Financial Analysis, Planning and Forecasting Theory and Application By Alice C. Lee San Francisco State University John C. Lee J.P. Morgan Chase Cheng F. Lee Rutgers University Chapter 3 Discriminant Analysis and Factor Analysis: Theory and Method 1

Outline  3.1Introduction  3.2Important concepts of linear algebra Linear combination and its distribution Vectors, matrices, and their operations Linear-equation system and its solution  3.3Two-group discriminant analysis  3.4k-group discriminant analysis  3.5Factor analysis and principal-component analysis Factor score Factor loadings  3.6Summary  Appendix 3A. Discriminant analysis and dummy regression analysis  Appendix 3B. Principal-component analysis 2

3.2Important concepts of linear algebra  Linear combination and its distribution  Vectors, matrices, and their operations  Linear-equation system and its solution 3

3.2Important concepts of linear algebra (3.1) (3.1′) (3.2a) (3.2b) 4

3.2Important concepts of linear algebra (3.2b′) (3.2b′′) 5

3.2Important concepts of linear algebra (3.3) 6

3.2Important concepts of linear algebra (3.2b′′′) 7

3.2Important concepts of linear algebra 8

Step 1: Multiply A’ by B Step 2: Multiply C by A Linear Equation System and its Solution 9

3.2Important concepts of linear algebra 10

3.2Important concepts of linear algebra (3.5) (3.6) 11

3.2Important concepts of linear algebra (3.7) (3.8) 12

3.2Important concepts of linear algebra 13

3.2Important concepts of linear algebra 14

3.2Important concepts of linear algebra 15

3.2Important concepts of linear algebra 16 Note to instructor: The numbers with red circle are different from those in the text.

The simultaneous equation (a) can be written as matrix form as equation (b) Then we can solve this equation system by matrix inversion. 17 Extra Example to show how simultaneous equation system can be solve by matrix inversion method

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We know 20 Please note that this is one of three methods can be used to solve simultaneous equation system. Other two methods are substitution method and Cramer rule method. These two methods have been taught in high school algebra. In practice, matrix method is the best method to solve large equation systems, such as portfolio analysis (see Chapter 7).

Cramer’s Rule 21

3.3Two-group Discriminant Analysis where B = DD′, between-group variance; C = Within-group variance; A = Coefficient vector representing the coefficients of Eq. (3.8); E = Ratio of the weighted between-group variance to the pooled within variance. (3.12) 22

3.3Two-group discriminant analysis TABLE 3.1 Roster of liquidity and leverage ratios For two groups with two predictors and a “dummy” criterion variable Y. Group 1Group 2 [N 1 =6][N 2 =8]

3.3Two-group discriminant analysis (3.13) (3.14) Var(x 1i )a 1 + Cov(x 1i, x 2i ) a 2 = Cov(x 1i, y i ) (3.15a) Cov(x 1i, x 2i ) a 1 + Var(x 2i )a 2 = Cov(x 2i, y i ) (3.15b) 24

3.3Two-group discriminant analysis 25

3.3Two-group discriminant analysis 26

3.3Two-group discriminant analysis 27

3.3Two-group discriminant analysis 28

3.3Two-group discriminant analysis 29

3.3Two-group discriminant analysis. 30

3.3Two-group discriminant analysis 31

3.3Two-group discriminant analysis 32

3.3Two-group discriminant analysis (3.16) 33

3.4k-group discriminant analysis (3.17) (3.18) 34

3.4k-group discriminant analysis (3.20a) (3.20b) (3.20c) (3.20r) 35

3.4k-group discriminant analysis (3.21a) (3.21b) Where = Prior probability of being classified as bankrupt, = Prior probability of being classified as non-bankrupt, = Conditional probability of being classified as non- bankrupt when, in fact, the firm is bankrupt, = Conditional probability of being classified as bankrupt when, in fact, the firm is non-bankrupt, = Cost of classifying a bankrupt firm as non-bankrupt, = Cost of classifying a non-bankrupt firm as bankrupt. 36

3.5 Factor analysis and principal-component analysis  Factor score  Factor loadings 37

3.5 Factor analysis and principal-component analysis (3.22) (3.23) (3.24) 38

3.6 Summary  In this chapter, method and theory of both discriminant analysis and factor analysis needed for determining useful financial ratios, predicting corporate bankruptcy, determining bond rating, and analyzing the relationship between bankruptcy avoidance and merger are discussed in detail. Important concepts of linear algebra-linear combination and matrix operations- required to understand both discriminant and factor analysis are discussed. 39

(3.A.1) (3.A.2) where Appendix 3A. Discriminant analysis and dummy regression analysis 40

Appendix 3A. Discriminant analysis and dummy regression analysis (3.A.3) (3.A.2a) 41

Appendix 3A. Discriminant analysis and dummy regression analysis 42

(3.A.4) (3.A.5) Appendix 3A. Discriminant analysis and dummy regression analysis 43

(3.A.6) (3.A.7) (3.A.8) Appendix 3A. Discriminant analysis and dummy regression analysis 44

BA = ECA.(3.A.9) (1 + E)BA = E(B + C)A or (3.A.10) Appendix 3A. Discriminant analysis and dummy regression analysis 45

(3.A.11) (3.A.12) (3.A.l’) (3.A.13) Appendix 3A. Discriminant analysis and dummy regression analysis 46

(3.A.l4a) (3.A.l4b) (3.A.l5) Appendix 3A. Discriminant analysis and dummy regression analysis 47

(3.A.l6) Appendix 3A. Discriminant analysis and dummy regression analysis. 48

Appendix 3B. Principal-component analysis 49

（ 3.B.1 ） （ 3.B.2 ） （ 3.B.3 ） Appendix 3B. Principal-component analysis 50

（ 3.B.4 ） Appendix 3B. Principal-component analysis 51

Appendix 3B. Principal-component analysis. 52

（ 3.B.5 ） （ 3.B.6 ） Appendix 3B. Principal-component analysis 53

（ 3.B.7 ） （ 3.B.8 ） （ 3.B.9 ） Appendix 3B. Principal-component analysis 54