Portfolio construction
Kelly criterion
The bet size that makes your money grow fastest over the long run, if your estimates of return and risk were exactly right.
The idea in plain words
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.
How it is calculated
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) ÷ bHow to use it
- 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.
Drawing conclusions
- 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.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). One ticker: US500 with 8% return and 18% volatility has f* = 0.08 ÷ 0.0324 = 2.5 times the capital. For the four tickers, full Kelly wants 5.4× leverage; capped at the 2× maximum it holds 80% US500, 10% GER40 and 111% gold: 14.8% expected return, 23.4% volatility.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- 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.
A little history
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.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.