bankroll management

I had just been trying to figure out how much I have to rebuild by bankroll to before I can safely play $1/2 NL. My model is a drift-diffusion model, where the "drift" is created by good play, which gives you a certain winrate, parameterized by v, and the "diffusion" is created by suckouts (going both ways), etc., and parameterized by s (for sigma). Over a certain time t (=number of hands / 100), you will make the following amount of money x with 95% probability:

x = v*t +/- 2*s*sqrt(t)

You can then use this formula to estimate your maximum unlucky losses as a winning player, or maximum lucky wins as a losing player. The results are:

x_{min} = -s^2/v
x_{max} = s^2/|v|

The interesting part is when you plug in some real numbers. Over a sample of nearly 5k hands, I got s=86 bb/100 and v=20 bb/100 for 25NL online. If those numbers were my "true" numbers, then I'd expect to lose no more than 370 bb's (=$92.50) at least 97.5% of the time. However, if I'm on a hot streak, and s stays the same but v=5 bb/100, say, then I could lose up to $370. Conversely, if I'm actually a very mediocre 25NL player, breaking roughly even before rake, but have been lucky, with v=-2 bb/100, then I could win up to $3698 before coming back down to earth. Pretty sick, huh? The lesson is it takes a lot of hands to determine whether you're a winning player.

Applying these numbers to live play, it's clear that even if I had the same winrate at $1/2NL in bb/100 before rake, then you probably have to subtract another 5 bb/100 for the extra live rake (10% capped live instead of 5% online), and subtract another few bb/100 for worse table selection (it's not as easy to find the biggest fish of all at Foxwoods, or quickly switch tables). So I think v=10 would be a good goal for now. So how much could I lose by chance before I started to come back? Well, according to this analysis: $1480. Since that's where my bankroll is at right now, I might in principle be able to still play live $1/2 NL. However, I think I should still build up a cushion in case these assumptions are too optimistic. It seems like a starting bankroll of twice that, or roughly $3000, should allow for a healthy margin of error. In the meantime, I've estimated the standard deviation of limit play with the same big blinds is roughly 40% of that -- so my max possible losses (with 97.5% likelihood) at a roughly equal winrate of 5 bb/100 would be $237. Sounds pretty safe -- limit ftw! :-)