Title: Financial Analysis, Planning and Forecasting Theory and Application
1Financial Analysis, Planning and
Forecasting Theory and Application
Chapter 4
Application of Discriminant Analysis and Factor
Analysis in Financial Management
 By
 Alice C. Lee
 San Francisco State University
 John C. Lee
 J.P. Morgan Chase
 Cheng F. Lee
 Rutgers University
2Outline
 4.1 Introduction
 4.2 Credit analysis
 4.3 Bankruptcy and financial distress analysis
 4.4 Applications of factor analysis to select
useful financial ratios  4.5 Bond ratings forecasting
 4.6 Bond quality ratings and the change of
quality ratings for the electric utility
industry  4.7 Ohlsons and Shumways methods for Estimating
Default Probability  4.8 Summary
 Appendix 4A. Jackknife method and its application
in MDA analysis  Appendix 4B. Multiperiod Logistic Regression
34.2 Credit analysis

(4.1)  where
 Yi Index value for the ith account
 ith firms quick ratio
 ith firms total sales/inventory ratio
 and A and B are the parameters or weights to be
determined.
44.2 Credit analysis
54.2 Credit analysis

(4.4b) 
 Where
 Variance of X1
 Variance of X2

 Covariance between X1 and X2
 Difference between the average of X1s for
good accounts  and the average of X1s for bad accounts and
 Difference between the average of X2 for
good accounts  the average of X2 for bad accounts.
64.2 Credit analysis
 TABLE 4.1 Status and index values of the
accounts
Account Number Account Status Yi
7 Bad 0.81
10 Bad 0.89
2 Bad 1.30
3 Bad 1.45
6 Bad 1.64
12 Good 1.77
11 Bad 1.83
4 Good 1.96
1 Good 2.25
8 Good 2.50
5 Good 2.61
9 Good 2.80
74.2 Credit analysis
84.2 Credit analysis
94.3 Bankruptcy and financial distress analysis
 Discriminant Model (Y is the value of zscore)
 (4.5)
 TABLE 4.2 Mean ratios of bankrupt / nonbankrupt
firms  From Altman, E. I., Financial ratios,
discriminant Analysis, and the prediction of  corporate bankruptcy, Journal of Finance 23
(1968), p. 596, Table I. Reprinted  by Permission of Edward I. Altman and Journal of
Finance.  Zscore gt2.99 nonbankrupt sector Zscore lt
1.81 bankruptcy Zscore between 1.81 and 2.99
gray area.
Ratio Definition Bankrupt Group Mean Nonbankrupt Group Mean
X1 Working capital / total assets 0.061 0.414
X2 Retained earnings / total assets 0.626 0.355
X3 EBIT/ total assets 0.318 0.153
X4 Market value of equity / book value of total debt 0.401 2.477
X5 Sales / total assets 1.500 1.900
10Empirical
 When we apply Equation (4.5) to calculate
financial Zscore, the model should be defined as
 Here we use JNJ in 2005 as an example,
 Then, the zscore for JNJ is 1.2(0.3233)1.4(0.71
47)3.3(0.2353)0.6(8.8683)1.0(0.8706) 8.3567
Ratio Definition JNJ
X1 Net Working capital / total assets ( current asset current liability ) / total assets 0.3233
X2 Retained earnings / total assets 0.7147
X3 EBIT/ total assets 0.2353
X4 Market value of equity / book value of total debt 8.8683
X5 Sales / total assets 0.8706
114.3 Bankruptcy and financial distress analysis
Class Size of Sample Definition
1. PPO 2(1.8) Serious problempotential payoff. An advanced problem bank that has at least 50 percent chance of requiring financial assistance in the near future.
2. SP 14(12.7) Serious problem. A bank whose financial condition threatens ultimately to obligate financial outlay by the FEIC unless drastic changes occur.
3. OP 94(85.5) Other problem. A bank with some significant weakness, with vulnerability less than class 2, but still calling for aggressive supervision and extraordinary concern by the FEIC.
Total 110(100)
From Sinkey, J.F., A multivariate statistical analysis of the characteristics of problem banks, Journal of Finance 30 (1975), Table 2. Reprinted by permission. From Sinkey, J.F., A multivariate statistical analysis of the characteristics of problem banks, Journal of Finance 30 (1975), Table 2. Reprinted by permission. From Sinkey, J.F., A multivariate statistical analysis of the characteristics of problem banks, Journal of Finance 30 (1975), Table 2. Reprinted by permission.
124.3 Bankruptcy and financial distress analysis
 TABLE 4.3 Profile analysis for problem banks
Financial Ratio 1969 1970 1971 1972
Loans/Assets
1. Problem bank 53.9 55.4 56.9 56.0
2. Nonproblem bank 49.3 48.9 47.8 47.8
Loans/Capital plus Reserves
1. Problem bank 648.3 692.2 768. 9 838.6
2. Nonproblem bank 564.5 562.5 562.4 577.5
Operating Expense/Operating Income
1. Problem bank 83.9 85.5 89.3 94.1
2. Nonproblem bank 78.5 78.6 81.8 82.4
Loan Revenue/Total Revenue
1. Problem bank 64.7 65.8 68.8 69.8
2. Nonproblem bank 59.3 59.2 59.9 59.6
Other Expenses/Total Revenue
1. Problem bank 15.8 16.0 16.3 16.4
2. Nonproblem bank 12.3 13.0 13.2 13.7
134.3 Bankruptcy and financial distress analysis
Year Type I Error Type II Error Total Error
1969 46.36 25.45 35.91
1970 42.73 27.27 35.00
1971 38.18 24.55 31.36
1972 28.15 21.36 24.76
144.3 Bankruptcy and financial distress analysis

(4.6)  where
 0 Unsecured loan,
 1 Secured loan
 0 Past interest payment due,
 1 Current loan
 0 Not audited firm,
 1 Audited firm
 0 Net loss firm
 1 Net profit firm
 Working Capital/Current Assets
 0 Loan criticized by bank examiner,
 1 Loan not criticized by bank examiner.
154.3 Bankruptcy and financial distress analysis

(4.7)  where
 Agents balances/Total assets a measure
of the firms accounts receivable
management  Stocks at cost (preferred and
common)/Stocks at market
(preferred and common) measures investment
management  Bonds at cost/Bonds at market measures the
firms age  (Loss adjustment expenses paid
underwriting expenses paid) /
Net premiums written a measure of a firms funds
flow from insurance operations  Combined ratio traditional measure of
underwriting profitability and
 Premiums written direct/Surplus
a measure of the firms sales
aggressiveness.
164.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment
4 Earnings/Sales .88 .63 .75 .81
7 Earnings/Net Worth .79 .94 .95 .95
12 Earnings/Total Assets .93 .89 .85 .87
13 Cash Flow/Total Assets .92 .85 .84 .84
14 Cash Flow/Net Worth .50 .88 .79 .93
15 EBIT/Total Assets .89 .85 .77 .84
16 EBIT/Sales .89 .61 .70 .77
17 Cash Flow/Total Capital .94 .90 .85 .93
174.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment Factor 1?Return on Investment
18 Earnings/Total Capital .94 .90 .88 .94
19 Cash Flow/Sales .79 .59 .87 .74
41 EBIT/Net Worth .79a .92 .95 .97
47 Cash Flow/Total Debt .81 .73 .84 .70
48 Earnings/Total Debt .87 .78 .86 .73
53 Operating Funds/Total Assets .88 .82 .45 .82
54 Operating Funds/Net Worth .25 .75 .63a .86
55 Operating Funds/Total Capital .83 .81 .33 .88
184.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 2?Financial Leverage Factor 2?Financial Leverage Factor 2?Financial Leverage Factor 2?Financial Leverage Factor 2?Financial Leverage Factor 2?Financial Leverage
2 Net Worth/Total Assets .80 .85 .82 .69a
5 LongTerm Debt/Total Assets .87 .85 .85 .87
11 LongTerm Debt/Net Worth .88 .90 .91 .93
29 LongTerm Debt/Net Plant .85 .81 .80 .81
30 LongTerm Debt/Total Capital .89 .92 .94 .91
31 Total Debt/Net Worth .79 .85 .83 .71a
32 Total Debt/Total Assets .81 .85 .79 .74
50 Total Debt and Preferred Stock/Total Assets .79 .85 .78 .68
194.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 3?Capital Intensiveness Factor 3?Capital Intensiveness Factor 3?Capital Intensiveness Factor 3?Capital Intensiveness Factor 3?Capital Intensiveness Factor 3?Capital Intensiveness
3 Sales/Net Worth .66 .85 .70a .78
6 Sales/Total Assets .78a .81 .75 .79
19 Cash Flow/Sales .44 72a ? ?
20 Current Liabilities/Net Plant .81 .49 .81 .43a
22 Current Assets/Total Assets .88 .46 .84 .41
26 Sales/Net Plant .94 .78 .91 .79
27 Sales/Total Capital .85 .91 .86 .83
204.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 4?Inventory Intensiveness Factor 4?Inventory Intensiveness Factor 4?Inventory Intensiveness Factor 4?Inventory Intensiveness Factor 4?Inventory Intensiveness Factor 4?Inventory Intensiveness
1 Working Capital/Sales .72a .44 .69a .81
20 Current Liabilities/Net Plant .33 .71 ? ?
21 Working Capital/Total Assets .40 .76 .46 .85
22 Current Assets/Total Assets .39 .83 .45 .84
24 Current Assets/Sales .92 .74 .92 .74
25 Cost of Goods Sold/Inventory .91 .92  .94 .93
28 Inventory/Sales .87 .93 . 94 .93
214.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 5?Cash Position Factor 5?Cash Position Factor 5?Cash Position Factor 5?Cash Position Factor 5?Cash Position Factor 5?Cash Position
42 Cash/Total Assets .91 .93 .89 .81
43 Cash/Current Liabilities .84 .88 .83 .87
44 Cash/Sales .93 .86 .88 .89
46 Cash/Fund Expenditures .91 .86 .88 .89
224.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 6?Receivables Intensiveness Factor 6?Receivables Intensiveness Factor 6?Receivables Intensiveness Factor 6?Receivables Intensiveness Factor 6?Receivables Intensiveness Factor 6?Receivables Intensiveness
23 Quick Assets/Total Assets .52 .89 .68a .89
33 Receivables/Inventory .94 .84 .80a .82
34 Inventory/Current Assets .75a .70 .64 .76
35 Receivables/Sales .72a .83 .81 .83
37 Quick Assets/Sales .58 .86 .78 .88
40 Quick Assets/Current Liabilities .40 .76 .46 .81
45 Quick Assets/Fund Expenditures .55 .85 .75 .87
234.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 7?ShortTerm Liquidity Factor 7?ShortTerm Liquidity Factor 7?ShortTerm Liquidity Factor 7?ShortTerm Liquidity Factor 7?ShortTerm Liquidity Factor 7?ShortTerm Liquidity
21 Working Capital/Total Assets ? ? .73 .35
36 Inventory/Working Capital ? ? .79 .16
38 Current Liabilities/Net Work ? ? .55a .80
39 Current Assets/Current Liabilities .91 .64 .90 .61
40 Quick Assets/Current Liabilities .77 .37 .76 .31
49 Current Liabilities/Total Assets ? ? .64a .78
51 Net Defensive Assets/Fund Expenditures .55 .74 .75 .52a
244.4 Applications of factor analysis to select
useful financial ratios
 TABLE 4.4a Crosssectional comparison of
financial ratios and factor loadings
defining eight financial ratio categories for
industrial firms (Cont.)  From Johnson, W.B., The crosssectional
stability of financial ratio patterns, Journal
of Financial and Quantitative  Analysis 14 (1979), Table 2. Reprinted by
permission of W. Bruce Johnson and JFQA.  a Indicates variables having a withinsample
crossloading of between 0.50 and 0.70 on one
other factor.
Factor Loadings Factor Loadings Factor Loadings Factor Loadings
1972 1972 1974 1974
Primary Primary Primary Primary
Ratio Number Ratio Name Mfg. Retail Mfg. Retail
Factor 8?Decomposition Measures Factor 8?Decomposition Measures Factor 8?Decomposition Measures Factor 8?Decomposition Measures Factor 8?Decomposition Measures Factor 8?Decomposition Measures
56 Asset Decomposition .68 .74 ? ?
58 Equity Decomposition .84 .84 .86 .87
60 Noncurrent Items Decompostion .83 .78 .87 .85
61 Time Horizon Decompostion ? ? .62 .70
254.5 Bond ratings forecasting
 TABLE 4.4b Crosssectional congruency
coefficients for eight financialratio dimensions
for 1974
Factor Retail Firms Factor Retail Firms Factor Retail Firms Factor Retail Firms Factor Retail Firms Factor Retail Firms Factor Retail Firms Factor Retail Firms
Factor Primary Manufacturing Firms One Two Three Four Five Six Seven Eight
One ? Return on Investment .95 ?.41 ?.13 ?.05 .14 .05 ?.25 ?.26
Two ? Financial Leverage ?.40 .95 .11 ?.17 ?.17 ?.05 .45 .07
Three ? Capital Intensiveness ?.15 ?.00 .84 .28 ?.14 ?.16 .55 .04
Four ? Inventory Intensiveness ?.13 ?.02 ?.27 .87 ?.01 .15 .08 .08
Five ? Cash Position .20 ?.15 ?.21 .00 .88 .46 ?.29 .15
Six ? Receivables Intensiveness .01 ?.06 ?.42 .11 .29 .92 ?.24 .10
Seven ? Shortterm Liquidity .19 ?.34 ?.17 .30 .38 .39 .76 ?.01
Eight ? Decomposition Measures ?.20 .16 .06 .06 .01 .05 .27 .84
From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA. From Johnson, W.B., The crosssectional stability of financial ratio patterns, Journal of Financial and Quantitative Analysis 14 (1979), Table 3. Reprinted by permission of W. Bruce Johnson and JFQA.
264.5 Bond ratings forecasting
274.5 Bond ratings forecasting
 (5) Defensive Interval QA/Operating Expense
Minus Depreciation, Depletion, Amortization (6)
Capital Expenditure/Sales
284.5 Bond ratings forecasting
 From Chen, K. H., and T. A.
 Shimerda, An empirical analysis
 of useful financial ratios, Financial
 Management (Spring 1981),
 Exhibit 5. Reprinted by permission.
 Ratio not included in the final factors of the
PEMC studies.  Ratio not in the 48 ratios included in the
PEMC study.
294.5 Bond ratings forecasting
304.5 Bond ratings forecasting
 TABLE 4.7 Variable means, test of significance,
and important ranks
Bond Rating Bond Rating Bond Rating Bond Rating Bond Rating Function Ranks Function Ranks Function Ranks
Variable AA A BAA BA B FRatio One Two Three
X1 0.000 0.077 0.520 1.000 1.000 ? 1 6 2
X2 1.634 1.581 1.260 1.058 0.486 25.45 2 2 5
X3 1.869 1.657 1.275 1.354 1.250 13.97 3 3 1
X4 1.138 0.606 0.560 0.511 0.707 6.05 6 1 6
X5 0.091 0.162 0.154 0.151 0.215 4.06 5 4 4
KX6 0.099 0.075 0.066 0.075 0.069 2.68 4 5 3
From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level. From Pinches, G.E., and K.A. Mingo, A multivariate analysis of industrial and bond ratings, Journal of Finance 28 (March 1973), Table 3. Reprinted by permission. Significant at 0.001 level Significant at 0.01 level Significant at 0.05 level.
314.6 Bond quality ratings and the change of
quality ratings for the electric utility industry
 The multivariateanalysis technique
developed by Pinches and Mingo for analyzing
industrial bond ratings has also been used to
determine bond quality ratings and their
associated changes for electric utilities.
Pinches, Singleton, and Jahakhani (1978) (PSJ)
used this technique to determine whether fixed
coverages were a major determinant of electric
utility bond ratings. Bhandari, Soldofsky, and
Boe (1979) (BSB) investigate whether or not a
multivariate discriminant model that incorporates
the recent levels, past levels, and the
instability of financial ratios can explain and
predict the quality rating changes of electric
utility bonds. 
 PSJ (1978) found that fixed coverage is the
only (and not the dominant) financial variable
that apparently influences the bond ratings
assigned to electric utility firms. Other
important variables are the climate of
regulation, total assets, return on total assets,
growth rate or net earnings, and construction
expenses/total assets.2 The major finding of
BSBs study is that the MDA method can be more
successful in predicting bond rating changes than
it had been predicting the bond ratings
themselves. These results have shed some light
for the utility regulation agency on the
determinants of bond ratings and the change of
bond ratings for electric utility industries.
324.7 Ohlsons and Shumways methods for Estimating
Default Probability
 X1 Natural log of (Total Assets/ GNP Implicit
Price Deflator Index). The index assumes a
base value of 100 for 1968  X2 (Total Liabilities/Total Assets)
 X3 (Current Assets Current
Liabilities)/Total Assets  X4 Current Assets/ Current Liabilities
 X5 One if total liabilities exceeds total
assets, zero otherwise  X6 Net income/total assets
 X7 Funds provided by operations/total
liabilities  X8 One if net income was negative for the last
two years, zero otherwise and  X9 (Net income in year t Net income in t1) /
(Absolute net income in year t Absolute
net income in year t1).
334.7 Ohlsons and Shumways methods for Estimating
Default Probability



(4.8)  Where , P the
probability of bankruptcy.
344.7 Ohlsons and Shumways methods for Estimating
Default Probability


(4.9)  Where , P the
probability of bankruptcy 
 X1 Net Income/Total Assets
 X2 (Total Liabilities/Total Assets)
 X3 The logarithm of (each firms market
capitalization at the end of year prior
to the observation year / total market
capitalization of NYSE and AMEX market)  X4 Past excess return as the return of the
firm in year t1 minus the valueweighted
CRSP NYSE/AMEX index return in year t  1 and  X5 idiosyncratic standard deviation of each
firms stock returns. It is defined as the
standard deviation of the residual of a
regression which regresses each stocks
monthly returns in year t 1 on the
valueweighted NYSE/AMEX index return for the
same year.
354.8 Summary
 In this chapter, we have discussed
applications of two multivariate statistical
methods in discriminant analysis and factor
analysis. Examples of using twogroup
discriminant functions to perform credit
analysis, predict corporate bankruptcy, and
determine problem banks and distressed PL
insurers were discussed in detail. Basic
concepts of factor analysis were presented,
showing their application in determining useful
financial ratios. In addition, the combination
of factor analysis and discriminant analysis to
analyze industrial bond ratings was discussed.
Finally, Ohlsons and Shumways methods for
estimating default probability were discussed. 
 In sum, this chapter shows that multivariate
statistical methods can be used to do practical
financial analysis for both managers and
researchers.
36Appendix 4A. Jackknife method and its application
in MDA analysis
37Appendix 4A. Jackknife method and its application
in MDA analysis
TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions TABLE 4.A.1 Original and jackknifed (standardized) discriminant functions
Discriminant Function Discriminant Function Discriminant Function Discriminant Function Discriminant Function Discriminant Function
1 1 2 2 3 3
Variable Coefficient Jackknifed Coefficient Coefficient Jackknifed Coefficient Coefficient Jackknifed Coefficient
X1 ?.936 ?.882 .131 .102 ?.365 ?.361
X2 .528 .461 1.073 .863 ?.216 ?.017
X3 .360 .352 ? .758 ?.541 ?.493 ?.516
X4 .023 .041 ?1.284 ?.888 .006 .012
X5 ?.283 ?.171 ?.529 ?.544 .335 .421
X6 .327 .302 ?.280 ?.067 ?.340 ?.320