J.L.Kelly, in his seminal paper A New Interpretation of Information Rate (Bell System Technical Journal, 35, 917-926 see below) asked the interesting question: how much of my bankroll should I stake on a bet if the odds are in my favor? This is the same question that a business owner, investor, or speculator has to ask themself: what proportion of my capital should I stake on a risky venture?
Kelly did not, of course, use those precise words — the paper being written in terms of an imaginary scenario involving bookies, noisy telephone lines, and wiretaps so that it could be published by the prestigious Bell System Technical journal.
Assuming that your criterion is the same as Kelly's criterion — maximizing the long term growth rate of your fortune — the answer Kelly gives is to stake the fraction of your gambling or investment bankroll which exactly equals your advantage. The form below allows you to determine what that amount is.
The BJ Math site used to contain a great collection of papers on Kelly betting, including the original Kelly Bell Technical System Journal paper. Unfortunately it is now defunct, and only contains adverts for an online casino. However, you can find much of the content through the Wayback Machine archive. The Internet Archive also contains a copy of Kelly's original paper which appeared as A New Interpretation of Information Rate, Bell System Technical Journal, Vol. 35, pp917-926, July 1956. (If this link breaks — as it has done several time since this page was written — try searching for the article title).
We based the above calculations on the description given in the book Taking Chances: Winning With Probability by John Haigh, which is an excellent introduction to the mathematics of probability. (Note that there is a misprint in the formula for approximating average growth rate on p359 (2nd edition) and the approximation also assumes that your advantage is small. There is a short list of corrections which can be found through John Haigh's web page).
Note that although the Kelly Criterion provides an upper bound on the amount that should be risked, there are sound arguments for risking less. In particular, the Kelly fraction assumes an infinitely long sequence of wagers — but in the long run we are all dead. It can be shown that a Kelly bettor has a 1/3 chance of halving a bankroll before doubling it, and that you have a 1/n chance or reducing your bankroll to 1/n at some point in the future. For comparison, a “half kelly” bettor only has a 1/9 chance of halving their bankroll before doubling it. There's an interesting discussion of this (not aimed at a mathematical reader) in Part 4 of the book Fortune's Formula which gives some of the history of the Kelly criterion, along with some of its notable successes and failures.
Jeffrey Ma was one of the members of the MIT Blackjack Team, a team which developed a system based on the Kelly criterion, card counting, and team play to beat casinos at Blackjack. He has written an interesting book The House Advantage, which examines what he learned about managing risk from playing blackjack. (He also covers some of the measures put in place by casinos to prevent the team winning!)