Risk of ruin question

[uclaben]

I'm wondering if anyone here can answer this math question - I follow math well but I have no idea how to calculate this. Bonus points for anyone that can show me how to set it up in Excel, etc.

Given a standard limit hold-em buy-in (not bankroll), what is the risk of going broke in any given session?

Let's define the variables like this:

Buy-in: 20 BB
Win rate: 3 BB/100 (I mostly play live, so I think of it as 1 BB/hour at 33.3 hands/hour)
Variance/SD: You tell me, let's just assume a standard fill-in number

Time Period: 1 hour
2 hours
4 hours
8 hours
x hours

So the question is, at X hours, what is the likelihood that this player will have gone broke at some point leading up to that time demarcation? By the way, the practical application of this, as I imagine it, is "vacation" bankroll management: how much do I need to bring to be reasonably confident in playing a certain game in Vegas over an hour, day, weekend, etc.

Thanks in advance to anyone who puts in time on this. Appreciate it.

[Paul2432]

Calculate a z-score that corresponds to losing 20B. This can be done as follows:

z = -(20+(WR)(n))/((S.D)(n^0.5)

RoR = NORMSDIST(z)

Where:

n=hours played
WR=hourly winrate
SD=hour SD
NOMRSDIST is an Excel function.

For example, if your win rate is 1 BB/hr and your SD is 10 BB/hr and you play 4 hours, then:

z = -(20+4)/(10x4^0.5)=-24/20=-1.25

RoR = Normsdist(-1.25)= 10.6%.

So, if you play 4 hours, you have about a 10% chance of being down 20BB.

I think there are some technical assumptions that I have glossed over that may weaken the above. Maybe phzon or brucez can chime in.

Paul

[pzhon]

The main problem is that you estimated the probability that you would be down 20 BB at the end of the period, but we want the probability that you bust out at some point, which is significantly higher. It's a lot harder to estimate that.

There are some questions about using the normal/Brownian approximation. If you have enough hands for your end result to be roughly normal, that does not imply your downswings are roughly the same as for a Brownian motion. I'm not convinced that this makes a big difference, but with only on the order of 100 hands, the normal approximation itself is suspect.

[uclaben]

[ QUOTE ]
The main problem is that you estimated the probability that you would be down 20 BB at the end of the period, but we want the probability that you bust out at some point, which is significantly higher.

[/ QUOTE ]
This is correct. I'm looking for the probability that at any point in your session you will be 20BB down from where you started.

[ QUOTE ]
It's a lot harder to estimate that. There are some questions about using the normal/Brownian approximation. If you have enough hands for your end result to be roughly normal, that does not imply your downswings are roughly the same as for a Brownian motion. I'm not convinced that this makes a big difference, but with only on the order of 100 hands, the normal approximation itself is suspect.

[/ QUOTE ]
Remember when I said I follow math? I stand corrected. Thanks for weighing in...let me know if you figure anything out. I think this is an interesting question because of the application in the OP; when I'm on vacation in Las Vegas I often want to take a shot at a bigger game, but I'd like to know my chances of busting out in any given time period.

[BruceZ]

[ QUOTE ]
The main problem is that you estimated the probability that you would be down 20 BB at the end of the period, but we want the probability that you bust out at some point, which is significantly higher.

[/ QUOTE ]

In fact it is always at least twice the endpoint value.

This post gives a short-term risk of ruin formula intended to compute a "vacation" bankroll, as the OP is looking for. This is from Blackjack Attack by Don Schlesinger. Do not use the other method of doubling the endpoint value as that is only accurate when the EV is small compared to the bankroll; however, it does provide a lower bound on the risk of ruin.

Here are some results from this formula for EV = 1 bb/hr, SD = 10 bb for 1 hour, and a trip bankroll of 20 bb. These are rounded to the nearest percent. I wouldn't put alot of stock in these numbers for such short intervals for the reason that Pzhon mentioned. I give them here mainly so that you can confirm your understanding of the formula. Then you can apply it to somewhat longer intervals. Schlesinger gives a blackjack example for 16 hours at 100 hands/hr, and he says that the formula should err on the conservative side.

1 hour: 4%
2 hours: 13%
4 hours: 26%
8 hours: 39%

[uclaben]

Bruce - thanks, this is great. I messed with the Schlesinger formula in Excel to model a number of different scenarios. If anyone is interested, the results are below - variables are win rate, starting bankroll, and number of hours played. I left SD at 10bb/hr. If anyone wants me to reproduce the numbers at a different level of variance, I'd be happy to do so.

Thanks again.

winrate=1bb/hr

starting br
# of hours
20 50 100
1 3.7% 0.0% 0.0%
2 12.8% 0.0% 0.0%
4 25.7% 0.7% 0.0%
8 38.6% 4.5% 0.0%
12 45.2% 8.7% 0.1%
15 48.4% 11.4% 0.3%
20 52.1% 15.1% 0.9%
30 56.4% 20.4% 2.2%
50 60.6% 26.3% 4.9%


winrate=1.5

starting br
# of hours
20 50 100
1 3.3% 0.0% 0.0%
2 11.5% 0.0% 0.0%
4 23.0% 0.6% 0.0%
8 34.2% 3.4% 0.0%
12 39.8% 6.4% 0.1%
15 42.5% 8.4% 0.2%
20 45.5% 11.0% 0.5%
30 48.9% 14.5% 1.2%
50 51.9% 18.1% 2.5%


winrate=2bb/hr

starting br
# of hours
20 50 100
1 3.0% 0.0% 0.0%
2 10.3% 0.0% 0.0%
4 20.4% 0.4% 0.0%
8 30.1% 2.5% 0.0%
12 34.7% 4.7% 0.0%
15 36.9% 6.0% 0.1%
20 39.2% 7.8% 0.3%
30 41.7% 10.0% 0.6%
50 43.6% 12.0% 1.1%