In gambling games such as poker, blackjack or video poker where the player is trying to make money not only is it important to be playing with a positive expectation but also much care must be taken not to lose all of one’s money. As the saying goes - “It takes money to make money”. This requires winning players to maintain a certain amount of money in reserve to handle losing streaks and is commonly known as the player’s “bankroll”. This article will explore how to estimate the amount of money that is required to be kept as a poker bankroll in order to play so as reduce the probability of going broke to an acceptable level.

Risk of Ruin Formula (warning serious math content ahead)

The risk of ruin formula is given below. The formula is based on a random walk with upwards drift and has some assumptions that although not exactly correct are good enough that it still gives a good approximation of a poker player’s risk of ruin. These assumptions include: a) normal distribution of results b) constant win rate and standard deviation. If the actual parameters vary too far from these assumptions care must be taken as the formula is likely to underestimate the actual risk of ruin.

Risk of Ruin = e ^ (-2WB / (S ^ 2))

e = Constant (2.718281828)
W = Win Rate, measured in $ pr. hour
S = Variance (has a unit that doesn’t really make any logical sense)
B = Bankroll, measured in $
^ = Power symbol (e.g. 3^2 = 9; 3^4 = 81)
/ = Division Symbol (e.g. 24/6 = 4; 56/7 = 8 )

Reference : D. Schlesinger, Blackjack Attack, RGE publ., Oakland 1997 (2nd ed. 2000)

Note: A player’s win rate and standard deviation is usually obtained from a Poker Database program such as “Poker Tracker”.

Example

Given a winning poker player with win rate of $30/hour, standard deviation of $600 and bankroll of $15000:

Risk of ruin = e ^ (-2*30*15000/(600^2))
Risk of ruin = 8.208500%

Although this is a precise answer given to 6 decimal places it is only an approximation and probably would be rounded up to give a 10% risk of ruin in practice.

What Risk of Ruin should you accept?

One can never eliminate the risk of going broke, that is reducing the risk of ruin to 0%. Whilst 5% might be acceptable to a part time player, a professional whose only source of income is from poker should use something like a conservative 1% risk of ruin. Also given the error likely in measuring the win rate and standard deviation and also changing table conditions one should maintain a higher bankroll than calculated.

This article was written by Scott McIntosh of ReviewPoker Rooms which has a Bankroll Calculator based on the formula given above.

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