Kelly criterion

La idea en pocas palabras

Bet too little and you grow slowly; bet too much and the losses compound into ruin. In between there is a size that maximises the long-run growth rate of wealth. For a portfolio of tickers with normal-like returns it is the inverse covariance matrix times the expected excess returns, which usually means leverage.

Cómo se calcula

Maximise expected return − ½ × variance (the growth rate of a continuously rebalanced portfolio). Long only by default; the result is scaled down to the maximum leverage if it exceeds it. The part not invested is cash.

Growth rate  g(w) ≈ wᵀ(μ − r_f) − ½ wᵀΣw
Kelly:  w* = Σ⁻¹(μ − r_f)
One ticker:  f* = (μ − r_f) ÷ σ²     Coin flip: f* = p − (1 − p) ÷ b

Cómo usarlo

• As an upper bound: never bet more than Kelly, because beyond it you get more risk and less growth.
• Prefer Fractional Kelly for real use.
• Watch the "full Kelly would use …× leverage" note in the study.

Sacar conclusiones

• Full Kelly expects drawdowns of 50% or more along the way, even when the estimates are right.
• If it is capped by the maximum leverage, the cap, not Kelly, is setting your size.

Ten cuidado con

• Overestimate returns by a little and Kelly overbets by a lot; overbetting is far worse than underbetting.
• It assumes you can rebalance continuously and that the estimates hold.

Un poco de historia

John L. Kelly Jr. at Bell Labs, "A New Interpretation of Information Rate" (1956), about gamblers with a noisy private wire. Edward Thorp used it for blackjack (Beat the Dealer, 1962) and then in his hedge fund.