One of the things to come out of the post-crisis theory bashing was the notion that modern porfolio theory is bunk and instead, something simplistic like a bar-bell strategy should be broadly applied to your investing. Nassim Taleb has advocated this approach, and he is very influential. He has suggested parking much of your wealth in cash and the rest in a very risky investments that have great upside.
I have been unable to reconstruct a coherent argument for this view. Others have tried. For example this post by Cory Mitchell seems as good as any summary:
The argument goes something like this: no one can predict where the market is going, so you should not even try. There are a few things you do know, however. In the long run, the market tends to go up. Also, downturns or crashes occur from time to time–periods of time when stocks fall in value. What you can do, is allocate your portfolio in such a fashion that you gain more heavily during the good times so that over the long-run the downturns don’t hurt as much.
Needless to say this advice differs from the standard heuristic because one deliberately eschews the middle ground: investments that offer only moderate upside but presumably, only moderate risk.
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| The bar-bell portfolio promises safety plus upside. Or does it? |
The terminology bar-bell is borrowed from a special case of portfolio construction for bond portfolios. A bar-bell bond portfolio usually refers to holding a very short maturity bond and a very long maturity bond. Long maturity bonds will fluctuate in price more than short, so this may be seen as a similar strategy. However Taleb's use of the term is broader, applied to the set of all investments.
Here I am going to gently push back on Taleb's reasoning and I hope in some small way the entire category of such arguments. These break down into the following categories:
For those who are technically inclined my scratch working demonstrates an interesting mathematical fact. A bar-bell portfolio of zero coupon bonds maximizes modified excess return.
If you've never heard of modified excess return, that's because nobody uses it nor has any particular reason to. I had to reverse engineer a strange quantity.
Whether or not you want to check my mathematics, it is often the case that attempts to avoid optimization for some philosophical reason or another, usually predicated on vague robustness arguments, are often poor logic because the heuristics end up optimizing something anyway. Moreover:
(a) The thing that gets accidentally optimized isn't necessarily sensible
(b) The person is unaware of this fact.
Ignorance isn't bliss.
2. Anti-intellectualism is not novel ... please knock it off.
The second category of argument is a less sympathetic error. It is hard to forgive Taleb for rolling out various anti-intellectual tropes including the false image of the academic whose intelligence is divorced from common sense - such as empirical observation. Taleb claims that academics have been blinded by the bell curve and are otherwise guilty of rigid thinking. This ignores the fact that most of his targets were seminal thinkers - the very opposite of rigidity - and were keenly aware of the assumptions they make (as compared with Taleb who, as pointed out above, is very obviously not aware of what he is accidentally optimizing when suggesting the bar bell).
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. Taleb's suggestion 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) is ridiculous, made more so by the paucity of his attempt at an alternative bar-bell approach.
But every now and then anti-intellectualism sells. The reasons are more complex than either of us probably appreciate. These were analyzed by Hofstader in his classic work Anti-intellectualism in American Life and have been debated every since. It always seems to simmer, waiting for a moment to emerge such as a financial crisis. Nicholas Lemann titled his review of Hofstadter's work "The Tea Party is Timeless", suggesting another manifestation. As Lemann points out, the difference between intelligence and intellect seems to slide by.
To Hofstadter, intellectualism is not at all the same as intelligence. It is a distinctive habit of mind and thought that actually forbids the kind of complete self-assurance we often associate with very smart people.
Let's examine Taleb's technique a little more closely. One thing he does while bashing theorists is recycle essential elements of that same theory in order to convince the layman he is clever. One such notion is convexity and it makes for an interesting study. Convexity is the "abstract" concept Taleb can't help but ruminate on endlessly - but is it actually a weapon against theory?
Hardly. To suggest that convexity is overlooked is beyond ludicrous. 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. Option pricing theory, which fills multiple encyclopedic length treatises, 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 Taleb's populist revisiting of Jensen's Inequality Jensens's inequality is a careful statement about the implications of convexity. This makes is quite useful - and conveys in one line more insight than we will find in a hundred populist finance books. It is 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).
Irony piles on irony when Taleb attempts 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 due to subordination of equity shareholders by debt.
Who pointed out in clear terms the convexity of stocks? Why that would be another target of Taleb, Robert Merton. The Merton model makes convexity of stocks obvious to all, except those who are too busy or too disparaging of theory to notice.
Taleb might be well advised to address the theory head on, not the man. Here he is disparaging Merton:
Here Taleb careens from anti-intellectualism to something far more sinister - but casual references to Asperger's (elsewhere autism) are so beyond the pale I can't help readers who are sympathetic to this style of argument. Shades of Hitler.
3. You can't add words
Let us dispense with these distractions and turn to the relatively simple logic of the situation at hand.
If you take some stocks that are "safe" and some that have "upside" and combine them in a portfolio then you will get... a pretty bad investment allocation actually.
Here I am going to gently push back on Taleb's reasoning and I hope in some small way the entire category of such arguments. These break down into the following categories:
- [Generalist] Optimization is dangerous as it leads to over-optimization
- [Anti-intellectualism] Harry Markowitz relied too much on Gaussian distributions.
- [English as algebra] Taking some "safe" stocks and some "upside" stocks yields a portfolio that is both safe and has upside.
I will treat these in the same order. A summary of the counter-argument is as follows:
1. The bar bell portfolio does optimize something, whether you realize it or not.
2. Defer to Richard Hoftstadter
3. If English natural language reasoning was in any way reliable we wouldn't need mathematics and by the way this example provides as good a case against it as you are likely to find (because 3 above seems quite reasonable to the lay ear)
1. A bar-bell bond portfolio optimizes (by accident)
For those who are technically inclined my scratch working demonstrates an interesting mathematical fact. A bar-bell portfolio of zero coupon bonds maximizes modified excess return.
If you've never heard of modified excess return, that's because nobody uses it nor has any particular reason to. I had to reverse engineer a strange quantity.
Whether or not you want to check my mathematics, it is often the case that attempts to avoid optimization for some philosophical reason or another, usually predicated on vague robustness arguments, are often poor logic because the heuristics end up optimizing something anyway. Moreover:
(a) The thing that gets accidentally optimized isn't necessarily sensible
(b) The person is unaware of this fact.
Ignorance isn't bliss.
2. Anti-intellectualism is not novel ... please knock it off.
The second category of argument is a less sympathetic error. It is hard to forgive Taleb for rolling out various anti-intellectual tropes including the false image of the academic whose intelligence is divorced from common sense - such as empirical observation. Taleb claims that academics have been blinded by the bell curve and are otherwise guilty of rigid thinking. This ignores the fact that most of his targets were seminal thinkers - the very opposite of rigidity - and were keenly aware of the assumptions they make (as compared with Taleb who, as pointed out above, is very obviously not aware of what he is accidentally optimizing when suggesting the bar bell).
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. Taleb's suggestion 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) is ridiculous, made more so by the paucity of his attempt at an alternative bar-bell approach.
But every now and then anti-intellectualism sells. The reasons are more complex than either of us probably appreciate. These were analyzed by Hofstader in his classic work Anti-intellectualism in American Life and have been debated every since. It always seems to simmer, waiting for a moment to emerge such as a financial crisis. Nicholas Lemann titled his review of Hofstadter's work "The Tea Party is Timeless", suggesting another manifestation. As Lemann points out, the difference between intelligence and intellect seems to slide by.
To Hofstadter, intellectualism is not at all the same as intelligence. It is a distinctive habit of mind and thought that actually forbids the kind of complete self-assurance we often associate with very smart people.
Now where have we seen assumptions about very smart people used in a cynical way recently? Here's a sample 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). But 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 unreasonable applicability of mathematics to the universe.
Aside: Dear cab driver. I'm involved in financial markets and I forget basic facts all the time. That's sort of okay because I studied mathematics and I can re-derive things as needed.
Aside: Dear cab driver. I'm involved in financial markets and I forget basic facts all the time. That's sort of okay because I studied mathematics and I can re-derive things as needed.
Let's examine Taleb's technique a little more closely. One thing he does while bashing theorists is recycle essential elements of that same theory in order to convince the layman he is clever. One such notion is convexity and it makes for an interesting study. Convexity is the "abstract" concept Taleb can't help but ruminate on endlessly - but is it actually a weapon against theory?
Hardly. To suggest that convexity is overlooked is beyond ludicrous. 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. Option pricing theory, which fills multiple encyclopedic length treatises, 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 Taleb's populist revisiting of Jensen's Inequality Jensens's inequality is a careful statement about the implications of convexity. This makes is quite useful - and conveys in one line more insight than we will find in a hundred populist finance books. It is 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).
Irony piles on irony when Taleb attempts 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 due to subordination of equity shareholders by debt.
Who pointed out in clear terms the convexity of stocks? Why that would be another target of Taleb, Robert Merton. The Merton model makes convexity of stocks obvious to all, except those who are too busy or too disparaging of theory to notice.
Taleb might be well advised to address the theory head on, not the man. Here he is disparaging Merton:
“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)
Here Taleb careens from anti-intellectualism to something far more sinister - but casual references to Asperger's (elsewhere autism) are so beyond the pale I can't help readers who are sympathetic to this style of argument. Shades of Hitler.
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 - having derived the most famous convexity equation in finance. The Black Scholes PDE and its solution, for which the Nobel was awarded, related the convexity of the value of an option (one side of the equation) to its time value (the other side). For Taleb, who has made no such contribution, to argue that he appreciates convexity more than the next guy is ... just a little silly.
Merton's capital model is one of those easily criticized gadgets because like many excellent models it is stylized. The bashing is a shame because even the existing, imperfect lens provides an assessment of funds buying out of the money put options (Taleb's suggestion). 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 in terms of value, as compared to debt insurance for example. (See Coval et al 2008). But I digress.
3. You can't add words
Let us dispense with these distractions and turn to the relatively simple logic of the situation at hand.
If you take some stocks that are "safe" and some that have "upside" and combine them in a portfolio then you will get... a pretty bad investment allocation actually.
But wait you say. Taleb's rationale is that if you park 85 percent of your wealth in cash then you can lose at most 15 percent. Isn't that a nice lower bound?
Not really. No.
Shall 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?! Don't spent it all at once.
Not really. No.
Shall 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?! Don't spent it all at once.
Taleb's bar-bell doesn't seem terribly safe to me. And sometimes it is just clearly wrong.
Here's a really simple counter-argument to bar-bell thinking. Consider a world with only three investment alternatives:
Here's a really simple counter-argument to bar-bell thinking. Consider a world with only three investment alternatives:
a) Cash.
b) A low fee 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?
Well no! That would clearly be worse than a portfolio using (a) and (b) only because the ratio of upside to fees is better and you can replicate (c) using (a) and (b).
Notice that we'd be clearly 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.
Admittedly this is just an example. But it isn't an unrealistic example and if the bar-bell heuristic fails us here why would we trust it elsewhere?
Maybe if we change the example so that the high risk option (c) is more attractive post fees we can save the bar-bell rule? Sadly no.
The bar-bell doesn't make sense even if risky bets are good value
Here are the two separate pieces of advice which have been rather successfully promulgated. Let's not conflate them:
- It is wise to bet on unexpected outcomes, because markets underprice them
- 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.
Thousands of fools to place their money on horses with large odds and have since the beginning of the bookmaking profession. 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 that's an aside.
Thousands of fools to place their money on horses with large odds and have since the beginning of the bookmaking profession. 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 that's an aside.
My opinion, or yours, as to whether longshots have positive expected return doesn't change anything. You still shouldn't use a bar-bell, and we shall prove it. Instead you should lean on the heuristic par excellence if you prefer heuristics to optimization: invest in everything.
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
Why every investment (within reason) should be in your portfolio
Outline of argument:
- A new, entirely independent investment opportunity should be included in your portfolio, unless it is a negative expectation bet.
- 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.
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.
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.
Invest in everything that is not a losing bet.
Ignore "heuristics" that run up against this rule - they run counter to marginal returns.
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.
The standard of pop-finance is so low. Mention bell curves a few times and people get all riled up - invariably people not distinguishing "least squares" or "minimum variance" from "gaussian".
As it happens this too is irrelevant. Equity returns are surprisingly close to gaussian but you don't need a Bell Curve to justify diversified portfolios. In fact you can lean on entirely distribution free approaches if you want to, such as Universal Portfolio Theory. If you are looking for improvements over Markowitz start with Stochastic Portfolio Theory by Bob Fernholz.
And if you really must try to diminish Markowitz' achievement, at least let him do it because he's much more knowledgable than Taleb and self-deprecating. Here's what I mean...
An historical aside
Taleb's attacks Harry Markowitz and in the process, fails to acknowledge prior work as we have noted. Markowitz on the other hand, had the good grace to give credit where it was due. Taleb's research is so bad he fails to notice this. He's swinging a punch at the wrong guy. To strange to be true?
Let's consult the Journal of Investment Management, 2006. There we find a paper titled "de Finetti scoops Markowitz". It suggests that Markowitz might not be the true originator of mean variance portfolio optimization. This ought to be gold for Taleb, or anyone looking to throw mud. It would give him ammunition for denigrating Markowitz' work. Too bad he didn't find it.
So who wrote this article? A quick check reveals that the author was ... Harry Markowitz, one and the same. 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.
The standard of pop-finance is so low. Mention bell curves a few times and people get all riled up - invariably people not distinguishing "least squares" or "minimum variance" from "gaussian".
As it happens this too is irrelevant. Equity returns are surprisingly close to gaussian but you don't need a Bell Curve to justify diversified portfolios. In fact you can lean on entirely distribution free approaches if you want to, such as Universal Portfolio Theory. If you are looking for improvements over Markowitz start with Stochastic Portfolio Theory by Bob Fernholz.
And if you really must try to diminish Markowitz' achievement, at least let him do it because he's much more knowledgable than Taleb and self-deprecating. Here's what I mean...
An historical aside
Taleb's attacks Harry Markowitz and in the process, fails to acknowledge prior work as we have noted. Markowitz on the other hand, had the good grace to give credit where it was due. Taleb's research is so bad he fails to notice this. He's swinging a punch at the wrong guy. To strange to be true?
Let's consult the Journal of Investment Management, 2006. There we find a paper titled "de Finetti scoops Markowitz". It suggests that Markowitz might not be the true originator of mean variance portfolio optimization. This ought to be gold for Taleb, or anyone looking to throw mud. It would give him ammunition for denigrating Markowitz' work. Too bad he didn't find it.
So who wrote this article? A quick check reveals that the author was ... Harry Markowitz, one and the same. 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.
Oh yeah, de Finetti really invented the concept of risk neutral pricing too ... but by all means believe there were no intelligent thinkers in between Sextus Empiricus and Nassim Taleb if you must.
An optimization as simple as falling off a log
I'm not done.
I'm trying to find a way to convey to you my level of mathematical disgust I have for the lack of mathematical intuition that is required to even suggest that the bar-bell is a good idea. In my humble opinion this belies a terrible lack of geometric ability of any kind. Perhaps an example will get towards why this so deeply offends my mathematical sensitivities.
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?
An optimization as simple as falling off a log
I'm not done.
I'm trying to find a way to convey to you my level of mathematical disgust I have for the lack of mathematical intuition that is required to even suggest that the bar-bell is a good idea. In my humble opinion this belies a terrible lack of geometric ability of any kind. Perhaps an example will get towards why this so deeply offends my mathematical sensitivities.
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 (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.
Do you see why I am upset now?
It is, I claim, geometrically obvious the bar-bell idea is bad advice. Moreover anything in a much broader category of similar advice is equally obviously bad because the boundaries between securities are imagined. To believe in the bar-bell heuristic is akin to believing that the Black Death cannot cross a national border.
To see why let's quickly solve the toy problem by thinking about allocating our wealth to the outcomes, not the securities themselves. 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. 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, as already noted. 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.
And it is also beside the point, as already noted. 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 (ahem, Taleb) 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.
To be continued .... the final nail in the coffin
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.
[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.)
