# The Definitive Guide to Minimum Bankroll Requirements for Poker

## Introduction

Every serious poker player, every professional poker player, and every aspiring professional poker player will at some point ask the same question.

“How big a bankroll do I need?”

Sometimes it is phrased differently.

- “How big a bankroll do I need to play that game?”
- “How big a bankroll do I need to go pro?”
- “How big a bankroll do I need to play at those stakes?”
- “How do I know if my bankroll is big enough?”

It is the same question. And we all ask it at one point or another. We usually get the same answers too.

“Twenty times the buyin for no-limit hold’em. Three hundred big bets for limit poker.”

It’s a rule of thumb. And it’s a pretty good one, like “don’t draw to an inside straight.”

But a rule of thumb does not apply to every situation.

- Is the pot is laying fifty-to-one?
- Will your opponent will call off their entire stack?
- Are there cards that you can bluff with?
- What are the odds you have the best hand already?
- What other cards could give you the best hand?

Every situation is different. And if you are a serious poker player you know how to figure out if you should draw to an inside straight in each of these situations.

A rule of thumb about bankrolls is the same, it does not apply to every situation.

- Are you less risk-averse or more risk-averse
- Are you are looser and more aggressive or tighter and more passive?
- Are your opponents are looser and more aggressive or tighter and more passive?
- Are you playing a game with relatively high variance (like no-limit hold’em) or low variance (like limit Omaha hi-low)?
- Are you are playing professionally and taking money out of your bankroll for rent?
- Or are you are playing recreationally and adding to your bankroll from your paycheck?

Every situation is different. And if you are a serious poker player you should know how to figure out what your required bankroll is in each of these situations.

This guide will show you how.

## Table of Contents

- Introduction
- Table of Contents
- About the Author
- What is a Bankroll?
- Why Do Bankrolls Matter?
- What Is Your Bankroll?
- How Big of a Bankroll Do You Need?
- The Bankroll Requirement Formula
- Conclusion

## About the Author

Before we start, I want to talk about myself a little bit in case you want to know where this is coming from. If you don’t care about me you can skip ahead to “What is a bankroll”.

in 2002 I earned a Bachelor of Science in Computer Science from the Massachusetts Institute of Technology.

In 2006 I quit my job as a software engineer to be a professional poker player. I played professionally for about seven years.

Even though I was a solid winner, I was perpetually broke and constantly worried about my bankroll. Eventually, I went back to my career in software but I held on to the desire to understand poker bankrolls and to return to the freedom of being a professional poker player.

In 2016 I found the last piece of the puzzle and discovered that there really were answers to all my questions about my bankroll.

In October 2018, I put everything I had learned about bankroll management into an app that I called Neverbust.

Neverbust implements everything you’re about to learn about tracking and managing your poker bankroll (and it is free to start).

## What Is a Bankroll?

Before you can figure out your bankroll requirements, you need to know what a bankroll is.

Your bankroll is the amount of money you have to play poker. If your bankroll goes to zero you can’t play anymore. You are bust.

Going bust doesn’t have to be permanent. Most players have some way to replenish their bankroll. For some it’s a trip to the ATM. For others, it’s a year of saving up for their next gambling vacation. For others, the only way to replenish their bankroll is to go out and get a job.

## Why Do Bankrolls Matter?

Bankrolls matter because if you don’t have a bankroll you can’t play poker. How much your bankroll matters depends on how much playing poker matters to you.

If you are a recreational player, your bankroll is important because if you don’t have a bankroll you can’t play poker and playing poker is fun.

If you are a professional player, your bankroll is important because if you don’t have a bankroll you can’t play poker and playing poker is not just fun, it is your job, livelihood, and source of income.

Bottom line, if poker is important to you then your bankroll is also important.

## What Is Your Bankroll?

Your bankroll is the amount of money that you are willing and able to lose playing poker

Your bankroll may be less money than you have set aside for poker. If you lost 95% of the money you set aside for poker, would you be willing to gamble with the last 5%?

Your bankroll may be more money than you have set aside for poker. Tommy Angelo says his bankroll includes “all the money he can borrow.”

## How Big of a Bankroll Do You Need?

The answer to the question “How big of a bankroll do I need?” is the same as the answer to every question in poker.

“It depends.”

Lucky for you, I know what it depends on and soon you will too.

There is a simple formula for calculating your bankroll requirements. But in order to understand it, we need to look at what it means and where it comes from.

What happens if your bankroll is too small? At any moment, you might hit a run of bad luck, go on a downswing, and lose a bunch of money. If your downswing is as big as or bigger than your bankroll, you’ll lose all your money. You’ll be bust. That is what we are trying to avoid.

I have some bad news. There is absolutely no bankroll that will guarantee that you will never go bust. No matter how big your bankroll is, there is some chance that you are about to start a downswing big enough to drive it to zero.

The good news is that you can decide how much risk you want to take of going bust. It can’t be zero, but it can be as close to zero as you want. If you know that number, and you know the characteristics of the game you play, there is a simple formula for figuring out how big a bankroll you need.

## The Bankroll Requirement Formula

The mathematical formula for figuring out how big a bankroll you need is called the risk-of-ruin equation.

`P(bust) = ( 2 / (1 + mu / (sqrt(mu^2 + sigma^2))) - 1)^(s / (sqrt(mu^2 + sigma^2)))`

- `P(bust)` is the probability of busting, aka your bust risk,
- `s` is your starting bankroll,
- `sigma` is standard deviation, and
- `mu` is your win rate.

Once you know these values, all you need to do is plug them into the equation and you’ll know exactly how large a bankroll you need. So let’s figure out what these values are and how to find them.

### Bust Risk

Your bust risk is the risk you are willing to take of going bust.

The less risk you are willing to take, the larger a bankroll you will need.

If you play relatively low stakes and have a good job that you can use to replenish your bankroll, you may be willing to take a significant risk of going broke.

If you play poker professionally, for high stakes, and don’t have any other marketable skills, you will probably want your bust risk to be very small.

No bankroll can completely eliminate your bust risk. So you need to decide how much of a risk you are willing to take.

### Win Rate

Your win rate is the amount that you win, on average, per hour of play.

#### Estimating Win Rate

It is impossible to know with certainty what your win rate is. The best we can do is estimate it. And the best estimate of how much you will win per hour is how much you have won per hour.

`bar x = X/T`

- `X` is the total amount won,
- 'T' is the total number of hours player, and
- `bar x` is the estimate of your win rate.

#### Losing Players

If you are a losing player you will go bust no matter how big your bankroll is. If you have a large bankroll it will take a little longer. If your variance is high you have a better chance of hitting a lucky streak first and losing less than average or even winning for a while.

But if you lose money on average, it is only a matter of time before you lose all the money in your bankroll.

#### Breakeven Players

If you are a breakeven player you will go bust no matter how large your bankroll is.

No matter how large your bankroll is, you will eventually have a downswing that is at least that big. And when it happens, you will go bust.

#### Winning Players

If you are a winning player you have a chance of not busting.

For every finite amount you will eventually have a downswing at least that big. To survive you must grow your bankroll fast enough that when that downswing happens your bankroll is big enough to survive it.

#### Net Win Rate

When figuring out your bankroll requirement, your win rate isn’t what’s really important. What is really important is how quickly your bankroll is growing (or not growing).

If you lose, on average, five hundred dollars per month at poker, but you take six hundred dollars from your paycheck every month and add it to your poker bankroll, then your bankroll will grow one hundred dollars per month, on average.

If you win, on average, five hundred dollars per month at poker, but you take six hundred dollars from your bankroll every month for rent, then your bankroll will shrink one hundred dollars per month, on average.

If your bankroll is shrinking on average, it will eventually get to zero.

If your bankroll is staying the same on average, you will go bust when you hit a downswing that is at least as large as your bankroll.

This is another reason amateur and casual players don’t have to worry about their bankrolls. If you are adding a significant amount of money to your bankroll on a regular basis, it drastically lowers your risk of going bust.

In contrast, if you are playing at stakes that are high compared to your disposable income, your bankroll is much more important. You cannot add significant amounts from other income.

The situation is even more extreme for a professional poker player. They are actually taking money away from their bankroll, as income. As a result, their win rates and bankrolls need to be much higher.

#### Estimating Your Net Win Rate

Expenses are typically measured on a monthly basis. So in order to calculate your hourly net win rate you need to also know how many hours you play per month.

`bar x = (X/T) - (E / H)`

- `X` is the total amount won,
- `T` is the total number of hours played,
- `E` is the amount taken out of your bankroll each month, and
- `H` is the number of hours played per month.

### Standard Deviation

Standard deviation is a measure of how much your results fluctuate.

At a regular job, your pay doesn’t fluctuate at all. Target pays fifteen dollars per hour. If you spend an hour stocking shelves at Target they’ll pay you fifteen dollars, every time, for every hour.

Poker isn’t like that. Today you might be dealt huge hands over and over again and win every pot. Tomorrow you might be have the same big hands but lose every pot to even bigger hands.

The more your results fluctuate, the larger your bankroll needs to be.

Your fluctuations will depend on the stakes you play at, the type of game, your style of play, and your opponents’ style of play.

- Higher stakes games have more fluctuations than lower stakes games.
- No-limit games have more fluctuations than limit games.
- Loose-aggressive play creates more fluctuations than tight-passive play.
- Loose-aggressive opponents create more fluctuations than tight-passive opponents.

#### Variance

Another term for these fluctuations is variance.

As a math term, variance is equal to the standard deviation squared. It has some interesting properties and makes most formulas a little simpler.

I prefer to use standard deviation because the numbers have an easily comprehensible meaning.

- Sixty-eight percent of the time, your results will be within one standard deviation of your expected value.
- Ninety-five percent of the time they will be within two standard deviations.
- Ninety-nine point seven percent of the time they will be within three standard deviations.

(This is known as the 69-95-99.7 rule.)

#### Estimating Standard Deviation

It is impossible to know with certainty what your standard deviation is. The best we can do is estimate it.

##### Sample Standard Deviation

For our purposes, the best estimate of standard deviation would be the standard deviation of your results after applying Bessel’s correction.

`sigma^2\ = (1/(N-1))sum_(i=1)^N(x_i - bar x)^2`

- `sigma` is your standard deviation per hour,
- `N` is the number of hours you have played,
- `x_i` is the result of the `i`-th hour, and
- `bar x` is your win rate

The problem with this method is that few live poker players have records of their results on an hour-by-hour basis. We could stop every hour to record our results, but I don’t like being interrupted or distracted during a poker game. Besides, Kenney Rogers said never count your money when you’re sitting at the table.

#### Weitzman Variance Estimator

You can use our session result to estimate your hourly standard deviation by using an equation developed by Max Weitzman.

The Weitzman variance estimator was first published on Usenet in the group rec.gambling.poker.

I discovered it
on page 62
of *Gambling Theory and Other Topics*
by Mason Malmuth.

`sigma^2 = (1/N)sum_(i=1)^N(x_i^2/t_i) - (bar x^2/N)sum_(i=1)^N t_i`

- `sigma` is your standard deviation per hour,
- `N` is the number of hours you have played,
- `x_i` is the result of the `i`-th hour, and
- `bar x` is your win rate

With the Weitzman variance estimator you can estimate your standard deviation from your session results.

## Conclusion

Most people are satisfied with a simple rule of thumb to calculate their bankroll requirements. Most people will never be successful as poker players.

If you want to calculate your exact bankroll requirement for the games you play, the way you play them, the people you play against, and your specific risk tolerance you have everything you need to do that now.

If you want a tool you can use to record your results that will calculate your bankroll requirements for you (and a lot more), sign up for an account on Neverbust. It’s free to start.