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Randomly pick a (long only) portfolio from all possible portfolios over the S&P 500. The expected performance of this portfolio should equal the actual performance of the S&P 500.

In contrast, consider investing in the S&P 500 as a zero sum game, where a zero outcome means that a portfolio provides the return of the S&P 500. In this game players compete for the highest return on investment. As success in the stock market game is not random, players can be more and less skillful. Professional players such as portfolio managers, professional day traders or insiders in average should outperform a naïve investor.

So as the average naïve investor underperforms an average professional investor, he or she also underperforms the S&P500 (otherwise the game wouldn't be zero-sum). Is this conclusion correct?

To me it seems counter-intuitive. A naïve investor following basic investment rules such as "buy the dip" or "sell in may and go away", picking stocks based on value metrics and whatever recommendations there are for beginners would underperform. Where in contrast an investor ignoring all recommendations and behaving not goal-oriented at all, which leads to a random portfolio, could expect an average return.

However, modeling a stock market as zero sum game deviates from reality. Not all investors play for the highest ROI: they might also be in the game for the fun of gambling, or (as a company) buying stocks in the context of a merger. Or they might be forced to sell when receiving a margin call. But still I would assume that the vast majority of stock transactions are performed rationally towards the goal of maximizing the ROI, leaving little room for naïve players to outperform the market.

Is there any literature discussing such ideas?

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    $\begingroup$ I saw a recent post syndicateroom.com/articles/random-portfolio-will-outperform-vcs saying just that. $\endgroup$ Oct 11, 2021 at 15:45
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    $\begingroup$ I have a hard time following your reasoning. Professional investors have an extremely hard time outperforming the market, so why talk about naive players. Credit Suisse shows there is hardly any player or strategies that outperform the S&P500. In the words of Warren Buffett, "I have talked to huge pension funds, and I have taken them through the math, and when I leave, they go out and hire a bunch of consultants and pay them a lot of money. Just unbelievable." $\endgroup$
    – AKdemy
    Oct 11, 2021 at 19:12
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    $\begingroup$ S&P Dow Jones Indicesalso has good data. The Canadian example is an extreme outlier. As of Dec 31, 2020, 98.63% of funds underperformed the S&P/TSX Composite. However, US data does not look particularly good either. Letting monkeys throw darts to decide where to invest also doesn't do worse than most fund managers. $\endgroup$
    – AKdemy
    Oct 11, 2021 at 19:12
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    $\begingroup$ It depends whether the errors of the naive are random and offset each other's (as EMH people tend to think) or whether they go in the same direction and affect prices (which is closer to the behavioral finance view). $\endgroup$
    – nbbo2
    Oct 11, 2021 at 19:28
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    $\begingroup$ As most stocks don't outperform bills (papers.ssrn.com/sol3/papers.cfm?abstract_id=2900447), whether a random portfolio would perform in line with the market presumably depends on holding period, rebalancing, etc. $\endgroup$
    – user42108
    Oct 11, 2021 at 20:21

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Thanks a lot for the comments, very helpful! I studied Bessembinder’s findings and indeed a random investor cannot expect the performance of the underlying index. The stock market is positively skewed, so the median performance of a random portfolio is worse than the mean performance. However, still the mean performance of a random portfolio matches the performance of the index. The positive skewness as well explains why most fund managers underperform, when they pick few stocks and so probably miss the big winners.

However, given that the stock market is a game of skill where players seek return on investment. Then a skilled (or professional) investor should outperform an unskilled (or naïve) investor. And so, not only the median but as well the mean performance of an unskilled investor is worse than the underlying index.

As a conclusion, the random beats the unskilled investor. Possible explanations:

  • Cognitive Bias. Rubinstein, Tversky and Heller examined a treasure hiding game. They found that players tend to prefer certain locations, though according to game theory, they should choose the location with equal probability. So, in this game, a skillful player, taking advantage from the cognitive bias, would beat the naïve player, but not a random player. Cognitive bias of various kinds certainly exists in the stock market too.
  • Risk mitigation. Investors not only strive for high returns but also avoid risks. So according to their individual tradeoff they might – for example – pick stocks with a low beta or diversify their portfolios, rationally accepting a lower than mean return on their investment. Viewed this way, the stock market is not a zero-sum game: some players seek outperformance, while other players seek low risk. Both groups can (for the most part) achieve their goal.
  • Ethical investment. Similarly, investors might exclude certain stocks from their universe through ethical reasons and accept a lower than mean return.
  • Taxation. Investors are taxed differently, according to their residency, their personal income and so on. A strategy optimized for a high ROI after tax might not be optimal before tax.
  • Transaction costs might motivate an investor to diversify less and change the relative composition of their portfolio when buying stocks or cashing out.
  • ETFs sell stocks when they leave the underlying index.
  • Business strategies: Merger and acquisitions, management stock options, buyback programs…

And so on, the stock market is certainly complex. Basically, from my perspective, these insights resolve the apparent paradox of a successful random strategy:

  1. The stock market is not the only game where a random strategy leads to mean outcome.
  2. Skill and actual return are loosely coupled: a good strategy can underperform because of skewness and bad luck. Accordingly, it is a simplification to judge a strategy by its outcome.
  3. An investor who, measured by her ROI, underperforms the market, might overperform measured by her individual goals.
  4. Divide the approx. 40 trillion market capitalization of the S&P 500 based on the investor skills into two roughly equal halves. Let the owners of the portfolios in the first half be more skillful than the owners of the portfolios in the second half. Then your mean ROI might outperform the market even if you are in the second half, because players in the first half might pursue different goals.

Further reading (my homework): Game theory

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