The Kelly Criterion tells you how much capital to risk when you have a statistical edge. It chooses the fraction of wealth that maximises expected long run compounded growth, rather than the biggest expected profit on one bet. Its most important lesson for traders is that betting too much can turn a profitable strategy into a losing one.
The Kelly Criterion formula
For a repeated bet that wins with probability p, earns b dollars of net profit per dollar staked on a win, and loses the entire stake otherwise, the growth optimal fraction of current capital is:
f* = (b × p − (1 − p)) / b = p − (1 − p) / b
Here f* is the fraction of your bankroll to stake. If it is zero or negative, the formula does not justify betting on that outcome.
Why this fraction? After a win, wealth is multiplied by 1 + bf; after a loss, by 1 − f. Kelly maximises g(f) = p ln(1 + bf) + (1 − p) ln(1 − f), the expected logarithmic change in wealth per round.
This textbook formula assumes known, stable odds, repeated opportunities, no fees and no losses greater than the stake. Market returns rarely satisfy those assumptions exactly.
Worked example: a 60% winning bet
Imagine a coin that lands heads 60% of the time and pays even money, so p = 0.60 and b = 1. Kelly gives f* = 0.60 − 0.40 = 0.20, meaning a 20% stake.
With a $100 bankroll, you initially risk $20. After each round, calculate the next stake from your updated bankroll rather than repeatedly staking a fixed $20.
Original EverythingQuant practice: what if the payoff is larger?
A bet wins 30% of the time and pays $4 of profit for every $1 staked. What is its Kelly fraction?
f* = 0.30 − 0.70 / 4 = 0.125, so the growth optimal stake is 12.5% of current capital under the stated assumptions.
Why overbetting is dangerous
A positive edge does not rescue an oversized position. Large losses shrink the capital base used for future compounding, so maximising expected dollars and maximising growth are different objectives.
| Betting strategy | Capital staked | Expected log growth |
|---|---|---|
| Half Kelly | 10% | +1.50% |
| Full Kelly | 20% | +2.01% |
| Twice Kelly | 40% | −0.24% |
At twice Kelly, the expected one round profit is still 8% of the bankroll, but expected log growth is negative. Two consecutive losses cut capital by 64%, compared with 36% at full Kelly.
The exact point where overbetting becomes unprofitable depends on the probabilities and payoffs. The larger lesson is that position sizing can destroy a genuine trading advantage.
Fractional Kelly in quant trading
Full Kelly is optimal for the mathematical model, not automatically for a real portfolio. Quants must estimate win probabilities and payoff distributions, while markets introduce costs, changing regimes, correlated positions and losses beyond a simple binary payoff.
Fractional Kelly reduces the calculated exposure: f = α × f*, where α might be 0.5 for half Kelly. This sacrifices some theoretical growth in the ideal model in exchange for smaller positions and often more tolerable drawdowns.
For multiple correlated strategies, apply the same log wealth idea to the portfolio as a whole, maximising E[ln(1 + portfolio return)] subject to realistic position and risk limits. Adding independently calculated Kelly stakes can substantially underestimate combined exposure.
Three Kelly Criterion projects for quants
Each of these can become a short Python notebook with a clear question, reproducible results and an honest limitations section.
1. The overbetting simulator
Simulate 10,000 bankroll paths of 250 repeated bets. Compare half, full and twice Kelly using median final wealth, worst drawdown and the fraction of paths that lose more than half their starting capital.
Plot the growth curve against stake size and use a fixed random seed for a fair comparison.
2. Kelly with uncertain probabilities
Estimate the win probability from small training samples, then size new bets using full Kelly, half Kelly and a conservative capped strategy. Compare performance on unseen outcomes to expose how overconfidence changes growth.
Extend the experiment using a Bayesian estimate or a changing underlying win rate.
3. Portfolio Kelly optimiser
Generate correlated asset returns and numerically maximise expected portfolio log wealth. Compare unconstrained weights with leverage caps, trading costs and a fractional allocation.
Require positive wealth in every simulated scenario. Evaluate out of sample growth and drawdown rather than presenting an in sample optimum as a tradable strategy.
Common Kelly Criterion questions
Does Kelly prevent ruin?
No. Even correctly sized positions can suffer severe drawdowns, and real trading can involve leverage, execution gaps or larger losses than the model assumes.
Is half Kelly always the right choice?
No. It is a common risk reduction choice, not a universal optimum. The appropriate fraction depends on estimation uncertainty, correlation, loss limits and your willingness to accept drawdowns.
Further reading and sources
- J. L. Kelly Jr., A New Interpretation of Information Rate, Bell System Technical Journal, 1956.
- MacLean, Thorp and Ziemba, Good and Bad Properties of the Kelly Criterion, 2010.
- Ben Lynn, The Kelly Criterion, a worked mathematical explanation.