# Long and fat tails

I don’t read much business journalism, but I do generally like Joe Nocera’s writing in the New York Times.  I thought his long article on risk management in last Sunday’s Times Magazine was quite good.  In particular, I love seeing an article in a nontechnical magazine that actually talks about probability distributions in a reasonably careful and detailed way.  If you want to understand how to think about data, you’ve got to learn to love thinking about probabilities.

The hero, so to speak, of the article is Nassim Nicholas Taleb, author of the book The Black Swan.  According to the article, he’s been trying to convince people of the inadequacies of a standard risk-assessment method for many years, mostly because the method doesn’t pay attention to the tails of the probability distribution.  The standard method, called VaR, is meant to give a sort of 99% confidence estimate of how much money a given set of investments is at risk of losing.  Taleb’s point, apparently, is that even if that’s right it doesn’t do you all that much good, because what really matters for risk assessment is how bad the other 1% can be.

That’s an important point, which comes up in other areas too.  When people compare health insurance plans, they look at things like copays, which tell you what’ll happen under more or less routine circumstances.  It seems to me that by far the more important think to pay attention to in a health plan is how things will play out in the unlikely event that you need, say, a heart transplant followed by a lifetime of anti-rejection drugs.  (In fact, extreme events like these are the reason you want insurance at all: if all you were going to need was routine checkups and the like, you’d be better off going without insurance, investing what you would have spent on premiums, and paying out of pocket.)

I do have a couple of observations about the article:

1. Nocera, quoting Taleb, keeps referring to the problem as the “fat tails” of the probability distribution.  I think he means “long tails,” though, which is pretty much the opposite.  A probability distribution with fat tails would be one in which moderately extreme outcomes were more likely than you might have expected.  That wouldn’t be so bad.  A distribution with long tails is one in which very very extreme outcomes have non-negligible probabilities.  The impression I get from the rest of the article is that this is the problem that Taleb claims  is the source of our woes.

2. The article is strong on diagnosis, but I wish it had said more about treatment.  Given that standard methods don’t handle the tails very well, what should risk-management types do about it?  I fear that the article might leave people with the impression that there’s no way to fix methods like VaR to take better account of the tails, and that we shouldn’t even bother trying to quantify risk with probability theory.  That’s certainly not the correct conclusion: There’s no coherent way to think about risk except by modeling probability distributions.  If the existing models aren’t good enough, we need better models.  I hope that smart quants are thinking about this, not just giving up.

3. Taleb seems like he’s probably very smart and is certainly not a nice guy.

### Ted Bunn

I am chair of the physics department at the University of Richmond. In addition to teaching a variety of undergraduate physics courses, I work on a variety of research projects in cosmology, the study of the origin, structure, and evolution of the Universe. University of Richmond undergraduates are involved in all aspects of this research. If you want to know more about my research, ask me!

## 8 thoughts on “Long and fat tails”

1. Here’s a recommendation: http://examinedlife.typepad.com/johnbelle/2009/01/pretend-i-said-something-funny-about-sir-mixalot-and-fat-tails.html . In the comments it becomes clear that VAR is not so useless a measure as people are now making it out to be.

Also, I wasn’t aware of the distinction you describe between long and fat tails. I thought only the terms “fat” and “heavy” tails were used, to indicate the rare-but-more-frequent-than-expected part of the distribution.

The “Long Tail”, on the other hand, is now used by Web2.0 proponents to indicate the exact opposite of the distribution, the most frequent events. The confusion arises because Anderson, in his Wired Long Tail article, used a graph with the random variable (popularity of stuff) on the y-axis. If you switch the x and y axes like that (with respect to normal histograms) the “tail” becomes the other side of the distribution of course.

2. The whole long vs. fat thing is probably just me being obtuse. Here’s how it seems to me. A “tail”, more or less by definition, is the part of a probability distribution containing a given small percentage of the total probability. So on the usual sort of a graph, the area of the tail is fixed. If one probability distribution has a “fatter” tail than the other, then to keep the area conserved it has to be shorter, and vice versa.

3. Rex says:

Ted,

I believe that your explanation is on point. As I understand it, “fat tails”, “heavy tails” and “long tails” all relate to kurtosis and, in most instances, are used to describe leptokurtic distributions. As you state, the first two are synonymous and simply mean that there are more data points in the tails of your distribution than you would expect to find in a normal distribution. By comparison, “long tails” mean that you have a few rare data points that are WAY OUT THERE–for instance, a couple of data points that might be 4 or more standard deviations from the mean. So, your tails might get “fat” or “heavy” at around 2 to 3 standard deviations from the mean, and they also might be “long” if they contain values that are 4 or more standard deviations from the mean. Or, they might just be “long” because they contain a few extreme values, but not enough of them to be “fat”.

By way of simple example, let’s say that you have 100 daily data points in a time series. You then calculate the average and standard deviation of the 100 data points. In a normal distribution, you would expect to find 5% of the data points more than +/- 1.96 standard deviations from the mean. If the skew is 0, that means 2.5% beyond the + and – 1.96 tails. If the distribution is skewed, all 5% might be in one tail. If we had 10% of our data points more than +/- 1.96 standard deviations from the mean, we’d have “fat” or “heavy” tails. If one of those data points was also -4 standard deviations from the man, and one was +4 standard deviations from the mean, we’d also have “long” tails.

Regards,

Dave

4. Life indemnity is usually a wager, between you and the insurance company, if you’re going to snuff it or not. You only win when you snuff it.

5. This is definitely a blog worth following. Youve got a great deal to say about this subject, and so much knowledge. I think that you know how to make people listen to what you have to say, especially with an issue thats so important. Im glad to know this blog. Two big thumbs up, man!

6. I’d have to say risk management is huge. Successful investors are always the ones with excellent risk management to steer the risk more in their direction, and not just float the cloud of luck, because that could won’t last long. Just like share trading, you must protect your losses somewhat and not just throw mountains of money on a stock and pray it goes up.

7. s says:

In a sense I agree with you, but we can go further in thinking.

8. Fantastic points you’ve made here with this article. I read it several times in order to ensure I’d been reading you correctly. I value your position and I shall be guaranteed to check back here.