Preparing for the Worst Incorporating Downside Risk in Stock Market Investments

, by Vinod, Hrishikesh (Rick) D.; Reagle, Derrick

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ISBN: 9780471234425 | 0471234427
Cover: Hardcover
Copyright: 10/22/2004

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Stock market investors have very different reactions to downside versus upside risk. This book begins by explaining the current treatment of stock market risk and methods of lowering that risk. The authors then show that many types of asymmetry of stock returns or investor reactions cause the existing theory to fail. They present the theory of downside risk and utility theory to account for the asymmetry, showing how the previous model can be adjusted for downside risk.

List of Figures

xiii

List of Tables

xvii

Preface

xix

1. Quantitative Measures of the Stock Market

(28)

1.1. Pricing Future Cash Flows,

(5)

1.2. The Expected Return,

(3)

1.3. Volatility,

(5)

1.4. Modeling of Stock Price Diffusion,

(8)

1.4.1. Continuous Time,

(1)

1.4.2. Jump Diffusion,

(1)

1.4.3. Mean Reversion in the Diffusion Context,

(1)

1.4.4. Higher Order Lag Correlations,

(1)

1.4.5. Time-Varying Variance,

(2)

1.5. Efficient Market Hypothesis,

(4)

1.5.1. Weak Form Efficiency,

(1)

1.5.2. Semi-strong Form Efficiency,

(1)

1.5.3. Strong Form Efficiency,

(1)

Appendix: Simple Regression Analysis,

(3)

2. A Short Review of the Theory of Risk Measurement

(26)

2.1. Quantiles and Value at Risk,

(10)

2.1.1. Pearson Family as a Generalization of the Normal Distribution,

(4)

2.1.2. Pearson Type IV Distribution for Our Mutual Fund Data,

(1)

2.1.3. Nonparametric Value at Risk (VaR) Calculation from Low Percentiles,

(1)

2.1.4. Value at Risk for Portfolios with Several Assets,

(2)

2.2. CAPM Beta, Sharpe, and Treynor Performance Measures,

(5)

2.2.1. Using CAPM for Pricing of Securities,

(1)

2.2.2. Using CAPM for Capital Investment Decisions,

(1)

2.2.3. Assumptions of CAPM,

(1)

2.3. When You Assume

(2)

2.3.1. CAPM Testing Issues,

(2)

2.4. Extensions of the CAPM,

(6)

2.4.1. Observable Factors Model,

(1)

2.4.2. Arbitrage Pricing Theory (APT) and Construction of Factors,

(2)

2.4.3. Jensen, Sharpe, and Treynor Performance Measures,

(2)

Appendix: Estimating the Distribution from the Pearson Family of Distributions,

(3)

3. Hedging to Avoid Market Risk

(20)

3.1. Derivative Securities: Futures, Options,

(8)

3.1.1. A Market for Trading in Futures,

(3)

3.1.2. How Smart Money Can Lose Big,

(1)

3.1.3. Options Contracts,

(2)

3.2. Valuing Derivative Securities,

(5)

3.2.1. Binomial Option Pricing Model,

(1)

3.2.2. Option Pricing from Diffusion Equations,

(2)

3.3. Option Pricing Under Jump Diffusion,

(2)

3.4. Implied Volatility and the Greeks,

(3)

3.4.1. The Role of Δ of an Option for Downside Risk,

(1)

3.4.2. The Gamma (Γ) of an Option,

(1)

3.4.3. The Omega, Theta, Vega, and Rho of an Option,

(2)

Appendix: Drift and Diffusion,

(2)

4. Monkey Wrench in the Works: When the Theory Fails

(26)

4.1. Bubbles, Reversion, and Patterns,

(5)

4.1.1. Calendar Effects,

(1)

4.1.2. Mean Reversion,

(2)

4.1.3. Bubbles,

(1)

4.2. Modeling Volatility or Variance Explicitly,

(3)

4.3. Testing for Normality,

(5)

4.3.1. The Logistic Distribution Compared with the Normal,

(2)

4.3.2. Empirical cdf and Quantile-Quantile (Q-Q) Plots,

(1)

4.3.3. Kernel Density Estimation,

(2)

4.4. Alternative Distributions,

(13)

4.4.1. Pareto (Pareto-Levy, Stable Pareto) Distribution,

(1)

4.4.2. Inverse Gaussian,

(1)

4.4.3. Laplace (Double-Exponential) Distribution,

(1)

4.4.4. Azzalini's Skew-Normal (SN) Distribution,

(4)

4.4.5. Lognormal Distribution,

(1)

4.4.6. Stable Pareto and Pareto-Levy Densities,

(5)

5. Downside Risk

101

(18)

5.1. VaR and Downside Risk,

101

(9)

5.1.1. VaR from General Parametric Densities,

103

(1)

5.1.2. VaR from Nonparametric Empirical Densities,

104

(3)

5.1.3. VaR for Longer Time Horizon (τ > 1) and the IGARCH Assumption,

107

(2)

5.1.4. VaR in International Setting,

109

(1)

5.2. Lower Partial Moments (Standard Deviation, Beta, Sharpe, and Treynor),

110

(5)

5.2.1. Sharpe and Treynor Measures,

113

(2)

5.3. Implied Volatility and Other Measures of Downside Risk,

115

(4)

6. Portfolio Valuation and Utility Theory

119

(36)

6.1. Utility Theory,

119

(10)

6.1.1. Expected Utility Theory (EUT),

120

(5)

6.1.2. A Digression: Derivation of Arrrow-Pratt Coefficient of Absolute Risk Aversion (CARA),

125

(1)

6.1.3. Size of the Risk Premium Needed to Encourage Risky Equity Investments,

126

(1)

6.1.4. Taylor Series Links EUT, Moments of f(χ), and Derivatives of U(χ),

127

(2)

6.2. Nonexpected Utility Theory,

129

(6)

6.2.1. A Digression: Lorenz Curve Scaling over the Unit Square,

129

(2)

6.2.2. Mapping from EUT to Non-EUT within the Unit Square,

131

(4)

6.3. Incorporating Utility Theory into Risk Measurement and Stochastic Dominance,

135

(12)

6.3.1. Class D1 of Utility Functions and Investors,

135

(1)

6.3.2. Class D2 of Utility Functions and Investors,

135

(1)

6.3.3. Explicit Utility Functions and Arrow-Pratt Measures of Risk Aversion,

136

(1)

6.3.4. Class D3 of Utility Functions and Investors,

137

(1)

6.3.5. Class D4 of Utility Functions and Investors,

137

(2)

6.3.6. First-Order Stochastic Dominance (1SD),

139

(1)

6.3.7. Second-Order Stochastic Dominance (2SD),

140

(1)

6.3.8. Third-Order Stochastic Dominance (3SD),

141

(1)

6.3.9. Fourth-Order Stochastic Dominance (4SD),

142

(1)

6.3.10. Empirical Checking of Stochastic Dominance Using Matrix Multiplications and Incorporation of 4DPs of on-EUT,

142

(5)

6.4. Incorporating Utility Theory into Option Valuation,

147

(1)

6.5. Forecasting Returns Using Nonlinear Structures and Neural Networks,

148

(7)

6.5.1. Forecasting with Multiple Regression Models,

149

(1)

6.5.2. Qualitative Dependent Variable Forecasting Models,

150

(2)

6.5.3. Neural Network Models,

152

(3)

7. Incorporating Downside Risk

155

(26)

7.1. Investor Reactions

155

(5)

7.1.1. Irrational Investor Reactions,

156

(1)

7.1.2. Prospect Theory,

156

(2)

7.1.3. Investor Reaction to Shocks,

158

(2)

7.2. Patterns of Downside Risk,

160

(3)

7.3. Downside Risk in Stock Valuations and Worldwide Investing,

163

(8)

7.3.1. Detecting Potential Downturns from Growth Rates of Cash Flows,

163

(1)

7.3.2. Overoptimistic Consensus Forecasts by Analysts,

164

(2)

7.3.3. Further Evidence Regarding Overoptimism of Analysts,

166

(1)

7.3.4. Downside Risk in International Investing,

166

(1)

7.3.5. Legal Loophole in Russia,

167

(1)

7.3.6. Currency Devaluation Risk,

167

(1)

7.3.7. Forecast of Currency Devaluations,

168

(1)

7.3.8. Time Lags and Data Revisions,

169

(1)

7.3.9. Government Interventions Make Forecasting Currency Markets Difficult,

170

(1)

7.4. Downside Risk Arising from Fraud, Corruption, and International Contagion,

171

(10)

7.4.1. Role of Periodic Portfolio Rebalancing,

172

(1)

7.4.2. Role of International Financial Institutions in Fighting Contagions,

172

(1)

7.4.3. Corruption and Slow Growth of Derivative Markets,

173

(1)

7.4.4. Corruption and Private Credit Rating Agencies,

174

(1)

7.4.5. Corruption and the Home Bias,

175

(3)

7.4.6. Value at Risk (VaR) Calculations Worsen Effect of Corruption,

178

(3)

8. Mathematical Techniques

181

(26)

8.1. Matrix Algebra,

181

(13)

8.1.1. Sum, Product, and Transpose of Matrices,

181

(2)

8.1.2. Determinant of a Square Matrix and Singularity,

183

(2)

8.1.3. The Rank and Trace of a Matrix,

185

(1)

8.1.4. Matrix Inverse, Partitioned Matrices, and Their Inverse,

186

(1)

8.1.5. Characteristic Polynomial, Eigenvalues, and Eigenvectors,

187

(2)

8.1.6. Orthogonal Vectors and Matrices,

189

(1)

8.1.7. Idempotent Matrices,

190

(1)

8.1.8. Quadratic and Bilinear Forms,

191

(1)

8.1.9. Further Study of Eigenvalues and Eigenvectors,

192

(2)

8.2. Matrix-Based Derivation of the Efficient Portfolio,

194

(1)

8.3. Principal Components Analysis, Factor Analysis, and Singular Value Decomposition,

195

(4)

8.4. Ito's Lemma,

199

(2)

8.5. Creation of Risk-Free Nonrandom g(S,t) as a Hedge Portfolio,

201

(1)

8.6. Derivation of Black-Scholes Partial Differential Equation,

202

(3)

8.7. Risk-Neutral Case,

205

(2)

9. Computational Issues

207

(40)

9.1. Sampling, Compounding, and Other Data Issues in Finance,

207

(6)

9.1.1. Random Sampling Implicit in the Available Data,

209

(1)

9.1.2. Sampling Distribution of the Sample Mean,

209

(1)

9.1.3. Compounding of Returns and the Lognormal Distribution,

209

(2)

9.1.4. Relevance of Geometric Mean of Returns in Compounding,

211

(1)

9.1.5. Sampling Distribution of Average Compound Return,

212

(1)

9.2. Numerical Procedures,

213

(5)

9.2.1. Faulty Rounding Stony,

213

(1)

9.2.2. Numerical Errors in Excel Software,

214

(1)

9.2.3. Binary Arithmetic,

214

(1)

9.2.4. Round-off and Cancellation Errors in Floating Point Arithmetic,

215

(1)

9.2.5. Algorithms Matter,

215

(1)

9.2.6. Benchmarks and Self-testing of Software Accuracy,

216

(1)

9.2.7. Value at Risk Implications,

217

(1)

9.3. Simulations and Bootstrapping,

218

(12)

9.3.1. Random Number Generation,

218

(2)

9.3.2. An Elementary Description of Simulations,

220

(3)

9.3.3. Simple iid Bootstrap in Finance,

223

(1)

9.3.4. A Description of the iid Bootstrap for Regression,

223

(7)

Appendix A: Regression Specification, Estimation, and Software Issues,

230

(4)

Appendix B: Maximum Likelihood Estimation Issues,

234

(6)

Appendix C: Maximum Entropy (ME) Bootstrap for State-Dependent Time Series of Returns,

240

(7)

10. What Does It All Mean?

247

(6)

Glossary of Greek Symbols

253

(4)

Glossary of Notations

257

(4)

Glossary of Abbreviations

261

(4)

References

265

(12)

Name Index

277

(4)

Index

281

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Preparing for the Worst Incorporating Downside Risk in Stock Market Investments

Summary

Author Biography

Table of Contents

Supplemental Materials