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Kelly Criterion (2007)

by kwikiel on 11/19/18, 7:07 AM with 96 comments

• by freddie_mercury on 11/19/18, 11:04 AM

  The Kelly Criterion was the subject of an incomprehensibly bitter argument in the 1970s/1980s. Paul Samuelson, considered by many to be the greatest economist of the 20th century, believed the Kelly Criterion was wrong. And not just wrong but SO WRONG that anyone who believed it was an idiot.

  The kind of idiot who could only understand single syllable words. So he wrote a paper in the Journal of Finance and Banking in words of only a single syllable saying why no one should use the Kelly Criterion.

  http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Con...

• by soVeryTired on 11/19/18, 9:47 AM

  The the main flaw of the kelly criterion (along with a number of other results in investment theory like Markowitz allocation) is that in practice it's extremely difficult to know the distribution of the result you're betting on.

  The mean and variance of a prospective investment are not observable. But more to the point, if you try to use some sort of proxy like a sample mean or standard deviation, you'll get inconsistent results over time. We're a long way from the clean, simple, i.i.d world that theorists like to play in.

• by evrydayhustling on 11/19/18, 12:53 PM

  One thing I find really interesting about the Kelly Criterion is that it exposes a very stealthy and fundamental "rich get richer" phenomenon.

  Most real-life risks have a minimum and maximum investment amounts, meaning that you can't just size the bet exactly as Kelly says. So if your wealth is low, you cannot rationally participate in many risky but positive-expected-value investments.

  Simply put, the poor can't take many worthwhile risks (think college!) without rising ruin (and sub-optimal growth). Conversely, the rich can come closer to maximizing EV in many risky markets at once, increasing income and growth while even decreasing variance.

• by lordnacho on 11/19/18, 9:04 AM

  Ex hedge fund guy here. You can extend this into continuous space and use that to tell you how much leverage you should have, given some Sharpe ratio.

  Results may surprise you (it's a lot for even a modest Sharpe). But also most practitioners aren't going to use the full number. If you've overestimated it you are always worse off on the right side of that.

• by haliax on 11/19/18, 1:21 PM

  The Kelly Criterion is the subject of an absolutely incredible book by William Poundstone called "Fortune's Formula".

  In the course of discussing the formula, the book takes you through the birth of the MIT blackjack team, the genesis of statistical arbitrage, and mini biographies of people like Claude Shannon and Ed Thorpe. I can't recommend it highly enough.

• by aidenn0 on 11/19/18, 7:48 PM

  The most interesting thing to me about the Kelly criterion is that it demonstrates that the martingale system[1] is a bad strategy even if the odds are in your favor!

  While it's immediately obvious that the martingale is bad if the odds are in the house's favor, it's less obvious that you are likely to go bankrupt with the martingale even if the odds are slightly in your favor (assuming the house's bankroll is much greater than yours).

  1: https://en.wikipedia.org/wiki/Martingale_(betting_system)

• by avvt4avaw on 11/19/18, 10:20 AM

  This massively oversells the usefulness of the Kelly Criterion. The opening lines are

  > One should buy stock when it is undervalued. What I have always wondered about is how much stock one should buy. A few months ago I stumbled upon the answer which is given by the Kelly criterion.

  But the rest of the post analyses a mathematical game which has nothing to do with buying stocks, and is in fact only useful in theoretical situations where you know the precise distribution of outcomes.

• by anonu on 11/19/18, 12:34 PM

  Always a good topic to discuss. Many applications to high-frequency trading due to the probabilistic nature of outcomes.

  Here are 2 previous HN discussions:

  https://news.ycombinator.com/item?id=13143821

  https://news.ycombinator.com/item?id=2504222

• by krackers on 11/19/18, 9:25 AM

  I've only barely looked into the Kelly Criterion, but can someone explain the intuition behind maximizing the expected value of the log of your wealth? Trying the same derivation mentioned in the article but without the logarithm: the expected value comes out to 0.5×(1 + 1.1×f) + 0.5×(1 − f) = 1 + 0.05f which would make it seem that betting the entire fraction always maximizes your expected value. But why does this reasoning break down in the long term, and why does maximizing the log seem to make it work?

• by n4r9 on 11/19/18, 4:19 PM

  It's the expected growth rate, so yes it would vary in a real-world instance. I wondered for a while why they've taken the logarithm, but I think it's just because growth models are normally defined exponentially (with the rate being a parameter inside the exponential) - it shouldn't make a difference to the result.

  As for varying, that Samuelson article says:

  > For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine - and turn out so much less that my tastes for risk will force me to shun your mode of play.

• by pyrex41 on 11/19/18, 3:14 PM

  The reasoning behind the Kelly Criterion was explored recently in a more broad context, showing that the logarithmic utility is not required: https://aip.scitation.org/doi/10.1063/1.4940236

  Taleb has a good discussion here: https://medium.com/incerto/the-logic-of-risk-taking-107bf410...

• by bedhead on 11/19/18, 2:53 PM

  There was a hedge fund manager named Mark Sellers who blew up his fund in 2008 by following the Kelly Formula exactly, which had told him to put 90% of his fund in a single stock, which was a small offshore oil/gas driller. It was real money too, over $200 million.

• by pyrex41 on 11/19/18, 3:16 PM

  Most of the examples of Kelly criterion application are either concrete bets with discrete payoff/loss odds and values, or assumed to be normally distributed. This paper discusses how extremely skewed outcomes (eg, stock options) should affect the Kelly calculation: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2956161

• by OscarCunningham on 11/19/18, 9:11 AM

  I think the Kelly criterion doesn't apply as widely as people think it does. Its derivation is based on on maximising the growth rate of your fortune. But this is equivalent to assuming that money has logarithmic utility for you. If you don't value money in a logarithmic way then you shouldn't use the Kelly criterion.

  Personally I feel that my utility function is sublogarithmic. If I'm just spending on myself then beyond a certain point additional money makes me absolutely no happier. Note that the usual justification of progressive taxation also assumes sublogarithmic utility. So based on this we should be more conservative than Kelly.

  On the other hand, if I plan to give money to charity then my utility function is almost linear. Big charites can absorb a lot of money without becoming less effective. So in this case you should be maximally aggressive, betting everything at every opportunity.

  Sometimes people say that because the Kelly criterion maximises growth rate it will be the best "in the long run" even if your utility function isn't logarithmic. But I've never seen any evidence of this. Does anybody know of a toy model where you can prove the Kelly criterion is optimal even if your utility is linear?

• by dafty4 on 11/19/18, 6:33 PM

  "If all the economists in the world were placed end to end they would not reach a conclusion" -Isaac Marcoson (attrib. 1933 by O.O. McIntyre)

  http://www.systemicrisk.ac.uk/sites/default/files/downloads/...

• by rlander on 11/19/18, 2:06 PM

  A similar (but more useful) position sizing strategy is the Optimal F formula, described by Ralph Vince in his book Portfolio Management Formulas. But its real value is in showing you your 'cliff of death' curve: how close you can get to bankruptcy given your position sizing.

  In my opinion, position sizing is more way more important (and less understood) than market timing.

• by praptak on 11/19/18, 9:43 AM

  The correct number of individually picked stocks you should buy is most probably zero: httpedmarkovich.blogspot.com/2013/12/why-i-dont-trade-stocks-and-probably.html?m=1

• by bigpicture on 11/19/18, 8:32 PM

  Wow, this is the 2nd time in two weeks that something has appeared on my probability homework and then made the front page of HN almost immediately.

  Coincidence?

  Who else is taking Stat 110?