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Exhibit 2 Smooth Approach As the area to the left of the red vertical line shows, the smooth scenario-based approach does a better job of estimating the probability of tail events than the mean-variance model.
Scenario-Based Distribution Lognormal Distribution

can be handled in a scenario-based model. For example, in scenarios that represent normal times, returns on different equity markets could be modeled as moving somewhat apart from each other while scenarios that represent global crises could model the markets as moving downward together.
Ultrasonic Statistical Technology

Finally, we make use of new technology. Because it could take thousands of scenarios to adequately model return distributions, a disadvantage of the scenario-based approach has been that it requires large amounts of data to be stored and processed. Even with the advances in computer hardware, the conventional approach of representing scenarios with large tables of explicit numbers remained problematic. That is, until recently. The phenomenal speed of computers has given rise to the field of probability management, an extension of data management to probability distributions rather than numbers. The key component of probability management is DIST—Savage’s Distribution String, which can encapsulate thousands of trials as a single data element. The use of Distribution Strings greatly saves on storage and speeds processing time—a Monte Carlo simulation consisting of thousands of trials can be performed on a personal computer in an instant. While not all asset-management organizations are prepared to create the Distribution Strings needed to drive the geometric-mean-CVaR optimization, some outside vendors, such as Morningstar’s Ibbotson Associates, can fulfill this role. Another facet of probability management is interactive simulation technology, which can run thousands of scenarios through a model before the sound of your finger leaving the Enter key reaches your ear. These supersonic models allow much deeper intuition into the sensitivities of portfolios and encourage users to interactively explore different portfolios, distributional assumptions, and potential “black swans.”

–90 –80 –70 –60 –50 –40 –30 –20 –10

0% 10

Annual returns of large-company stocks 1926–2008.

the 5th percentile of the return distribution. For example, as we mentioned, the 5th percentile of the distribution shown in Exhibit 2 is negative 25.8%, so its 5% VaR is 25.8%. This means there is a 5% chance of losing $2,850 or more on a $10,000 investment. CVaR, however, accounts for possible losses beyond VaR; it is the expected or average loss of capital should VaR be breached. Therefore, CVaR is always greater than VaR. The 5% CVaR for the distribution shown in Exhibit 2 is 35.8%, or $3,580, on a $10,000 investment.
Scenarios Versus Correlation

are related. In fact, the R-squared statistic of a simple linear regression model for two series of returns is equal to the square of the correlation coefficient. For many pairs of asset classes, however, a linear model misses the most important features of the relationship. For example, during normal times, non-U.S. equities are considered to be good diversifiers for U.S. equity investors. But during global crises, all major equity markets move down together. Furthermore, suppose that the returns on two asset-class indexes were highly correlated, but instead of including direct exposures to both in the model, one was replaced with an option on itself. Rather than having a linear relationship, we now have a nonlinear relationship that cannot be captured by a correlation coefficient. Fortunately, these sorts of nonlinear relationships between returns on different investments

Next, we model the distribution of returns differently. In mean-variance analysis, the covariation of the returns of each pair of asset classes is represented by a single number, the correlation coefficient. This is mathematically equivalent to assuming that a simple linear regression model is an adequate description of how the returns on the two asset classes

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Morningstar Advisor - April/May 2010

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