Morningstar Advisor - February/March 2009 - 30

Spotlight Exhibit 2: Cracks in the Bell Standard risk models assume S&P 500 returns follow a bell-shaped distribution, even though the index has experienced more than 10 declines of at least –13%. 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% 5 10 Histogram shows the frequency of monthly returns for the S&P 500 from January 1926 to November 2008. With these standard models, the primary measure of risk is standard deviation. If returns follow a normal distribution, the chance that a return would be more than three standard deviations below average would be a trivial 0.135%. Since January 1926, we have 996 months of stock market data; 0.135% of 996 is 1.34—that is, there should be only one or two occurrences of such event. But the record of the stock market tells a different story. The monthly returns of the S&P 500 have been more than three standard deviations below average 10 times since 1926. In other words, the standard models assign meaninglessly small probabilities to extreme events that occur five to 10 times more than the models predict. We can illustrate the problem further by overlaying a lognormal model of returns over a histogram of monthly total returns on the S&P 500 (Exhibit 2). The model says that declines of more than negative 13% have almost no chance of happening—yet they have occurred at least 10 times since 1926. An Alternative Approach: Log-Stable Distributions Mandelbrot’s model to stock prices and obtained promising results.5 Until recently, however, the work of Mandelbrot and Fama had been largely ignored.6 In his dissertation, Fama assumed that the logarithm of stock returns followed a fat-tailed distribution called a “stable Paretian distribution,” or stable distribution.7 Hence, we refer to the resulting distribution of returns as a “log-stable distribution.” We can illustrate an example of Fama’s work by using the same S&P 500 histogram in our earlier exhibit but with a log-stable distribution curve overlaying it instead of a lognormal curve.8 The log-stable model (Exhibit 3) fits the empirical distribution much closer than the lognormal both at the In the early 1960s, Benoit Mandelbrot, a mathematician teaching economics at the University of Chicago, was advising a doctoral student named Eugene Fama. Mandelbrot had developed a statistical model for percentage changes in the price of cotton that had “fat tails.” That is, the model assigned nontrivial probabilities to large percentage changes. In his doctoral dissertation, Fama applied 5 For an account of the work of Mandelbrot and Fama during this period, see Benoit Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets, New York: Basic Books, 2004. 6 The idea of using fat-tailed distributions to model asset returns is starting to gain some traction. FinAnalytica was founded to provide investment analysis and portfolio construction software based on Mandelbrot and Fama’s work. Morningstar added distribution charts and forecasting models based on it to Morningstar EnCorr. 7 Strictly speaking, the assumption is that the logarithm of one plus the return in decimal form follows a stable Paretian distribution. 8 This chart can be produced in Morningstar EnCorr Analyzer using the log-stable feature. 30 Morningstar Advisor February/March 2009

Morningstar Advisor - February/March 2009

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