bankroll

[cruzer1169]

i am trying get a feel on theories on how big a bankroll you need if poker is your only source of income.I know depends on style of play, and the limit you play. If you play 15-30, or 20-40. How much do you need?
Comments

[Jimbo]

I would think prudence would dictate you have 30 or 40 thousand for your poker bankroll to play full time at those limits. Your tolerance for risk (within reason) is nearly as important as a mere computation. As is your willingness to move down in limits if you go on a bad run. In this case a much smaller bankroll can be considered safe. I believe Mason outlines bankroll computations in his book Gambling Theory and other topics on pages 40 thru 63.

[BruceZ]

The formulas in that book will significantly underestimate bankroll requirements and risk of ruin. This is because they do not take into account the chance of going broke before reaching the point at which at which a given standard deviation loss is maximum. Use these formulas instead:

B = -(sigma^2/2u)ln(r)

r = exp(-2uB/sigma^2)

where u is your hourly rate
sigma is your hourly standard deviation
r is your desired risk of ruin
B is your bankroll

This assumes you will play at a given limit until you are broke. If you are willing to step down in limit when you lose, bankroll requirements can be reduced significantly.

[Bob T.]

I would guess that those are the same formulas that are used in the bankroll tool in Pokerstat. Does that mean that if we were going to play a single limit, we should have about twice the bankroll that pokerstat recommends.

Thanks,
Bob T.

[BruceZ]

I'm not familiar with pokerstat. If it uses the formulas I have given, you should not have to increase the bankroll. If it uses the book formulas, then you may have to more than double the bankroll for a 5% risk of ruin as you have observed. Note using a bankroll half as big results in a risk of ruin 4 times as great or 20%. Actually the example I worked out gave a risk of ruin of 26% since you had to more than double the bankroll. Let me know how which formula the software uses, I'm curious.

[Jimbo]

Quote BruceZ"This assumes you will play at a given limit until you are broke. If you are willing to step down in limit when you lose, bankroll requirements can be reduced significantly." Gee I wish I had said that! Bruce, why am I to assume you are correct and Mason is wrong? A comparison of the two formulas would help in that regard.

[BruceZ]

Mason alludes to the shortcoming of his calculation in his book, so there is not a disagreement about it. Unfortunately, we now know that the consequences of this shortcoming are very large, and an equally simple formula exists which avoids this shortcoming. Here is an explanation I wrote ealier which contains the comparison you are looking for:

In Gambling Theory and Other Topics, there is a section on bankroll requirements. At the end of the section it says that if you desire a risk of 5% to substitute 1.64 standard deviations in the equations for bankroll instead of 3 standard deviations. These formulas seriously understimate bankroll requirements and risk of ruin. Using the text example, if you use the bankroll computed by the book's formula, your risk of ruin will be 26% rather than 5%, and you would need to more than double this bankroll in order to have a 5% risk of ruin.

The equations in GTAOT cannot be used to compute the required bankroll for a given risk of going broke such as 5%. These equations assume that we will play until the number of hours for which our 1.64 sigma loss is a maximum, and they give us the value that our loss will exceed 5% of the time at that number of hours. This does not take into account the times that we lose this amount before we ever get to that number of hours. This has been shown to occur over 10% of the time, so it is actually the dominant factor. There will also be times that we make it to this number of hours without going broke, and then go broke after that.

I have posted the correct formula for computing risk of ruin and bankroll requirements on several occasions, and these appear again in the example below. This formula is well known to the blackjack community. It has been extensively verified; and it has been offered by at least one mathematician at an international conference on gambling and risk taking. Another twoplustwo poster also derived a similar formula which gives almost identical results, and has also produced a derivation of the exact formula I have given. In Blackjack Attack by Don Schlesinger, where this formula appears, there is a detailed discussion of the problems with the type of calculation made in GTAOT which has been made by many people. When the proper risk of ruin formulas are used, the bankroll requirements change dramatically:

GTAOT FORMULA FOR 5% RISK (combining equations, u = hourly rate):
bankroll = (1.64sigma)^2/4u
u = $30/hr, sigma = $650/hr gives bankroll of $9470

CORRECT RISK OF RUIN FORMULA:
bankroll = -(sigma^2/2u)ln(.05)
u = $30/hr, sigma = $650/hr gives bankroll of $21,095

risk for GTAOT 5% bankroll of $9470 would be
r = exp(-2uB/sigma^2) = 26%

Now to be fair, the book does state that you need to increase the bankroll over the amount given by the book's formulas, and it gives various reasons for this; however, as you can see, it needs to be increased much more than the 20-30% estimated in the text.

[Jimbo]

Thanks for the great explanation Bruce.

[D.J.]

I'm going to skip all the technical stuff and just say to play full time you'd need 15-20K, I just believe that if you can't make it w/this much, you don't need to be playing poker professionally.

-D.J.

[Bozeman]

Hmmm, so bankroll should be b=c*sigma^2/u and Mason found c=.6724 while the correct derivation gives c=1.497

Why do you use sigma=$650 here, Bruce? Does this come from data or simulations?

Craig

[BruceZ]

Why do you use sigma=$650 here, Bruce?

The $650 came from an example in the book for a lowball draw player. I was just working this example with both equations to show the disparity. The numbers are not important; the disparity will always be the same factor for a given risk of ruin.

so bankroll should be b=c*sigma^2/u and Mason found c=.6724 while the correct derivation gives c=1.497

Exactly, so you can multiply all the numbers in the 5% table by 1.497/.6724 = 2.23 to get the correct bankroll.

BTW, I neglected to put the general bankroll equation in this post. The ln(.05) in the equation is just for a 5% risk of ruin. The general equation is

bankroll = -(sigma^2/2u)ln(r)

where r is the risk of ruin.

[Mason Malmuth]

Hi Everyone:

What Bruce is saying is correct. However, even though I give 5 percent numbers in the book, I personally never use them. Instead, I always think in terms of bankroll requirements at three standard deviations. This is because I plan for play poker (and other gambling) for a very long time and will expect to come up against 5 percent situations many times. Furthermore, and I think that Bruce agrees, at three standard deviations the problem that he allures to becomes much smaller and is easily dominated by something else which I refer to in the book as a non-self weighting effect. By the way, after many years of using my published numbers, I have found them to be very accurate (at three standard deviations).

Best wishes,
Mason

[Mason Malmuth]

Hi Bruce:

Even though I mention it in my post below, I think it's important that it be mentioned here again, and this is consistent with the correspondence that we had a while back. At three standard deviations the effect that you talk about becomes fairly minimal, while at 1.64 standard deviations it is important. That's because it's much less likely for a path that leads you to being down three standard deviations at the computed time to have gone below that amount at a previous time than a path that leads you to being down 1.64 standard deviations.

However, what does happen in poker is that game conditions can change dramatically from game to game. The maximum likelihood estimator for the standard deviation is assumed to always come from the same (identical) distribution. Since this is obviously not the case, your standard deviation for a particular game will sometimes be less than the estimate, and sometimes more. This has the statistical effect of reducing the sample size and I refer to it in GTAOT as a non-self weighting effect and recommend that the estimated bankroll numbers (at three standard deviations) be increased by 10 to 20 percent. This should take care of everything.

Note that in blackjack, since the dealer must always employ the same strategy, as long as penetration is roughly the same and you use the same count/bet scheme, this non-self weighting effect will be far smaller than it is in poker.

Best wishes,
Mason

[BruceZ]

Mason,

At three standard deviations the effect that you talk about becomes fairly minimal, while at 1.64 standard deviations it is important.

At 3 standard deviations, the impact of this effect on bankroll is smaller but even more significant because the impact on risk of ruin will be even greater! A 3 sigma risk of ruin is 0.13%. We would require an increase in bankroll of 48% over what is computed by the book equations (see below). But this implies that the book equations will cause the risk of ruin to be increased to (.0013)^(1/1.48) = 1.1%, which is an increase by a factor of 8.6 and far cry from a 3 sigma risk of ruin.

That's because it's much less likely for a path that leads you to being down three standard deviations at the computed time to have gone below that amount at a previous time than a path that leads you to being down 1.64 standard deviations.

That's true because the risk of ruin is smaller. The real issue is the risk of going broke at a previous time vs. the risk of being down at the endpoint, comparing both for the same risk of ruin. This actually increases as we go from 1.64 to 3 sigma. It can be proven that we are always at least twice as likely to go broke before reaching the endpoint than to be broke if we play to the endpoint (going in the hole along the way if need be).

This has the statistical effect of reducing the sample size and I refer to it in GTAOT as a non-self weighting effect and recommend that the estimated bankroll numbers (at three standard deviations) be increased by 10 to 20 percent.

Using the book equations, we would need a 48% increase before we even account for any variation in standard deviation (at 3 sigma). Any adjustment for the non-self weighting effect would be on top of this.

Showing work: /forums/images/icons/smile.gif
Both of our equations are of the form c*sigma^2/2u for a given risk of ruin. My value of c is -ln(r)/2 where r is the risk of ruin. The book value of c is n^2/4 where n is the number of standard deviations. For 3 sigma, my c is 3.32, while the book value is 2.25. 3.32/2.25 = 1.48.

-Bruce

[Mason Malmuth]

Hi Bruce:

I just want to make sure that everyone understands this. In a perfect world, using my three sigma numbers, you really have about a 1 percent chance of going broke, while three sigmas implies that you should have approximately a 0.13 percent chance of going broke. Thus you should agree that the published numbers in GTAOT are pretty good for a safe bankroll requirement assuming you have a good handle on exactly what your win rate and standard deviation are.

Now your standard deviation should converge to a fairly accurate number quickly, but that's not the case with your win rate. So the inherent error in the method which my book uses is from a practical standpoint insignificant compared to this problem. Then you throw in what I call the non-self weighting effect, and you need a larger number than what the book computes. But the book clearly states that this is the case.

By the way, and this is something that I don't think I have ever written before, it's my experience, and I know this opinion is also shared by some experts who I have talked to over the years, that you almost never have quite the win rate that you think you do. In poker your opponents will adjust to you, in blackjack pressure from the pit might cause you to occasionally adjust down your betting, in sports, what use to work may not work as well any more, and so on. So this is also another reason to increase the numbers in the tables I give (or just assume a little lower win rate than what you think is correct). Always keep in mind that a good statistician errors on the conservative side.

A final comment is that when I first published these numbers I was told by many good players that they were rediculously too high. But I know from my own experience that they have proved accurate (at three standard deviations) and many of the people who originally said they were too high have come back to me at a later date and conceded that they were wrong.

Best wishes,
mason

[Phat Mack]

</font><blockquote><font class="small">In reply to:</font><hr />
Now your standard deviation should converge to a fairly accurate number quickly, but that's not the case with your win rate

[/ QUOTE ]

I've noticed this from my own records, but can't conceptualize why. Why does the SD converge so much more quickly?

[BruceZ]

Mason,

I agree that the 3 sigma numbers really correspond to a 1% chance of going broke. Although the point of 3 sigma was to virtually gurantee that we don't go broke, 1% is still probably sufficiently convervative for most people. I also agree that the 5% numbers should not be used since they really correspond to a 26% risk of ruin which I venture to say nobody wants. For computing the required bankroll for any risk of ruin, the formulas I gave at the top of this thread should be used.

Now your standard deviation should converge to a fairly accurate number quickly, but that's not the case with your win rate. So the inherent error in the method which my book uses is from a practical standpoint insignificant compared to this problem.

At 3 sigma we incurred a 1% hit off the top on our risk of ruin due to this error. If in addition we overestimate our hourly rate or underestimate our variance by 18%, then we will incur an additional 1% hit. If we only know our hourly rate to within 50%, for example if I think I'm a 1 bb/hr winner but I'm actually only a .5 bb/hr winner, then our risk of ruin would increase to 4.4%. This is possible, and it is a larger increase than due to the computation error, but it is not very uplifting since it just says that if we don't know our hourly rate accurately then our risk of ruin can be much higher than we expect. In any case, we really don't need the extra 1% error added on to this uncertainty, and the new formulas avoid this. Also notice that I could not have done this calculation of the effect of underestimating hourly rate without using the new formulas to compute the actual risk of ruin. Had I used a baseline of 0.13%, then I would have gotten a very different answer.

One minor correction on the above post:
Both of our equations are of the form c*sigma^2/2u for a given risk of ruin. My value of c is -ln(r)/2 where r is the risk of ruin. The book value of c is n^2/4 where n is the number of standard deviations. For 3 sigma, my c is 3.32, while the book value is 2.25. 3.32/2.25 = 1.48.

That should read "Both of our equations are of the form "c*sigma^2/u" in order for the rest of the paragraph to be correct.

-Bruce

[rockoon]

Is there a formula that takes into account monthly expendatures?

Clearly someone with a $2000 monthly nut needs a larger bankroll for the same risk-of-ruin as someone with a $1000 monthly nut, all other things being equal.

[Tommy Angelo]

I am almost 45 and let's say I live to age 90. That's 45 years. Right now I need $50,000 per year to live my lifestyle. Later, I'll need $100,000 per year, and in the later years (because of inflation) it'll be more like $200,000.

Rounding quickly, we could say that I need to make about three million dollars, the slow way, at a poker table, between now and my grave. I wake up everyday stuck three million. From that view, the difference between having a bankroll of five thousand or fifty thousand doesn't seem all too significant.


Tommy

[Jimbo]

True Tommy but you might get lucky and not live to be 90. /forums/images/icons/smile.gif

[Mason Malmuth]

Hi Tommy:

But it is very different. A bankroll of $5,000 might mean that it is very unlikely that you will ever get unstuck. But a bankroll of $50,000, assuming you play well, gives you a pretty good shot at getting even.

Best wishes,
Mason

[BruceZ]

Use the same formulas, but just reduce your hourly rate by the amount of your expenses. So if you play 160 hours a month, and you have 2K in expenses, then reduce your hourly rate by 2000/160 = $12.50/hr. This would give your risk of ruin if you play for a long time. If you are only going to play for a shorter length of time, then you can just keep your living expenses separate from your bankroll, and make sure you have enough to cover the period. In either case, you should start with enough in reserve to cover the initial period of time until you can be confident of booking a win. Some have suggested a 6 month reserve of living expenses separate from bankroll.

[BruceZ]

In either case, you should start with enough in reserve to cover the initial period of time until you can be confident of booking a win. Some have suggested a 6 month reserve of living expenses separate from bankroll.

Actually, in theory it isn't necessary to have this separate reserve if you use these formulas to compute your bankroll and subract your expenses from your hourly rate. You just have one pile of money from which you play poker and live, and your expenses are like a big rake.

Say your expenses are 2K/month or $12.50/hr. You want to play poker for the rest of your life winning $30/hr with an hourly standard deviation of $300. This standard deviation may be too high for this win rate, but we'll be conservative. You really have an hourly rate of $30-$12.50 = $17.50. If you wanted to play with a 5% chance of ever being on the rail, you would need -[(300)^2/(2*17.50)]*ln(.05) = $7700.

[gaylord focker]

If you put money in the bank, and are able to get 2% on it, you shouldnt have to worry about inflation. Hence the problem with keeping all of your money under your matress.

[BruceZ]

From that view, the difference between having a bankroll of five thousand or fifty thousand doesn't seem all too significant.

Nothing could be further from the truth. Even making small increases to your bankroll causes your risk of going broke to decrease exponentially; that means a little means a whole lot. Let's say you win $50/hr. Your expenses will take half of that so you really win $25/hr. I'll assume taxes are included. Let's give you an hourly standard deviation of $400 (let me know if you want some other numbers). With a $5000 bankroll, your risk of going broke without any other sources of income are 21%. If you had a $50,000 bankroll, your risk of going broke are, get this, 0.00001%. Multiplying your bankroll by a factor of 10 raises your risk of ruin to the 10th power (meaning it gets real small since it's a number less than 1 to begin with).

The above may not be realistic becasuse the standard deviation is too low for that hourly rate. Say your hourly standard deviation is $800. That might be the case if you play 50-100 for 1 sb/hr with an hourly sd of 8 bb. Then your risk of going broke with a $5000 banroll would be 68%, but with a $50,000 bankroll it is 2%.

[Ed Miller]

I just wanted to say that I think BruceZ rules. That is all.

[AmericanAirlines]

Hi Tommy,
I think the difference comes in time frames. If I have a 50K roll *now*, I may be able to live and eat a little better *now* than with 5K.

The 3 million stuck you talk about is in the future time frame... but your body demands it's survival need in the current time frame.

Sincerely,
AA

[AmericanAirlines]

Hi BruceZ,
Is your primary career something math, accounting or science?

Your analysis seem to be firmly rooted in the math of probability.

So I was just curious how you arrived at that bent of mind.

Sincerely,
AA

[BruceZ]

Is your primary career something math, accounting or science?
Your analysis seem to be firmly rooted in the math of probability.
So I was just curious how you arrived at that bent of mind.


I'm an electrical engineer, but I also have degrees in math and physics. I had a concentration in probability and statistics for my math degree. My graduate work in electical engineering involved a ton of probability theory. As for my bent mind, that's a different matter.

[Tommy Angelo]

"Nothing could be further from the truth. Even making small increases to your bankroll causes your risk of going broke to decrease exponentially; that means a little means a whole lot."

Great post, Bruce. Of course you are right, and thanks for putting it in terms that my girlfriend can never understand. :-)

Tommy

[Zeno]

Tommy,

These important concepts can be shown visually with the use of a graph or diagrams. Especially the concept of " a little means a whole lot". Sometimes this makes important ideas easier to grasp and understand. Explanations to girlfriends are, at times, precarious.

I view this as a harmless suggestion. Your stance may be otherwise. Good Luck.

-Zeno

[rockoon]

--&gt; So if you play 160 hours a month, and you have 2K in expenses, then reduce your hourly rate by 2000/160 = $12.50/hr.


I'm not sure this works. There is no deviation on monthly expenses.

??

[**MR.MANHATTAN**]

TOMORROW NEVER COMES......HAHAHAAHAH