Posted by Steve Heston on July 05, 1997 at 14:30:49: In Reply to: Re: Expected log(growth) < log(expected growth) posted by Richard Reid on July 04, 1997 at 21:40:33:1) What is log(expected growth)?
This is just the logarithm of the expected value of G. Although the expected value of ln(G) is p*N*ln(1+f)+(1-p)*N*ln(1-f), the expected value of G is roughly (1+f)^(p*N)*(1-f)^((1-p)*N)*exp(.5*N*var), where var is the variance of a single bet.
2) What is the expected utility of G?
Nobel prize winning economists Von Neumann and Morganstern characterized preferences under uncertainty by taking the expected value of a function of your bankroll (called the "utility" function). A 'Kelly' bettor maximizes the expectation of ln(G), while a risk-neutral bettor maximizes _the expectation of_ G. But you might want to maximize the expected square root of G, or the cube root, or -1/G. Even though the "Kelly" bettor will eventually be richest, different people may want to maximize different things.
- ?Steve Mike Lea 14:19:49 7/07/97 (6)
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