22A

Multiple Discriminant Analysis

s we have seen, bankruptcy—or even the possibility of bankruptcy—can cause significant trauma for a firm’s managers, investors, suppliers, customers, and community. Thus, it would be beneficial to be able to predict the likelihood of bankruptcy so that steps could be taken to avoid it or at least to reduce its impact. One approach to bankruptcy prediction is multiple discriminant analysis (MDA), a statistical technique similar to regression analysis. In this extension, we discuss MDA in detail and illustrate its application to bankruptcy prediction.1 Suppose a bank loan officer wants to segregate corporate loan applications into those likely to default and those unlikely to default. Assume that data for some past period are available on a group of firms that includes companies that went bankrupt as well as companies that did not. For simplicity, we assume that only the current ratio and the debt/ assets ratio are analyzed. These ratios for our sample of firms are given in Columns 2 and 3 at the bottom of Figure 22A-1. The Xs in the graph represent firms that went bankrupt; the dots represent firms that remained solvent. For example, Firm 2, which had a current ratio of 3.0 and a debt ratio of 20%, did not go bankrupt. Therefore, its current ratio and its debt/assets ratio are marked with a single dot in the twodimensional graph; this dot is labeled “A” and is shown in the upper left section of the graph. Firm 19, which had a current ratio of 1.0 and a debt ratio of 60%, did go bankrupt, so an X is used to mark its current ratio and debt/assets ratio. This X is labeled “B” and is shown in the lower right section of Figure 22A-1. The objective of discriminant analysis is to construct a boundary line through the graph such that firms on one side of the line are unlikely to become insolvent whereas those on the other side are likely to go bankrupt. This boundary line is called the discriminant function, and in our example it takes this form: Z ¼ a þ b1 ðCurrent ratioÞ þ b2 ðDebt ratioÞ Here Z is called the Z score, the term a is a constant, and b1 and b2 indicate the effects of the current ratio and the debt ratio on the probability of a firm going bankrupt. Although a full discussion of discriminant analysis would go well beyond the scope of this book, some useful insights may be gained by observing the following six points. 1

A

This section is based largely on the work of Edward I. Altman, especially these three papers: (1) “Financial Ratios, Discriminant Analysis, and the Prediction of Corporate Bankruptcy,” Journal of Finance, September 1968, pp. 589–609; (2) with Robert G. Haldeman and P. Narayanan, “Zeta Analysis: A New Model to Identify Bankruptcy Risk of Corporations,” Journal of Banking and Finance, June 1977, pp. 29–54; and (3) John Hartzell and Matthew Peck, “Emerging Market Corporate Bonds, A Scoring System,” Emerging Corporate Bond Research: Emerging Markets, Salomon Brothers, May 15, 1995. The last article reviews and updates Altman’s earlier work and applies it internationally. 1

2

Web Extension 22A: Multiple Discriminant Analysis

FIGURE 22A-1

Discriminant Boundary between Bankrupt and Solvent Firms

Current Ratio 4 Good: Low Probability of Bankruptcy Bad: High Probability of Bankruptcy Discriminant Boundary, Z

3

A

2

1

B

–0.3611

20

40

60

80 Debt/Assets Ratio (%)

Firm Number (1) 1 2(A) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19(B)

Current Ratio (2) 3.6 3.0 3.0 3.0 2.8 2.6 2.6 2.4 2.4 2.2 2.0 2.0 1.8 1.6 1.6 1.2 1.0 1.0 1.0

Debt/Assets Ratio (3) 60% 20 60 76 44 56 68 40 60 28 40 48 60 20 44 44 24 32 60

Did Firm Go Bankrupt? (4) No No No Yes No Yes Yes Yesa Noa No No Noa Yes No Yes Yes No Yes Yes

Z Scorea (5) –0.780 –2.451 –0.135 0.791 –0.847 0.062 0.757 –0.649 0.509 –1.129 –0.220 0.244 1.153 –0.948 0.441 0.871 –0.072 0.391 2.012

Probability of Bankruptcy (6) 17.2% 0.8 42.0 81.2 15.5 51.5 80.2...