Financial Analysis, Planning and Forecasting Theory and Application

1
Financial Analysis, Planning and
ForecastingTheory and Application
Chapter 3
Discriminant Analysis and Factor Analysis
Theory and Method

• By
• Alice C. Lee
• San Francisco State University
• John C. Lee
• J.P. Morgan Chase
• Cheng F. Lee
• Rutgers University

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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 3A. Discriminant analysis and dummy
  regression analysis
• Appendix 3B. Principal-component analysis

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3.2 Important concepts of linear algebra

• Linear combination and its distribution
• Vectors, matrices, and their operations
• Linear-equation system and its solution

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3.2 Important concepts of linear algebra

• (3.1)
• (3.1')
• (3.2a)
• (3.2b)

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3.2 Important concepts of linear algebra

• (3.2b')
• (3.2b'')

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3.2 Important concepts of linear algebra

• (3.3)

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3.2 Important concepts of linear algebra

• (3.2b''')

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3.2 Important concepts of linear algebra
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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
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3.2 Important concepts of linear algebra
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3.2 Important concepts of linear algebra

• (3.5)
• (3.6)

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3.2 Important concepts of linear algebra

• (3.7)
• (3.8)

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3.2 Important concepts of linear algebra
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3.2 Important concepts of linear algebra
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3.2 Important concepts of linear algebra
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3.2 Important concepts of linear algebra
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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)

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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.

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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)

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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis
.
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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis
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3.3 Two-group discriminant analysis

• (3.16)

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3.4 k-group discriminant analysis

• (3.17)
• (3.18)

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3.4 k-group discriminant analysis

• (3.20a)
• (3.20b)
• (3.20c)
• (3.20r)

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3.4 k-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.

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3.5 Factor analysis and principal-component
analysis

• Factor score
• Factor loadings

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3.5 Factor analysis and principal-component
analysis

• (3.22)
• (3.23)
• (3.24)

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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.

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• (3.A.1)
• (3.A.2)
• where

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• (3.A.3)
• (3.A.2a)

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

• (3.A.4)
• (3.A.5)

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• (3.A.6)
• (3.A.7)
• (3.A.8)

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• BA ECA. (3.A.9)
• (1 E)BA E(B C)A
• or
• (3.A.10)

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• (3.A.11)
• (3.A.12)
• (3.A.l)
• (3.A.13)

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• (3.A.l4a)
• (3.A.l4b)
• (3.A.l5)

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Appendix 3A. Discriminant analysis and dummy
regression analysis

• (3.A.l6)

.
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Appendix 3B. Principal-component analysis
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Appendix 3B. Principal-component analysis

• (3.B.1)
• (3.B.2)
• (3.B.3)

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Appendix 3B. Principal-component analysis

• (3.B.4)

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Appendix 3B. Principal-component analysis
.
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Appendix 3B. Principal-component analysis

• (3.B.5)
• (3.B.6)

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Appendix 3B. Principal-component analysis

• (3.B.7)
• (3.B.8)
• (3.B.9)