Book: Fortune’s Formula
Author: William Poundstone
Key takeaways: The book is primarily about the Kelly Formula, which is summarized below. The author tells stories and anecdotes from several walks to life (gangsters to scientists) to make this formula come alive. I thought that the geometric mean is a very useful construct as well.
One of the best known simple formulae for betting is known as martingale, which is simply doubling down until you win.
Ed Thorp found that many of the exciting new securities (e.g. S&P futures) were over-valued.
As an aside, Claude Shannon, one of smartest people described in the book, thought that a smart investor should understand where (s)he has edge and invest in only those opportunities.
Kelly Formula
The %age of your bankroll you should bet = Edge/Odds
If heads gives you $2 and tails is a loss of $1, Edge = ($2 x 50%) + (-$1 x 50%) = $0.50, and Odds = $2 (the amount that you win, IF you win, for every $1 wagered), so the bankroll you should bet = $0.50 / $2 =25%.
Edge is how much you expect to win, on the average, assuming you could make this wager over and over with the same probabilities. Odds means the public or tote-board odds. It measures the profit if you win. All this said, it is important to remember that these #s are the very edge of what one should do, and one shouldn’t be seduced by the “optimal” or some such nomenclature that many will give them. The ratio is the line between being very aggressive and being a lunatic (as a bet larger than the Kelly size decreases returns while increasing volatility). Like Thorp said, the real test of an aggressive position is whether one can sleep at night. Also, we need to remember that Kelly criterion is useful when the profits are being reinvested (again, very useful when investing in markets).
The Kelly system is not married to any model of how the market is “supposed” to behave; it works in both log-normal random walk and one with flea-like jumps, for example.
Geometric Means
Bernoulli said that risky ventures should be evaluated by the geometric mean of outcomes. Geometric mean is almost always less than the arithmetic mean, and thus, is a more conservative way of assessing risky propositions. One way to visualize the geometric mean is to imagine a wheel with various discrete outcomes and related probabilities. Note that if even one of the outcomes is zero, the geometric mean becomes zero (but arithmetic mean doesn’t). This IS a very specific scenario though where one is making the same bet over and over again. Having said that, one can argue that every day (/month/quarter/year) one chooses to hold a security, (s)he is making a similar (if not the same) bet again and again, so this concept is still somewhat applicable.
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