Bankroll Management. Samples

   Hey guys(and girls in case any of you read :D )

Recently i remembered about nice forum post made by Pzhon (coach @ Pokerstrategy). It's about bankroll management mainly. Here's the link to it. I'll copy/paste it here and will add some of my own thoughts and a bit of explanations about math :) (Quoted text is in italics)

The bankroll you need depends on your ROI, which games you are playing, and your personal risk tolerance. A simple, consistent formula is 

Bankroll = (comfort x SD^2)/(win rate)

comfort is something which depends on your risk tolerance and your ability to move down in stakes if you hit a bad streak. A comfort level of 2 is generally considered aggressive. A comfort level of 4 is conservative. You can use the same comfort level across many forms of advantage gambling. One other meaning of the comfort level is that your instantaneous risk of ruin, your chance to go broke if you stay in your current game, is about 1/7^comfort.

SD = standard deviation measures how spread out the results are of each tournament. For single table tournaments, this does not depend much on your playing style. In a 50-30-20 structure, your standard deviation is going to be about 145%-155% of a buy-in. In larger tournaments, the standard deviation is larger, and it becomes more sensitive to your playing style and how much you win. In a 45-player tournament, the standard deviation may be 280%-330% of a buy-in. In a 180 player tournament, the standard deviation may be 500%-800% of a buy-in.

Your win rate should be expressed in the same units as the standard deviation, say in buy-ins. You can include rakeback and bonuses. If you regularly take money out of your bankroll, then you should reduce your win rate by these withdrawals. 

For example, if you use a target comfort level of 3, and you have an ROI of 5% in STTs, then you should have a bankroll of 3 x 1.55^2 / 0.05 = 144 buy-ins. With the same level of risk tolerance and in the same games, if your ROI is 10%, then you only need 72 buy-ins. If you have an ROI of 40% in 180 player tournaments with a standard deviation of 6 buy-ins, then you should have 3 x 6^2 / 0.40 = 270 buy-ins. MTT buy-ins are much easier to lose than STT buy-ins.

If your bankroll falls to below the target bankroll, that is ok, you don't have to move down immediately. However, if your bankroll falls to less than half of the target amount, then you should move down. Playing with less than half of the target bankroll means that playing should be viewed as an expense. 

Well, i don't have what to add about BRM, but here's some things about sample sizes:

It is normal not to know exactly what your ROI is. You can make tentative calculations, and update them as you get more information. A rough 95% confidence interval for your ROI after n tournaments is your observed ROI +- 2 SD / Sqrt(n)(2.6*SD/Sqrt(n) for 99% confidence interval and 3.1*SD/sqrt(n) for 99.8% confidence interval). You might want to play enough that this confidence interval does not include 0 before moving up. Be prepared to drop down again if you do not have good results at the higher level. 

Some of you might not understand what 95% confidence interval(CI) means, so i'll try to explain it. Let's say that we have played 1,000 games and we find our 95% CI. This means that in 95% of 1,000 games sessions our ROI will be in that interval. Quite simple, huh? :D (Ofc, don't hesitate to ask if i made it unclear to you :) )

So, now let's look at some examples. I'd also like to note that because it's difficult to know your exact SD in 45s/180s, i'll just use the average of provided SDs (so, 305% for 45s and 650% for 180s).

Let's start with 45mans.

Well, let's say that we have 10% ROI(which is not really decent, but pretty good) after 3,000 games. So, to find our 95% CI, we have to calculate 2*SD/Sqrt(n), which is 2*305%/Sqrt(3,000)=11.1% Oh wow, so 95% CI of our ROI is 10%+-11.1% or [-1.1%, 21.1%] Well, that's quite large interval and our real ROI for sure can be really different from our current 10%.

Let's see what's our 99.8% CI with same sample and ROI. 

3.1*305%/Sqrt(3,000) = 17.3%  now it's really huge, 10%+-17.3% or [-7.3%, 27.3%] is a damn big interval so our sample size is damn small.

Now, let's find out what sample size would be pretty good, i'd say it's quite good if the interval is going to be ROI+-2%. I'll use 95% CI here

2*305%/Sqrt(n)=2%

610%/2%=Sqrt(n)

305%=Sqrt(n)

n = 305^2 = 93,025  Well, you can see that it's a damn huge sample size and i guess no one has it. 

Few more examples:

95% CI after 3,000 games is going to be ROI+-11.1% 

95% CI after 10,000 games is ROI+-6.1%

95% CI after 50,000 games is ROI+-2.73%

What about 180s? I'll provide only results without calculations, cause math is the same only SDs differs :)

95% CI after 3,000 games is going to be ROI+-24% 

95% CI after 10,000 games is ROI+-13%

95% CI after 50,000 games is ROI+-5.8%

Here's a real life example:

Username	Games Played	Av. Profit	Av. Stake	Av. ROI	Total Profit	Form	Ability /100	Network	Filter
Bigniux	11,093	$1.5	$9	17%	$16,690	LLLLLLLL	N/A	PokerStars	E180-180 S5-15	x

95% CI is going to be 17%+-12.3% or between ~5% and ~29% which is well, quite huge but maths prove that i'm at least small winner most likely :)

So, to conclude, i'll just say that you should make your own conclusions from the numbers :D Try not to underestimate variance and don't take your results for granted, unless you have huge sample :)

Good luck at the tables,

Ignas ;)