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.)

4 comments:

  1. But it's not like Taleb is running around advocating a Bar Bell approach to investing. He hates to talk about investing. I've also heard him advocate an index of the entire market. (Check out the EconTalk interviews http://www.econlib.org/cgi-bin/fullsearch.pl?query=nassim+taleb&x=17&y=15&andor=and&sel=32
    My Taleb takeaway is a broad swath of language and ideas around the notion of Mediocristan and Extremistan.
    I would really like to hear your 'Fat Tony' version of this post.

    I only want to add that with both Taleb and the Long Now Foundation, I feel that much of what they are about is, in the McLuhan sense of the word, probing–introducing ideas that make us think.

    ReplyDelete
  2. It is hard to take an unsigned paper seriously... Taleb for instance always signs what he writes...

    Unsigned rants lack personal conviction and suggest cowardice or malice... Stand up guys take responsibility for their words

    ReplyDelete
  3. Merry Christmas to you too, jonc.

    As I am a nobody I don't see why my personal details should be relevant to your comprehension, or lack thereof. You can find me on linked-in if you really GAS.

    But hey, thanks for mistaking this for a paper. I'm flattered. Maybe I could publish it in the Journal of Trivial Yet Adequate Responses to Pseudo-Scientific Claptrap.

    Please take another swing old chap, and try to get closer to the mark

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  4. Woowit,

    Taleb is not asking us to think. He is offering neologisms that vandalize existing statistical terminology (usually conflating different concepts). Taleb invites his reader to take a great big short cut.

    And don't get me started on Mediocristan and Extremistan. It is a bloody ridiculous dichotomy. For example, did Google take on a linear or a non-linear problem? Ans: page popularity is about as non-linear as it gets. Did Google use linear or non-linear mathematics? Ans: Linear. Classic linear algebra.

    Maybe we should ban numbers as they can too easily be used as linear transformations - clearly not appropriate for a non-linear world

    Peter

    ReplyDelete