The math of bankroll management

Bankroll management is one of those topics that gets repeated so often it starts to sound like superstition. “Twenty buy-ins for cash. A hundred buy-ins for MTTs. Move down when you bust.” That advice is roughly right, but it is mostly handed down without any explanation of why — which means you don’t know when to deviate from it.

The truth is that bankroll management is just a particular application of probability theory, and once you understand the math, the rules of thumb you have heard your whole poker life start to make sense (and you can adjust them confidently for your own situation).

This is a longer post. Get a coffee.

The question we are answering

Bankroll management is the answer to this question:

Given my expected win rate, the variance of my game, and the size of my bankroll, what is the probability that I will go broke before I realize my expected value?

That probability is called your risk of ruin (RoR). When people talk about needing “20 buy-ins” for a cash game, what they are implicitly saying is: 20 buy-ins is a bankroll where your risk of ruin is low enough that you’re comfortable.

But low enough for whom? At what win rate? In what game? With what variance? Those questions matter a lot, and the standard rules of thumb gloss over them.

The two numbers you need

To compute your risk of ruin, you only need two inputs:

1. Your expected win rate, usually expressed in big blinds per 100 hands (bb/100) for cash, or as an ROI percentage for tournaments.
2. The standard deviation of your results, in the same units.

For online No-Limit Hold’em cash, a “winning low-stakes regular” looks something like:

• Win rate: 4 bb/100
• Standard deviation: 100 bb/100

(Both numbers depend on stake and game, and standard deviation in particular tends to be higher in PLO and looser games.)

For online MTTs, a “winning regular” might look like:

• ROI: 15%
• Standard deviation per tournament: ~150% of buy-in

These numbers are illustrative. Your actual numbers come from your tracking software over a large sample — at least 100,000 hands for cash, at least 5,000 tournaments for MTT.

The risk-of-ruin formula

For a cash game with normal-ish distribution, the standard risk-of-ruin formula is:

RoR = exp( -2 * B * WR / SD^2 )

Where:

• B is your bankroll, in big blinds
• WR is your win rate, in bb per hand (so 4 bb/100 = 0.04)
• SD is your standard deviation, in bb per hand (so 100 bb/100 = 1.0)

Let’s plug in some numbers. Suppose you have a bankroll of 20 buy-ins (= 2000 big blinds at a 100bb-deep table), with the win rate and SD above:

• B = 2000
• WR = 0.04
• SD = 1.0

RoR = exp( -2 * 2000 * 0.04 / 1.0^2 ) = exp(-160) ≈ 0

So a 4bb/100 winner with 20 buy-ins has essentially zero risk of ruin. Great. The “20 buy-ins” rule of thumb works out for a solid winner.

But what about a marginal winner — 1bb/100, same SD?

RoR = exp( -2 * 2000 * 0.01 / 1.0^2 ) = exp(-40) ≈ 4 × 10⁻¹⁸

Still essentially zero. Good.

What about a break-even player who thinks they’re a 4bb/100 winner but is actually a 0.5bb/100 winner?

RoR = exp( -2 * 2000 * 0.005 / 1.0^2 ) = exp(-20) ≈ 2 × 10⁻⁹

Still vanishingly small. OK.

Here is where the formula starts to bite. What about a player who’s actually a 0bb/100 break-even player (which is more common than people think)?

RoR = exp( 0 / 1.0 ) = exp(0) = 1

100% risk of ruin. You will eventually go broke. The bankroll is irrelevant — if your true win rate is zero or negative, bankroll management cannot save you. The only thing it does is delay the result.

This is the single most important takeaway of bankroll management math: bankroll management is only meaningful for winning players. For losing players, it merely controls how slowly you bleed out.

What about tournaments?

Tournaments have higher variance for two reasons. First, the payout structure is top-heavy: most of your EV comes from rare deep runs. Second, you don’t get to recover variance by reloading — when you bust, you bust.

The ROI-based RoR formula is messier, but the intuition is clear: tournament bankrolls need to be much larger relative to buy-in than cash bankrolls. The “100 buy-ins” rule of thumb for MTTs corresponds to roughly the same risk-of-ruin level as “20 buy-ins” for cash, given typical win rates and variance.

If you play turbo or hyper-turbo tournaments, variance is even higher, and 200 buy-ins is more appropriate. If you play deep-stack slow-structured tournaments, you can sometimes get away with 50–75.

The shot-taking question

A common scenario: you’re playing $0.50/$1 and you have $1500. Should you take a shot at $1/$2?

The math: a single $200 buy-in at $1/$2 is 13.3% of your roll. If you lose it, you’re down to $1300, still 6.5 buy-ins at $1/$2 — under the “20 buy-in” guideline, but not catastrophic. Lose two and you’re at $1100 (5.5 buy-ins). At that point you should drop back.

Two principles for shots:

1. Define your stop-loss before you sit down. “I will play one buy-in, max. If I lose it, I go back to $0.50/$1 for a week before trying again.” Write it down. Stick to it.
2. Take shots when you’re playing well, not when you’re stuck. Stuck-shot-taking is the #1 way regs blow up. If you’re already $400 down at $0.50/$1 today, do not move up to “get it back.” That is gambling, not poker.

What about win rate uncertainty?

Here is the most-ignored complication: you don’t actually know your win rate.

Your “win rate” from a sample of, say, 50,000 hands is just an estimate of your true win rate, with a confidence interval. For 50,000 hands of NLHE, that interval is typically ±2.5 bb/100. So if your observed win rate is 4 bb/100 over 50,000 hands, your true win rate is somewhere between 1.5 and 6.5 bb/100 with 95% confidence.

This is enormous. A 4 bb/100 winner has near-zero risk of ruin at 20 buy-ins. A 1.5 bb/100 winner still has very low RoR but their EV per hour is dramatically lower. And if you’re being honest with yourself, your true win rate could be at the low end of that interval.

The implication: be conservative with your bankroll, because you don’t know your win rate as well as you think you do. If the standard advice says 20 buy-ins, consider 30. The extra cushion costs you nothing except some lost EV from playing a stake lower than you “could.” That cost is usually small. The cost of being wrong about your win rate and going broke is much higher.

A practical framework

If you take nothing else from this post, take this framework:

1. Cash NLHE: 25–30 buy-ins for the stake you play. Move up when you have 25 buy-ins of the next stake. Move down (firmly, no exceptions) when you have fewer than 15 of your current stake.
2. MTTs: 100 buy-ins for slow structures, 150 for regular, 200+ for turbos.
3. PLO: 40+ buy-ins. Variance is meaningfully higher than NLHE.
4. Mixed games or unfamiliar formats: pad by 50% until you have a real sample of your win rate.

Set these numbers before you start a session. Do not change them mid-downswing. Variance is normal. A 20-buy-in downswing happens to every winning player in their career, sometimes more than once. If you’ve calibrated your roll correctly, you survive it.

A note on the psychological side

The math is one half of bankroll management. The other half — and the harder half — is the discipline to follow it.

Most players who go broke don’t go broke because they had bad math. They go broke because they knew the right move was to drop down a stake and they didn’t, because dropping down felt like admitting a downswing was real. Or they took a shot at higher stakes when they were stuck because they wanted to get unstuck in one session.

Both of those decisions are emotional, not mathematical. The math can’t fix them. What can fix them is having very specific written rules, set when you’re calm, that you follow when you’re not calm. “If my roll falls below X, I move down to stake Y for a minimum of one week” is a rule you can actually follow. “I’ll move down if I run bad enough” is not.

If you have not written down your rules, your bankroll management plan is not real, no matter how well you understand the math.

Wrapping up

Bankroll management is not magic. It’s a probability calculation, and like all probability calculations, it gives you a precise answer to a precisely-stated question. The question — will I go broke? — has a number behind it, and that number is a function of your win rate, your variance, and your roll.

Know your numbers. Pad your assumptions. Write your rules. And then move on and play the game.