Here is what I have concluded:

(1) The answer to the lack of correlation is that the link between short and long term performance of these funds is a PATH DEPENDENT PROCESS. That is, the correlation (or lack thereof) depends on the precise pattern of daily returns. Sometimes the correlations are close, so an investor gets returns that track relatively well with his/her “intuitive” guess that a 2 times leveraged bullish fund would return about twice the market’s return. Sometimes–often–the correlations are not close, and even perverse (the OPPOSITE of the “intuitive” expectation), like my experience.
(2) The particular characteristics of path dependency that resulted in the greatest amount and number of counterintuitive results was NUMBER AND FREQUENCY OF ADVERSE EXTREME DAILY MOVEMENTS. That is, for a BULLISH leveraged fund, the more big down days (adverse daily market movements), the worse the performance relative to expectation.
(3) One possible explanation for these perverse results is that the funds are not dynamically adjusted intraday for market exposure. To understand this adverse effect, consider a hypothetical extreme daily market movement of -25% for a 2 times leveraged bullish fund. At the end of the day, such a fund would be down 50%. At or after the market close, its holdings are readjusted to return it to a double leveraged position ON AN ASSET BASE THAT IS 1/2 what it was the day before. Suppose further the following day the market recovers, in percentage terms, and is therefore up 25%, so the fund is up 50%. Where is the investor now? He’s still down 25%. Where is the unleveraged investor, by comparison? She’s down only 6.25% (1 times .75 times 1.25). The hapless leveraged investor, then is down 4 TIMES what the unleveraged investor is down, rather than down 12.5%, his intuitive expectation of DOUBLE the underlying market movement. The leverage investor gets clobbered because the market recovery IN PERCENTAGE TERMS (25% down and then 25% up) is NOT a market recovery for the investor. His recovery took place on position that was down by 50% from its starting point. So, IN DOLLAR TERMS, he gets back only half the number of dollars he lost.
(4) To summarize, bad news (big adverse daily market moves)is REALLY bad news for the leveraged fund shareholder, considerable worse than his intuitive expecation. Throw in a considerable number of extreme adverse daily market moves during a given holding period, and the result can become devastating.
(5) The reverse is also true, that is, under the right conditions (an extreme number of large POSITIVE daily price movements), a leveraged fund can outperform the intuitive expectation.
(6) All of these funds by Pro Funds and Direxion fund groups are burdened with high expense ratios, reducing the investor’ s returns under any market return scenario.
(7) As a result of my own experience and my analysis of hypothetical returns under simulated market conditions, I would advise anyone to be extremely cautious about trafficking in any of these leveraged funds. Unless you are very good or lucky about predicting short term market movements, the likelihood you will lose money is high. The longer you do it, the more likely it is you will lose money.
(8) The creation and promotion of these products is yet another indictment of EVERYONE is the so-called financial services businesses. The sell-side (brokerage and investment banking) and buy-sides (investment management, including mutual funds) of the business are both set up to TAKE investors’ money, not to MAKE investors money. An investor is well advised to be suspicious of EVERYONE who has something to sell him–broker, investment banker, newsletter writer, managed futures and CTA’s hedge funds, etc. NO EXCEPTIONS.
(9) With all due respect, I believe Mr. Fitzgerald’s praise of these funds is misguided. THEY ARE EXCEPTIONALLY DANGEROUS. I WILL BET ANYONE MONEY THAT THE OVERWHELMING NUMBER OF INVESTORS WHO COMMIT MONEY TO THEM WILL LOSE.
(10) I don’t have time to point out all the errors in Mr. Fitzgerald’s ridicule of the EMH hypothesis, but from my study the evidence is OVERWHELMING that markets are generally efficient most of the time. That is, quite simply, why it is so difficult for anyone to beat the market over time on a risk-adjusted basis. Mr. Fitzgerald: if markets are so inefficient, I would assume you personally have made lots of money by taking advantage of these inefficiencies with your own money, right? How about proving your thesis by showing your readers the money you have made? SHOW US THE MONEY!!
Why don’t you run a mutual fund or a hedge fund if you have such exceptional talent and ability? In 30 years’ experience I have NEVER SEEN MUCH EVIDENCE THAT ANYONE CAN CONSISTENTLY BEAT THE MARKET ON A RISK-ADJUSTED BASIS. Of course it’s a big world out there, and there may be a small number of exceptions to this rule, but even if there are, they are few and far between. (By the way, I have an MBA in finance, and have considerable professional experience on both the sell and buy sides of the market, including being a former partner in a multi-billion dollar registered investment advisory firm.)

Particularly when they funds are double-short, or “ultra” short as they say, it can easily happen that over the long term they go DOWN when the underlying stocks go down (rather than up when the underlying stocks go down).

That doesn’t happen on a day-to-day basis, just cumulatively. For example, take a look at URE and its inverse SRS. Both are down over the last 12 months.

I have a book on options that is about 20 years old and it contains statistical data on the fat-tails stuff. So Taleb is not the first observer of the phenomenon.

Secondly, fat tails have nothing at all to do with whether or not past is prologue in the markets. You can have a distribution that is the result of utterly random processes that has a fat tail. (And “randomness” is very close to “efficiency”, as a practical matter, in that with it one can’t benefit from any market knowledge.)

Finally, the log-normal distribution is popular in part because it is stable, in the sense that it can easily be shown that successively applying a short-term log-normal distribution many times generates a long-term distribution that is also log-normal.

But a Mr. Levy showed, many years ago, that other distributions (fat-tailed among them) have the same desirable property (and that kind of stability is indeed a property that has a sort of “fractal” meaning to it). All of the Levy distributions are associated with pure randomness, which is to say that they are compatible with the efficient markets idea (though they are not directly based on it).