Kelly, Drawdowns and the Danger of a Confident Forecast
The Kelly criterion is the mathematically fastest way to grow a bankroll — if you know the true probability. You never do. Here is what happens when you are wrong, and why the right stake is often far smaller, or zero.
Original educational writing based on the cited research. Grey-bordered notes report what studies found; gold-bordered notes are our interpretation. All prices and teams in examples are hypothetical.
Why Kelly maximises long-run growth
In 1956 John Kelly asked how fast a gambler with an informational advantage could grow capital, and showed that the answer is reached by maximising the expected logarithm of wealth rather than expected wealth Kelly (1956). Logarithms matter because returns compound: losing half your bankroll requires doubling the remainder to recover. A strategy that maximises the average outcome of each bet can still drive the typical bankroll to zero; maximising expected log wealth maximises the long-run growth rate instead.
Growth per bet: g(f) = p × ln(1 + f(o − 1)) + (1 − p) × ln(1 − f)Setting the derivative of g to zero gives the Kelly fraction for a single binary bet:
f* = (p × o − 1) / (o − 1)Worked example · Kelly on a hypothetical price
A hypothetical quote of 2.30 and a probability you know to be 48%.
- Expected value
0.48 × 2.30 − 1+10.4% per unit - Kelly fraction
(0.48 × 2.30 − 1) / (2.30 − 1) = 0.104 / 1.308.0% of the bankroll - Growth at Kelly (8%)
g(0.08) = 0.48 ln(1.104) + 0.52 ln(0.92)≈ +0.413% per bet (log) - Growth at half Kelly (4%)
g(0.04)≈ +0.311% per bet — about 75% of the maximum - Growth at double Kelly (16%)
g(0.16)≈ 0.004% per bet — essentially zero
Growth is flat near the top and falls away sharply beyond it. Staking about twice Kelly (16.0% here) eliminates long-run growth entirely, even though every single bet still has +10.4% expected value.
The catch: Kelly assumes you know p
Kelly is exact only when the probability is exactly right. In sport, p comes from a model or a judgement, and both carry error. Suppose you believe 48% but the truth is 45%. The true Kelly stake would be (0.45 × 2.30 − 1)/1.30 ≈ 2.7%, so your 8% stake is about three times too large.
| Stake | Growth per bet |
|---|---|
| 2% of bankroll | +0.044% |
| 4% of bankroll | +0.036% |
| 8% (Kelly on the wrong 48%) | −0.134% |
Every one of those bets still has positive expected value (0.45 × 2.30 − 1 = +3.5%). Yet staking Kelly on the overconfident estimate shrinks the bankroll over time. Over-betting a real edge can be as damaging as having no edge at all.
A simple teaching version of the idea: let the true probability be 50% at odds of 2.20 (Kelly = 8.3%), and let your estimate be noisy with a standard deviation of 5 points. Searching over multipliers of Kelly-on-your-estimate, the one with the highest average true growth in our seeded simulation is about 0.5 — half Kelly. That is an illustration under one assumed noise model, not a universal rule: more noise means more shrinkage.
Full versus fractional Kelly
Fractional Kelly stakes a fixed share of the Kelly fraction. For small edges, staking a fraction c of Kelly delivers roughly c(2 − c) of the maximum growth rate: half Kelly keeps about three quarters of the growth (75.1% in the example above) while halving the size of each swing. That trade-off is the main reason practitioners favour fractions of Kelly MacLean, Thorp & Ziemba (2010) Thorp (2006).
A simulation: flat stakes, Kelly and the wrong forecast
The Kelly and Drawdown lab simulates 400 synthetic bankrolls of 100 units over 500 even-money bets (odds 2.00), with every strategy facing the same outcomes. In the first scenario the model believes 55% but the truth is 50% — a 5-percentage-point overestimate (10% relative to the true probability) and, in reality, zero edge.
| Strategy | Stake | Median final bankroll | Median worst drawdown | Paths with a 50%+ drawdown |
|---|---|---|---|---|
| Flat 1% of start | 1 unit | 98.0 | 22% | 2% |
| Fixed 2% of current | 2% | 86.9 | 42% | 29% |
| Full Kelly on 55% | 10% | 6.6 | 98% | 100% |
| Half Kelly, capped at 5% | 5% | 48.4 | 79% | 98% |
| Half Kelly on a 50% lower bound | 0% | 100.0 | 0% | 0% |
With no real edge, each staking rule in this reference run has a median below the start or stands still; individual paths can still finish ahead. Full Kelly on the false belief wipes out over 93% of the median bankroll. The rule sized from the lower bound of the estimate sees no edge and abstains, preserving the bankroll on every path.
| Strategy | Stake | Median final bankroll | 5th percentile final | Paths with a 50%+ drawdown |
|---|---|---|---|---|
| Flat 1% of start | 1 unit | 150.0 | 112.0 | 0% |
| Fixed 2% of current | 2% | 246.0 | 115.0 | 2% |
| Full Kelly | 10% | 1,223.4 | 27.0 | 100% |
| Half Kelly, capped at 5% | 5% | 652.9 | 97.5 | 62% |
| Half Kelly on the 52% bound | 2% | 246.0 | 115.0 | 2% |
When the edge is real and exactly known, full Kelly produces by far the largest median — and still every one of the 400 paths suffers a drawdown of half the bankroll or worse, and one path in eight finishes below where it started. Half Kelly gives up much of the median for a dramatically better worst case. Nobody knows in advance which scenario they are in.
Re-run both scenarios, change the seed, and try a smaller overestimate. Notice how quickly the ranking of strategies changes when the truth moves.
Lab 3 · Kelly and drawdown
Synthetic · deterministicChoose inputs and run the simulation. The default sets a model that believes 55% when the truth is 50%: a 5-percentage-point overestimate (10% relative) at even money.
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
How deep are the bad runs when the edge is real?
Scenario B is the optimistic case: a real 10% edge, estimated perfectly. The worst peak-to-trough fall of each simulated path still tells a sobering story.
| Strategy | Median worst drawdown | 90th percentile worst drawdown | Paths ending below start |
|---|---|---|---|
| Flat 1% of start | 10.8% | 17.5% | 1.0% |
| Fixed 2% of current | 25.3% | 36.6% | 1.5% |
| Full Kelly | 83.6% | 95.0% | 12.5% |
| Half Kelly, capped at 5% | 54.3% | 71.2% | 5.3% |
Under full Kelly, the typical path at some point loses more than four fifths of its peak value, and one in ten loses 95%. Even half Kelly's typical worst fall exceeds half the bankroll. These are the results with a perfectly known edge; any overestimate makes them worse. A useful exercise is to decide in advance the largest drawdown you would genuinely tolerate, then find the stake size that keeps the 90th-percentile drawdown inside it. For most people that stake is far below Kelly.
- Decide limits first: a maximum drawdown and a maximum share per bet, before looking at any opportunity.
- Size from the pessimistic estimate, not the central one.
- Cap the total exposure across bets that can lose together.
- Re-estimate after results, but never raise stakes to recover losses.
Correlation: when bets are not independent
Kelly's single-bet formula assumes each bet is independent. Bets on the same match rarely are. In a Poisson score model with expected goals of 1.6 for the home side and 1.0 for the away side, home win and over 2.5 goals have probabilities of 51.3% and 48.2%, but they occur together 30.0% of the time rather than the 24.7% independence would imply — a correlation of about 0.21. Staking full Kelly on each as if they were separate overstates your true exposure to one result. Both bets lose together 30.5% of the time, against 25.2% if they were independent — so a double loss, the event that hurts a bankroll most, is about a fifth more likely than a naive calculation suggests.
The same applies to several bets that share a team, a model error or a kick-off window. A practical rule: count correlated positions as one bet when sizing, and cap total exposure per match and per model.
Sensitivity analysis: how fragile is the stake?
| Probability | EV per unit | Full Kelly |
|---|---|---|
| 44% | +1.2% | 0.9% |
| 46% | +5.8% | 4.5% |
| 48% | +10.4% | 8.0% |
| 50% | +15.0% | 11.5% |
| 52% | +19.6% | 15.1% |
A two-point change in the probability moves the Kelly stake by about 3.5 points of bankroll. Since few football models are accurate to within two points on a single match, the input uncertainty is larger than the stake itself. A sensible analysis therefore reports the stake as a range, and asks what happens at the pessimistic end.
Why no action may be the correct conclusion
Kelly carries a lesson that is easy to miss: when the edge is zero or negative, the optimal stake is exactly zero. Uncertainty-aware versions make that outcome more common, because they size from what you can defend rather than what you hope. If the pessimistic end of your range has no edge, the robust stake is nothing.
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References
- John L. Kelly Jr. (1956). A New Interpretation of Information Rate.Bell System Technical Journal 35(4), 917–926Maximising expected log wealth gives the fastest long-run growth when probabilities are known.Original source (doi:10.1002/j.1538-7305.1956.tb03809.x) (opens in a new tab)
- Rose D. Baker, Ian G. McHale (2013). Optimal Betting Under Parameter Uncertainty: Improving the Kelly Criterion.Decision Analysis 10(3), 189–199Kelly stakes should be shrunk when probabilities are estimated.Original source (doi:10.1287/deca.2013.0271) (opens in a new tab)
- Enzo Busseti, Ernest K. Ryu, Stephen Boyd (2016). Risk-Constrained Kelly Gambling.The Journal of Investing 25(3), 118–134Explicit drawdown constraints on Kelly betting.Original source (doi:10.3905/joi.2016.25.3.118) (opens in a new tab)
- Leonard C. MacLean, Edward O. Thorp, William T. Ziemba (2010). Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria.Quantitative Finance 10(7), 681–687Good and bad properties of Kelly and fractional Kelly.Original source (doi:10.1080/14697688.2010.506108) (opens in a new tab)
Further reading
- Thorp (2006): The Kelly Criterion in Blackjack, Sports Betting, and the Stock MarketA practitioner's exposition of Kelly in blackjack, sports betting and investing.
- Walsh & Joshi (2024): Machine learning for sports betting: Should model selection be based on accuracy or calibration?A corrected example of how Kelly betting can still lose with a calibrated model.