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

5. Portfolio Modelling
   1. Terminology
   2. Short Sales
   3. Minimizing Risk
   4. Statistical Parameters of an N-Asset Portfolio
   5. Envelope of Financial Assets
6. Mean-Variance Analysis
   1. Principle of Two Fund Separation
   2. Efficient Frontier
   3. Mean-Variance Relation
   4. Mean-Variance Relation for 2-Asset Portfolios
7. 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
8. Bibliography