# 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

Contents
Letter from the Editor
Contributors
Inbox
How Do You Gauge and Measure Risk?
No Skinny-Dippers Here
Investment Briefs
Weapons of Mass Destruction?
Déjà Vu All Over Again
A Failure to Gauge Risk
Five Areas to Find Opportunities
Heavenly Returns
Greenspring Comes to the Rescue
How to Spot a Trustworthy REIT
Four Picks for the President
Find Succor in These Large Dividends
These Stocks Are Fiscally Fit
Mutual Fund Analyst Picks
50 Most Popular Equity ETFs
Undervalued Stocks
Most Popular Variable Annuities
New at Morningstar
You Gotta Look Sharpe
Morningstar Advisor - February/March 2009 - Intro
Morningstar Advisor - February/March 2009 - Cover2
Morningstar Advisor - February/March 2009 - 1
Morningstar Advisor - February/March 2009 - 2
Morningstar Advisor - February/March 2009 - Contents
Morningstar Advisor - February/March 2009 - 4
Morningstar Advisor - February/March 2009 - 5
Morningstar Advisor - February/March 2009 - 7
Morningstar Advisor - February/March 2009 - 8
Morningstar Advisor - February/March 2009 - Letter from the Editor
Morningstar Advisor - February/March 2009 - Contributors
Morningstar Advisor - February/March 2009 - Inbox
Morningstar Advisor - February/March 2009 - How Do You Gauge and Measure Risk?
Morningstar Advisor - February/March 2009 - 13
Morningstar Advisor - February/March 2009 - No Skinny-Dippers Here
Morningstar Advisor - February/March 2009 - 15
Morningstar Advisor - February/March 2009 - 16
Morningstar Advisor - February/March 2009 - 18
Morningstar Advisor - February/March 2009 - 19
Morningstar Advisor - February/March 2009 - Investment Briefs
Morningstar Advisor - February/March 2009 - 21
Morningstar Advisor - February/March 2009 - Weapons of Mass Destruction?
Morningstar Advisor - February/March 2009 - 23
Morningstar Advisor - February/March 2009 - 24
Morningstar Advisor - February/March 2009 - 25
Morningstar Advisor - February/March 2009 - 26
Morningstar Advisor - February/March 2009 - 27
Morningstar Advisor - February/March 2009 - Déjà Vu All Over Again
Morningstar Advisor - February/March 2009 - 29
Morningstar Advisor - February/March 2009 - 30
Morningstar Advisor - February/March 2009 - 31
Morningstar Advisor - February/March 2009 - 32
Morningstar Advisor - February/March 2009 - 33
Morningstar Advisor - February/March 2009 - A Failure to Gauge Risk
Morningstar Advisor - February/March 2009 - 35
Morningstar Advisor - February/March 2009 - 36
Morningstar Advisor - February/March 2009 - 37
Morningstar Advisor - February/March 2009 - Five Areas to Find Opportunities
Morningstar Advisor - February/March 2009 - 39
Morningstar Advisor - February/March 2009 - 41
Morningstar Advisor - February/March 2009 - 42
Morningstar Advisor - February/March 2009 - 43
Morningstar Advisor - February/March 2009 - 44
Morningstar Advisor - February/March 2009 - 45
Morningstar Advisor - February/March 2009 - 46
Morningstar Advisor - February/March 2009 - 47
Morningstar Advisor - February/March 2009 - Heavenly Returns
Morningstar Advisor - February/March 2009 - 49
Morningstar Advisor - February/March 2009 - 50
Morningstar Advisor - February/March 2009 - 51
Morningstar Advisor - February/March 2009 - Greenspring Comes to the Rescue
Morningstar Advisor - February/March 2009 - 53
Morningstar Advisor - February/March 2009 - 54
Morningstar Advisor - February/March 2009 - 55
Morningstar Advisor - February/March 2009 - How to Spot a Trustworthy REIT
Morningstar Advisor - February/March 2009 - 57
Morningstar Advisor - February/March 2009 - Four Picks for the President
Morningstar Advisor - February/March 2009 - 59
Morningstar Advisor - February/March 2009 - Find Succor in These Large Dividends
Morningstar Advisor - February/March 2009 - 61
Morningstar Advisor - February/March 2009 - These Stocks Are Fiscally Fit
Morningstar Advisor - February/March 2009 - 63
Morningstar Advisor - February/March 2009 - Mutual Fund Analyst Picks
Morningstar Advisor - February/March 2009 - 65
Morningstar Advisor - February/March 2009 - 66
Morningstar Advisor - February/March 2009 - 67
Morningstar Advisor - February/March 2009 - 50 Most Popular Equity ETFs
Morningstar Advisor - February/March 2009 - 69
Morningstar Advisor - February/March 2009 - 70
Morningstar Advisor - February/March 2009 - 71
Morningstar Advisor - February/March 2009 - Undervalued Stocks
Morningstar Advisor - February/March 2009 - 73
Morningstar Advisor - February/March 2009 - 74
Morningstar Advisor - February/March 2009 - 75
Morningstar Advisor - February/March 2009 - Most Popular Variable Annuities
Morningstar Advisor - February/March 2009 - 77
Morningstar Advisor - February/March 2009 - 78
Morningstar Advisor - February/March 2009 - New at Morningstar
Morningstar Advisor - February/March 2009 - You Gotta Look Sharpe
Morningstar Advisor - February/March 2009 - Cover3
Morningstar Advisor - February/March 2009 - Cover4
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