1. Financial Analysis, Planning and Forecasting Theory and Application
Chapter 3
Discriminant Analysis and Factor Analysis: Theory and Method By
Cheng F. Lee
Rutgers University, USA
John Lee
Center for PBBEF Research, USA

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

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

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

5. 3.2 Important concepts of linear algebra

6. 3.2 Important concepts of linear algebra
(3.3)

7. 3.2 Important concepts of linear algebra

8. 3.2 Important concepts of linear algebra

9. 3.2 Important concepts of linear algebra
Step 1: Multiply A’ by B
Step 2: Multiply C by A
Linear Equation System and its Solution

10. 3.2 Important concepts of linear algebra

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

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

13. 3.2 Important concepts of linear algebra

14. 3.2 Important concepts of linear algebra

15. 3.2 Important concepts of linear algebra

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

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

20. We know
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).

21. Cramer’s Rule

22. 3.3 Two-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)

23. 3.3 Two-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 1
Group 2
[N1=6]
[N2=8]
2.0
1.8
2.3
3.1
1.9
2.5
0.50
0.48
0.49
0.41
0.43
0.44 1
1.7
1.5
2.2
2.8
1.6
1.4
0.35
0.34
0.42
0.36
0.38
0.55
0.56

24. 3.3 Two-group discriminant analysis
(3.13)
(3.14)
Var(x1i)a1 + Cov(x1i , x2i) a2 = Cov(x1i , yi)
(3.15a)
Cov(x1i , x2i) a1 + Var(x2i)a2 = Cov(x2i , yi)
(3.15b)

25. 3.3 Two-group discriminant analysis

26. 3.3 Two-group discriminant analysis

27. 3.3 Two-group discriminant analysis

28. 3.3 Two-group discriminant analysis

29. 3.3 Two-group discriminant analysis

30. 3.3 Two-group discriminant analysis

31. 3.3 Two-group discriminant analysis

32. 3.3 Two-group discriminant analysis

33. 3.3 Two-group discriminant analysis
(3.16)

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

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

36. 3.4 k-group discriminant analysis
(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.

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

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

39. 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.

40. Appendix 3A. Discriminant analysis and dummy regression analysis
where

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

42. Appendix 3A. Discriminant analysis and dummy regression analysis

43. Appendix 3A. Discriminant analysis and dummy regression analysis

44. Appendix 3A. Discriminant analysis and dummy regression analysis

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

46. Appendix 3A. Discriminant analysis and dummy regression analysis

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

48. Appendix 3A. Discriminant analysis and dummy regression analysis

49. Appendix 3B. Principal-component analysis

50. Appendix 3B. Principal-component analysis

51. Appendix 3B. Principal-component analysis

52. Appendix 3B. Principal-component analysis

53. Appendix 3B. Principal-component analysis

54. Appendix 3B. Principal-component analysis