In statistics class, everyone learns about the bell curve and normal distributions.  Understanding the concept of standard deviation is essential when using the bell curve. According to Investopedia, standard deviation is “a measure that is used to quantify the amount of variation or dispersion of a set of data values.”  A layman’s definition according to MathIsFun is “a measure of how spread out numbers are.”

The normal-shaped curve depicted below simply shows that most values fall in the large part of the curve with half of the deviations on the left side of the curve and the other half on the right. The bell curve can be a useful tool in business. For example: suppose you make women’s shoes and need to calculate how many shoes of each size to produce. The average women’s shoe size in America is 8 with a standard deviation of around 1.5.  The bell curve tells you that 68.5% of women’s shoes should be sized between 6 ½ and 9 ½. Because you now know that most of your production should be making shoes between 6 ½ and 9 ½, you shouldn’t spend too many of your resources making shoes sized smaller than 3 ½ or larger than 12 ½ because only .3% of women need shoes that size.   

Unfortunately, sometimes the bell curve is used for analysis when the data doesn’t fit exactly within the curve. The conventional wisdom of investing uses the bell curve to give investors an expectation about volatility in their portfolio.  Using a hypothetical example, assume an investment averages a 10% annual return with a standard deviation of 15%.  The bell curve tells us that 68.5% of the time, the returns of this investment will fall within one standard deviation, or between 25% and -5%. Also, if investment returns follow a normal distribution, two standard deviations mean that 95.4% of the time, the annual returns for this investment will be somewhere between 40% and -20%. The problem with using the bell curve with investment returns it that the “fat tails” shown in the diagram are those outlier events, and these tails are much larger with respect to financial markets than the normal bell curve would lead you to believe.  This is called “tail risk” in investing because, in the real world, the risk of a rare event occurring happens more often than the bell curve would suggest.  

Benoit Mandelbrot describes this problem in his book “The Misbehavior of Markets.” He points out that by analyzing stock market return data from 1916-2003, the bell curve tells us that there should be 58 days when the Dow Jones Industrial Average moved more than 3.4%. Daily moves in the Dow which were larger than 3.4% occurred 1001 times during that period.  Also, the bell curve tells us that severe daily swings are almost impossible. The Dow should have experienced a daily swing of greater than 7% once every 300,000 years. It has happened 48 times from 1916-2003. Where would Black Monday, October 19, 1987 when the Dow dropped 22.6%, fall on the bell curve? It wouldn’t. The occurrence of Black Monday is statistically impossible. The number used to measure the odds of that event does not exist in mathematics. While we may never see an event like Black Monday again, the point is that it’s difficult to believe that any investor living through that day said, “I feel okay losing more than 1/5th of my net worth because today’s events are statistically impossible.”

If you and/or your financial advisor have built a portfolio using the expectations of the bell curve, the odds of significant volatility in the market are more common than you might be expecting.  How do you protect yourself from that volatility?  Contact us at or 214-922-9200 and we will show you how we do it.