Kelly Risk Analysis: Snatch the Pebble From My Hand, Grasshopper

The intelligent and thoughtful Felix Salmon makes a subtle and interesting error--an error that I would make on at least a monthly basis had Robert Waldmann not patiently explained all this to me in the winter of 1986--in discussing Kelly risk analysis:

RGE - Bill Gross on CPDOs: My favorite bit of the piece is a wonderful thing called a Kelly risk analysis, which shows that increased leverage doesn't always increase expected returns. It does to begin with, but eventually you reach a point at which increasing leverage decreases expected returns -- and it's no coincidence that that point of maximum leverage is more or less exactly where CPDO structures (and today's hedge funds) sit...

Pushing leverage beyond the Kelly point does not decrease expected return. Rather, it decreases the likelihood of organizational survival and the chance that you will be wealthy. If you are acting as one of many agents for a well-diversified principal, you will in general want to ignore the Kelly point and leverage yourself up to the gills. If your objective is, instead, to maximize your own chances of remaining in the game with boasting rights, you will position yourself at the Kelly point.

A stark way of seeing this difference is to think of the following situation: Matt Rabin from the office beneath mine comes up the stairs and offers me the following: I start with a stake $1. I can wager none, some, or all of my stake. He flips a fair coin. If it is tails, I lose my wager. If it is heads, I win twice my wager. We do this ten times in a row, with my stake growing or shrinking.

To maximize expected return--this is, after all, a very advantageous game for me--I should be my whole stake, and let it ride time after time. After 10 rounds, there is one chance in 1024 that I have $59,049 and 1023 chances in 1024 that I have zero, for an expected portfolio value of $57.67.

The Kelly point, by contrast, says that I should wager 1/4 of my current stake each round. If Matt flips ten heads, then I have only $57.67 instead of $59,049. And my expected final wealth is only $3.25 instead of $57.67. But my median final wealth is not $0 but is instead $1.80. I make money not 1/1024 of the time but 638/1024 of the time. And if Robert were here he could prove in five lines that as the number of rounds goes to infinity an agent wagering according to the Kelly criterion almost surely ends up wealthier than an agent choosing his wager from his or her stake according to any other rule. The Kelly point makes sense if you are risk averse (and if this portfolio is a major component of your wealth) or if organizational survival and relative organizational prosperity is your major goal.

If Matt showed up at my office and announced that we were going to play this game on each of the next 1024 days, I would have no trouble choosing to follow the bet-the-limit strategy rather than the Kelly strategy on each day. 1024 x $3.25 is only $3328, which is a lot less than 1024 x $57.67 = $59049. But what if Matt says that this is my one and only one day? The right way to think about it is that my marginal utility of wealth is surely pretty flat over the range of $50,000 or so, and so I ought to be risk-neutral in this particular situation. The right way to think about it is that I "buy" lots of lottery tickets of various types during my life, and that the right strategy is to maximize the expected value of each lottery ticket--not to apply the Kelly criterion to each situation individually. (Of course, the generalized Kelly criterion--maximizing the expected value of the log of your portfolio--for 1024 rounds is not to apply the Kelly criterion to each round independently.)

But I would find it hard. I would have a hard time giving the 1/1024 chance of winning $59049 its proper weight in the face of the 1023/1024 chance of suffering the humiliation of bankruptcy.

In the end, however, I would be the limit. The humiliation for an economist like me of violating the axioms of expected utility is much worse than the humiliation of losing my entire stake.

Kelly Criterion finger exercises at: http://spreadsheets.google.com/pub?key=p_zylRhg4towI71xZsP62Fg