Except that as you reduce the sample size, you expect to have some outperform and some underperform. If you 100x pick stocks at random, you will expect to see some fraction of those 100 picks way outperform.
Any time you reduce the sample size, you increase the variance, which gives the impression that skill is involved. In fact from just looking at a single distribution of outcomes it's not possible to tell if skill or luck is the cause.
There exist funds which have annualized a two-sigma return over SPY for over 20 years. If you model returns as approximating a normal distribution (e.g. just luck), and you model years achieving a return at least two standard deviations above the mean under a binomial distribution (i.e. number of years they've been exceptionally lucky), the likelihood of those track records existing are around 1 x 10^-37.
I would call that sufficient evidence to reject the null hypothesis that the returns are normally distributed, which is to say it's not luck. If you expand your sample size to all investment vehicles throughout history, there still haven't been anywhere nearly enough for such a track record to emerge by chance.
Elementary statistics is well equipped to distinguish between a distribution signifying luck and a distribution signifying skill. It's structurally the same as assessing normality, noise, randomness, etc.
This topic has actually been studied:performance of top performing funds can be based on a manager's skill, but after a certain point that skill reaches the end of the runway. The more skilled the manager, the larger the AUM they can still get returns for, but at some point it's just too much.
> For an average fund in the cross-section, we estimate a drop in alpha of 20 basis points if the fund doubles its size over one year. We also find a non-negligible impact of the size of the fund industry, although its magnitude is significantly smaller than the impact of individual fund scale. We reconcile our findings with existing empirical studies. Taken as a whole, our results lend considerable support to theoretical models that build on the premise of decreasing return to scale for active portfolio management.
> There exist funds which have annualized a two-sigma return over SPY for over 20 years
the problem is that _many_ funds have positive returns until, suddenly, they don't. It's basically as hard to pick a fund or money manager for the long run as it is to pick a stock.
Consider Neil Woodford[0], he beat the market for over twenty years and was considered the best investor in Britain. Then started a new set of funds, which went terribly. It'd have been reasonable to let him manage your money, but it would still not have worked out.
That doesn't refute the mathematics demonstrating that people can reliably do this with skill rather than luck. Eventually Brady's not going to be able to play football professionally either. But he still does, and when he can't do it anymore that won't indict his professional record.
The difference is that our ability to tell if Brady was a good football player 10 years ago doesn't depend on his results today. Whereas a fund manager can turn out to have had a bad strategy all along whose gains are only wiped out after many years of apparent good performance.
How many actual examples do you have of the same funds blowing up which outperformed for decades?
Buffett outperformed SPY for most of his 60 year career. Do you think he doesn't know what he's doing, and it was all luck, just because Berkshire Hathaway isn't doing as well as it used to?
They're not normally distributed. Almost any investment strategy has heavier tails than normal. So exceeding some number of sigma (assuming a normal distribution) does not imply an implausible amount of luck.
For example, a single investment in Amazon 20 years ago would outperform the market by many sigma. But the probability of an average fool having picked that particular stock 20 years ago was not 10^-(some large number). At least 1 in 100 fools would have picked Amazon.
That's a fair counterargument, but I don't think it holds up. I'm talking about a portfolio rather than a single asset. Technically we would want to model that using a log-normal distribution, but I think the example suffices. Can you think of a realistic example where someone would accidentally hold a dynamic portfolio that exhibits outperforming returns over 20 years?
Any time you reduce the sample size, you increase the variance, which gives the impression that skill is involved. In fact from just looking at a single distribution of outcomes it's not possible to tell if skill or luck is the cause.