Kelly criterion

用通俗的话解释

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.

计算方法

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

如何使用

• 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.

得出结论

• 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.

注意事项

• 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.

一点历史

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.