Using some simple math (and somewhat inaccurate math), let's calculate the chance of the null hypothesis being true: he has no skill and it is just luck, assuming 10% annualized volatility for the market and 8% annualized return. By my calculation, over the six year period, there's a 4.82e-18 chance that his returns are due to luck and the null hypothesis is true. Of course I'm simplifying a lot, but I think you get the idea.

Also, you don't need to do anything "particularly unique" to generate very impressive returns. No firm (besides maybe RenTec) is doing anything that is truly out there and on another level: employees join and leave, ideas get passed around, etc. And yet, there are many top tier firms that end up hitting out of the park, year after year.

People on HN always make these cheap throwaway comments about "expert coin-flippers" and "survivorship bias" when talking about finance. I'm not sure exactly why, but I think it comes from disdain for finance. I also think that the idea that some people are just better at generating wealth through the markets can be uncomfortable.

Anyways, the chance of his returns being luck is extremely small, any which way you cut it.

From the article:

"Its long/short equity fund gained an astounding 274 percent, thanks in large part to a 700 percent surge in the price of Tesla’s stock, which accounted for 37 percent of Worm’s publicly traded equities portfolio at the end of the third quarter."

This means that, outside of his TSLA position, the rest of his portfolio made about 25%. In a normal year that'd be impressive, but 2020 was a year where SPY was up 15% and there was insane volatility.

So basically this guy gets decent-to-good performance on 3/5 of his portfolio and put the other 2/5 into a blind gamble which turned out to pay off. The chances of that happening by luck aren't "4.82e-18."

Unlikely things happen all the time. If he lost money, this article wouldn't exist.

Also, my disdain for this guy is not due to finance. It's due to his investment strategy being a Seinfeld episode:

> I just totally gave up and said, "I'm going to do the exact opposite."

> Unlikely things happen all the time. If he lost money, this article wouldn’t exist.

Exactly! The person you replied to did the right calculation but completely threw away the context. The argument here is akin to p-hacking where all the investors in the world are the experiments and this article merely picked the one that got lucky.

Different scenario but similar argument, I roll a large set of dice, 5x rolls each die. If the set is large enough, one of the dice is likely to land all 6s. While that is unlikely, that doesn’t automatically mean the diced are unfair. It’s not just enough to reject the null hypothesis but you also need to prove the new hypothesis. Not to mention having a solid working theorem for why the alternate hypothesis is correct.

You can do a similar calculation about a particular shuffled card deck.

Yes, or the lottery. It's extremely unlikely that a given person will win, but someone's gonna win.

Sure, all I'm saying it is much more likely than not that his returns are due to skill.

I don't understand you. Why do you dismiss survivorship bias and then do your math without that in mind?

If I read you correctly, you calculated "how likely is it that someone gets these results?". But accounting for survivorship bias, shouldn't it be "how likely is it that someone gets these results?" ?

Maybe I'm reading something wrong here, if so I would be obliged if you could elaborate.

Didn't you write the same, twice?

The first statement should be read as "what are the chances that this particular person got results this good?"

The second statement should be read as "out of all investors active in the market during this period, what are the chances that one of them got results this good?"

Although I agree that survivorship bias is a cheaply constructed argument, it’s hard to know how good this fund really is when it’s only been in operation during the greatest bull run of our lifetime.

That is also true, though the portfolio manager did well during bear markets as well. Doesn't say tell us how his current portfolio would hold up, but he's probably as well equipped as anyone to navigate the environment (which maybe isn't saying much).

you mean like warren buffet?

Your comparison of his investing style to the broad market is incredibly far off. If you want to make calculations at least do it off a more accurate benchmark and not the S&P 500. Something like ARKK or at least QQQ.

ARKK? You want to use an actively managed fund as a benchmark? No one does this. You could use QQQ, with pretty much the same results.

Depends what he is investing in. "Innovative" growth companies? Then ARKK isn't a bad benchmark. QQQ is a lot more conservative compared to what he was investing in.

What do the volatility and return of the market have to do with anything?

Are you calculating the probability that he did achieve these returns by luck timing the market? That's obviously not what he did.

Picking and holding a stock that did extremely well by over the period is not a one in a quintillion event.

For a fund like this, more volatility = more opportunities to make money. Doing 25% when the benchmark does 15% is less impressive than when it does 10, 5, or is down over the same period.

(Not GP.)

This statement is correct, but doesn't address the problem with the OP's calculation.

This fund is up basically entirely on the strength of TSLA being up 700%. OP is basically considering two possibilities:

1. TSLA stock is a driftless geometric Brownian motion with a volatility matching that of the general market, and happened to get a 700% return purely by chance, or

2. The fund manager, due to his exceptional skill, knew that TSLA was going to be up 700%.

The OP is rejecting option (1) and then concluding that option (2) must be the case.

Of course in reality neither is the case and the OP's calculation is totally irrelevant.

But what matters is not the volatility of the market as much as the cross-sectional dispersion of stocks. If correlation was high and all the stocks hade beta one and were almost identical to the market there would be little opportunity for a stock-picking fund like this, whatever the volatility of the market.

The market's only moved in one direction over the period; the fund can make money on moves in both directions.

So the more volatile (within a period) the more money making moves there.

If this is the benchmark:

     _   _/\  /
    / \_/   \/

at a dollar per slash, it's up $2. A fund that bet (and realised) a $1 per slash made $8.

(Even with only long bets, they could make $5.)

The market has moved a lot. But the point is that if the fund goes 100% TSLA at the beginning and does nothing else it would have outperformed massively and the volatility of the market would be completely irrelevant.

What point is that? You're moving the goalposts, why would/should/do you think this fund did go '100% TSLA at the beginning and [do] nothinig else'?

That seems closer to what they did than timing the market, don't you think?

My point was that to "test" if a concentrated stock-picking fund can get that result by luck you don't look at how often the market with such and such return and volatility gets that performance or how often randomly trading the market would you get this performance.

You look at how rare is it that a concentrated portfolio of random stocks has a very good performance. The answer is "not that much".

you look at a normal distribution of returns using SPY mean return and SPY stdev as the parameters and then calculate the chance of his returns or better 6 years in a row.

There's a lot of things wrong with my calculation, but it was illustrative of how P < 0.05 in this case, no matter how you calculate it.

We agree at least that there are a lot of things wrong with your calculation.

Null hypothesis is: he is not generating alpha given the assumption the underlying distribution of his returns are the same as SPY. How would you calculate the chance of him generating alpha using only the fact (and no other data) that he generated an annualized return of ~37% (iirc) over a six year period?

So you’ve rejected the hypothesis that his portfolio has the same distribution of returns as the SPY. Looks reasonable, because his portfolio does not track the S&P 500. Congratulations!

The QQQ (Nasdaq 100) also generates alpha without any doubt then, as does the SPUU (leveraged S&P 500).

If you're able to tolerate an arbitrary amount of risk, you can potentially achieve absolutely absurd returns with very little skill. You just need to maximize your portfolio's volatility. Find the most volatile stock you can with an upcoming earnings announcement or other catalyst and invest all your money into it. If you didn't lose all your money, sell everything and invest all your money in another volatile stock with an upcoming catalyst. Rinse and repeat. This is how people win stock competitions.

If you're able to tolerate an arbitrary amount of risk, just go to the casino and keep betting on red with doubling stakes, borrowing along the way if necessary, until you win the amount you want.

isn't this also how people loose stock competitions?

Yeah, but it's kind of a Pascal's Wager:

Normal investment: maximize expected profit

Competition investment: maximize the probability of your profit being the best in the pool

Normal strategy: 100% chance of losing the competition. Extremely risky strategy: 99.99% of losing - who cares by how much? There's this 0.01% of winning and it's all that matters.

Losing doesn't cost anything because you aren't playing with real money. You get the same payout from -100% returns as you would with 20% returns.

The argument even applies to competitions where you play with your own real money. Any non-zero prize moves the slider a bit towards the riskier strategies.

You don't explain how you calculated that number but it seems absurd (presented with absurd over precision). For example, I would be astonished if there aren't individuals, probably 10s or 100s or even 1000s of individuals who followed an "all in on Tesla" strategy that would generate these types of returns. That suggests an absolute lower bound around 1e-8 (very conservatively) before doing any serious analysis.

You are setting up the experiment as if we took one guy and decided to follow him for 6 years to see if he could beat the market.

Instead a reporter found one out of a couple of million investors who did well the last 6 years and decided to interview him.

I don't see why you feel talking about survivorship bias is unsound in this case.

The best investment decisions will nearly always look obvious in hindsight.

You're saying GME wasn't obvious?

Yeah, but like many unexpected events, it should have been :-)

The article does not claim uniqueness, but focuses on the difference between basing investments on exceptionalism rather than value.

Stand out hedge fund performances typically don't translate even to the same or similar hedge funds at different times let alone scaling to the point of 'gonna shake up institutional investing'.