Ever since my first article on bankroll management was published here on pocketfives, I have been getting emails with questions about the specific reasons I chose the numbers I did for the article. Well you asked for it. It is fairly math intensive, so if that isn’t your thing, just skip this and check out my first article on bankroll management. The first article is simple and easy, and it has a pretty solid set of numbers that should keep you out of trouble. If you are a math geek, or you just really want to know why I picked the numbers I did, here’s all the math you need to know. <READMORE>

It all starts with a formula called The Kelly Criterion, which is a mathematical formula that allows us to find the percentage of our bankroll that we need to bet in any given situation in order to achieve the greatest return. The Kelly Criterion was devised by John L. Kelly 50 years ago, and since then it has been used from Wall Street to horse tracks and everywhere in between to make very smart people very large quantities of money.

The formula -

The Kelly Criterion uses edge/odds (E/O) to determine how much you should risk in any given situation.

Edge (E) = Average gain / amount risked

Odds (O) = The payout from the bet (like 2 to 1)

To calculate E we just take what we stand to win on average with each wager and put it over the amount we risked to win that amount. So if we are getting 10 to 1 odds on a coin flip then we have stand to make $5 per flip and we risk $1 to win that amount so we have an E of 5.

To calculate O we just take the payout of the bet which in this case is 10 to 1 giving us an O of 10. In this example 5/10 is our Kelly number and we should bet 5/10 (or ½) of our bankroll on each coin flip to achieve the highest long term profit margin.

To understand betting systems it’s easiest to start with a very simple example. I’ll use a coin flip as our first example here.

We’ll assume you are betting coin flips and getting two to one on each flip (We're assuming your opponent is so stupid he can't go outside without a helmet on). Each time the coin comes up tails you pay your opponent $1 which goes into his “New Crayons Fund”, and each time the coin comes up heads you get $2 which means he isn’t going to get new crayons any time soon.

Now we have to figure out how much to beat on each flip. Our E here is .5 because we stand to make fifty cents on each bet over a large sample. Our O is 2 because we are getting a payout of 2 to 1 on each bet, so E/O is .5/2 which is .25. The Kelly Criterion would have you betting one fourth your bankroll on each coin flip. If you start with $100 you would bet $25 on the first flip and then 1/4 of your bankroll for each bet after that. While this yields the highest rate of return possible over the long run it also causes huge fluctuations that are tough to deal with.

Because poker players rarely have the edge they think they have, and fluctuations can put you on tilt as well, I think about half of the Kelly bet is about right. This gives you ¾ of the ROI that the Kelly bet gives you, with only ¼ of the risk. So in a poker game where you are twice as likely to double up as you are to go broke the most you should risk is 1/8th of your bankroll. Finding a game where you are twice as likely to double up as you are to get broke is rare, so let's look at a more common example. We’ll use SNG's with an ROI of 20% and an ITM% of 40% which a solid player can usually manage to sustain.

In this case using $10 SNG's the amount risked is $11, and the average gain is $2.20, so your edge is a paltry 20% or 1/5th.

Your Odds (O) are tough to calculate, but assuming a reasonable distribution of payouts we'll say that you average a 3 to 1 payoff when you do make it into the money. That puts your odds at 3 to 1. This gives us the equation .2 / 3 which comes to 1/15. That means the Kelly Criterion says you should be betting 1/15th of your bankroll on SNG's, if you are a solid winning player at the level you are considering, to get the very highest rate of return. Of course I recommend higher numbers than that and I have good reasons for doing so.

At the Kelly Criterion betting level the chance that your bankroll will drop to a certain fraction of it's current level is exactly that fraction. If you play at exactly the Kelly Bet the chance of your bankroll being reduced to 1/10th of it's current amount is exactly 1 in 10. The chance of your bankroll being cut in half at some point would be exactly ½. This also assumes that you are never going to pull money out of your bankroll... Since most of you plan to spend some of that money at some point, you'll be pushing awfully hard on that bankroll and taking a lot of risks by playing with 15 buy-ins. You also have to constantly adjust if you play at the Kelly level because if you lose your first SNG you are below 15 buy-ins and must move down.

Another reason for playing below the Kelly level is that you are never exactly sure what your win rate is. You may have a large sample, but are the games exactly like they were before? Are you really certain that your win rate will continue to be as high as it was before? If you bet at the Kelly level, then any time you are overestimating your win rate your profit actually starts to drop and your variance shoots through the roof. If you play at ¾ of the Kelly level then when your win rate isn’t quite as high as you thought you will be playing around the exact Kelly number. When your win rate is higher than you thought then you will be playing at close to ½ Kelly which is quite acceptable and still yields a tidy profit. ½ Kelly is probably a little smarter because you will see much smaller swings in your bankroll and still see very solid profits if you are a winning player.

The most important part of this equation is that whenever you go above Kelly you make less money! It is easily proven mathematically, and any graphing calculator or home PC can easily make the graph for you to prove it. More risk and less money add up to your bankroll plummeting and pretty soon you're on the rail. Playing below the Kelly bet is safer, more profitable, and a lot less stressful. That may explain why so many players go broke by playing slightly above their bankroll, and it’s probably lucky for the rest of us that some excellent players have been taken out of our way simply because of bad bankroll management principles.

The math is somewhat counterintuitive for most people and can be tough to wrap your head around, but it’s worth doing if you are really serious about playing for a living and managing your bankroll wisely. And if you thought this article was a little much don’t even ask me about cash games and the Kelly Criterion because all the variables make things an absolute mess. I did those calculations once for the bankroll management article and I’m never going to do it again!

I’ll see you at the final table,
Fox

This article sponsored by PokerFox.net
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