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Mathematical Models in Portfolio Analysis

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Language:  English
This book explains portfolio modelling in financial mathematics as a consistent mathematical theory. Topics include mean-variance portfolio analysis and capital market theory.
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This book explains portfolio modelling in financial mathematics as a consistent mathematical theory with all steps justified. The topics include mean-variance portfolio analysis and capital market theory. The book contains many examples with solutions. Linear algebra rather than calculus is used as foundation for portfolio analysis; this approach is more conceptual and helps to avoid tedious calculations. The reader does not need much previous mathematical knowledge, only interest in mathematics and its financial applications because the book provides a general mathematical introduction.

  • Preface
  • Part 1: Mathematical Introduction

  1. Matrices and Applications
    1. Terminology
    2. Matrix Operations
    3. Determinants
    4. Systems of Linear Equations
    5. Positive Definite Matrices
    6. Hyperbola
  2. Orthogonal Projection
    1. Orthogonal Projection onto a Subspace
    2. Orthogonal Projection onto a Vector
    3. Minimal Property of Orthogonal Projection
  3. Random Variables
    1. Numerical Characteristics of a Random Variable
    2. Covariance and Correlation Coefficient
    3. Covariance Matrix
  4. Regression
    1. Euclidean Space of Random Variables
    2. Regression
    3. Regression to a Constant
    4. Simple Linear Regression
  • Part 2: Portfolio Analysis
  1. Portfolio Modelling
    1. Terminology
    2. Short Sales
    3. Minimizing Risk
    4. Statistical Parameters of an N-Asset Portfolio
    5. Envelope of Financial Assets
  2. Mean-Variance Analysis
    1. Principle of Two Fund Separation
    2. Efficient Frontier
    3. Mean-Variance Relation
    4. Mean-Variance Relation for 2-Asset Portfolios
  3. Capital Market Theory
    1. Risk-Free Asset
    2. Tangency Portfolio
    3. Market Portfolio. Capital Market Line
    4. Finding Market Portfolio and CML Equation
    5. Regression in Finance. Beta Coefficient
    6. Capital Asset Pricing Model and Security Market Line
  4. Bibliography
About the Author

Farida Kachapova

Farida Kachapova graduated from the Moscow State University, Russia, with a Master Degree in mathematics with distinction and completed a Ph.D. in mathematical logic in the same university in 1987. She also has a Graduate Diploma in Teaching from the University of Auckland, New Zealand.

Farida has industry experience from Russia and New Zealand and decades of experience in tertiary education. She was a mathematics lecturer in the Moscow Institute of Steel and Alloys, Russia, for 9 years. She moved to New Zealand in 1999. Since 2001 Farida is a senior lecturer in the Auckland University of Technology where she teaches courses in several areas of mathematics.

Farida is the author of over 80 research publications in the areas of mathematical logic, statistical mechanics, financial mathematics, and mathematics education.