. He makes around 0.62% per round. It is not spectacular, but it is positive and it accumulates.
The same edge. The same signal. One is ruined and the other compounds. The difference is size.
The arithmetic mean lies; the compounded one pays
The root of the misunderstanding is that there are two ways of averaging returns and only one of them shows up in your account.
- The arithmetic mean adds the results up and divides. It is what appears in the brochures and what mathematical expectation uses.
- The compounded mean, or geometric mean, is what actually multiplies your capital when you reinvest.
A brutal and simple example: a strategy that gains 100% one year and loses 50% the next, repeating the cycle, has an arithmetic mean of 25% a year and a compounded return of exactly 0%. You have not made a penny in your whole investing life while the brochure boasts of 25%.
The compounded mean is always smaller than or equal to the arithmetic one, and the distance between the two depends on a single thing: how much the result moves about. A good approximation is that compounded growth equals the average return minus half the square of the volatility. With Ana's trade: a mean of 5%, a swing of 45%, a penalty of about 10 points, a result of around minus 5%. It matches the earlier calculation.
Why the punishment grows faster than the prize
Here is the heart of the module. When you double the fraction risked, the average return doubles too: it is a linear relationship. But volatility doubles as well, and since the penalty depends on the square of volatility, that penalty is multiplied by four.
That mismatch explains the whole of the following table for the very same trade:
- Risking 10% of capital: a mean of 0.5%, a minimal penalty, compounded growth of around 0.4%.
- Risking 25%: a mean of 1.25%, compounded growth of 0.62%. This is the peak.
- Risking 50%: a mean of 2.5%, a penalty of another 2.5 points, compounded growth of exactly zero. You multiply by 1.25 and by 0.80, and you are back where you started.
- Risking 100%: a mean of 5%, compounded growth of minus 5%. Ruin guaranteed given time.
Look at the 50%: the manager who risks half the capital has twice the average return of the one who risks a quarter, and yet ends up with nothing. He is collecting more expectation and making less money. Past a certain point, raising the size makes both things worse at once: what you earn goes down and what you suffer goes up.
What this does not say
It is worth guarding the idea against two mistaken readings.
It does not say that risking little is free. Whoever risks 1% in the example above makes barely 0.05% per round: he survives, but he gets nowhere. There is a genuine trade-off here, not an instruction to shrink.
And it does not say that a negative edge can be fixed with size. If the expectation is unfavorable, any positive fraction loses money and the only optimal figure is zero. Size amplifies the sign you already had; it does not change it.
Real case: Victor Niederhoffer, October 27, 1997
For years Niederhoffer was one of the best-performing managers of his generation, with a good decade of very high returns and a celebrated book published barely a year earlier. His strategy included selling put options on the American index: collecting a steady premium in exchange for taking on the fall if the market collapsed. It is a trade with positive expectation on the overwhelming majority of days.
On October 27, 1997, as the Asian crisis spread, the Dow Jones lost 554 points, around 7%, and trading was halted. The short positions shot up in value, the margin calls arrived and Niederhoffer's funds were wiped out. It was not a long run of mistakes: it was one day.
What matters for this module is that his signal was not absurd. Selling volatility on an index has historically been profitable on average. What did not survive was the size: an exposure sized for the normal day and not for the day that comes round once every few years. When the adverse outcome is large enough relative to capital, the quality of the signal stops mattering, because there is no capital left to go on playing with.