The Crystal Ball

Nassim Taleb, in his books The Black Swan and Fooled by Randomness, argues vehemently against predictions as they can be dangerous, blinding us to other outcomes, having us make decisions based on poor (or no) information. In the 25-year special edition of The Economist a wonderful piece of irony unfolds. Taleb himself, unfathomably, makes some predictions for 2036. On page 90 there is an article Future imperfect which analyses the Economist’s record at making predictions. And for the grand finale, the magazine provides a set of financial forecasts for 2011 for various industries and countries.

I was surprised to find Taleb had written such an article. When at first I saw, quite accidentally during a Google search, that someone claimed he had made such predictions, I thought it was a joke. But there the article is, in the The Economist. I am still not convinced that perhaps Taleb is not the one playing the prank on The Economist and its gullible readers.

The first article mentions Taleb; forgets to mention Taleb would have abhorred all the predictions mentioned (whether they came true or not). The set of 2011 figures are the kinds of things Taleb most wants us not to predict: GDP and inflation (population is also included, but this is arguably more predictable). The errors in such predictions are large. What use are the numbers if we can place no confidence in them?

Taleb’s predictions are

1.       City-states will prevail rather than nation-states

2.       Currencies will either not exist or be pegged to a standard

3.       No more large, debt-laden, listed companies

4.       Most technologies more than 25 years old will survive and most of the younger ones will be dead

5.       There will be biological and electronic pandemics

6.       Religion will experience a revival

7.       Science will produce few gains in non-linear domains

8.       Academic economics will be regarded with disdain

This is a very eclectic mix of predictions, is it not? As with many predictions, I suspect it is really just a wishlist (with the possible exception of point 5). From this angle point 6 is very interesting. Taleb’s books are secular and he has remained very obscure regarding his feelings on religion. Would he like a revival? I would. But that is drifting off topic.

Taleb bases these predictions on the premise that one will expect a system that relies too heavily on prediction to break eventually and 25 years is a long enough time. I agree with TechnoOccult that the link between this premise and the predictions is threadbare at best. I would very much like to know the logic Taleb is using. I am worried that he has lost his objectivity. A podcast I heard a few months ago contained a ranting Taleb with little thought that he might not always be right.

The future is filled with black swans, by definition unpredictable. Every one of these predictions could, in principle, be thrown off by a well-placed black swan. Is this not what Taleb warned us against? We are not just talking about a system breaking; we are speculating about how and what will take its place. Perhaps Taleb is becoming overconfident, fooling himself into thinking that he is cultivating his deep-seated insecurity, while making ever bolder ideological statements. I would be truly sad if this is the case. Of course we need to wait 25 years to see if he is right, by which time we might be eulogising him.

Some references

Taleb’s books:

• Taleb, N.N., 2007a. Fooled by Randomness 2nd ed., Penguin.
• Taleb, N.N., 2007b. The Black Swan, Penguin

The TechnoOccult article (which for some reason only mentions five predictions) that alerted me to all this:

• Finley, K., 2010. 5 Predictions from Nassim Taleb for 2036. TechnoOccult. Available at: http://technoccult.net/archives/2010/11/29/5-predictions-from-nassim-taleb-for-2036/.

The podcast I mentioned:

• EconTalk, 2010. Taleb on Black Swans, Fragility and Mistakes. Library of Economics and Liberty. Available at: http://www.econtalk.org/archives/2010/05/taleb_on_black_1.html.

Power laws: Mandelbrotian is the new Gaussian?

Benoit Mandelbrot, the pioneer of fractals, died last year. I only found out months later. His death should have been announced as a great tragedy to the whole world. Mandelbrot I also only recently found out also made contributions to financial mathematics. I would like to discuss one aspect, which may not be original Mandelbrot, but which Taleb names after him.

The old

It has been well publicised that normal distributions do not work. At least they do not work for modelling financial markets. It is my impression that, despite knowing this, everybody continued to use them. Because the quants, the people who were supposed to ‘know’ said it was okay, the normal distribution became a kind of placebo.

These methods have, however, always had their critics. Taleb (who wrote Fooled by Randomness and The Black Swan) is one of them. The latest financial crisis proved them right. But still, the normal distribution is what everybody knows, and I think it will take a while before ‘better’ practices become widespread.

The new

One alternative to normal distributions that has been suggested, and which Taleb argues we should use, albeit with caution, is that of power laws. He calls such distributions Mandelbrotian, to challenge the conventional Gaussian. Power laws have, of course, been around for a very long time (they are hardly new). Zipf’s law is a specific example in the field of linguistics that was proposed in 1935 and the specific example we will be looking at was researched even earlier. I am not quite sure how much work on power laws has been done in terms of financial markets nor am I certain exactly what Mandelbrot’s contribution was.

Power laws are interesting creatures, showing themselves seemingly everywhere. They occur in nature where they give the frequency of different sizes of earthquakes, of the sizes of craters on the moon and the extinctions of species. These are some examples mentioned by John Gribbin in his book Deep Simplicity.

Things such as the distribution of city sizes and the disparity of wealth also seem to follow power law distributions. The ubiquity of this distribution not only in nature, but in human behaviour, makes it seem like some kind of universal law (which we cannot help but obey). Such an interpretation should be avoided of course. With interest in power laws high, you are more likely to hear about confirming evidence (this is what people will be looking for). Eventually we will find important examples that are not governed by power laws or we may find the power law distribution is a poorer estimate than some other law.

Definition

There is a general definition of a power law distribution, but I will merely define a special case, the Pareto distribution which has the distribution function

with support x > k and a > 0. There is a more general definition for which some of the properties I discuss below would only hold asymptotically. The Pareto distribution is also the source of the infamous 80-20 or Pareto law (not really a law). I have yet to establish exactly how this was derived but watch this space.

Why power laws are cool

Self-similarity

They display self-similarity. Self-similarity is a property of fractals (remember that Mandelbrot is famous for his work on fractals). Taleb calls these distributions scalable (as opposed to the normal distribution, which is not scalable). It follows from the following property:

So, for example, if a = 1, doubling x always lowers the probability of seeing a higher value by half. This also demonstrated by the fact that if you fix a new minimum wealth level you get a Pareto distribution with the same exponent:

More concretely, if X represented wealth levels, the inequality among the rich would be the same as that among the poor. So if your wealth is x_m, half the population will have twice that wealth, and half of them four times that and so on.

Infinite moments

For certain (common) cases they have infinite variance or even infinite mean. For a <= 2, the variance is infinite. For a <= 1, so is the mean. In these cases we will not be able to rely on the Central Limit Theorem (or the Law of Large numbers in the second case). This makes the maths harder and moves us irrevocably away from normality.

Fat tails

They have fat tails. This means that extreme values are more likely than for the normal distribution. The structure of the tails is given by the self similarity of the distribution as shown earlier. For the normal distribution doubling the wealth level would lower the probability of higher observations by much more than half and much more rapidly further up the tail we go.

  

Estimation difficulties

They are hard to fit. Determining the exponent to use is apparently not easy and the resulting distribution is very sensitive to use. Taleb gives a good illustration.

Can they help us?

The danger is that everyone will jump on the new bandwagon. I can see the headlines twenty years from now. Math whizzes cause greatest financial crash in history. Math models fail us again. These Mandelbrotian distributions are just another fraud. Taleb warns that we cannot model everything, that even appropriate mathematics is of but limited use. In any event we should use models with caution. I am very interested in how far we can take them. I am definitely not yet done with power laws. Expect more posts on this topic.

Some references

Gribbins links power laws and chaos theory (both linked to fractals):

• Gribbin, J., 2004. Deep Simplicity, New York: Random House, Inc.

 Taleb gives some useful examples and a mathless discussion:

• Taleb, N.N., 2007. The Black Swan, Penguin.

 From Wikipedia¹: 

• Wikipedia, 2011a. Pareto distribution. Wikipedia. Available at: http://en.wikipedia.org/wiki/Pareto_distribution.
• Wikipedia, 2011b. Pareto principle. Wikipedia.
• Wikipedia, 2011c. Power law. Wikipedia. Available at: http://en.wikipedia.org/wiki/Power_law_distribution .
• Wikipedia, 2011d. Zipf’s law. Wikipedia. Available at: http://en.wikipedia.org/wiki/Zipf’s_law.

1 Yes I am going to shamelessly reference Wikipedia. It is useful and usually accurate. This is a blog, not an academic article. I am aiming for clarity of thought rather than accuracy of assumptions.

The inefficiency of Christmas

It’s that time of year. Again (*sigh*). Though I do not restrict my celebration of Jesus Christ to Christmas time, I find it quite appropriate that my first post should relate to his coming (I am a Christian, after all). In many ways though, Christmas annoys me (those hackneyed carols that start playing in August already, for example). I would, however, like to mention some of the financial asymmetries, if you will, that Christmas causes.

The Christmas rush
It is an anecdotal fact that there is a spurt of buying (perhaps from around November) leading up to Christmas (then followed by a lull). It follows from elementary economic principles that prices should increase at this time. There is social pressure to buy Christmas gifts and few seem to have the foresight to buy well ahead of time. So we are all suckers. This surge in demand is (at least partly) fuelled by the Christmas bonus giving people additional disposable income (although the original causality may well lie in the opposite direction).

This rush from consumers causes a rush from businesses that need to increase their staffing levels around Christmas (and may need to pay higher wages), so unemployment may be somewhat less at Christmas time. But who only wants to be employed once a year? Some businesses (notably some selling only Christmas accessories such as tree decorations for which there appear to be no demand at any other time) are only open before Christmas. They must find floor space, which may be unused (or underutilized) during the rest of the year. Business retailing systems (software, equipment, etc.) must be configured to be able to handle peak demand (not average demand), which probably occurs around Christmas and can be substantially higher than average demand. This makes for a (mostly) underutilized system that costs more than one that only needs to cater for something just higher than average demand (were sales not skewed in this manner).

The January lull
The counter to the Christmas rush is the drop in spending in January (in fact it has often been related to me that pawnshops are filled as people sell items they bought with their Christmas bonuses that they now find they can no longer afford). Naturally, prices should fall. The ‘rational’ (the reason for the quotation marks will be explained later) consumer would not buy at Christmas time, but during January, when prices are low. Incidentally, a friend of mine told me he has thought about trying to move his family’s Christmas a week later to capitalise on this (his comment crystallised the idea for this post – which had been in my head some time – and got me to start writing). I do not think he will succeed. The social pressure is too strong. It is this social pressure that stops the expected economic equilibrium wherein sales in January and December are even (aside from some randomness) from occurring. If people were ‘rational’ (homo economicus) more and more people would buy in January rather than in December, until the demand in both months were the same.

It is interesting to note that the Canadian article I mention below suggests that some shift of sales toward January is taking place (at least in that country), but the drop is still quite high and I personally need more evidence. They think the reason for the shift is the use of gift cards (like cash, but can only be spent at certain places). Gift cards are usually redeemed in January. This is one very interesting possible way of fooling the human psyche, which needs a gift in December, but bringing about an equilibrium outcome by moving actual expenditure to after Christmas.

Useless gifts
In my undergraduate economics class I was presented with the difference between gifts in kind (the ones we wrap) and gifts in cash. From a purely economic standpoint the latter is better (not least because money and effort is not wasted on ultimately useless wrapping paper). Cash can become anything. That is it can be what the person needs or wants (that which maximises his utility). Yet we keep giving presents, even though half the time they end up unused or are rewrapped and end up under another Christmas tree. Presents are thoughtful, considerate. They have intrigue and suspense. They prevent people from buying dishwashing liquid with their cash gift (however, I suspect most people compartmentalise gift money and set it aside for luxuries even though this is not rational either).

The market slow
Market activity slows down in the holiday season around Christmas (at least it does in South Africa). This means the market behaves differently at this time. It becomes less liquid, possibly less efficient. I have spoken to a hedge fund manager who says no one seems to work in the holiday period. I have not examined any studies on the matter, however, and it is not clear how significant the effect is. It is interesting to note that (in America at least) markets tend to rise the most in January. This is known as the January effect. Perhaps there is a connection with the market slow (but I have not seen it yet).

Is it rational?
Are these ‘aberrations’ in our behaviour rational? There are certainly good psychological reasons for why we act this way. I am not a psychologist (though I am very interested in what motivates people) so I will not discuss that any further. But I will say that it seems rationality is not a concept with a well-defined (or well-accepted definition) – this is why I used quotation marks earlier. And in our treatment of rationality we always try to determine what should be (normative) rather than what is (positive). All I can say is that I can imagine another way things could be and try to understand why reality differs.

Why do I care?
Firstly I am curious and these patterns of behaviour are interesting. Secondly, they may affect my work. That is, they make modelling and statistical analysis more complex and treacherous. It is not really possible to compare a set of (say, half-yearly) financial statements of any company that contains December with those that do not. The spike in spending (and the lull later) is an example of a seasonal pattern in a time series. I know there are statistical methods for removing seasonality (I have not used them), but it would make statisticians across the world happier if they did not have to do so. Because, essentially, distributions differ at different times of the year one also needs more observations for reliable inference (making more assumptions could lower the observations needed, but the point is there is added effort).

Furthermore the different behaviour of the market during the holiday season means the same trading rules may not work. Prices may need to be analysed with different reference points. It may not even be possible to get useful work done, which means money is lost. The hedge fund manager, earlier, seemed to think there was little to be gained from trading in the holiday season (though I think he also liked the excuse to take a holiday).

Final thoughts
The same things (to a lesser degree) affect other widely and collectively observed occasions, such as Valentine’s day, Saint Patrick’s day and public holidays. Birthdays are, of course, an exception as they are spread throughout the year (and I can think of no good reason why this spread should not be approximately uniform). Things would certainly be easier, more efficient, and more ‘objectively’ rational were Christmas more like birthdays. However, Christmas cannot practicably be distributed in this manner. Each person cannot (or rather does not want to) have their own Christmas date. Leap years are, of course, another pesky aberration that confuse the amount of time that should be assigned to any year in modelling or statistical exercises.

The ‘problem’ could be fixed if people stopped giving gifts on Christmas (including 'gifts' to themselves). My family stopped years ago. We still enjoy a good (too big) meal and each others’ company. The gifts are superfluous. They are only a symbol, after all, of Christ’s gift and that is not something we should only celebrate once a year. Of course, doing away with religion is another possible ‘solution’ (assuming Christmas is not already too deeply entrenched even in secular society), but talk about throwing the baby out with the bathwater.

Some references

An article on the January effect.

• Harford, T. (2007). Why the Stock Market Rises in January …. Slate. Retrieved from http://www.slate.com/id/2156844/

For a discussion on rationality and the difference between positive and normative economics (among many other things). You could also read an economics textbook, but that would be less fun.

• Taleb, N. N. (2007). Fooled by Randomness (2nd ed.). Penguin.

A Canadian article on shopping patterns.

• Zhang, Y. (2003). Consumer holiday shopping patterns. Retrieved from http://www.statcan.gc.ca/pub/11-621-m/11-621-m2004019-eng.htm