Drawdowns and risk of ruin
Measure drawdowns properly, read their distribution from simulation, and set stop rules and affordability limits — then see how a five-point overestimate turns Kelly staking into ruin.
Learning objectives
- Compute the maximum drawdown of a bankroll path and the gain needed to recover from it.
- Read median, 90th-percentile and tail drawdown statistics from a staking simulation.
- Show how overestimating p by five points changes drawdown and ruin for flat, fixed, Kelly and lower-bound staking.
- Write a pre-committed stop rule and an affordability limit for a simulated virtual-Credit bankroll.
- Explain what risk-constrained Kelly adds and what it assumes.
Drawdown: the number you live through
A drawdown is the fall from the highest bankroll reached so far, as a share of that peak. The maximum drawdown is the deepest such fall over a whole path. Time under water — the number of picks until a new peak — matters too, because long recoveries are where discipline breaks.
DD_t = (peak_t − W_t) / peak_t, MaxDD = max over t of DD_tGain needed to recover = 1 / (1 − d) − 1| Drawdown | Gain needed to return to the peak |
|---|---|
| 20% | +25.0% |
| 30% | +42.9% |
| 50% | +100% |
| 75% | +300% |
| 90% | +900% |
In this lesson ruin means the simulated bankroll falls below 1% of its starting value, the threshold the Kelly and Drawdown lab uses. With proportional staking a bankroll never reaches exactly zero, so a practical ruin level is needed.
Drawdowns are a distribution, not a number
One simulated path tells you little. The reference scenario below repeats 500 even-money picks over 400 paths (seed 1). The bettor believes p = 0.55; the truth is p = 0.50, so the real edge is zero. The lower bound of the bettor's interval is 0.50. All five rules face the same outcome sequence on each path.
| Rule | Stake | Median final | Median max DD | 90th pct max DD | P(DD ≥ 50%) | P(below start) | P(ruin) |
|---|---|---|---|---|---|---|---|
| Flat 1% of start | 1% | 98.0 | 22.2% | 39.9% | 2.3% | 51.2% | 0.0% |
| Fixed 2% of current | 2% | 86.9 | 41.7% | 60.8% | 28.5% | 60.0% | 0.0% |
| Full Kelly on 0.55 | 10% | 6.6 | 97.9% | 99.8% | 100% | 85.5% | 16.3% |
| Half Kelly, capped at 5% | 5% | 48.4 | 78.7% | 92.7% | 97.8% | 72.3% | 0.0% |
| Half Kelly on lower bound | 0% | 100 | 0% | 0% | 0% | 0% | 0% |
With zero true edge, any positive stake has negative log growth, and bigger stakes lose faster. The simulation agrees with theory: G(0.10) at p = 0.50 is −0.00503 per pick, and exp(500 × G) = 0.081, close to the simulated median of 6.6 out of 100. For 5% the theory gives 0.535 (simulated 48.4); for 2%, 0.905 (simulated 86.9).
Now suppose the 55% had been right. In the same setup full Kelly's median rose to 1,223.4, but its median maximum drawdown was still 83.6% and every path suffered a 50% drawdown; half Kelly reached a median of 652.9 with a 54.3% median drawdown. Even a correct Kelly bettor lives through deep falls.
Stop rules and affordability limits
A stop rule is decided before you start. Useful examples for a simulated bankroll: cap any single pick at a fixed share; pause and re-audit the model if the bankroll falls 25% from its peak; and stop a test after a planned number of picks rather than when results look good or bad. A stop rule does not raise EV. It limits damage while a model may be wrong and forces a checkpoint to re-estimate.
Losing streaks are normal. In 4,000 simulated sequences of 500 fair coin-flip picks, 86.0% contained a run of seven or more losses. Seven straight losses at a 10% stake leave 47.8% of the bankroll; ten straight at 2% leave 81.7%.
Risk-constrained Kelly
maximise G(f) subject to P(bankroll ever falls below α × start) ≤ βThis is a better way to choose a stake than an arbitrary fraction if the probabilities are right. It does not show that drawdown constraints protect a bettor whose estimates are wrong — and the table above shows that a wrong estimate is the bigger danger. Combine a risk constraint with lower-bound probabilities, not with optimistic point estimates.
Worked example
Worked example · Reading drawdown from a path
A simulated virtual-Credit bankroll records these values after each pick: 100, 110, 125, 118, 96, 88, 104, 131, 120, 140. Your pre-committed stop rule says: pause and re-audit when the bankroll is 25% below its peak.
- Running peak
100 → 110 → 125 (held until 131, then 140)Peak of 125 after pick 2 - Deepest trough below that peak
Lowest value before a new high: 88Trough 88 at pick 5 - Maximum drawdown
(125 − 88) / 125 = maxDrawdown(path)29.6% - Gain needed from the trough
125 / 88 − 1+42.0% - Time under water
From the peak at pick 2 to the new high of 131 at pick 75 picks - Stop-rule check
Drawdown at 96 = 23.2%; at 88 = 29.6%Trigger fires at pick 5 - Later dip
(131 − 120) / 1318.4%, well inside the limit
The path ended at a new high of 140, so pausing at pick 5 would have cost upside in this instance. That is the nature of insurance: a stop rule is judged across all paths, not by the one you happened to see.
Practice · Five points of overconfidence
Open the Kelly and Drawdown lab with the preset (estimated 55%, actual 50%, lower bound 50%, odds 2.00, 500 picks). Record each rule's median ending bankroll, median maximum drawdown, chance of a 50% drawdown and chance of ruin. Then set actual to 55% and compare.
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.
Show what you should find
Reference run (400 paths, seed 1): full Kelly stakes 10% — median 6.6, median drawdown 97.9%, 100% of paths hit a 50% drawdown, ruin 16.3%. Half Kelly capped at 5% — median 48.4, median drawdown 78.7%, 97.8% hit 50%, no ruin. Fixed 2% — median 86.9, drawdown 41.7%, 28.5%. Flat 1% — median 98.0, drawdown 22.2%, 2.3%. Lower-bound half Kelly stakes 0% and stays at 100. With actual 55%: full Kelly's median rises to 1,223.4 but its median drawdown is still 83.6%; half Kelly reaches 652.9 with 54.3%. Your lab may use a different path count; the pattern should match.
Common mistakes
- ✗ Judging risk by the average ending bankroll.Look at medians, 90th-percentile drawdowns and tail probabilities. Averages are pulled up by a few lucky paths.
- ✗ Using a stop rule as a reason to raise stakes and ‘win it back’.Stop rules pause and trigger a re-audit; they never raise EV. Chasing increases exposure when your model is most likely wrong.
- ✗ Believing ruin requires a losing edge.Over-staking ruins positive-edge bankrolls too, and a modest overestimate of p (55% believed, 50% true) produced 16.3% ruin under full Kelly.
- ✗ Assuming a drawdown-constrained stake protects you from model error.Risk-constrained Kelly assumes the probabilities are known. Pair it with conservative (lower-bound) probabilities.
When it fails
Supporting research
- Busseti, Ryu & Boyd (2016): Risk-Constrained Kelly GamblingThe Journal of Investing · Peer-reviewed theoryRisk-constrained Kelly improves the growth–drawdown trade-off in simulation when the return distribution is known.
- MacLean, Thorp & Ziemba (2010): Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteriaQuantitative Finance · Peer-reviewed theoryKelly can be very risky in the short run; fractional Kelly lowers risk.
- Baker & McHale (2013): Optimal Betting Under Parameter Uncertainty: Improving the Kelly CriterionDecision Analysis · Peer-reviewed theoryEstimation error in probabilities justifies smaller stakes than Kelly on the point estimate.
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