Question for Bruce Z

[Timer]

I stumbled upon an old thread discussing bankroll requirements and I was going through your formulas. I'm having trouble with a couple of them, and I was wondering if you could shed some light.

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

>>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

The above two formulas I understand and I can come up with the correct answer.

However the following formula is giving me some problems:

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

First of all what is exp? Is it 5%? If it is then I come up with .0672. If it is something else could you please explain?

In one of the threads you say:

>> 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.

I don't quite understand the premise. It appears you have two risk of ruins here. Namely, 5% and 26%. If your willing to take a 5% chance of going broke then what does the 26% mean? Does it mean that you actually have a 26% chance of going broke?

I'm just a bit confused. Thanks for any help.

[Saborion]

Where did you find this thread if I may ask?

[Timer]

[ QUOTE ]
Where did you find this thread if I may ask?

[/ QUOTE ]

To be perfectly honest with you I have no idea. I think someone posted a link to it over on RGP and I clicked on it and there it was in a browser tab. Anyway, I believe this is the link.

http://tinyurl.com/j0jk

[Bozeman]

GTAOT is Gambling Theory and Other Topics, for those who don't follow acronyms. Since Mason is more of a poker player than a mathematician, he made a seemingly small error that can have significant effects. Bruce showed that the bankroll calculation which was supposed to have a 5% RoR actually has a 26% RoR.

exp(x) is e^x, where e is a mathematical constant derived from limit x->0 of (1+x)^(1/x) (natural log is log base e), and e ~ 2.72

But Bruce will explain it much better,
Craig

[BruceZ]

However the following formula is giving me some problems:

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

First of all what is exp?

It is the exponential function e^x where e is approximately 2.7183. exp is the inverse of the natural logarithm ln. If y = ln(x) then x = exp(y). You can find exp on some caclulators, or excel. Otherwsise you can just use 2.7183^x.

I don't quite understand the premise. It appears you have two risk of ruins here. Namely, 5% and 26%. If your willing to take a 5% chance of going broke then what does the 26% mean? Does it mean that you actually have a 26% chance of going broke?

GTAOT has a table labeled bankroll requirements for a 5% risk of ruin, and these are derived from his formula which you have quoted. These bankrolls will actually give a 26% risk of ruin as seen by the formulas I gave. That is, you will have a 26% chance of going broke with these bankrolls.