risk, bankroll and optimal betting (math warning ON)

[BillC]

We've just finished an article that addresses bankroll, risk of ruin, and other risk parameters. It addresses and interrelates the topics in the subject. I believe that it properly addresses bankroll requirements, which are improperly addressed on this forum and by Malmuth in GTAOT (plus he didn't send me the book). The relevant result is:

Let k be your Kelly fraction in blackjack (which is equivalent to a utility function, which in turn specifies your risk tolerance). The value of k is equivalent to having a risk of ruin tolerance of exp(-2/k). This in turn is equivalent to always having a bankroll B of v/kr, where v is the variance s^2.

Assume r/s is fixed constant, where r is your hourly win rate and s is the standard deviation. Theoretically, if you always choose a game with r, s so that B=v/kr, your bankroll is the same (as a random process) as a k times Kelly bettor in blackjack with win rate r/s and unit standard deviation.

Example: Let bb=Big Bet. Assume a benchmark range of games with pure ratio s/r=10 (e.g. s=10 bb and r=1 bb). Thus v/r=100 bbs. It is optimal (w.r.t. geometric rate of bankroll growth) if you always have 100 bbs (this is k=1 or full Kelly betting). Having k=1 is optimal but entails wild swings (e.g. you will eventually get halved with probabilty .5) Most blackjack teams are more risk averse and set k between .25 and .4. For poker, k should probably be smaller, e.g. k=1/6, because of lack of certainty about r,s. If k=.25, your bankroll will have to be 400 BBs;
for k=1/6, your bankroll will have to be 600 bbs. Some experts counsel even smaller values of k, e.g. k=1/10. Some futures traders suggest k=1/6 (they call it optimal f ). For holdem, 300 bbs is not really enough for most folks.

Your bankroll is all your assets minus expenses. It is not your separate "gambling bank". For the latter artificial sort of bankroll, using a value of k closer to 1 is recommended (e.g. k=.5), B=200 bbs.

These are facts from the paper. If you want to learn more more about Kelly betting, utility, etc. a good place to start is bjmath.com. Also see R. Epstein's book. The paper will be posted soon. I will email you a copy if you were at least a math or related major. It will be over your head otherwise. The material is based on a diffusion model from stochastic calculus. nigelc21@hotmail.com