Morningstar Advisor - December/January 2010 - 25

Applying risk premium data based on a portfolio of primarily healthy companies with a small slice of potentially distressed companies acknowledges the less-than-100% chance that a subject firm will be perfectly healthy for the indefinite future. K Warren Miller is a senior quantitative equity analyst with Morningstar. James P. Harrington is director of business valuation research in Morningstar’s Financial Communications business. Magdalena Mroczek is a business valuation analyst with Morningstar. References Altman, Edward I., “Corporate Distress Prediction Models in a Turbulent Economic and Basel II Environment,” NYU Working Paper No. FIN-02-052 (September 2002). Bemmann, Martin, “Improving the Comparability of Insolvency Predictions,” Dresden Economics Discussion Paper Series No. 08/2005 (June 23, 2005). Cantor, Richard Martin and Mann, Christopher, “Analyzing the Tradeoff between Ratings Accuracy and Stability,” Journal of Fixed Income (September 2006). Morningstar, Inc., “Stock Grade Methodology for Financial Health,” Morningstar Methodology Paper (March 26, 2008). Accuracy Ratio Explained More than 100 years ago, U.S. economist Max O. Lorenz first used cumulative accuracy profiles to analyze inequalities in wealth distribution. Lorenz’s idea for representing inequality, commonly called a Lorenz curve, was straightforward. He plotted the cumulative percentage of people against the cumulative percentage of wealth they owned. Cumulative Percentage of Wealth 100 3 50 Cumulative Percentage of People The 45-degree line (called the line of perfect equality) in the graph represents an equal wealth distribution. At any point on this line, the bottom x percent of people own x percent of the wealth—the bottom 10% of people own 10% of the wealth, the bottom 20% own 20% of the wealth, the bottom 50% own 50% of the wealth (Point 1 !), and so on. In other words, everyone has the same amount of wealth. In contrast, the green line—a Lorenz curve—is a representation of an unequal wealth distribution. At Point 2 @, the bottom 50% of people own 5% of the wealth, implying that the top 50% own 95% of the wealth. The most unequal wealth distribution is represented by the black line (the line of perfect inequality). At Point 3 #, one person would own 100% of the wealth; the rest of the population would own 0%. But Lorenz curves don’t measure inequality; they only depict it. To measure the amount of inequality of a distribution, statisticians use the Gini coefficient (named after Italian statistician Corrado Gini). The Gini coefficient is a ratio of the area between the line of perfect equality and a Lorenz curve of an unequal distribution (Area A in the graph) and the total area between the line of perfect equality and the line of perfect inequality (Area A plus Area B). A 4 (A 1 B) 5 Measure of Inequality 5 Gini Coefficient If inequality increases as a Lorenz curve bows out further toward the lower right corner of the graph, it follows that the Gini coefficient increases, too. Perfect inequality would have a ratio of 1; perfect equality would have a ratio of 0. In the context of credit-scoring models, such as Morningstar’s Distance to Default, the Gini coefficient can be used to measure accuracy. After all, the ideal credit-scoring model would maximize the inequality of bankruptcy distribution. In this context, economists call the Gini coefficient an accuracy ratio. The higher a model’s accuracy ratio, the better it was at predicting defaults. On the Web This article is based on the paper “Comparing Models of Corporate Bankruptcy Prediction: Distance to Default vs. Z-Score,” written by Warren Miller. It can be downloaded from our White Paper Bank on MorningstarAdvisor.com. MorningstarAdvisor.com 25

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Morningstar Advisor - December/January 2010

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