Kelly criterion
Kelly gives the stake that maximises long-run compound growth for a known edge. It is also the most over-applied formula in finance, because it assumes the edge is known — so this page reports the growth curve and the over-betting cliff alongside the number.
| Multiple | Fraction | Stake | Log growth | Verdict |
|---|---|---|---|---|
| 0.25× | 4.38% | $1,094 | 0.0080 | Under-betting |
| 0.50× | 8.75% | $2,188 | 0.0137 | Under-betting |
| 0.75× | 13.13% | $3,281 | 0.0171 | Under-betting |
| 1.00× | 17.50% | $4,375 | 0.0183 | Growth-optimal |
| 1.50× | 26.25% | $6,563 | 0.0136 | Over-betting |
Growth is flat near the optimum, so half Kelly gives up very little growth for roughly half the variance — which is why practitioners rarely bet full Kelly.
f* = (p(b+1) − 1)/b · equivalently p − q/b
g = p ln(1+fb) + q ln(1−f) · expected log wealth per bet
f = k·f* · k < 1 trades growth for stability
f > 2f* ⇒ g < 0 · past twice Kelly, growth turns negative
A clear path from inputs to results.
Kelly maximises long-run compound growth given a known edge. It is also the most over-applied formula in finance, because the edge is never known — so this page reports the growth curve and the over-betting cliff, not just the number.
How to use
- Enter the probability of winning.
- Enter the net odds: how much you win per unit staked.
- Set the bankroll.
- Choose the Kelly multiple — 0.5 is the usual practical choice.
- Check that the edge is positive before sizing anything.
The formula
Reference: Kelly (1956); Thorp (2006) on the Kelly criterion in markets.
Worked example
55% win rate at 1.2:1 odds, $25,000 bankroll
- Win probability 55%.
- Net odds 1.2:1.
- Bankroll $25,000.
- Half Kelly.
- 1
Edge: 0.55 × 1.2 − 0.45 = 0.21 per unit staked.
- 2
Full Kelly: (0.55 × 2.2 − 1) / 1.2 = 17.5% of bankroll.
- 3
Half Kelly: 8.75%, or $2,187.50.
- 4
Log growth at half Kelly is slightly below the full-Kelly maximum.
- 5
Betting above 35% (twice Kelly) turns growth negative despite the edge.
Full Kelly 17.5% · half Kelly 8.75% ($2,188) · over-betting threshold 35%.
Interpreting the result
Growth is flat near the optimum, which is the practical case for fractional Kelly: half Kelly gives up very little growth for roughly half the volatility and a much smaller chance of a catastrophic drawdown. The asymmetry at the top end is brutal — twice Kelly is zero growth, and beyond that the long-run outcome is ruin.
Assumptions & limitations
- The win probability and payoff are known and constant across bets.
- Bets are independent and can be repeated indefinitely.
- No transaction costs, taxes or position limits.
- The bankroll is fully available and cannot be replenished.
Frequently asked questions
What if Kelly says the fraction is negative?
There is no edge, and the correct stake is zero. A negative Kelly is the formula telling you to take the other side or walk away, not to bet a small amount.
Does Kelly apply to stock portfolios?
Yes, in continuous-time form it becomes the growth-optimal portfolio, which for a single risky asset is the excess return divided by variance. With multiple assets it is the mean-variance tangency portfolio.
Why do practitioners use half Kelly?
Because the inputs are estimates. If you overestimate your edge, full Kelly is already over-betting; half Kelly gives a margin of safety and halves the variance for a small growth cost.
Why is over-betting so dangerous?
Because log growth is concave and turns sharply negative past the optimum. At twice Kelly the expected growth rate is exactly zero, and any further increase guarantees a declining bankroll over time.
Methodology last reviewed 2026-10-18. Results are informational and not financial or engineering advice — see our disclaimer.