# Morningstar Advisor - February/March 2009 - 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`

# 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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