The overwhelming majority lose in a zero-sum trading game.

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b0B
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Joined: Fri Oct 19, 2018 11:39 am

Re: The overwhelming majority lose in a zero-sum trading game.

Post by b0B » Sat Dec 08, 2018 11:46 am

JoMoney wrote:
Thu Dec 06, 2018 6:19 pm
Many (perhaps most) gamblers would lose to the house even if casinos had zero edge on their games, simply because most have poor bankroll management and take on too much risk pretty much guaranteeing eventual ruin / bankroll depletion when a bad sequence occurs, it is inevitable if too much risk is repeatedly taken.
The Kelly Criterion gives a mathematical formula to determine how much of the bankroll can be bet to maximize growth rate and eliminate the risk of ruin... It will also will tell you the amount to bet on a game with zero expected advantage(EV) is nothing.

Beating the market is a zero-sum game (you can't have a portfolio that achieves above market returns without one that underperforms). If we presume that markets are efficient (even though I don't think they are), then there is no positive expected value from trading, it's just trading different levels of risk/risk preferences. How much risk (relative to the market) the active trader is assuming can be measured through "active share" and seeing how much it deviates from the market return. Most mutual funds don't have very high active share, and the returns typically show a bell curve with the index being somewhere around the middle of the distribution. If you look at the tails (on either side) I'm sure you will find a lot more 'active share'. You would also likely see a lot more left-tail underperforming funds than outperforming funds on the right side of the distribution, and this can be explained by both high fees and by taking too much risk in the zero-sum trading game.
If markets are not efficient, it's still a zero-sum trading game, but it's even worse for traders with no information - if there are traders who do have information and asymmetrical advantages, then the zero information trader has negative expected value.
So in an efficient market (zero EV) Kelly Criterion would suggest you bet nothing on active risk.
In an in-efficient market where you have no special information (negative EV) Kelly Criterion would also suggest you bet nothing on active (actually it would suggest you bet on the guy who has the information, but you presumably don't know who that is).
Yes, Kelly Criterion involves the same kind of math. It's all about maximizing geometric mean in an iterated situation where returns are multiplied. (That's why I referred to a product of random variables earlier.) There's lots of applications of this concept.

By the way, in my example, I probably don't really need the "actives" to behave randomly, since there is plenty of randomness in the market itself.

afan
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Joined: Sun Jul 25, 2010 4:01 pm

Re: The overwhelming majority lose in a zero-sum trading game.

Post by afan » Sat Dec 08, 2018 4:30 pm

So each day, each active trader has exactly the same expected (arithmetic mean) return as the passives, but the results are more dispersed, so the geometric mean is lower. Iterate this for many days, and an ever increasing majority of the active investors trail the market.
This assumes that the portfolios of the active investors are more volatile than the market as a whole, leading to greater differences in arithmetic vs geometric means. But there is no mandate that the active portfolios are more volatile, even if truly composed randomly. If the active traders intentionally create low volatility portfolios then they will have smaller differences between arithmetic and geometric means than the market portfolio.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

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pezblanco
Posts: 482
Joined: Thu Sep 12, 2013 8:02 pm

Re: The overwhelming majority lose in a zero-sum trading game.

Post by pezblanco » Sat Dec 08, 2018 6:51 pm

afan wrote:
Sat Dec 08, 2018 4:30 pm
So each day, each active trader has exactly the same expected (arithmetic mean) return as the passives, but the results are more dispersed, so the geometric mean is lower. Iterate this for many days, and an ever increasing majority of the active investors trail the market.
This assumes that the portfolios of the active investors are more volatile than the market as a whole, leading to greater differences in arithmetic vs geometric means. But there is no mandate that the active portfolios are more volatile, even if truly composed randomly. If the active traders intentionally create low volatility portfolios then they will have smaller differences between arithmetic and geometric means than the market portfolio.
afan wrote:
Fri Dec 07, 2018 6:58 pm
One would argue that, with respect to the realized risk adjusted returns, the stock picking by active investors is random. So it is reasonable to model their behavior as random deviations from the market. That does not mean they intend it to be random. They may well use highly systematic approaches. The systematic approaches do not increase risk adjusted returns. All they generate is (random) noise.

This does not lead one to the original conclusion that all but one active investor must go to zero, with the entire ownership of actively managed stocks vesting in one owner.

As for the talk of the "math" showing it. I have yet to see any math.

Note that the deviation of the arithmetic from the geometric mean applies just as much to the passive side of the market. It arises because of the way the two are calculated and the varying values of the stock market. It has nothing to do with active vs passive.
I outlined a proof of the result using the inevitablitliy of hitting the absorbing state in a Markov Chain type argument. If one is only interested in that, the result stated by Bob is true and trivial. Estimates of hitting times to that state are more interesting ...

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