How to Use the Bankroll Formula and Standard Deviation in Poker Forecasting

Forecast based on actual past performance

If we use a WR of 4bb/100, SD of 70 and 370K hands (120K played and 250K in the next year) the expected range of win rates based on past performance and required Bankroll can be forecast. We know the SD should fall as a player become more consistent, so this is a worst case scenario.

SD = 70
3SD = 210
Hands = 120,000 (370,000)
Hands/100 = 1200 (3700)
SQRT (Hands/100) = 34.6 (60.8)
3SD/SQRT (Hands/100) = 210/34.6 = 6.06 bb/100 (210/60.8 = 3.45 bb/100)
Range = 2 x 6.06 = 12.12 bb/100 (2 x 3.45 = 6.90 bb/100)

Therefore the expected range of future performance based on 120,000 hands, a WR of 4bb/100 and a SD of 70 is equal to -2.06/100 to 10.06bb/100 (4bb/100 +/- 6.06bb/100).

If we go through the same process using the figures in brackets for the end of the 250,000 hands (370,000 in total played) over the next 12 months, assuming the same WR and SD, the figures work out at a range of 6.90 bb/100 or 0.45bb/100 to 7.45bb/100 around a mean WR of 4bb/100.

Bankroll Formula

The SD can be used in the Bankroll Formula to find the required amount of Bankroll needed to achieve long-term success without going bust.

There are a number of different comfort levels of risk. A comfort level of 3 is normal, 4 is very safe and 2 is where you accept more risk, so we will use a CL of 3 in this example.

BR = Comfort Level x SD^2 / Win Rate

BR = 3 x (70*70)/4

BR = 3 x 4900 / 4

BR = 3675 bb

BR = $918 or 37 buyins at NL25

The current WR and high SD is suggesting 37 buyins as a safe Bankroll to use.

However if the player intends to improve skills and reduce the SD to 50, the BR falls to $469 or 19 buyins, $375 (15 buyins) with a 5bb/100 WR and $312 (13 buyins) for a 6bb/WR.

If you wish to accept more risk and use a CL of 2, the BR figures are $612 (25 buyins), $312 (13 buyins), $250 (10 buyins) and $208 (9 buyins) respectively.

It is a useful exercise to do on a regular basis to make sure you have sufficient bankroll and performance is improving.