**Fortune's Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street**, by William Poundstone

I just ordered this book from Amazon -- apparently it is in vogue among some of the big institutional investors :-)

The optimization problem described is interesting from an academic point of view, and also as a good rough guide for investors, but of course it assumes that certain quantities, such as correct probabilities of future outcomes, are knowable. That may be the case in casino gambling, but in investing one can only make rough guesses based on past performance. I recall being puzzled at a section on portfolio optimization in the information theory book by Cover and Thomas, but now I see the connection.

BTW, on the topic of finance books, I highly recommend this biography of Fischer Black, which I should have reviewed here long ago. Fischer was yet another outsider (his background was in theoretical physics) to finance who made an important contribution. Unlike Kelly, he was accorded mainstream recognition (professorship at Chicago and partnership at Goldman) during his career. The most impressive thing about Black was his ability to think deeply and independently -- beyond the conventional wisdom. There are some very intriguing passages in the book about his views on money and banking which are, I think, quite unconventional to mainstream economics.

See here for an interesting review of Fortune's Formula by Berkeley math professor E. Berlekamp -- himself a former manager of Jim Simon's Medallion Fund and a collaborator of Claude Shannon!

In a paper published in 1956, John L. Kelly of Bell Labs formulated the asset-allocation problem in terms of an idealized model for which he derived some quantitative results. He used colorful racetrack terminology reminiscent of the classic Damon Runyon movie Guys and Dolls: Suppose that one goes to the racetrack with an available bankroll, B. Suppose further that one knows for each horse the correct probability that it will win the next race. Suppose further that the betting odds are at least slightly inconsistent with this information. And finally, suppose that each race is merely one of a very long sequence of betting opportunities. Kelly found criteria for deciding how much one should then bet on each horse in each race.

Kelly observed that, under similar idealized assumptions, the same formulation could also be applied to investments. In the idealized model, the portfolio manager has an accurate probability distribution on the future performance of each asset in the universe of potential investments. Kelly's methodology then provides a quantitative specification of how big a position to take in each of the candidate assets. Not surprisingly, the fraction of one's portfolio to be invested in any asset that has a negative expected rate of return will be zero. Most assets with positive expected rates of return will merit the investment of some positive fraction of the portfolio. Among assets with similar expected rates of return, those whose returns are relatively stable will be weighted more heavily than those whose future returns have significant risks of substantial losses, even when these risky investments also have some chance of large gains. All of these qualitative features of Kelly's performance criteria concur with conventional wisdom. What distinguishes Kelly's work from that of his predecessors is his quantitative specificity and the fact that he succeeded in proving that, under his assumptions, in the very long run the bankroll of an investor who followed his criteria would eventually surpass the bankroll of anyone following any other strategy.

Kelly also derived a formula for the rate at which this bankroll would grow. This formula is related to a fundamental information-theoretic notion that Claude Shannon (now widely considered to be the father of the information age) had introduced in 1948. Shannon had shown that noise on a communication channel need not impose any bound on the reliability with which information can be communicated across it, because the probability of transmitting a very long file inaccurately can be made arbitrarily small by using sufficiently sophisticated coding techniques, subject to a constraint that the ratio of the length of the source file to the length of the encoded file must be less than a number called the channel capacity. Kelly showed that the asymptotically optimum asset allocation could be determined by solving a system of equations that maximized the log of one's capital. In his horse-track jargon, Kelly also showed that the resulting optimal compound growth rate could be viewed as the capacity of a hypothetical noisy channel over which the bettor was getting the information that distinguished his odds from those of the track. Kelly's betting system, expressed mathematically, is known as the Kelly criterion.

The title of Kelly's paper, "A New Interpretation of the Information Rate," highlighted his discovery of a situation in which Shannon's celebrated capacity theorem applied even though no coding was contemplated. The paper, which appeared in the Bell System Technical Journal, initially attracted a modest audience among information theorists but went unnoticed by economists and professors of finance courses in business schools. Perhaps it would have received more attention if it had had another title. "Information Theory and Gambling" was the title that Kelly himself used for an earlier draft of his paper, but that title was rejected by AT&T executives.