The statistical and investment views of Nassim Nicholas Taleb, author of the blockbuster book The Black Swan, are easy to summarize. First, most probability distributions are fat-tailed, not thin-tailed like the normal distribution. That renders useless or wrong the application of almost all conventional statistical techniques. Second, almost everybody underestimates the likelihood of an extreme tail event, a “Black Swan.” That means there is an opportunity to make money in the long run with a strategy of buying very deep out-of-the-money put options.
His first view is without doubt correct, notwithstanding the apparently unstoppable, widespread overuse of those statistical techniques. His second view is highly debatable and almost impossible to test.
A few years ago Taleb sent me a copy of his book, Statistical Consequences of Fat Tails, following a brief email exchange – though the book is freely available on the internet. In an Advisor Perspectives article later, I promised that I would cover the book in a future article. But I only recently read it. The first thing that surprised me was how heavily mathematical it is – real mathematics, that is, not the kind you find in journals on quantitative finance. Consequently, I will only be able to discuss most of its contents in a general sense in this non-technical article. I’ll get to that in a moment.
My brushes with Taleb
In the spring of 2005, I was living in the touristy town of Sintra, Portugal, writing a book I had tentatively titled The Big Investment Lie. I had no idea where it was going and had little hope of getting a publisher. But it was something I was inspired to do because of an experience I had just had.
Luckily, I did get a publisher several months later, the excellent Berrett-Koehler Publishers. But BK required me to complete a few tasks before it would agree to publish. One was that I had to compare my book to at least five other published books in a similar category and explain how my book added to those.
The most obvious comparison was with Fooled by Randomness by Nassim Nicholas Taleb. It was not easy to explain how my book added something to the message of that book because much of both books were along the same theme, and Taleb’s book covered the ground well.
A few months after my book was published another book came out: The Black Swan by Nassim Nicholas Taleb. With that book, Taleb became a best-selling author, while I didn’t (my book’s sales weren’t bad but it was far from a best-seller). So, in addition to being a Taleb admirer because both Fooled by Randomness and The Black Swan were good reads, I was a bit envious.
Much later I discovered that Taleb’s doctoral dissertation, in actual mathematics (not, like too many in the finance field, in a quasi-mathematical or pseudo-mathematical field like quantitative finance), was advised by French mathematician Hélyette Geman. Hélyette is the wife of Don Geman, another distinguished academic mathematician who was my friend and fellow student in graduate school.
A third key commonality between myself and Taleb is that we both think that almost all academic finance on the subject of investment is utter nonsense. One typical Taleb pronouncement with which I whole-heartedly agree, in a footnote in Statistical Consequences of Fat Tails, is this:
…an abhorrent approach called ‘risk parity’ largely used to raise money via pseudotheoretical and pseudoacademic smoke…
Hear, hear!
Not that I agree with Taleb on everything. When I followed up on the writings of a younger colleague of his – perhaps a protégé – whose praises Taleb had sung, I found little of value. And as I’ll note in a moment, there are at least one or two things in Statistical Consequences of Fat Tails that don’t ring entirely true to me.
Statistical consequences of fat tails
One consequence of fat tails is that aggregate empirical measures like the sample standard deviation and even the mean are dominated by the small part of the sample that lie on the tails. As a result, sample statistics can be extremely unreliable measures.
Note, for example, an anomaly I described in my previously noted article, “Understanding Fat Tail Returns.” For a particular 40-coin-toss experiment, it can be calculated using probability theory that the exact average winnings of a large number of bettors each betting $10 on the sequence would be $69.67. But if that number were estimated by averaging the winnings of a sample of “only” 10,000 bettors, the estimate could range from $50 to $106. And if the sample contained only 500 bettors, the estimate produced by the average could range from $21 to $237. To be assured of coming close to the right figure you’d need a sample of 10 million bettors.
Obviously, a sample of 10 million is impractical for most applications. But if the probability distribution being sampled is fat-tailed – as most are – an estimate derived from the much smaller sample that can only be obtained in reality could be very, very far off base.
And this is only an example of how far off a sample average could be – a sample standard deviation could be even further off. And that means that statistical distillates that require standard deviation, such as the t-statistic, and the “statistical significance” that is derived from it, could be even more misleading.
Taleb believes, in fact, that it was a mistake made in the early days of the development of the field of statistics to choose standard deviation as a measure of central tendency. He thinks mean average deviation (MAD) should have been the choice.
Statistical Consequences of Fat Tails has several chapters about phenomena that yield very fat – or long – tails, including the incidence of violence, of wars, of pandemics. In each case, he says, the likelihood of an even worse incident can’t be estimated from past data, because the more data you gather, the farther out on the tail of the underlying probability distribution (if there even is one) the data will creep. Hence, past extremes are likely to understate future extremes.
I was gratified to note that Taleb says that if you allow volatility to be stochastic (to vary randomly), unlike the constant variance assumed in the standard geometric Brownian motion model, the tails become fat. I sometimes add stochastic volatility to Brownian motion in programming random walk models, and have believed, though without trying to prove it, that while it models the evidence of price changes better, it also creates fat tails. I’m glad to see that Taleb finds my conjecture to be true, though I did not see a rigorous proof of it in his book.
A note of skepticism on the lognormal distribution
Taleb presents a rigorous test to define whether a probability distribution has fat tails or does not. But then he implies of the lognormal distribution – which is the distribution always implicitly assumed for ordinary rates of return by the standard geometric Brownian motion model – that it is sometimes fat-tailed and sometimes not; that it behaves like a Gaussian (normal) distribution when the variance, s2, is small, but like a fat-tailed distribution when s2 is large. I don’t see how this could be true, given his mathematical test of fat-tailedness. Either the lognormal meets the test or it doesn’t; the value of s2 shouldn’t matter.
He further says that “many cases of Paretian-ity [fat-tailedness] are effectively lognormal situations with high variance; the practical statistical consequences, however, are smaller than imagined.”
I don’t think those consequences are smaller than imagined. I have pointed out that if Rolf Banz in his very-much-cited 1981 study of small stock returns had taken into account the lognormal distribution of returns, he would not have detected any small stock alpha.1
Taleb’s investment philosophy
As noted above, the second of Taleb’s major precepts, along with his contention that most probability distributions in real life are fat-tailed rather than thin-tailed like the normal distribution, is that the vast majority of people underestimate the likelihood of extreme tail events, i.e. “Black Swans.” Therefore, he apparently believes, it is possible for the very few (like himself) to take advantage of this underestimation by betting on such extreme events to occur. The bet will usually not pay off, but when it does, it will pay off very big.
This philosophy was fueled by his childhood experience in Lebanon, where an idyllic country turned very suddenly into a horror, causing his family to emigrate quickly, losing almost everything – and also by the fact that his early big bet on an unprecedented negative Black Swan event in the stock market paid off in the millions when the market dropped more than 20% in one day on October 19, 1987.
This must be his theory because his two investment companies, Empirica Capital, founded in 1999, and Universa Investments, established in 2007, follow this principle. Empirica shut down in 2005 after a string of low returns in the years 2001-2005, though perhaps not because of those low returns but for other reasons, because strings of low returns are to be expected with such a strategy, and its return was high in 2000, the year of a stock market plummet.
Universa, however, has experienced good returns over time with the same strategy. Nevertheless, it is almost impossible to determine whether the strategy worked even in the past, let alone whether it will work in the future.
Taleb does not define himself in terms of his investment knowledge or skill, though he does revel in his one-time and perhaps continuing role as an option trader and participant in the community of option traders. But his true choice of self-description is that of a philosopher, or to be precise, a “flaneur” (his choice of word), which is variously defined as “an idle man-about-town” (Merriam-Webster) and “a man who saunters around observing society” or “a passionate observer who is at the centre of the world and yet remains inconspicuous” (Oxford Reference).
Inconspicuous, perhaps not. Nonetheless, more power to him.
Economist and mathematician Michael Edesess is an adjunct professor and visiting faculty at the Hong Kong University of Science and Technology. In 2007, he authored a book about the investment services industry titled The Big Investment Lie, published by Berrett-Koehler. His new book, The Three Simple Rules of Investing, co-authored with Kwok L. Tsui, Carol Fabbri and George Peacock, was published by Berrett-Koehler in June 2014.
1 See “Alphas and the Choice of Rate of Return in Regressions” in Edhec-Risk Institute Research Insights June 2014.
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