Risk of ruin when central limit theorem hasn't kicked in

[Izverg04]

I want to be able to calculate risk of ruin for non-normally distributed results.

The best example where I need this is calculating bankroll in video poker with positive expectation. If the return is reasonably large, let's say >1%, then the number of hands N over which we expect the largest negative swings is not yet large compared to 40,000, the average number of hands to get to the jackpot, thus the central limit theorem does not yet apply.

After searching this forum thoroughly, the best source I found is this article referenced by BruceZ in an old thread.

I understand the abstract, that they calculate the bounds on the risk of ruin using the first 4 moments of result probability distribution, but I get so lost thereafter that I can't even find the answer. If anyone is able to translate their results to me, I would really, really apreciate it. I understand that this is a fairly advanced question, but reading this forum give me hope that there are a few top-notch experts here.

[pzhon]

I think you should check whether the quoted result of Feller satisfies your needs: Using Newton's method, determine the unique value rho_0 on (0,oo)\{1} so that (for Deuces Wild)

P(no hand)rho_0^-1
+ P(trips)rho_0^0
+ P(straight or flush)rho_0^1
+ P(full house)rho_0^2
+ ...
+ P(natural royal flush)rho_0^799
=1

Then the probability of winning at least W units before losing L is between

lower bound:
(rho_0^L - 1)/(rho_0^(L+W+799-1) - 1)

and

upper bound:
(rho_0^(L+1-1) - 1)/(rho_0^(L+W+1-1) - 1)
= (rho_0^L - 1)/(rho_0^(L+W) - 1)

As an example, for full-pay Deuces Wild, rho_0 is about 0.999345. Suppose you want to know the probability of busting before at least doubling up if you start with 1000 bets (5000 coins). The lower bound is about .572 and the upper bound is about .658.

Much of the paper develops "easy" estimates for rho_0 when you can't solve for it directly. These result in slightly weaker estimates than Feller's bounds, so if the above doesn't seem tight enough, this paper won't help.

The paper also discusses estimates of the average length of time before you win or lose.

[Izverg04]

Thanks, your explanation really helped me along.

The game in question, full-pay Deuces Wild has EV=0.00762, and Var=25.8. Taking long-term risk of ruin RoR=e^(-8), I get the following bankroll requirement using the normal approximation:

BR=4*Var/EV=13540

Using Feller's method I get a better estimate, which is actually an exact number since upper bound=lower bound

BR = ln(RoR)/ln(rho0)
BR = 12080

For this game, normal approximation is already fairly good, because EV is fairly small.

Unfortunately, not all examples I have in mind have this integer payout scheme, so I'll still try to read through the rest of the paper.

[Izverg04]

On the second thought, I have my problem completely solved. I could easily make a spreadsheet for any result distribution, calculate rho_0 numerically and immediately get the value of bankroll. Great! Fantastically simple method.