Antifragile: Things That Gain from Disorder

by Taleb, Nassim Nicholas

  I noticed as a trader—and obsessed over the idea—that correlations were never the same in different measurements. Unstable would be a mild word for them: 0.8 over a long period becomes −0.2 over another long period. A pure sucker game. At times of stress, correlations experience even more abrupt changes—without any reliable regularity, in spite of attempts to model “stress correlations.” Taleb (1997) deals with the effects of stochastic correlations: One is only safe shorting a correlation at 1, and buying it at −1—which seems to correspond to what the 1/n heuristic does.

  Kelly Criterion vs. Markowitz: In order to implement a full Markowitz-style optimization, one needs to know the entire joint probability distribution of all assets for the entire future, plus the exact utility function for wealth at all future times. And without errors! (We saw that estimation errors make the system explode.) Kelly’s method, developed around the same period, requires no joint distribution or utility function. In practice one needs the ratio of expected profit to worst-case return—dynamically adjusted to avoid ruin. In the case of barbell transformations, the worst case is guaranteed. And model error is much, much milder under Kelly criterion. Thorp (1971, 1998), Haigh (2000).

  The formidable Aaron Brown holds that Kelly’s ideas were rejected by economists—in spite of the practical appeal—because of their love of general theories for all asset prices.

  Note that bounded trial and error is compatible with the Kelly criterion when one has an idea of the potential return—even when one is ignorant of the returns, if losses are bounded, the payoff will be robust and the method should outperform that of Fragilista Markowitz.

  Corporate Finance: In short, corporate finance seems to be based on point projections, not distributional projections; thus if one perturbates cash flow projections, say, in the Gordon valuation model, replacing the fixed—and known—growth (and other parameters) by continuously varying jumps (particularly under fat-tailed distributions), companies deemed “expensive,” or those with high growth, but low earnings, could markedly increase in expected value, something the market prices heuristically but without explicit reason.

  Conclusion and summary: Something the economics establishment has been missing is that having the right model (which is a very generous assumption), but being uncertain about the parameters will invariably lead to an increase in fragility in the presence of convexity and nonlinearities.

  FUHGETABOUD SMALL PROBABILITIES

  Now the meat, beyond economics, the more general problem with probability and its mismeasurement.

  How Fat Tails (Extremistan) Come from

  Nonlinear Responses to Model Parameters

  Rare events have a certain property—missed so far at the time of this writing. We deal with them using a model, a mathematical contraption that takes input parameters and outputs the probability. The more parameter uncertainty there is in a model designed to compute probabilities, the more small probabilities tend to be underestimated. Simply, small probabilities are convex to errors of computation, as an airplane ride is concave to errors and disturbances (remember, it gets longer, not shorter). The more sources of disturbance one forgets to take into account, the longer the airplane ride compared to the naive estimation.

  We all know that to compute probability using a standard Normal statistical distribution, one needs a parameter called standard deviation—or something similar that characterizes the scale or dispersion of outcomes. But uncertainty about such standard deviation has the effect of making the small probabilities rise. For instance, for a deviation that is called “three sigma,” events that should take place no more than one in 740 observations, the probability rises by 60% if one moves the standard deviation up by 5%, and drops by 40% if we move the standard deviation down by 5%. So if your error is on average a tiny 5%, the underestimation from a naive model is about 20%. Great asymmetry, but nothing yet. It gets worse as one looks for more deviations, the “six sigma” ones (alas, chronically frequent in economics): a rise of five times more. The rarer the event (i.e., the higher the “sigma”), the worse the effect from small uncertainty about what to put in the equation. With events such as ten sigma, the difference is more than a billion times. We can use the argument to show how smaller and smaller probabilities require more precision in computation. The smaller the probability, the more a small, very small rounding in the computation makes the asymmetry massively insignificant. For tiny, very small probabilities, you need near-infinite precision in the parameters; the slightest uncertainty there causes mayhem. They are very convex to perturbations. This in a way is the argument I’ve used to show that small probabilities are incomputable, even if one has the right model—which we of course don’t.

  The same argument relates to deriving probabilities nonparametrically, from past frequencies. If the probability gets close to 1/ sample size, the error explodes.

  This of course explains the error of Fukushima. Similar to Fannie Mae. To summarize, small probabilities increase in an accelerated manner as one changes the parameter that enters their computation.

  FIGURE 38. The probability is convex to standard deviation in a Gaussian model. The plot shows the STD effect on P>x, and compares P>6 with an STD of 1.5 compared to P>6 assuming a linear combination of 1.2 and 1.8 (here a(1)=1/5).

  The worrisome fact is that a perturbation in σ extends well into the tail of the distribution in a convex way; the risks of a portfolio that is sensitive to the tails would explode. That is, we are still here in the Gaussian world! Such explosive uncertainty isn’t the result of natural fat tails in the distribution, merely small imprecision about a future parameter. It is just epistemic! So those who use these models while admitting parameters uncertainty are necessarily committing a severe inconsistency.2

  Of course, uncertainty explodes even more when we replicate conditions of the non-Gaussian real world upon perturbating tail exponents. Even with a powerlaw distribution, the results are severe, particularly under variations of the tail exponent as these have massive consequences. Really, fat tails mean incomputability of tail events, little else.

  Compounding Uncertainty (Fukushima)

  Using the earlier statement that estimation implies error, let us extend the logic: errors have errors; these in turn have errors. Taking into account the effect makes all small probabilities rise regardless of model—even in the Gaussian—to the point of reaching fat tails and powerlaw effects (even the so-called infinite variance) when higher orders of uncertainty are large. Even taking a Gaussian with σ the standard deviation having a proportional error a(1); a(1) has an error rate a(2), etc. Now it depends on the higher order error rate a(n) related to a(n−1); if these are in constant proportion, then we converge to a very thick-tailed distribution. If proportional errors decline, we still have fat tails. In all cases mere error is not a good thing for small probability.

  The sad part is that getting people to accept that every measure has an error has been nearly impossible—the event in Fukushima held to happen once per million years would turn into one per 30 if one percolates the different layers of uncertainty in the adequate manner.

  1 The difference between the two sides of Jensen’s inequality corresponds to a notion in information theory, the Bregman divergence. Briys, Magdalou, and Nock, 2012.

  2 This further shows the defects of the notion of “Knightian uncertainty,” since all tails are uncertain under the slightest perturbation and their effect is severe in fat-tailed domains, that is, economic life.

  ADDITIONAL NOTES, AFTERTHOUGHTS, AND FURTHER READING

  These are both additional readings and ideas that came to me after the composition of the book, like whether God is considered robust or antifragile by theologians or the history of measurement as a sucker problem in the probability domain. As to further reading, I am avoiding the duplication of those mentioned in earlier books, particularly those concerning the philosophical problem of induction, Black Swan problems, and the psychology of uncertainty. I managed to bury some math
ematical material in the text without Alexis K., the math-phobic London editor, catching me (particularly my definition of fragility in the notes for Book V and my summary derivation of “small is beautiful”). Note that there are more involved technical discussions on the Web.

  Seclusion: Since The Black Swan, I’ve spent 1,150 days in physical seclusion, a soothing state of more than three hundred days a year with minimal contact with the outside world—plus twenty years of thinking about the problem of nonlinearities and nonlinear exposures. So I’ve sort of lost patience with institutional and cosmetic knowledge. Science and knowledge are convincing and deepened rigorous argument taken to its conclusion, not naive (via positiva) empiricism or fluff, which is why I refuse the commoditized (and highly gamed) journalistic idea of “reference”—rather, “further reading.” My results should not depend, and do not depend on a single paper or result, except for via negativa debunking—these are illustrative.

  Charlatans: In the “fourth quadrant” paper published in International Journal of Forecasting (one of the backup documents for The Black Swan that had been sitting on the Web) I showed empirically using all economic data available that fat tails are both severe and intractable—hence all methods with “squares” don’t work with socioeconomic variables: regression, standard deviation, correlation, etc. (technically 80% of the Kurtosis in 10,000 pieces of data can come from one single observation, meaning all measures of fat tails are just sampling errors). This is a very strong via negativa statement: it means we can’t use covariance matrices—they are unreliable and uninformative. Actually just accepting fat tails would have led us to such result—no need for empiricism; I processed the data nevertheless. Now any honest scientific profession would say: “what do we do with such evidence?”—the economics and finance establishment just ignored it. A bunch of charlatans, by any scientific norm and ethical metric. Many “Nobels” (Engle, Merton, Scholes, Markowitz, Miller, Samuelson, Sharpe, and a few more) have their results grounded in such central assumptions, and all their works would evaporate otherwise. Charlatans (and fragilistas) do well in institutions. It is a matter of ethics; see notes on Book VII.

  For our purpose here, I ignore any economic paper that uses regression in fat-tailed domains—as just hot air—except in some cases, such as Pritchet (2001), where the result is not impacted by fat tails.

  PROLOGUE & BOOK I: The Antifragile: An Introduction

  Antifragility and complexity: Bar-Yam and Epstein (2004) define sensitivity, the possibility of large response to small stimuli, and robustness, the possibility of small response to large stimuli. In fact this sensitivity, when the response is positive, resembles antifragility.

  Private Correspondence with Bar-Yam: Yaneer Bar-Yam, generously in his comments: “If we take a step back and more generally consider the issue of partitioned versus connected systems, partitioned systems are more stable, and connected systems are both more vulnerable and have more opportunities for collective action. Vulnerability (fragility) is connectivity without responsiveness. Responsiveness enables connectivity to lead to opportunity. If collective action can be employed to address threats, or to take advantage of opportunities, then the vulnerability can be mitigated and outweighed by the benefits. This is the basic relationship between the idea of sensitivity as we described it and your concept of antifragility.” (With permission.)

  Damocles and complexification: Tainter (1988) argues that sophistication leads to fragility—but following a very different line of reasoning.

  Post-Traumatic Growth: Bonanno (2004), Tedeschi and Calhoun (1996), Calhoun and Tedeschi (2006), Alter et al. (2007), Shah et al. (2007), Pat-Horenczyk and Brom (2007).

  Pilots abdicate responsibility to the system: FAA report: John Lowy, AP, Aug. 29, 2011.

  Lucretius Effect: Fourth Quadrant discussion in the Postscript of The Black Swan and empirical evidence in associated papers.

  High-water mark: Kahneman (2011), using as backup the works of the very insightful Howard Kunreuther, that “protective actions, whether by individuals or by governments, are usually designed to be adequate to the worst disaster actually experienced.… Images of even worse disaster do not come easily to mind.”

  Psychologists and “resilience”: Seery 2011, courtesy Peter Bevelin. “However, some theory and empirical evidence suggest that the experience of facing difficulties can also promote benefits in the form of greater propensity for resilience when dealing with subsequent stressful situations.” They use resilience! Once again itsnotresilience.

  Danchin’s paper: Danchin et al. (2011).

  Engineering errors and sequential effect on safety: Petroski (2006).

  Noise and effort: Mehta et al. (2012).

  Effort and fluency: Shan and Oppenheimer (2007), Alter et al. (2007).

  Barricades: Idea communicated by Saifedean Ammous.

  Buzzati: Una felice sintesi di quell’ultimo capitolo della vita di Buzzati è contenuto nel libro di Lucia Bellaspiga «Dio che non esisti, ti prego. Dino Buzzati, la fatica di credere»

  Self-knowledge: Daniel Wegner’s illusion of conscious will, in Fooled by Randomness.

  Book sales and bad reviews: For Ayn Rand: Michael Shermer, “The Unlikeliest Cult in History,” Skeptic vol. 2, no. 2, 1993, pp. 74–81. This is an example; please do not mistake this author for a fan of Ayn Rand.

  Smear campaigns: Note that the German philosopher Brentano waged an anonymous attack on Marx. Initially it was the accusation of covering up some sub-minor fact completely irrelevant to the ideas of Das Kapital; Brentano got the discussion completely diverted away from the central theme, even posthumously, with Engels vigorously continuing the debate defending Marx in the preface of the third volume of the treatise.

  How to run a smear campaign from Louis XIV to Napoleon: Darnton (2010).

  Wolff’s law and bones, exercise, bone mineral density in swimmers: Wolff (1892), Carbuhn (2010), Guadaluppe-Grau (2009), Hallström et al. (2010), Mudd (2007), Velez (2008).

  Aesthetics of disorder: Arnheim (1971).

  Nanocomposites: Carey et al. (2011).

  Karsenty and Bones: I thank Jacques Merab for discussion and introduction to Karsenty; Karsenty (2003, 2012a), Fukumoto and Martin (2009); for male fertility and bones, Karsenty (2011, 2012b).

  Mistaking the Economy for a Clock: A typical, infuriating error in Grant (2001): “Society is conceived as a huge and intricate clockwork that functions automatically and predictably once it has been set in motion. The whole system is governed by mechanical laws that organize the relations of each part. Just as Newton discovered the laws of gravity that govern motion in the natural world, Adam Smith discovered the laws of supply and demand that govern the motion of the economy. Smith used the metaphor of the watch and the machine in describing social systems.”

  Selfish gene: The “selfish gene” is (convincingly) an idea of Robert Trivers often attributed to Richard Dawkins—private communication with Robert Trivers. A sad story.

  Danchin’s systemic antifragility and redefinition of hormesis: Danchin and I wrote our papers in feedback mode. Danchin et al. (2011): “The idea behind is that in the fate of a collection of entities, exposed to serious challenges, it may be possible to obtain a positive overall outcome. Within the collection, one of the entities would fare extremely well, compensating for the collapse of all the others and even doing much better than the bulk if unchallenged. With this view, hormesis is just a holistic description of underlying scenarios acting at the level of a population of processes, structures or molecules, just noting the positive outcome for the whole. For living organisms this could act at the level of the population of organisms, the population of cells, or the population of intracellular molecules. We explore here how antifragility could operate at the latter level, noting that its implementation has features highly reminiscent of what we name natural selection. In particular, if antifragility is a built-in process that permits some individual entities to stand out from the bulk in a challenging situation, t
hereby improving the fate of the whole, it would illustrate the implementation of a process that gathers and utilises information.”

  Steve Jobs: “Death is the most wonderful invention of life. It purges the system of these old models that are obsolete.” Beahm (2011).

  Swiss cuckoo clock: Orson Welles, The Third Man.

  Bruno Leoni: I thank Alberto Mingardi for making me aware of the idea of legal robustness—and for the privilege of being invited to give the Leoni lecture in Milan in 2009. Leoni (1957, 1991).

  Great Moderation: A turkey problem. Before the turmoil that started in 2008, a gentleman called Benjamin Bernanke, then a Princeton professor, later to be chairman of the Federal Reserve Bank of the United States and the most powerful person in the world of economics and finance, dubbed the period we witnessed the “great moderation”—putting me in a very difficult position to argue for increase of fragility. This is like pronouncing that someone who has just spent a decade in a sterilized room is in “great health”—when he is the most vulnerable.

  Note that the turkey problem is an evolution of Russell’s chicken (The Black Swan).

  Rousseau: In Contrat Social. See also Joseph de Maistre, Oeuvres, Éditions Robert Laffont.