Mathematical Portfolio Theory and Analysis

Description

Designed as a self-contained text, this book covers a wide spectrum of topics on portfolio theory. It covers both the classical-mean-variance portfolio theory as well as non-mean-variance portfolio theory. The book covers topics such as optimal portfolio strategies, bond portfolio optimization and risk management of portfolios. In order to ensure that the book is self-contained and not dependent on any pre-requisites, the book includes three chapters on basics of financial markets, probability theory and asset pricing models, which have resulted in a holistic narrative of the topic. Retaining the spirit of the classical works of stalwarts like Markowitz, Black, Sharpe, etc., this book includes various other aspects of portfolio theory, such as discrete and continuous time optimal portfolios, bond portfolios and risk management. The increase in volume and diversity of banking activities has resulted in a concurrent enhanced importance of portfolio theory, both in terms of management perspective (including risk management) and the resulting mathematical sophistication required. Most books on portfolio theory are written either from the management perspective, or are aimed at advanced graduate students and academicians. This book bridges the gap between these two levels of learning. With many useful solved examples and exercises with solutions as well as a rigorous mathematical approach of portfolio theory, the book is useful to undergraduate students of mathematical finance, business and financial management. Preface Contents About the Authors List of Figures Mechanisms of Financial Markets 1.1 Types of Markets 1.2 Market Players 1.3 Financial Instruments 1.3.1 Bonds 1.3.2 Stocks 1.3.3 Derivatives Fundamentals of Probability Theory 2.1 Finite Probability Space 2.2 General Probability Space 2.3 Two Important Distributions 2.3.1 The Binomial Distribution 2.3.2 The Normal Distribution 2.4 Some Important Results 2.5 Least Squares Estimation 2.6 Exercise Asset Pricing Models 3.1 The Binomial Model of Asset Pricing 3.2 The gBm Model 3.3 Exercise Mean-Variance Portfolio Theory 4.1 Return and Risk of a Portfolio 4.2 Estimation of Expected Return, Variance and Covariance 4.3 The Mean-Variance Portfolio Analysis 4.3.1 Minimum Variance Portfolio for Two Risky Assets 4.3.2 Minimum Variance Portfolio for n Risky Assets 4.3.3 The Efficient Frontier for Portfolio of n Risky Assets 4.3.4 The Efficient Frontier for Portfolio of n Risky Assets and a Riskfree Asset 4.4 Capital Asset Pricing Model 4.4.1 Capital Market Line 4.4.2 Security Market Line or CAPM 4.4.3 Pricing Aspects 4.4.4 Single Index Model 4.4.5 Multi-index Models 4.5 Arbitrage Pricing Theory 4.6 Variations of CAPM 4.6.1 Black's Zero-Beta Model 4.6.2 Brennan's After-Tax Model 4.7 Portfolio Performance Analysis 4.8 Exercise Utility Theory 5.1 Basics of Utility Functions 5.2 Risk Attitude of Investors 5.3 More on Utility Theory 5.4 Exercise Non-Mean-Variance Portfolio Theory 6.1 The Safety First Models 6.1.1 Roy's Safety First Criterion 6.1.2 Kataoka's Safety First Criterion 6.1.3 Telser's Safety First Criterion 6.2 Geometric Mean Return 6.3 Semi-variance and Semi-deviation 6.4 Stochastic Dominance 6.4.1 First-Order Stochastic Dominance 6.4.2 Second-Order Stochastic Dominance 6.4.3 Third-Order Stochastic Dominance 6.5 Portfolio Performance Analysis 6.6 Exercise Optimal Portfolio Strategies 7.1 Discrete Time Optimization 7.2 Continuous Time Optimization 7.3 Continuous Time Optimization with Consumption 7.4 Exercise Bond Portfolio Optimization 8.1 Basics of Interest Rates 8.2 Bond Pricing 8.3 Duration 8.4 Duration for a Bond Portfolio 8.5 Immunization Using Duration 8.6 Convexity 8.7 Convexity for a Bond Portfolio 8.8 Applications 8.9 Exercise Risk Management of Portfolios 9.1 Value-at-Risk 9.2 VaR of a Portfolio 9.3 Decomposition of VaR 9.4 Methods for Computing VaR 9.4.1 Historical Simulation Approach 9.4.2 Delta-Gamma Method 9.4.3 Monte Carlo Simulation 9.5 Determination of Volatility 9.6 Exercise Bibliography