Determining the size bankroll you will need can be quite a challenge. First of all, what are your poker goals? The bankroll needed to play on weekends is much different than the bankroll needed to go pro. It is actually quite impossible to have someone come up with a one size fits all number and yet it's not hard to find those 'one size fits all' numbers in many poker books. To determine the size bankroll you need it is important that you have some numbers handy as well as your goals set out. The question has come up a lot about how much is needed to go pro and so the following is geared toward that end. Just remember, that if all you want to do is play $3/$6 every other weekend or so – you probably don’t need any more of a bankroll than a decent job. Some of the number you’re going to need are your hourly rate, your standard deviation (SD) and you're going to have to make a determination how much risk you are willing to take that you won't go broke due to poor short term luck. Truth is, you can never be 100% sure that you won't go broke due to short term fluctuations. The good news is you can pick any other percent you want for instance, you can calculate a bankroll requirement to assure that you won't go broke due to short term luck 99.999999999999999999999999999% of the time if you like - you just can't pick 100%. This is called the risk of ruin and is actually expressed opposite of the % change of not going broke i.e.: if you want to ensure that you don't go broke 99% of the time, then your risk of ruin is 1% or 0.01.
Standard deviation is a number that most people don't calculate and can be tedious to do so but fortunately for now I'm going to just use 10 times your win rate which is a standard approximation. At the end of these calculations I’ll walk you through how to calculate your standard deviation on your own.
So... ready? Bankroll needed = -(SD^2/2*hourly win rate)ln(risk of ruin). Let's say you are playing $10/$20 (which is probably the minimum for making a living wage) and your win rate over time is 1BB/hour or $20/hr. Let's also say that you are willing to accept a 5% risk of ruin. Punching these new numbers into our formula we get: -(200^2/2*20)ln(.05) or -40,000/40*ln(.05) = $2996 (rounded to the nearest dollar). 5% is pretty risky though - it means you'll go broke 1 time in 20 which is not a good plan if this is your only source of income. 1% is a more practical and advisable number to work with. Let's try it: -40,000/40*ln(.01) = $4605 (rounded to the nearest dollar). So assuming that you are a winning player with a standard deviation of 10x your win rate - you can expect to play forever with a 99% confidence with a bankroll of $4605. The bonus is - if you are winning and adding to your bankroll this confidence number goes up quickly. i.e.: a 99.9% confidence requires a bankroll of only $6908. I say only because your likelihood of not going broke has increased 1000% but your bankroll only had to increase 50%. Note that $6908 is approximately 345 big bets. Did you ever wonder where that magic 300BB number came from?
Keep in mind that your standard deviation is important and if it is off significantly, this can really affect these numbers. For what it's worth, my standard deviation is way off the tenfold my win rate number. You should calculate standard deviation yourself if you want to be certain. So how do you do that? Standard deviation is the square root of variance and variance can seem a bit difficult to calculate if you haven't done it before so I'm going to put an example in here so anyone that wants can have an example to work off. Let's say you played 10 sessions and lo and behold (because you're so great) you won them all. Here were your results:
1. +100 - 8 hrs
2. +300 - 8 hrs
3. +200 - 8 hrs
4. +200 - 12 hrs
5. +100 - 12 hrs
6. +250 - 10 hrs
7. +400 - 10 hrs
8. +50 - 10 hrs
9. +300 - 12 hrs
10.+100 - 10 hrs
Now if you wanted to calculate your standard deviation you would probably just think to yourself, "Hell, I made so much money I'll just hire a mathematician!" which isn't a bad idea but since this is imaginary money you'd have to get an imaginary mathematician and they aren't too useful so... let's assume that you want to do it yourself - you know, to build character.
First, add up all your results. This equals $2,000. Now add up all your hours. This equals 100 hours which we’ll also refer to as ‘T’. To determine your hourly earn simply divide your earn by the number of hours played: $2,000/100hrs = $20/hr
Then it gets a little tricky. Add up each win squared divided by the hours; in other words:
100^2/8 + 300^2/8 + 200^2/8 + 200^2/12 + 100^2/12 + 250^2/10 + 400^2/10 + 50^2/10 + 300^2/12 + 100^2/10 =
10000/8 + 90000/8 + 40000/8 + 40000/12 + 10000/12 + 62500/10 +160000/10 + 2500/10 + 90000/12 + 10000/10 =
1,250 + 11,250 + 5,000 + 3,333 + 833 + 6,250 + 16,000 + 250 + 7,500 + 1,000 = 52,666 which we'll call 'x'
Not to worry, that was the tough part - it gets easier from here:
Variance = (1/number of sessions)x - ((hourly earn)^2/number of sessions)(T) =
(1/10)(52,666) - ((20)^2/10)(100) =
5,267 - (400/10)(100) =
5,267 - 40 = 5,227
Recall that the standard deviation is the square root of variance so we need the sqrt of 5,227 which is $72 (to the nearest dollar). Note that this standard deviation is only 3.5 times your hourly earn; here our ten times approximation would yield serious differences (over 7-fold). Your standard deviation should be calculated using (according to Mason Malmuth who is a guy you can trust when it comes to math) you should have at least 30 sessions to accurately determine your standard deviation but when you get these 30 sessions - you'll know how to calculate it now.
