Tuesday, June 15, 2010

The Story of “New Normal” and the Fat Tail

Prior to 2007, few people had heard of Nassim Taleb and his Black Swan. Some people had heard of fat tails, but most were quant modelers and academics with seemingly boundless mathematical knowledge. Additionally, the predictability of the distribution curves was an easy sell when three standard deviations encompasses nearly all outcomes, four, five, and six standard deviations were hardly worth considering.

But the question that continues to arise is whether or not there is a "new normal" and, if so, is it more dangerous than the "old normal"? Will the "new normal" carry greater volatility and risk? The short answer is probably.

As we know, volatility can come in many forms but it is always descriptive of a range of outcomes. Whether it is implied, historical or expected, volatility is described by a percent annualized standard deviation. In other words, if over the last year a $100 stock traded with a 25% volatility, we can be reasonable sure that 68% of that time it traded between $75 and $125. Similarly, if we expect the stock to have a 25% range over the next year, we are expecting 68% of outcomes to fall between $75 and $125.

With the flash crash a recent memory and the near collapse of the financial system (the verdict is still out on that one, in my opinion) having occurred in the not so distant past, I asked myself, "Has there really a broad increase in volatility over the years or is it just a passing fancy?"

Obviously, I needed a lot of data to answer this question; fortunately, we have this thing called the internet. So I pulled the historical prices for the S&P 500 from Yahoo! from Jan 1962 to present day, put them on a spread sheet and went to work.

The first thing I thought was that if volatility is increasing we should see an increase in the standard deviation of daily percentage returns, meaning that overall the range of outcomes probable on a daily basis should expand. With over 12,000 data points, I decided some summary was in order so I broke the information down into annual groups.

Along the bottom of the chart there are the average daily percentage returns for each year. This is fairly constant with average returns in the +/- 1.5% range for each year.

The spiky reddish orange line is the standard deviation of the close to close changes for each year. Not unexpectedly, the largest increases in range occur in down years (2008, 2002, 1987, etc.). Additionally, there are higher highs and higher lows which are suggestive of a long term up trend. That trend can be seen easily with the overlay of the linear regression line (royal blue). Since 1962, the standard deviation of daily returns has more than doubled on an annual basis from .0575% to 1.25%. That's a fair sized increase and certainly evidence of more volatile markets. Remember, this is only close to close and does not take into account any increases in intraday volatility.

Now knowing the mean (average) daily return and standard deviation for each year gives us all the information we need to understand the annual distributions of returns for any given year. Unfortunately, that only applies to normal distributions and the stock market has never been so kind as to keep things simple. To really reinforce my thesis of increasing volatility I wanted to know more about the distributions.

This is about skew and kurtosis. These are descriptive statistics that provide you with additional information about the distribution in question. Briefly, skew can be positive or negative with the direction reflecting which side the fatter tail is on. Positive skew reflects a plethora of outcomes just below or to the left of the mean, with outliers, when the occur tending to be far right of the mean. Negative skew reflects the majority of outcomes just above or to the right of the mean, with outliers tending to occur at the extreme of negativity or far to the left of the mean. The broad stock markets tend to trade with a slight negative skew, with the majority of outcomes resulting in a small percentage gains, but the large moves, when they occur produce large daily losses.

Our other statistic, kurtosis, can also be positive or negative. Positive excess kurtosis indicates that the majority of outcomes are clustered around the mean with a number of both positive and negative outliers fattening the tails of the distribution. As a result, there are fewer median outcomes than would be found in a normal distribution or bell curve, a condition known as leptokurtosis. Negative excess kurtosis, or platykurtosis, is characterized by few outliers (thin tails) and a roughly even probability of outcomes around the mean. The broad markets tend to exhibit leptokurtosis.

My intention here was to determine whether over time the tails, and in particular the downside tail was increasing; were getting fatter by means of increasing kurtosis and decreasing (more negative) skew. However rather than examine each year individually, I grouped them together 6 years at a time. This was somewhat arbitrary, but it allowed for an easy to deal with 8 data sets.

Before we get into the results, take a look at the above chart for a little more understanding on skew and kurtosis. The period from May 1998 through May 2004 (red line) most resembles a normal distribution with the lowest skew and second lowest kurtosis of the 8 periods. It is compared to the April 1986 to April 1992 which had the highest kurtosis and most negative skew.

Hopefully you can see that the red hand drawn mean line virtually bisects the red distribution, which would be typical of a normal distribution. Whereas the blue median line shows the largest number of outcomes just to the right or the median with more bumps on the far left (negative skew). Additionally, the blue distribution is much narrower while also having more outliers (leptokurtosis). This chart isn't indicative of anything, but will hopefully help you better visualize the numbers in the table below.

Although I would not consider this conclusive, the four most recent periods (April 1986 to present) have a higher average excess kurtosis and a more negative skew compared to the prior 24 years. Obviously this is heavily influenced by the inclusion of the April 1986 to April 1992 data, which was unquestionably extreme.


Never the less, the table does add additional weight to the theory that increasing volatility with fatter tails is the new normal. I would feel remiss however if I didn't propose some fundamental explanation as to why this might occur.

First, globalization has increased correlations between countries, businesses, and other entities. As we saw with our own financial crisis and now the crisis in Europe, when things start going wrong they go wrong for more people. It is like a pebble falling off a cliff into a pool of water. It used to be that when something failed it would fall by itself, maybe dragging a few smaller rocks along as well causing some nominal ripples. Now, the little pebbles are chained to rocks which are chained to boulders, and the ripples are more like the waves caused when a chuck of glacier falls into the ocean. No matter the level of regulation, if you feel that helps, this interconnectedness is not going to change.

Second, electronic trading and penny increments have diminished liquidity, in my opinion. While I admit this is an arguable point, back in the day, when stocks were quoted in fractions, size would build up on the bid and the offer because they were relatively wide spreads. Even if there is a thousand bid and offered on a stock that is only a penny wide, it is much scarier to execute a large order of 10,000 shares or more. Before, you might find 10-15,000 up on a market that was $50 - $50.25, now with 1000 at each penny you are not sure where you will get the order filled and once the buying or selling begins, algorithms are likely to jump on board in the same direction putting additional pressure on the stock. Furthermore the same computers that are following the momentum are also canceling liquidity providing orders on the other side. This also is not going to change.

This is not a doom and gloom piece, this is about knowing and understanding that the markets are evolving. So keep those puts pumped, don't take relative calm for granted, and enjoy the glacial waves.

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