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The Two Sigma Dow01/06/09 09:03:19 AM
by Mike Carr, CMT
Standard deviations are a common tool used in financial analysis as a measurement of risk. It is the mathematical principle behind Bollinger bands, a widely used technical indicator.
|To calculate Bollinger bands, technical analysts find the 20-period moving average of price. Although any number of periods can be used, 20 is the default value in almost all software packages. It can also be calculated for any time frame, from minutes to years, so that is why we refer to periods in this description of how to calculate the indicator. After finding the moving average, two standard deviations of that average are calculated, and the three lines are commonly drawn on a chart along with the stock price (Figure 1).|
|FIGURE 1: DJIA, MONTHLY, AND BOLLINGER BANDS. The green line shows the level where prices would be two standard deviations greater than their recent average. The red line indicates the point where they would be two standard deviations below the blue line, which is the average price for the last 20 months. The two ovals indicate where prices have had sharp declines in 2000–02 and in 2008.|
|Graphic provided by: Trade Navigator.|
|The standard deviation has some interesting mathematical properties. One is that two standard deviations should contain approximately 95% of the price movement. This is based upon the assumption that stock prices follow a statistical model known as the standard distribution. The details of this model are not critical, but it is important to understand that there is an underlying theory supporting the use of standard deviations as a proxy for risk.|
|The term "sigma" describes the area outside the standard deviation. Two sigma means an event that is two standard deviations away from the average. Bollinger bands are thus useful to identify two sigma events. They offer a quick way to visually assess when these rare events occur. Prices should be expected to close higher than the upper band about 2.5% of the time and lower than the bottom band about 2.5% of the time. The question we ask is what happens after the markets experience these rare events. The answer is surprising, whether prices are in the upper or lower area of unusual performance, that prices are generally lower over the next few months.|
|We used the Dow Jones Industrial Average (DJIA) to conduct a test because it has the longest available history. Looking at all closes from 1900 to 2008 gives us more than 1,200 months of data, and we should expect to see about 30 times that the average closed more than two standard deviations above or below its average price. We found there were 88 closes above the upper band, and 35 below the lower band — significantly more than expected but not surprising, since we know that the market moves more based upon human emotions rather than mathematical laws. The results are summarized in Figure 2. The table shows the percentage of time the market closed higher some months after the unusual price action.|
|FIGURE 2: PERCENTAGE OF TIME THE MARKET CLOSED HIGHER AFTER CLOSING MORE THAN TWO STANDARD DEVIATIONS ABOVE OR BELOW ITS 20-MONTH AVERAGE. Closes above the Bollinger band seem to show exhaustion in the stock market and are followed by periods of generally declining prices. When the price falls below the lower Bollinger band, it seems to indicate that things are accelerating to the downside.|
|Graphic provided by: Trader Navigator.|
|Prices have now closed more than two standard deviations below average for four consecutive months. This is the longest such decline since 1903, when prices remained that oversold for six months. After breaking out of that slump, prices consolidated for six months before moving 50% higher in the following six months. If history is a guide, then we may see a period of consolidation in the stock market during the first half of 2009.|
Mike Carr, CMT
Mike Carr, CMT, is a member of the Market Technicians Association, and editor of the MTA's newsletter, Technically Speaking. He is also the author of "Smarter Investing in Any Economy: The Definitive Guide to Relative Strength Investing," and "Conquering the Divide: How to Use Economic Indicators to Catch Stock Market Trends."
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