Morningstar Advisor - February/March 2009 - (Page 31) Exhibit 3: It’s a Fat-Tailed World, After All A log-stable distribution does a good job of modeling the empirical returns of the S&P 500, especially at the center and the tails. 200 180 160 140 120 100 –29% 80 60 40 20 –29% –21% –13% –5% 3% 11% 19% 27% 35% 43% –25% –21% –17% –13% 13 5 10 Histogram shows the frequency of monthly returns for the S&P 500 from January 1926 to November 2008. center and the tails. In particular, note the close match between the density curve and the histogram between negative 13% and negative 29%. The tails of a stable distribution are so fat that its variance is infinite. In other words, the concepts of standard deviation and variance are not defined for stable distributions. You might find the idea of an infinite variance counterintuitive, because it is possible to calculate a standard deviation for any finite set of data. However, the underlying mathematical distributions that we use to model asset returns assign probabilities over the range from negative infinity to positive infinity.9 Some distributions that cover this infinite range assign so little probability out in the tails that variance can be defined. These are “thin-tailed” distributions, the normal or bell-shaped distribution being the best-known example. Other distributions assign so much probability to the tails that variance is infinite. Such is the case with stable distributions. The manner in which a stable distribution assigns probability to its tails is very close to what is known as “power law.” When a distribution of a loss follows a power law, a plot of logarithm of the magnitude of loss (x) versus the logarithm of the probability of the loss turning out to be x or worse is a downward-sloping straight line. Therefore, while the probability of loss decreases with the magnitude of loss, it does so gradually. In Exhibit 4, we plot the magnitude of loss versus the logarithm of the probability of Exhibit 4 Power Law Tails: Unlike a normal distribution, a stable distribution approaches the straight line of a power law, indicating that it has “fat tails.” In (Prob[X<x]) 2 Higher Probability –2 Power Law Stable –6 Normal Higher Loss –4.5 –4.0 –3.5 –3.0 –2.5 –2.0 –1.5 –1.0 In (|x|) 9 That is the probability distribution of one plus the return on an asset return in decimal form. The lowest possible return on an unleveled position in an asset is negative 100%, which is negative 1 in decimal form. Adding one we get 0. The logarithm of 0 is –∞. MorningstarAdvisor.com 31 http://www.MorningstarAdvisor.com
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