Morningstar Advisor - February/March 2009 - 32

Spotlight loss for a normal distribution, a stable distribution, and a power law distribution. The line for the normal distribution curves down, indicating that it has thin tails. In contrast, the line for stable distribution approaches the straight line of the power law because it is very similar to a power law for large losses. These results show that the log-stable distribution does a good job of modeling the empirical returns distribution of the S&P 500. The better fit of the log-stable distribution demonstrates that the S&P 500 has fatter tails than predicted by the lognormal model. It also calls into question commonly used portfolio construction techniques such as the mean-variance optimization, which relies on the assumption of a finite variance. If the log-stable model does such a better job in describing the distribution of asset returns, why has it not received more acceptance? There are several possible reasons. First, the mathematics is challenging. Second, the variances and all higher moments of stable Exhibit 5 Role of Time: The log-stable model indicates that there’s a 4% to 5% probability that the S&P 500 will lose 50% or more over extended time periods. The lognormal model puts the odds much lower. Probability of Drop of 50% or More 6% Hard Eight The S&P 500 has suffered eight peak-to-trough declines of more than 20%. Peak Trough Decline % Recovery August 1929 August 2000 December 1972 October 2007 August 1987 November 1968 December 1961 May 1946 June 1932 September 2002 September 1974 November 2008 November 1987 June 1970 June 1962 November 1946 83.41 44.73 42.64 40.89 29.58 29.16 22.28 21.76 January 1945 October 2006 June 1976 To Be Determined May 1989 March 1971 April 1963 October 1949 Table shows the worst cumulative peak-to-trough declines in percentage terms since December 1925. Based on monthly returns. random variables are infinite. The lack of a finite variance means that most portfolio theories and most portfolio construction techniques are invalid, including those based on alternative risk measures such as “downside risk.” Finally, there is no single obvious way to estimate the parameters of stable distributions as there is with normal distributions. Risk Measures versus Risk Models an investment, advisors would benefit by beginning to think about a more complete risk model. A complete risk model allows investors to consider three questions about a potential decline in value simultaneously: r How likely might a decline occur? r How long might it last? r How bad might it get? It is already common practice in some segments of the financial-services industry to use a risk model to measure “value at risk”—that is, how bad a loss might be over a given length of time and with a given probability. As you can appreciate through our study of historical stock market declines, time horizon is a key dimension of risk not explicitly addressed by standard risk measures. A complete risk model can be used to explicitly take time horizons into account. Log-Stable g For advisors, the lesson here is not that they should throw away the standard ways of summarizing risk using measures such as standard deviation and downside deviation.10 Nor should advisors run to embrace Fama’s log-stable models. Instead, we think advisors should understand the limitations of standard risk measures and have a basic understanding of what Mandelbrot’s and Fama’s work says about describing risk. Rather than solely relying on a few summary statistics to characterize the risks of 4% 2% Lognormal 5 10 15 20 25 30 35 40 45 50 Number of Years 10 In recognition that return distributions may not be symmetric, measures such as skewness and kurtosis are sometimes presented alongside standard deviation. However, like variance, these measures are not defined for stable Paretian distributions. 32 Morningstar Advisor February/March 2009

Morningstar Advisor - February/March 2009

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