# Math time!

• Moderator
Moderator
Joined: 24.06.2008
Hi everyone

I have been busy with w34z3l's 6max walkthrough assignments and I have to calculate my Risk of Ruin. I have derived a mathematical formula from the book 'The Education of a Modern Poker player'. This book is quite advanced so the math is as well. I also have mentioned the formula in my blog.

Some elements I don't know in the formula and hope someone could follow the steps and ask for clarification on some points. I believe together we can make this formula work! Good luck and enjoy the ride

Formula

Winrate: 12.38/100
Std dev: 84.36
ROR: 1%
R = ROR
B = Bankroll (\$50)
b = winrate = 12.38/100bb
-k = constant - from here comes the trouble
exp(-kB) = R(B) - not sure what 'exp' consists of
c = 1 (Because we know that if our bankroll is zero, we're certain to go bust as we are bust already. This means that R(0) = 1 and hence, c - 1)
P(b) = 2μ/ σ2 = (118.9*2) / (84.36*2) - def not sure if I did this correct! If someone can tell me where/how to get the real numbers, let me know

Bankroll: \$50
ROR: R(B) = R(B) = 1%(50)
Splitting bankroll: \$B1 & \$B2 = B1 + B2 = B = \$25 + \$25 = \$50
R(B1) and R(B2) = R(25) + R(25) = R(B1 + B2) = R(B1) R(B2) = R(25) R(25) = 1 – (0.25 * 0.25) = 37.5%

R’(B1 + B2) = R’(B1)* R'(B2) and R’(B1 + B2) = R’(B1)* R’(B2) =
R’(B1) / R'(B1) = R’(B2) / R’(B2) – This holds only when both equations are equal (which is; \$25)

Constant = -k R(B)
Solution: R(B) = c exp(-kB)
When go bust: R(0) = 1, and therefore c = 1 and R(B) = exp(-kB)
So k must be positive!

R(B) > 0 – always since c = 1 and is inherent to (-kB), which is inherent to exp(-kB), which is inherent to c exp(-kB), which is R(B)! - Right?

Let’s say we start with \$50B, this is \$50B+b (where \$b is the amount won/loss; so: b<0 or b>0)
Let’s say we win \$5 with probability P(b), the ROR must be integral over all the possible amounts we might win, namely:

R(B) = ∫_(-00)^00 P(b)R(B+b)db - You see we don’t know k but we do know what R(B) is. P(b) is given in the book as: 2μ/ σ2 - Also here, the sign ∫ is an 'integral' in mathematics. This integral shows in the book 2 zeros on top and 2 zeros on the bottom of it. Not sure what this means

Final formula: R(B) = exp(-2𝛍B /𝛔₂ = 1%(\$50) = exp(-237.8*50 / 168.72)=-67%
100-67%=33. Therefore, I need a bankroll of 33 BI

Sooo, I hope you enjoyed it and do get something out of the formula that makes you want to puzzle immediately

Good luck everyone and thanks for helping me out!
• 11 replies
• Moderator
Moderator
Joined: 27.01.2013
So this is a bankroll management formula? I didn't understand it, it looks weird
• Moderator
Moderator
Joined: 24.06.2008
Haha yeah man. The book states that in order to retain a 1% chance of going bankrupt, you need to have a certain amount of buy-ins. Apparently, something like this is then the formula Let's just hope someone saw something like this before
• Super Moderator
Super Moderator
Joined: 02.09.2010
Hi, Hesticus
I think that somewhere you need to insert your experienced win rate.

Bear in mind that your true winrate will only be known after playing for some time, so that you build up a reasonable hand sample. As you gain experience your win rate changes (hopefully for the better).

So you never really know your "true" win rate.

But from what you show here, it looks like 50 BI is a reasonable number for beginners.

Hopefully somebody knows...
VS
• Moderator
Moderator
Joined: 24.06.2008
Hi Vorpal,

Thanks so much for your response.

Yes, true. In case of my experienced winrate, I would just have to do a 5/100bb, right?
I also will have a fixed buy in for me. The book says that 37 BI is the average. I put myself 40 BI. I think this should work. Additionally, I have coaching.

Hopefully indeed who knows! Maybe the math isn't really applicable. Maybe the information is incomplete, we will see..

Cheers,
Hesticus
• Bronze
Joined: 22.11.2009
All looks very suspicious if you ask me.
• Bronze
Joined: 19.06.2014
You calculations is incorrect, seems you don't know that exp(x) is another way to write e^x and e is mathematical constant(like pi) roughly equal 2.72. Formula for RoR is RoR = e ^ (-2WB / (S ^ 2)), where W -winrate, B - bankroll, S - standart deviation.
So if you want to know what bankroll you need to have with desired RoR, then we can get from the previous formula that it is
B=-(S^2)*ln(RoR)/2W. I put this formula into wolfram for you here:
with numbers you told ( W=12 bb/100h, S = 85 bb/100h, RoR = 0.01) we got Bankroll of 1386bb or ~14BI. You can also put another numbers there if you want.
If you think that it is too low comparing to what usually recomended, that is because 12bb/100 is super high winrate for midstakes regular (Good regs on midstakes on stars have smth about 5bb/100) and on higher limits standart deviation tends to be higher. For microstakes(especially 2,5,10 NL) 12bb/100 is a not that high and quite achivable, but begining player in most cases would have much lower WR. Also you need to have at least 100k hands to start evaluating you winrate(if it is not extreme high or low). And assuming you actively work on your game your winrate wouldn't be constant across this sample anyway. Standart deviation in NLHE 6 max depends on your and your opponents game style and usually somewhere between 75-120 bb/100h.
If you will put winrate 5bb/100, and STD 85 bb/100 you'll get 33BI for 1% RoR.
• Moderator
Moderator
Joined: 24.06.2008
Hi Altairsky,

Thank you soo much for your answer! This is even better and looks also less complicated. One question though: What here means 'ln'?
The constant is in this case also that the equation will hold for all values (Standard dev, RoR and Winrate), right? Maybe I will dig into the theory behind this constant.

I have to say, I play 2NL and have under 100K hands. I was already thinking about it, but then this calculation wouldn't be applicable to me yet, would it? In any case, assuming I will play for at least 100K-200K hands and I will keep having a 5/100bb and see that 33BI should work. That is, assuming these remain 'constant'

Thanks for your help, I can see you put great effort in this!

Cheers,
Hesticus
• Bronze
Joined: 04.01.2010
• Moderator
Moderator
Joined: 24.06.2008
Ah thanks, that is the constant!

I will dig into this coming week. Maybe I find a way to explain this in plain language in my blog

Cheers,
Hesticus
• Bronze
Joined: 19.06.2014
Logarithm(Log) is a function inverse to exponentiation, for example if you have 9=3^x and you want to know x then x=log(3)9=2.
Ln is common denomination for log(e), it is called natural logarithm, for example if A=e^x, then x=lnA.
Not sure what constant you're speaking about. If it is k in your 1st post, then k=2W/S^2. If you want to derive this part of formula or whole formula from scratch (don't have your book, but from what you've posted seems that it is done there by some extent) or at least understand how it is done, then you need to take some course of calculus. You probably don't have proficiency to do it now. But anyway when you have made formula don't see why you need that.
If by constant you mean e, then it is not some poker bankroll related number, but it is fundamental mathematical constant. You can read about it on Wikipedia.

The formula is really only applicable if you now your "true" winrate and std. By true I mean the numbers that taken from the infinite sample. If your sample is finite (obviously), then more hands you have, bigger the probability that the numbers would be closer to their true values. If for example you have 100 hands, you can calculate your winrate, but the probability that it has nothing to do with your true winrate is higher than probability that it has. 100k hands is just rule of thumb, it is very possible to have 100k and even 500k upswings and downswings due to pure variance.

If you want to be rigorous with your brm you also need to expect higher winrates on micros than on higher limits and adjust your brm accordingly. I believe on NL2 and NL5 you could have winrate 20+bb/100, but that would be hardly possible higher.

I also took mathematical approach to my BRM, but go futher than just using poor RoR formula and derived method to calculate BRM that will give theoretically the highest speed of moving through the limits while preserving desired risk that I would go on the limit lower at the highest point of a given limit. It would take too much time to explain how I had done this and requires advanced math, but this what I'd got and what I'm using:

This is calculated with assumptions of 5bb/100 expected WR on a limit, that std grows with the rate of 5bb/100 by each limit and 5% desired risk of droping to lower limit at the highest point of higher limit. Also I haven't done math for NL 2 and 5 and put numbers that feels reasonable for me. Basically I go on a limit when reach \$ in first column and go to lower limit when rich \$ in second column.
• Moderator
Moderator
Joined: 24.06.2008
Hi Altairsky,

I like the way you explain and approach my thinking. I agree now that it wouldn't make much sense to dig into the theory behind the formula. I can accept this with ease as I had basic econometrics during my Bachelors for a financial course and teachers here said as well that you don't really need the theory behind the formula. Managers these days most of the time also don't know the algorithm and only use solely the formula (of a certain subject in a certain case) to make a judgement.

I shall take your formula for granted and have a very relieved feeling that the math behind the RoR is solved On top of that, you give a very clear explanation what we can use and what not, and why we shouldn't take certain steps within the formula, which sheds light on the psychological part as well.

Of course, if you decide to take us all on a journey to explain how you made your BRM, let us know and pass your link

Cheers,
Hesticus