Saturday, October 8, 2011

Fuck Off Taleb: Congress Shouldn't Heed a Quack in the Intellectual Tradition of Sextus Empiricus


A certain professor at the Stanford Business School has been taunting me of late, knowing full well where my weak points lie. He forwarded me this last night with the subject line "Guess Who?"
"I’m not interested in money; I’m not interested in finance, I'm comfortable enough as it is. I don’t need it. Finance should be a footnote in my bio, not a central component. Why should I waste time in finance when my influence as an intellectual is so high?

This latest gem is reported in a Bloomberg article by Seth Lubove and Miles Weiss and yes, it is Nassim Taleb. It was a successful provocation on this occasion for as said nameless professor is well aware, I have have a soft spot for financial intellectuals, the hijacking of quantitative finance  and the extraordinary success a handful of loud mouths have had in determining the perception of mathematics. The unnamed professor (who might not want his name mentioned in a low-brow blog like this) has insisted I write a book with called "Fuck Off ...", and that prompts the title of this post. I'm still open to other less inflamatory suggestions but suffice to say Nassim Taleb is very good at annoying people. Comments like the above go a long way toward explaining this. 

But they don't go all the way. My personal fascination with the Taleb phenomenon does not revolve solely around his relentless pomposity or inventive egotism, though it does seem to help. In the spirit of Hofstadter's Law you have to give the guy some credit. What comes out of Taleb's mouth continues to surprise even when you think you have already heard so much you will not be surprised by anything, and even when you take this into account as well, if you know what I mean. (I'm sure there is some vindication of Black Swan struggling in this regress of doubt, somewhere). In this case it was the 'footnote' comment and use of 'intellectual' that got under my skin. I'll return to 'intellectual' later but foremost, the irony in the footnote comment. 

                                               Post-Probabilism

To be a footnote in anything more permanent than a bio you write yourself someone other than a journalist has to read your work, make sense of it, and tie it into something of merit. That doesn't seem to be happening for this particular overexposed thinker. Taleb's attempts to formalize his thoughts are scattered far and wide but represent an astonishingly low citation to download ratio. In Taleb's case, few have been able to build upon what he has to say and nobody has been able to to so in a technical sense. The reason is simple: Taleb doesn't make sense. 

True, many have quoted Taleb, referred to him in general terms, or intimated that there is some underpinning theory - and this vague  anti-probability bandwagon is one of the more surprising post-crisis phenomena. But the element of surprise isn’t always enough. By necessity this populist crusade is a comedic battle pitched between important sounding prose and the terse but plainspoken language of mathematics: a verbal assault on one of the great abstract constructions, its relevance to the real world, and its role in financial calamity. Or if you prefer sporting analogies, recall the image of gymnast Brian Meeker overstepping the springboard and ploughing headlong into the pommel horse. 

But to Taleb, probability is "outdated". Seriously. That is the word he used to describe probability in a review of fellow post-probabilist Elie Ayache's book The Blank Swan on Amazon. (Taleb may have removed the post after reading this article).  In passing, the notion that probability is outdated is somewhat ironic, because some corners of modern finance have barely begun to use it. And Taleb is not presenting counter-establishment thinking at all, but lazy, unscientific banter. If there is an "establishment" that matters isn't particularly scientific, much less theoretical so Taleb's failure to acknowledge probability theory (or large parts of statistics) is not exceptional but closer to the norm for market participants - not to mention those that regulate them, rather more unfortunately, and certainly those that manage them. 

In fact pissing on probability theory is entirely standard behaviour in the middle to bottom tier of uneducated schmucks given license to trade - some of whome would rather re-invent tiny parts of theory and screw it up in the process than read a book (so they can revel in the seeming profundity of their 'paradoxical', 'novel' insights). It goes a little far though when, in encouraging everyone to eschew the only weapons that work, our 'thought leaders' influence our elected leaders who then invent hair-brained anti-intellectual policies designed to make finance safer (by curtailing the use of mathematics, usually, or adding layers of administration to its use). Probability, not committees, is what automates trading. Theory can help stem the leak, reduce cynical rent extraction, and streamline cumbersome business processes. 

You won't hear that from Taleb. Almost surely (as the probabilists say) you'll hear soft nonsense. Probability was created with the wrong type of intelligence, apparently. Intelligence, “in the sense of IQ tests and SAT scores” is not, we learn from Taleb, “as natural & ancestrally fit as wit”. By not natural, the philosopher explains, “I mean not Black-Swan robust, skills we call intelligence because of a certain construction, but that are not needed ecologically”. The modern day thinker is referring to mating probabilities and poses an important question for humorless nerds without dates. “Is ‘intelligence’ without wit & verbal brilliance really intelligence?” If not, it stands to reason, we should be using ecologically valid criteria to “define true intelligence” and more importantly, to discern “relevant subjects”. One might conclude for example that “painting, wit, music are more NATURAL than abstract mathematics”. 

This kind of ‘bolstering’ is almost as silly as Taleb's central, non-probabilistic message itself. (The flowery waffle is lame, but the attempts to make it rigorous are lamer). Taleb's asserts that estimates of probability should never be too small ... and we are to further infer, as best I understand, that "too small" corresponds to roughly one percent. It is oh so simple. When faced with decision making under uncertainty, otherwise known as life, we should skip over the probabilistic assessment and just ensure that we are betting on the outcome with the biggest upside! That is the 'non-probabilistic' approach, pure and simple, so I sympathize with academics who have stumbled across this and wondered, where is the hidden camera? Is post-probabilism, or pre-probabilism, an elaborate hoax intended to embarrass the media and make a salient point about the superficiality of public discourse?

Maybe. Taleb believes it is jiggy to forget the mean and act as if all probabilities are above a certain minimum threshold ... and that is lovely. But it isn't an impressive assault on probability theory and even it it were, the pioneers saw him coming. As de Finetti and Savage were aware, to name two, one defines probability in several ways and the notion of ‘personal probability’ might help Taleb sharpen his non-argument. That will be difficult but in boring, conventional statistics one can say sensible, logical things about behavior-inferred probability and as discussed in The Foundations of Statistics, for example, Taleb falls right into that wheelhouse. He just happens to be someone who sets a lower bound on his personal probabilities and  whether he thinks he is using probability or not is irrelevant. While some regard this as 'thinking outside the box', it actually places him in a special, pre-reserved idiot box. 

And while his position might sound reasonable for about oh, four seconds, it runs aground on basic notions like symmetry well before we come to empirical evidence for long-shot betting and option writing. The ironies pile high. True empirical sceptics should have more trouble getting around symmetry than the rest of us, because they are reluctant to use past data. So what winning probabilities do they assign to non-elite entrants in a marathon? More than one percent? We'll never know because confronting the obvious contradictions in Black Swan struggling is not on the priority list for post-crisis radicals. They prefer to get by with motherhood words like robust, heuristic, non-linear, ecological and so forth. Too bad a moment’s reflection reduces the Talebian position to head-in-the-sand wishful thinking falling for the world's oldest lure: long odds offered by bookmakers on events with very low probabilities. 

                       Taleb's Intellectual Mentor: Sextus Empiricus

So I sure hope this is a hoax. But joke or not Taleb's baby mush is the perfect food for our leaders, most of whom are so far removed from their studies that they can be classified as mathematically geriatric. He has, with a certain  flanerial flair, convinced pundits, think tanks and one British Prime Minister of his standing as an intellectual. And this would leave us with the problem of defining 'intellectual' had one polarizing intellectual, Noam Chomsky, not already provided a unique, mildly cynical but truly fitting one. 

Whatever you think of Chomsky, his definition of intellectual was made for Taleb. As he commented recently in the Boston Review, an intellectual is defined as someone who: 
  1. Makes public pronouncements about things outside their field 
  2. Fails to fully acknowledge, or even consider, symmetry
Chomsky was referring to higher, ethical symmetry of course: the manner in which we characterize actions of our enemies versus our own. It is a far more controversial topic than symmetries in everyday, mundane, applied probability where approximate symmetry is - dare I say - substantially easier to establish. Professional gamblers routinely estimate probabilities smaller than one in a million, with sufficient accuracy to get by, and no amount of pseudo-philosophical verbiage should be able to distract us from this. When one gets down to business Taleb adds absolutely nothing of value because despite the grandstanding he has not yet rediscovered even simple tricks (like Good-Turing estimates for the probability of novel things).  

Instead, Taleb summons the forgotten wisdom of the ancients and slots himself into, or rather above, two thousand years of skeptical thinking - an interesting vantage point but one from which one might easily overlook the last century or so of non-trivial intellectual accomplishment in many relevant fields. Things started to go wrong around the time of the Enlightenment which is a convenient if extreme anti-establishment slant these days. It makes responding to his 'intellectualism' just as frustrating as arguing with the ancients, because much of what has passed in between - including all of modern statistics - has been ignored. 

To that end Taleb models himself after ancient Sceptic Sextus Empiricus, and even refers to Sextus as his mentor. Sextus Empiricus makes for a better bedfellow than Turing or Laplace, given Taleb’s inter-millennial pose - it would almost be humble to choose someone more recent. We are to presume humans with insights as dazzling as Taleb don't come around too often. And that we're so far off track (with all this learning stuff) that only he can pull us back. Taleb probably assumes Prime Ministers are unlikely to check on whether Sextus actually said anything useful - just so long as he looks good on a dusty portrait.

But I did. And I now must wonder about Sextus' suitability as mentor given his seeming inability to string two rational thoughts together. In fact Sextus' logic was so poor that contemporary reprintings are accompanied by lengthy editorial apologies, begging us to consider Sextus despite tedious ruminations like the following:
What if someone says that ten is divided into one and two and three and four, ten is not divided into these things. For as soon as its first part, i.e. one, is removed - to grant this for the moment by way of concession - ten is no longer present, but rather nine - and in general something different from ten. Thus the subtraction and division of the rest is made not on ten but on the other things which alter at each subtraction. Perhaps, then, it is not possible to divide a whole into what are said to be its parts
Yup, that is Sextus trying to undermine Sesame Street. And if you read on you, like me, might start to wonder precisely what knowledge Taleb has gleaned from Sextus (I am still tossing up a complete reading against the opportunity cost - though I suppose I could try to read it and pass a kidney stone at the same time). No wonder Barnes and Annas, editors of the Cambridge reprinting I refer to, find it "difficult to believe that Sextus ever seriously searched for the truth”. His philosophical arguments aimed at the Dogmatists of his day are “not only ad hominem” but also “vastly unsympathetic”.  Sextus was a “quack”, to use their word (only partially out of context) and he “rarely considers how a Dogmatist might react to his sceptical objection, or what he might then say in replay to the reaction”. The commentators go so far as to say that “for someone who professedly continues the inquiry, Sextus shows little interest in intellectual exploration" (and that is why anyone purporting to denigrate mathematics or applied probability can take the ancient Sceptical school, and all their accumulated wisdom, and shove it up their arse). 

We should be careful about attributing Sextus' mediocrity to his tendency to step outside the bounds of his expertise, because apparently he was pretty mediocre in the field of medicine too. I shall not be sending my newborn child the way of Dr. Sextus Empiricus any time soon, or anyone following the Empiric tradition of medicine. In treating patients Sextus took a philosophical stance: one should rely on experience alone, and downplay rational enquiry, inference and theory. 

As I started to read Barnes and Annas lengthy apology for this poor sod I couldn't help noticing some parallels between Sextus and Taleb. Annas and Barnes confess that Sextus’ text is “transparently sophistical”, comprising “embarrassingly bad” arguments that “will not delay a half-competent philosopher for more than a minute”. They suggest his approach has "zero probative value". But the editors come up with an ingenious defense based on his equally illogical contemporaries. Sextus arguments are “no worse than, say, the arguments in Plato’s dialogues” and might not have been subject to such a relentless pounding had Sextus not expresses his poorest arguments clearly and distinctly. Shame on Sextus for not hiding the inadequacy of his logical arguments better! 

                   Hey Dudes! Let's all just forget about the mean!  
                              ( are you fucking serious Taleb? )

I feel that is a tribute we should also pay to the author of The Black Swan who has presented the “central idea of uncertainty” upon which we might “build an overall theory of decision making” in a manner that could not possibly be more straightforwardly daft: 
"But the idea behind Pascal’s wager has fundamental applications out-side of theology. It stands the entire notion of knowledge on its head. It eliminates the need for us to understand the probabilities of a rare event (there are fundamental limits to our knowledge of these); rather, we can focus on the payoff and benefits of an event if it takes place."
Aside: I happen to be an athiest but in principle I've no problem with founding a new philosophy on the single most moronic theological argument ever made (Pascal's Wager only works on those with a blind-spot for symmetry, but remembering Chomsky's definition, we expect it to find success with "intellectuals" of Taleb's ilk). 

But hang on a minute.  When it comes to ignoring the mean, as Taleb would have us do, I may require more convincing. That's because I'm pretty sure I've heard this one from just about every mug who has ever entered a racetrack. Mean-ignorers exist in great abundance and there must have been even more before the "concept" of the mean became prominent in that sphere. 

You probably think I'm joking, because the notion of considering the expected return on a bet and not just those big tasty 500/1 odds on offer seems rather obvious. Yet a glance over the history of Australian punting suggests the contrary: it required a lawyer named Don Scott to invent it. Scott unleashed a "value revolution", in which those placing good value bets started to make money. Who would have thought?

Great First Moments in Statistics

Taleb's achievement trumps Scott though, because he has managed not to invent the mean, but forget about it. We might churlishly suggest that Taleb has shunned the post 1957 "philosophy" of the great Don Scott, handicapping guru, preferring the ancient wisdom of Sextus Empiricus. 

There I go beating up on Sextus again. In fairness, something intelligent was surely written by the Sceptical school which stretched from 400 B.C. to 200 AD, roughly, and included Pyrrho,Timon, Aenesidemus, Agrippa and Menodotus. You'll have to find it yourself though. Sextus was, like many philosophers of his time, actually a doctor of medicine. Sextus attacked the arguments of his Dogmatic adversaries but could not apply the same scrutiny to his own arguments. So we can say by Chomsky's definition he was indeed an intellectual. I would agree that Sextus is a suitable mentor for Taleb, and more so than Bertrand Russell or Kurt Godel (to whom we would never expect Taleb to compare himself to, incidentally, unless of course, he already had).

            Taleb's Policy on Research into Systemic Risk: Don't Bother

No why does any of this matter? Only because Nassim Taleb, a modern day Sextus Empiricus of sorts, has for some time been advising the U.K. government and now, it would seem, seeks to guide the United States Congress. True, congress takes pseudo-science seriously on a regular basis so this is nothing new, but surely it need not move into pseudo-probability? Taleb feels otherwise and the House Financial Services Committee entertained what can only be described as his Sceptical report recently, on the effectiveness and possible side effects of the newly created Office of Financial Research. I suppose it took quite a lot 'Empirical' work treating patients before anyone noticed the circulatory system, but sending a sick financial system in the direction of “less is more” heuristics and “non-probabilistic risk measures” seems reasonable. 

The author of this fine document, Nassim Taleb, appeared before the committee arguing against efforts to measure systemic risk without a hint of self-consciousness, citing heady tombs like Lecturing Birds on Flying by Pablo Triana (see my prior post) and his own populist piffle. It is all beyond the pale but this if we are talking about the implications of ditching probability this merely scratches the surface. Outside of systemic risk, the topic of the hearing, downplaying quantitative methods will cost taxpayers another trillion dollars over the next decade, while they are fed folksy finance by trader-thinkers promoting experience, intuition, gut instinct and just about anything else that sounds good over theory or scientific method. That's because the only thing that will reduce the cost of finance is automation, and that means mathematical, probabilistic technology.

Taleb’s remarkable memo argues against any such thing, naturally, and alerts us to the generic danger of increasing knowledge. He predicts that we will fail to predict risk in the future because we have failed to predict risk in the past, a claim from a distant observer of fixed income markets, in particular, that might generously be classified as an inductive error were it not for the counterfactual premise: central banks have already tried using “thousands of PhDs on their staff” to build elaborate models for systemic risk. That premise, and hence the scene, is ridiculous but I suppose there are new measures by which we assess probabilistic insight these days. 

We will have to put up with this nonsense for quite some until someone writes a revolutionary business book about the mean, or creates a bigger distraction questioning real numbers. In the meantime, the mascot for intellectual post-probabilism has made it into the Thinkers 50 and if you are wondering how that came about you need only consult their methodology: a survey of “business people, consultants, academics and MBA students throughout the world”. No problem here I guess, except that the nutritional content of the more ‘conceptual’, 'abstract' business ideas exhibits something of a long tail. One might have to consume it all like plankton to reach any kind of tipping point. 

-----

Note: Taleb's report is titled "Report on the Effectiveness and Possible Side Effects of the Office of Financial Research" Section b includes the following: “Had the last crisis been predictable, or the risks been measurable, then central banks with access to all manner of information, and thousands of PhDs on their staff, would have been able to see it” [so there].

Saturday, October 1, 2011

Bar Humbug - The Talebian Bar-Bell Portfolio, Part I

One of the truly silly things to come out of the post-crisis theory bashing was the notion that the bar-bell strategy should be broadly applied to your investing. Nassim Taleb, amongst others, has advocated parking much of your wealth in cash and the rest in a very risky investments that have great upside. This differs from the standard (and very sensible) heuristic (invest in everything) because one deliberately eschews the middle ground: investments that offer only moderate upside but presumably, only moderate risk. 


The bar-bell portfolio promises safety plus upside. Nonsense. 
                 
The terminology is borrowed from a special case in fixed income. For those who are technically inclined and interested in the specific case of a bar-bell bond portfolios (where we hold a very short maturity bond and a very long one) my scratch working demonstrates they maximise a thing called modified excess return. But here I'm going to present much simpler, commonsense arguments requiring no stochastic calculus that illustrate why taking "safety plus upside" literally is just a little simple minded. 

I find the popularity of the bar-bell rather intriguing from a sociological perspective, because of the following: 
  1. It is very bad thinking - and very easily shown to be so. 
  2. Apparently people can be convinced anyway, through a combination of scattershot arguments ranging from "safety plus upside" to anti-intellectual nonsense.
  3. It suggests that Nassim Taleb (who has made his hay belittling the quantitative community and used the phrase "great intellectual fraud" when referring to Gauss - perhaps the greatest mathematician of all time) has no mathematical, geometric, or economic intuition whatsoever. 
Taleb has made no compelling argument beyond the "safety plus upside" verbal trick. Taleb's only stated rationale is that if you park 85 percent of your wealth in cash then you can lose at most 15 percent. This I will concede! It is hard to argue with that logic and we strengthen Taleb's result by noting that if you persist with this strategy for thirty years you will be sure to keep one percent of your original wealth, almost.  

                         A simple illustration that "bar-bell" thinking is flawed

Let's quickly piss on the bar-bell. Later we'll hang draw and quarter it. 

Consider a world with only three investment alternatives: 

        a) Cash. 
        b) A low fee index fund. 
        c) An identical fund with twice the leverage and three times the fees.

Should we put most of our money in cash (because it is safe) and a little in the leveraged index fund (because it has upside), eschewing the somewhat boring middle ground: the index fund? Perhaps a 90-10 allocation between cash and the leveraged fund? 

It is clear that in this example the simplistic "upside + safety" heuristic leads us astray. We'd be strictly better off holding 80 percent in cash and 20 in the index fund. We'll have more money in our pockets no matter what happens. The bar-bell heuristic is nonsense. 

Taleb, who has read this blog, has never responded to this most trivial counterexample. And there are bigger problems for him in what follows. I introduced fees (albeit realistic ones) as a device just to point out why blindly following verbal advice like "upside plus safety = good portfolio" is utter nonsense. But fees are not required in the demolition of this silliest of "strategies". 

We'll get to that but first, let's anticipate the attempts by Taleb to wriggle away. 

                        A possible distraction: betting on longshots

Here is where the bar-bell backers start making excuses. The longshots are better bets, they say. 

I say good. Its doesn't matter. You still shouldn't use a bar-bell. 

Here are the two separate pieces of advice which have been rather successfully promulgated. Let's not conflate them:
  1. It is wise to bet on unexpected outcomes, because markets underprice them
  2. It is wise to adopt a bar-bell portfolio eschewing a middle ground of only moderately risky investments
Taleb implicitly links the two. But Taleb says a lot of things without really thinking carefully. The former does not logically imply the latter. 

Should you believe 1 above? I don't care. Not in this post. Almost certainly you shouldn't, as an aside, because it runs contrary to the bulk of empirical studies. Nor can it be justified by the very same arguments that have led thousands of fools to place their money on horses with large odds. Indeed Taleb's revival of the longshot lure in the form of allegedly philosophical arguments can hardly be compelling unless an equally compelling case is made for why those very same arguments don't apply at the racetrack. One might be forgiven for thinking that Taleb is entirely oblivious to the longshot literature

But I can't emphasize this enough. You opinion as to whether you like longshots doesn't change anything. You still shouldn't use a bar-bell, and we shall prove it. 

Of course the implication 1 => 2 is again mildly intriguing because, like most half baked pseudo-science, it sounds perfectly reasonable. Yet if you were assuming that the world's foremost thinker on probability and uncertainty has thought about it carefully on your behalf, and you need not, think again. 


                                   Dispensing with other distractions  

Let's take the fight further into that strange verbal-math realm where business leaders, polititics, journalists and philosophers like to conduct business. For if I simply laid out the rationale for diversification I would be open to all sorts of charges, including the alleged circularity of using financial mathematics to defend financial mathematics. It is fair to say that the American mind has been closing to that possibility for some time and Taleb has much to answer for in that regard - he is a real master of distraction. 

There are many devices that might encourage us to set theory aside. For example, we hear that that academic theory is not robust because (to rephrase) it relies too heavily on small "p" modern platonism. Taleb hopes that we will be kindly disposed to any kind of half-baked discussion that so much as aspires to robustness, or just mentions "robust" many times. (I'll be sure to mention robust every other paragraph). But that may have been a tactical mistake by Taleb. It only encourages us to consider the robustness of the bar-bell advice to changing assumptions. 

Another device used by Taleb is old fashioned anti-intellectualism, in this context aimed at Harry Markowitz.  This has received so much airtime that some perceive an actual assault on modern portfolio theory - though I would characterize it more as an insult to modern portfolio theory and for that matter, the most rudimentary financial or geometrical intuition. We'll get to that but first, how did this soft nonsense get into the newspapers? It is true that the world does not have “clearly defined rules one picks up in a rulebook of the kind one finds in a Monopoly package but that is a broad brush argument against modeling of any kind and a rather poor one at that. So too suggestions that an “immediate result of Dr Markowitz’s theory” was the “near collapse of the financial system in the summer of 1998” (thanks to fellow Nobel Laureates Robert Merton and Myron Scholes) set the tone for the amount of thought which has gone into the bar-bell portfolio 'concept'. 

I guess nobody needs thought, however, when you have quality installments like this to numb the brain:
Now, I explained the point to a cab driver who laughed at the fact that someone ever thought that there was any scientific method to understanding markets and predicting their attributes. Somehow when one gets involved in financial economics, owing to the culture of the field, one becomes likely to forget these basic facts (pressure to publish to keep one’s standing among the other academics).
That was Nassim Taleb in Fooled by Randomness (page 241) and its the kind of rhetoric which is good enough for Malcolm Gladwell though not, I trust, the reader. No, we shan't be casting aside conventional financial advice merely because it seems staid or untrendy, or because one cab driver failed to fully come to terms with one of the most astonishing features of life in this universe: the universal applicability of reason and in particular, mathematics. 

Moving on a third trick up Taleb's sleeve, as far as the layman is concerned, is convexity. This is the "abstract" concept Taleb can't help but ruminate on endlessly, and in the eye of some beholders a new weapon against theory. But of course had convexity not been acknowledged in mathematical finance there would be almost no theory at all. Sections of textbooks would blank themselves out in shame. Even option pricing would more or less reduce to the time value of money. Linear terminal conditions translate to linear solutions. An important core of mathematical finance would be trivial. In short, Taleb introducing convexity to finance is like me introducing the concept of death to the medical profession and then complaining bitterly that doctors underestimate the importance of 'death theory'.

But naturally we have been encouraged by jounalists and British Prime Ministes to fawn over the  populist revisiting of Jensen's Inequality  - a rather more careful statement about the implications of convexity than we will find in a hundred populist finance books. Its a shame The Black Swan and Fooled by Randomness spend a lot of ink convincing us to see Jensen's with wide eyes (without acknowledgement of its existence)  then attempt to steer us away from insufficiently "convex" bets like stocks.  

For in arriving at a bar-bell strategy the convexity that Taleb hasn't noticed is truly astonishing. Like the fact that every stock is a convex bet (and therefore deserving more consideration from "convex" philosophers, surely). The Merton model makes this obvious to all, except those who are too busy or too disparaging of theory to notice. Too busy disparaging Merton the man, as it happens:
 “They all experiences problems during the crisis … that brought down their firm Long Term Capital Management. Note that the very same people who make a fuss about discussions of Asperger as a condition not compatible with risk-bearing and the analysis of nonexplicit off-model risks, with its corresponding dangers to society, would be opposed to using a person with highly impaired eyesight as the driver of a school bus”
                                             Nassim Taleb, The Black Swan (page 341)

Which brings us back to the anti-intellectualism. In the interest of accuracy I note that this abuse was aimed also at Myron Scholes, not just Robert Merton, but of course Scholes knew a little about convexity too having derived the most famous convexity equation in finance (one side of the Black Scholes PDE measures the convexity of the value of an option - the whole point is to relate this to the money dribbled away by betting on convexity). The Merton model on the other hand treats equity as a bet on the assets of the company and it is evidently a non-linear function thereof (cue an avalanche of recent pundits entranced by the term "non-linearity"). 

Merton's is one of those easily criticized gadgets and the task of improving it is beneath the critics, of course. That's a shame because even the existing, imperfect lens provides an assessment of funds buying out of the money put options. There were plenty of other ways to bet on calamity and a preponderance of evidence suggests that equity put options were never the pick of the crop. (See Coval et al 2008, for example). But I digress and these are all minor quibbles compared to the question of one's life savings. 

                                What's the beef with optimization?

Portfolio theory has come under fire from modern day sceptic Taleb because, I presume, it considers the long term prospects of an investment strategy and balances risk and return though explicit optimization - a sensible but shamelessly orthodox thing to do. The irony here is that quantitative philosophers like Taleb haven't stumbled across the use of optimization to design for robustness. Even brain-dead exercises such as varying different assumptions could surely be turned into some of those blessed "heuristics". Perhaps some Talebian zealots might want to perform some clandestine, heretical optimizations behind the scenes. The means justify the ends. The result might be more accurate heuristics for use by those who think heuristics are more robust than optimization (which is, I presume, a heuristic). 

But given the possibility of observing other people's optimizations it is even stranger that Taleb and his uncritical follows have not stumbled on the one heuristic that invariably falls out of many different optimizations: invest in everything! Diversification is the heuristic par excellence. To run against it sets a high watermark for lack of geometric intuition - not that we should be surprised. The counter-probability anti-optimization thinkers have used their gut instinct, apparently. And by this means arrived at the entirely opposite heuristic. 

So let me try again to present some mathematical intuition. Let's forget about fees momentarily, for they are a distraction.

                    Why every investment (within reason) should be in your portfolio

You've got to love cold hard logic. Taleb can write thousands of words and make references to wise mean of antiquity, elephants with long memories, and so forth. It's all for naught:
  1. A new, entirely independent investment opportunity should be included in your portfolio, unless it is a negative expectation bet. 
  2. It follows that every partially orthogonal investment opportunity should also be included. 
Here is a way to reason to the first conclusion. Imagine you start with a portfolio entirely in cash. You have the opportunity to move some of that cash into one or more risky investments, and you do so. Suppose, for example, that somebody offers you 6/1 odds on a coin flip. That's a terrific bet and there's no question you put at least one dollar on it. Then another dollar. You keep going until the marginal benefit of putting one more dollar starts to fall. It falls because at some point the amount you bet starts to be significant. In the limit you wouldn't bet your entire wealth on a coin flip, no matter what the odds.

You can imagine a bookmaker who starts winding in the odds as you shift your money. The actual odds stay the same but the effective odds for you come in (that depends on your personal preferences, risk profile, wealth, utility or however you wish to express it). He twiddles the nob. 5/1, then 4/1, then all the way to even money. Finally the effective odds on your marginal dollar are less than 1:1 so you stop.

That will be true for every bet you are offered. And by definition when a new opportunity comes along you will invest something in it (perhaps not much, but something) so long as the marginal effective odds start out well.

For example, suppose there are only two investments on offer other than cash. One is a guy offering 6/1 on a coin flip. The other is a woman offering 2/1 on a different coin flip. You will invest in both. It doesn't matter than one investment is seemingly strictly better and more exciting, because that's only true for the first dollar you invest in it. Eventually the odds come all the way into 2/1, then 6/4, then 5/4 and so forth - so clearly you should also be betting on the 2/1 coin flip as well until it too gets wound all the way in. 

                                    Beating up on the Wrong Guy?


But I know that won't impress journalists clinging to the belief that they have discovered a profound truth, or some academic conspiracy going back to Gauss. They have been lured away from theory by the Platonic pied piper and now it is impossible to debunk the bar-bell strategy for the same reason it is impossible to debunk creationism. Taleb's even less sophisticated groupies paint portfolio theory as a false fashion and for that, beat up on Gauss. Mention bell curves a few times and people get all riled up (invariably people not distinguishing "least squares" or "minimum variance" from "gaussian") but as it happens you don't need a Bell Curve to justify diversified portfolios. If you are really interested in distribution free optimization you should talk to Tom Cover about universal portfolios.

If you are looking for improvements over Markowitz start with Stochastic Portfolio Theory by Bob Fernholz. If you are looking for someone to blame, consult the Journal of Investment Management 2006. 

They will find a paper titled "de Finetti scoops Markowitz". A quick check reveals that the author was indeed the Harry Markowitz, one and the same father of modern portfolio theory, and not an imposter out for cheap laughs. Markowitz was referring to a 1940 paper on reinsurance decisions in which de Finetti solved a classic planning problem using the mean-variance approach. De Finetti's self-promotion skills left something to be desired, it has been suggested, as he apparently regarded this as one of his lesser works. Coming at the outset of the Second World War, de Finetti's timing was inexcusably poor and though his work was appreciated in certain European circles, it had little impact in the world of finance.

De Finetti hardly languished in obscurity, being one of the foremost probabilists of the twentieth century - some would say the most accomplished Italian mathematician of the modern era. Markowitz' paper, in passing, was written at the urging of Mark Rubinstein, professor of Finance at the University of California, Berkeley. Perhaps Markowitz and Rubinstein differ over whether de Finetti solved the full problem (where assets are correlated) or only a simpler problem in which assets are unrelated - but it is unequivocally agreed that he introduced the mean-variance approach. Almost seventy years on, de Finetti's countrymen have attempted some clarification and in a recent essay Falvio Pressacco and Paolo Serafini conclude that de Finetti's contribution to the general case of correlated assets is still an open question (but that de Finetti gave a "fully correct" procedure with "a plain extension" to the more general case). The authors point out that de Finetti made a convenient technical assumption, but show that this can be removed while respecting the simple original logic from the 1940 paper. With this technical clarification, de Finetti's approach not only solves the general case but also lays the groundwork for a whole class of modern optimization problems.


                                          The "Log Now" Foundation

If you'll forgive the aside I'm going to pick on one particular society that ought to know better than promotion of pseudo-science, especially when it comes to long term investing. Behold the Long Now Foundation a San Fransisco think tank established in “01996” that hosted Nassim Taleb a while back. The Log Now Foundation is building a monument scale, multi-millennial, all mechanical clock in the desert as an icon to long term thinking. It will tick for ten thousand years, unassisted. Fortunately they about future maintenance costs. 

And just as well. Adopting a bar-bell strategy might not be the way to fund projects for eternity. Perhaps the long history of probability was also downplayed by the Long Now Foundation as they hosted Mr Short Everything Now. 

[Picture a room full of people who are i) ready to listen to a talk about probability - some of them for the first time of their lives and ii) equally ready to entertain the possibility that they will discover some insight that was entirely lost on de Finetti, Markowitz and everyone else who has checked over their work. Picture a room full of people (I dare not comment on the demographic) scratching their chins and saying "good point old chap" when Taleb rattles off any number of atrocities (including, apparently, the fact that prediction markets can't be trusted because the odds change over time.]

I fear a memo from the lesser known "Log Now Foundation" is in order. The memo reads "Warning gentle folk. Optimization fleeing thinkers are not very good at balancing things. No, don't listen to nutters. No, don't park your life savings in a grossly sub-optimal portfolio because someone says that optimization is evil. We’d best build up some intuition for ourselves."  

True, the Log Now Foundation might be short on donors just now due to the perception of quantitative finance. But thousands of years from now it will still be around. The same might not be true for some long-shot betting funds and that is because it is helpful to consider the logarithm of your wealth. The Log Now Foundation's only message, by the way, is the compounding effect (something translating into Black Swan terminology as "scalability of one’s wealth", perhaps). 

The logarithmic scale is as useful in investing as it is in astronomy. In these coordinates we notice an interesting thing about long-shot betting strategies: several trips to negative infinity can be been made in a remarkably short period of time. 

                             An optimization as simple as falling off a log

There is something else about the bar-bell that deeply offends my mathematical sensitivities and I'd like to share it with the reader. To head in that direction consider a simple asset allocation example where an idealized stock has two futures. The setup is essentially a coin flip but we suppose the coin is loaded, as it were, and the stock will either rise by ten dollars with probability 2/3, or fall ten dollars with probability 1/3. A 'bond' (or cash position, if you will) is the only other investment opportunity, we suppose, and it will maintain the same value regardless. We further presume that you are a fund manager with the opportunity to buy or sell the stock and buy or sell the bond in any size. How should you allocate your capital?


There is a subtle point, at least in comparison to insufferably stupid rules of thumb (like 'optimization is bad'). The cash versus stock trade-off is equivalent to a different decision: how should you split the entirety of your wealth between the two possible future states if you think of them as simple bets, as shown in the table below. While we are at it, we should mention that the stock/bond decision is also equivalent to an infinite number of other problems for that matter (just assume they correspond to different combinations of the two possible outcomes) and if you think of the two securities as a somewhat arbitrary choice of basis (which they are) then you realize immediately they have little to do with the answer or, I should say, don't really help in an immediate manner when it comes to figuring out what to do. That is actually my only point, for now. 

But for fun let's quickly solve the toy problem by thinking about allocating our wealth to the outcomes, not the securities themselves. That just happens to be an easy way to do it in your head and it suggests that it is a clean choice of basis, not that this is central to my argument. Obviously the safest strategy is a fifty-fifty split. The market odds are “even money” for each of the two outcomes so you will neither win nor lose each time. In contrast, the most aggressive strategy places everything on the “up” state, whereby you double your money with 2/3’rds probability, and lose it all one time out of three - not the greatest recipe for long term survival.

Outcome
Probability
Return on $1
Stock up
2/3
$2
Stock down
1/3
$2
                                          A simple two state portfolio optimization

In between there is a compromise that maximizes the logarithm of your wealth by placing 2/3rds of your money in the “up” state.[1] That calculation is elementary calculus but perhaps the choice of log is not so obvious. The Log Now Foundation reminds us that our wealth grows by a random multiple each time we play. Should we wish to maximize the average long run yield over 1000 years this is equivalent to maximizing the product of many individual games. The log of a product is the sum of the logs, however, so on average this is equivalent to maximizing the log return each time we play.[2]  

Sceptics may object and one can certainly argue for a more conservative portfolio: such an allocation to the “up” state somewhere between 1/2 and 2/3’rds of our wealth. There are plenty of assumptions made in deriving the 2/3’rds allocation, certainly, and if errors in assessing the “up” probability are correlated from one time step to the next due to systematic biases in your methodology, or lack of methodology, then the case is strong. It hardly falls outside of theory however, or suggests we eschew optimization. And it is also beside the point: however you choose to alter this problem there is no privileged basis (except perhaps the basis I am working with). You can of course optimize the choice of stock and bond directly and you'll get the same answer but it is also clear that the stock and bond are rather arbitrary combinations of the more primal bets so there is no a priori reason why the best allocation should correspond to some “clean” combination of the investments on offer. We don’t expect a clean simple rule unless some other constraints start to bite.


The assumption in my setup is fairly benign: investments offered to you in the real world have messy relationships between them and are really different combinations of bets on the same things. As you can see from this example that's true even when there is a seemingly clean choice: bond or stock, because they are combinations of state bets. The details of the mathematics don't matter. If you pare back this most elementary example you see that the only real assertion is the one I make. In the real world, investments are even more intertwined than the toy example given. They are influenced by many of the same things including money supply, employment, consumption, oil prices, exchange rates, technology, climate and so on and so forth.


That's why it helps to ponder asset allocation for about thirty seconds longer than your meta-financial advisor might be prone to. Even if we believe in some miraculously straightforward relationship between odds and return for the state bets, a broad brush rule of thumb for portfolios along the lines of “put this security in, leave this out” will still run counter to mathematical intuition (to put it mildly) and should be treated with utmost suspicion.


      Next Time: The Miraculous Properties of the Bar-Bell (Three Possible Investments)


With only two investments to choose from it is a little hard to construct a bar-bell portfolio - there is nothing to leave out. So in my next rant we shall enlarge the problem to include three securities. A few obvious things will carry over however, from the two security example. It is evident that the amount we bet on states of the future will never be negligible because we can’t risk heading towards negative infinity on the log scale.  And it is therefore a priori unlikely that the optimal choice of investment will leave out a “middle” security, unless it is a rotten bet to begin with. That is, to belabor the point,  because the securities themselves are by no means a special basis.

Duh. 


[1] If you place a fraction w of your wealth in the up state each time (the problem scales so the answer cannot change with wealth) you wish to maximize 2/3 log(2w) + 1/2 log(2(1-w)) . This suggests we allocate w=2/3’rds of our wealth to the “up” state, assuming your author got the late night arithmetic correct. 
[2] The assumption of independence of returns lets us slip “on average” through the product.



[i] Nassim Taleb (Taleb, Nassim Taleb interview n.d.)