The 5% bankroll cap — Sharp's non-negotiable rule
Kelly tells you how much to stake given your edge. The 5% cap tells you the most you may stake no matter what Kelly says. Those are two different jobs, and the second one exists because the first one depends on a number you do not actually know. This lesson shows the drawdown arithmetic at 5%, 10%, and 20%, explains why the exception you are thinking of is never granted, and covers how to recompute the cap as the bankroll moves.
Kelly assumes you know your edge. You don't.
The Kelly formula is f* = (bp − q) / b, where b is the decimal payoff minus one, p is the true win probability, and q is 1 − p. It is provably growth-optimal, and the proof holds only if p is correct.
Watch what happens when it is not. At -110 (decimal 1.909, so b = 0.909), breakeven is 110 / 210 = 52.38%.
| Your true win rate | Full Kelly stake |
|---|---|
| 53% | 1.30% |
| 54% | 3.40% |
| 55% | 5.50% |
| 57% | 9.70% |
Now suppose you believe you are a 55% bettor and you are actually 53%. That is a two-point error in an estimate nobody can measure precisely over a realistic sample. Full Kelly at your believed edge is 5.50%. Full Kelly at your real edge is 1.30%. You are staking 4.2 times the correct amount, and Kelly's optimality property is gone.
The error is not symmetric either. Underbetting your edge costs you growth. Overbetting past a certain point costs you the bankroll, because compounding punishes large drawdowns far harder than it rewards equal-sized gains.
This is the entire argument for the cap. Kelly is a function of a quantity you estimate. The cap is a function of nothing. It holds when your estimate is wrong, which is exactly when you need it. Fractional Kelly — why ½ beats full almost every time makes the case for shrinking the fraction; the cap is the second layer.
The ruin math
Here is what a losing streak does to a bankroll at three staking levels, staking a fixed fraction of the current bankroll each time. All figures are exact arithmetic, not estimates.
| Consecutive losses | At 5% | At 10% | At 20% |
|---|---|---|---|
| 5 | −22.6% | −41.0% | −67.2% |
| 10 | −40.1% | −65.1% | −89.3% |
| 15 | −53.7% | −79.4% | −96.5% |
| 20 | −64.2% | −87.8% | −98.8% |
Now the part that matters more, because recovery is not symmetric with loss. To get back to even from a drawdown of d, you need a gain of d / (1 − d):
| Consecutive losses | Recovery needed at 5% | at 10% | at 20% |
|---|---|---|---|
| 10 | +67.0% | +186.8% | +831.3% |
| 20 | +179.0% | +722.5% | +8,573.6% |
Ten straight losses at 5% leaves you needing a 67% gain. Unpleasant, entirely achievable with a real edge and a normal season of volume. The same ten losses at 20% leaves you needing to grow the remaining stack more than eight-fold. That is not a bad run. That is a different account.
And ten straight losses is not a freak event. At a 55% win rate, the probability of a streak of ten or more consecutive losses somewhere inside a season is roughly 8.8% over 500 bets and 17.0% over 1,000 bets. Eight or more happens about 37% of the time over 500 bets. These streaks are scheduled, not surprising.
What the cap buys, honestly
Simulating a 500-bet season at a genuine 55% win rate on -110 prices, staking a fixed fraction of the current bankroll every time, over 20,000 runs:
| Stake | Median worst drawdown | Chance of a 50%+ drawdown | Median ending bankroll | Bottom 5% ending |
|---|---|---|---|---|
| 2.75% (half-Kelly) | 38.6% | 19.4% | 1.68× | 0.65× |
| 5% (the cap) | 61.5% | 80.1% | 1.98× | 0.35× |
| 10% | 89.9% | 100% | 1.25× | 0.04× |
| 20% | 99.9% | 100% | 0.01× | 0.00× |
Three things in that table are worth sitting with.
The cap is not comfort. A 61.5% median worst drawdown at a 5% flat stake is brutal. The cap does not make betting smooth. It makes it survivable, which is a lower bar and the only one that matters.
Past a point, more stake means less money. At 10% the median outcome is worse than at 5% despite the same edge, and at 20% the median run ends at 1% of the starting bankroll — with a real 55% edge working in your favor the entire time. Overbetting does not trade safety for return. It destroys both.
The 5% row is a worst case, not a plan. It assumes every bet is placed at the maximum. Half-Kelly on a genuine 5% edge lands at 2.75%, and that row is far calmer. The cap is a ceiling you rarely touch, not a target you aim at.
And if the edge was not there — the same simulation at a true 53% — the 5% stake ends at a median of 0.76× and the 10% stake at 0.18×. The cap does not save a bettor with no edge. It buys enough time to find out.
The "sure thing of the century" exception
Every bettor eventually finds the bet. The number is obviously wrong, the edge is enormous, the book is asleep, and 5% feels absurdly small.
The exception is never granted. Not once, not for anything.
The reason is not that big edges do not exist. They do, and Proptimizer surfaces them. The reason is that the confidence you feel is not evidence about the world. Every catastrophic bankroll story begins with a bet the bettor was sure about. The certainty is the common ingredient, not the edge.
There are mundane failure modes that analytical conviction does not protect against. The line is stale because of news you have not seen. The prop is mispriced because the book has a different player. Your devig assumption breaks on a heavy favorite. Someone gets scratched in warmups. A rule that admits exceptions for strong conviction fails precisely when conviction is highest.
If the edge really is that large, the correct response is not a larger stake. It is more bets at the cap. Volume at a maintained edge compounds. One oversized position does not.
Why the cap belongs in software
You will not enforce this at 11:40 p.m. after four losers. Willpower is the wrong substrate for a rule whose job is to work when judgment is degraded.
So put it in the tooling. One-Click Betting carries a configurable per-bet maximum, daily bet-count cap, weekly bankroll cap, and cool-down timer, and every Arbai agent ships with bankroll cap enforcement as one of four non-negotiable guardrails, alongside tilt-refusal middleware. The point is not that the software is smarter than you. It is that the software is not the one who just lost four in a row.
Two properties make a software cap work where a mental one fails: it applies before you see the betslip, and changing it is a deliberate act with friction. If you can raise your own limit in three seconds while tilted, you do not have a limit. Tilt — spot it before it costs you money covers why.
Recomputing the cap as the bankroll moves
The cap is 5% of the current bankroll, not 5% of what you started the season with.
Say you start at $5,000. Your cap is $250. You have a rough month and you are at $4,200 — the cap is now $210. Later you are at $6,300 and the cap is $315.
This is what makes fractional staking self-correcting. Stakes shrink automatically as the bankroll shrinks, which is why the drawdown table above bottoms out instead of hitting zero. Stakes grow as it grows, which is where compounding comes from. Skipping the downward adjustment is the more common and more expensive mistake.
Recompute on a fixed cadence — weekly is fine, daily is better — so a single result does not swing your sizing. And define bankroll honestly: it is the money set aside for betting, held in book balances and the account you fund them from. Not your net worth, and not money that has a job somewhere else.
The rule, in one line
No single bet exceeds 5% of the current bankroll. Half-Kelly sizes the bet; the cap overrides it whenever Kelly asks for more. Kelly is a recommendation built on an estimate. The cap is a constraint that does not care whether the estimate was right.