2. The Kelly criterion: nobody uses it in full

Risk 2: Position sizing

In 1956, John Kelly, a physicist at Bell Labs, published a paper on noisy communication channels that contained, almost in passing, the exact answer to the question raised in the previous module. Edward Thorp read it, realized it worked for blackjack and later for the markets, and built an entire career on it. It is worth knowing what that criterion says, and worth knowing even better why using it in full is a bad idea.

What question Kelly answers exactly

The Kelly criterion is not after the largest expected gain. If it were, the answer to any edge would be to stake everything. Nor is it after minimizing risk, because then the answer would be not to bet at all.

What it does is answer a very specific question: which fraction of capital, repeated indefinitely, grows wealth at the highest compounded rate possible. It is exactly the point we located at 25% in the previous module for that particular trade. Kelly is the general formula for that maximum.

The intuition is that the optimal fraction rises when the edge rises and falls when the swing in the results rises. In a simple bet where you win or lose the same amount, the Kelly fraction is literally the edge: the probability of winning minus the probability of losing.

A complete example: the biased coin

You have a coin that comes up heads 60% of the time. You bet on heads: you win what you staked or you lose it. You can repeat as many times as you like.

The Kelly fraction is 0.60 minus 0.40, that is, 20% of capital on each toss, recalculated on the live capital at every moment. With that fraction, compounded growth is roughly 2.0% per toss. A hundred tosses later, your capital has multiplied about 7.4 times.

Now the alternatives:

• Betting 10%, half of Kelly, growth falls to 1.50% per toss. With half the size you keep three quarters of the growth.
• Betting 5%, a quarter of Kelly, you keep around 44% of the growth with a quarter of the swing.
• Betting 40%, twice Kelly, growth turns slightly negative. With the same favorable coin and twice the conviction, you stop making money.

That last line deserves reading twice. Going past Kelly is not taking on more risk in exchange for more return: it is taking on more risk in exchange for less return. It is the one stretch of the financial world where both things get worse together, and it is where a great many people live without knowing it.

The path is unbearable

Even if you knew the edge with absolute certainty, betting full Kelly produces a journey almost nobody can stand. In the idealized model, whoever bets the full fraction has roughly a 50% probability of seeing his capital cut in half at some point along the way, and around 10% of seeing it fall to 10%. These are not failure scenarios: they are part of the optimal path.

Betting half of Kelly, that probability of ending up losing half the capital falls to roughly 12%. You have given up a quarter of the growth in order to divide by four the probability of the blow that would make you quit. That trade-off explains why professional practice almost always operates below Kelly.

The real problem: the edge is not known

All of the above assumes you know the coin comes up heads 60% of the time. In a market you do not know that: you estimate it from a finite, noisy sample, on a process that is changing as well.

Suppose you estimate 60% and stake the corresponding 20%, but the real edge was 55%. Your true Kelly was 10%, so you are betting twice what you should. The result: your compounded growth falls to practically zero. Five points of estimation error have eaten the entire return.

If the real edge were 52%, staking that 20% produces a compounded loss of around 1.2% per toss. The strategy still has a positive edge and it ruins you all the same.

Compare that with the symmetrical error. If you estimate 60% but stake only half, 10%, and the truth was 55%, you still grow at around 0.5% per toss. Falling short is forgiven; overshooting is not. That asymmetry, and not the elegance of the formula, is what governs practice.

There is one more layer, pointed out among others by Marcos López de Prado: the parameters that feed any sizing calculation are estimated on scarce data, and if they have also been obtained by trying many configurations until one of them shone, the estimated edge is inflated from the start. Feeding Kelly an inflated edge is exactly the scenario we have just seen.

Real case: Thorp, from blackjack to Princeton/Newport

Thorp applied Kelly first at the tables in Las Vegas in the early sixties, with a card-counting system that gave him a small, measurable edge, and later at his fund Princeton/Newport Partners, active from 1969 to 1988 and devoted above all to convertible and warrant arbitrage.

The figures Thorp himself publishes in his autobiography describe nearly two decades with annualized returns of around 15% net to investors and extraordinary consistency, with very few negative months over the whole life of the fund. That profile is not the profile of someone betting his theoretical maximum: it is that of someone who knew the formula better than anyone and systematically applied a fraction of it, on trades where the edge was small but reasonably measurable.

The lesson Thorp repeats is that the usefulness of Kelly lies not in the number it spits out but in the discipline of having to state your estimated edge before you size anything. If you cannot put a defensible figure on your edge, the problem is not which fraction to bet.