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Understanding RISK: Standard Deviation, Skew, & Kurtosis explained

June 27, 2005

 

We constantly talk about it in this newsletter and we constantly hear about it. What is it? RISK. No, not the board game...but rather the danger of loss inherent in any type of investment. Classic investment strategy tells us that the returns we receive from an investment are the trade off we get for taking on the risk of that investment. But how well do we really understand the concept and how it will affect the hard earned dollars, Euro, or yen we have invested?

Putting the sometimes difficult to understand statistical measures of risk into actual dollar amounts relevant to your investment in a trading system or managed futures product is essential. What exactly does a 20% annualized standard deviation mean, in dollar terms, for a $250,000 investment? Once an investor gets a better understanding of risk and can answer questions such as the preceding one, her chances of long term success are improved dramatically.

Why do you need to know what the risk is in dollar terms? For some reason, most investors go into an investment looking at percentages, but start looking at dollar amounts when and if the going gets tough. I can't tell you how many times I have seen an investor get into a trading system saying he can handle a 25% drawdown, but lose confidence in the approach once he loses $25,000 on a $100,000 investment. For the mathematically challenged: 25% of $100K = $25K. It's the same risk, yet people tend to "dollarize" on the downside, feeling the pain of -$25K a lot more than -25%.

So how do we get a handle on how big the dollar swing in our account will be? The first step is to look at that ubiquitous piece of statistics: standard deviation. Books have been written trying to explain standard deviation, but we'll take a crack at a short and concise definition here.

Standard Deviation:

One of the easiest ways to think of standard deviation is as a sort of expanded measure of an average. We all understand averages: The average man is 5'9", an average temperature 65 degrees, an average monthly return 1.3%, and so on. But despite the average man being 5'9" - we often see 6'2 and 5'4" men. In fact, the average roughly means that half of men will be shorter than 5'9" and half will be taller.

We rarely, if ever - see men 7' tall men, however - and the rareness of seeing a 7' or 4' man is the beginning of understanding standard deviation. The standard deviation tells us how rare or frequent a sighting of a 7' man would be. If we enlisted a stewardess - who sees thousands of people every day in airports, hotels, and on the plane - to keep a log of the height of every person she encounters every day, then plotted the results on a graph, with the x -axis being the height and the y axis being the number of times each height was seen, we would get the following graph.

 

 

In our example, there would have been many, man sightings of men 5'9" tall, many sightings (but a little less) of men 1" taller and 1" shorter than the average, many sightings (but again lower) of men 2" shorter and 2" taller, and so on until you get to a point (1 foot taller or shorter, for example) where there are significantly less sightings of people that much taller or that shorter than the average man's height. This level where the number of observations starts to tail off significantly is what standard deviation tells us.

For our example, the standard deviation of the height of men is 3". That tells us that the grand majority of men will be 3" taller or shorter than the average (5'9"). In fact, in a "normal distribution", which is just a fancy name for the bell shaped curve above, the math works out to tell us that:

• 65% of all American men will be between 5'6" and 6'0" in height.
• 95% of all American men will be between 5'3" and 6'3" in height.
• 99.7% of all American men will be between 5'0" and 6'6" in height.

The implications of the standard deviation statistic for investments and measuring risk should now be obvious. If we know the average return, and standard deviation of the returns for a particular investment, we can can see where 99.7% of returns should fall. Many investors therefore feel comfortable looking at the standard deviation of returns to gauge how much risk is in an investment.

Take the Compass trading system as an example. Compass has an average monthly return of 3.87% on an initial investment of $30,000, and a standard deviation of monthly returns of 15.33%. Using this data, we can calculate that 65% of monthly returns should be between -$3,569 and +$5,625. We can also calculate that the daily standard deviation of returns is 3.43%, or $1,028; meaning that 65% of days your account should swing one way or the other by about $1,000, and over 99% of days will swing no lower than -$3,000.

In its simplest form, this concept of "dollarizing" risk based on the standard deviation has its merits. For example, we can see in the table below that R-Mesa with a monthly standard deviation of about $6,500, will see monthly swings nearly three times greater than Eclipse eRL, despite both using a $30,000 capital base. Big swings are not necessarily a bad thing, as they can be positive swings - but investors should be aware that they could be negative swings. R-Mesa surprised many investors last month when it lost over $8,000 per full size contract last month - but some research into the system's standard deviation of monthly returns tells us that the loss was rather normal - falling about 1.5 standard deviations below the average monthly return.

Ziad Chahal, the ever diligent developer of Brix, Tzar, and other systems, shows the following chart on his website explaining how different daily standard deviations translate into possible drawdown numbers.

Problems with Standard Deviation:

There is one very big flaw in using the standard deviation as the basis for measuring investment risk. The standard deviation probabilities (i.e - 65% of observations should be xx or higher, 99% should be below yy) assume that the data is distributed "normally", meaning there is an equal number of observations above and below the mean, and that the number of observation decreases at an increasing rate the further away from the mean you get.

The problem is, not everything (and especially not financial investments) is distributed normally. Take the Black Monday crash of October 1987 as an example. If you had calculated a standard deviation of daily returns using the 10 years worth of data prior to Oct. 18, 1987, you would have concluded, among other things, that there was a one-in-one-trillion chance that the index would fall by more than 6% on the following Monday (or any future day, for that matter). Well, the S&P 500 plunged by more than 20% in a single session the next Monday, and one week later it fell another 8.3% proving once and for all that the stock market does not follow a normal distribution.

Skew & Kurtosis

Because the standard deviation can mask the risk of extreme events such as Black Monday, statisticians use two other measures called skew and kurtosis to view investment returns. Each of these measures tells us how different the distribution we're looking at (the investment returns) is from a normal distribution. The more it differs from a normal distribution, the less the importance of the standard deviation statistic.

The skew tells us whether the investment returns are shifted one way or another from the mean. A normal distribution has a skew of 0. A positive skew is a good thing in investing, as it tells us there is a greater number of returns greater than the average than there are returns less than the average. Of course, if there's a negative average where a few very big losing months pull down the average of a bunch of small positive months, a positive skew might not help you.

In contrast, you can see in the table below that two systems have negative skews. This doesn't mean they are bad systems, rather it tells us they have more months below the monthly average than they do above the monthly average. If the monthly average is still positive, however, this is usually just a case of a system having a few very big winning months and a lot of very small winning months that are below its average monthly gain.

Kurtosis tells us whether or not there is evidence of extreme values happening with greater frequency than we would expect from a normal distribution. A normal distribution has a kurtosis of 0. A large positive reading of kurtosis tells us there are more occurrences of outlier events than we would expect from a normal distribution. Going back to our example, if we got on a plane with a basketball team on board, our graph of heights would show an abnormally large number of readings several standard deviations away form the average. In the investing world, we often label returns with a positive kurtosis as having "fat tails" - referring to the "tails" or end of the curves on the bell curve graph.

"Fat tails" are troublesome for investors, as they are created by outlier events such as the -21% down day in US stocks in October of 1987. But a positive kurtosis reading isn't necessarily a bad thing, as it can be brought on by an abnormally high number of very large positive months. Compass, for example, has one of the highest kurtosis readings in the table below. However, the Compass systems had a +73% winning month in July of 2002 which was over 3 times greater than any other winning month. If you take that month out of its returns, the kurtosis reading is actually -1.05.

So before you get involved with the next hedge fund, trading system, or CTA program — make sure you look at the standard deviation, skew, and kurtosis to get a better feeling for the risks involved. At the very least, you will know what kind of swings to expect in a day, month, or year — and with a little extra work you can know whether or not to expect more returns greater than the average and whether there may be some extreme values on the horizon.

- Jeff Malec