Bayesian Methods in Finance

by Rachev, Svetlozar T.; Hsu, John S. J.; Bagasheva, Biliana S.; Fabozzi, Frank J.

• ISBN13: 9780471920830
• ISBN10: 0471920835
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-02-08
• Publisher: Wiley

Summary

This first-of-its-kind book explains and illustrates the fundamentals of the Bayesian methodology and their applications to finance in clear and accessible terms.

Author Biography

Svetlozar T. Rachev, PhD, Doctor of Science, is Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief-Scientist of FinAnalytica Inc.

John S. J. Hsu, PhD, is Professor of Statistics and Applied Probability at the University of California, Santa Barbara.

Biliana S. Bagasheva, PhD, has research interests in the areas of risk management, portfolio construction, Bayesian methods, and financial econometrics. Currently, she is a consultant in London.

Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance and Becton Fellow at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

Table of Contents

Preface	p. xv
About the Author's	p. xvii
Introduction	p. 1
A Few Notes on Notation	p. 3
Overview	p. 4
The Bayesian Paradigm	p. 6
The Likelihood Function	p. 6
The Poisson Distribution Likelihood Function	p. 7
The Normal Distribution Likelihood Function	p. 9
The Bayes' Theorem	p. 10
Bayes' Theorem and Model Selection	p. 14
Bayes' Theorem and Classification	p. 14
Bayesian Inference for the Binomial Probability	p. 15
Summary	p. 21
Prior and Posterior Information, Predictive Inference	p. 22
Prior Information	p. 22
Informative Prior Elicitation	p. 23
Noninformative Prior Distributions	p. 25
Conjugate Prior Distributions	p. 27
Empirical Bayesian Analysis	p. 28
Posterior Inference	p. 30
Posterior Point Estimates	p. 30
Bayesian Intervals	p. 32
Bayesian Hypothesis Comparison	p. 32
Bayesian Predictive Inference	p. 34
Illustration: Posterior Trade-off and the Normal Mean Parameter	p. 35
Summary	p. 37
Definitions of Some Univariate and Multivariate Statistical Distributions	p. 38
The Univariate Normal Distribution	p. 39
The Univariate Student's t-Distribution	p. 39
The Inverted x[superscript 2] Distribution	p. 39
The Multivariate Normal Distribution	p. 40
The Multivariate Student's t-Distribution	p. 40
The Wishart Distribution	p. 41
The Inverted Wishart Distribution	p. 41
Bayesian Linear Regression Model	p. 43
The Univariate Linear Regression Model	p. 43
Bayesian Estimation of the Univariate Regression Model	p. 45
Illustration: The Univariate Linear Regression Model	p. 53
The Multivariate Linear Regression Model	p. 56
Diffuse Improper Prior	p. 58
Summary	p. 60
Bayesian Numerical Computation	p. 61
Monte Carlo Integration	p. 61
Algorithms for Posterior Simulation	p. 63
Rejection Sampling	p. 64
Importance Sampling	p. 65
MCMC Methods	p. 66
Linear Regression with Semiconjugate Prior	p. 77
Approximation Methods: Logistic Regression	p. 82
The Normal Approximation	p. 84
The Laplace Approximation	p. 89
Summary	p. 90
Bayesian Framework For Portfolio Allocation	p. 92
Classical Portfolio Selection	p. 94
Portfolio Selection Problem Formulations	p. 95
Mean-Variance Efficient Frontier	p. 97
Illustration: Mean-Variance Optimal Portfolio with Portfolio Constraints	p. 99
Bayesian Portfolio Selection	p. 101
Mean and Covariance with Diffuse (Improper) Priors	p. 102
Mean and Covariance with Proper Priors	p. 103
The Efficient Frontier and the Optimal Portfolio	p. 105
Illustration: Bayesian Portfolio Selection	p. 106
Shrinkage Estimators	p. 108
Unequal Histories of Returns	p. 110
Dependence of the Short Series on the Long Series	p. 112
Bayesian Setup	p. 112
Predictive Moments	p. 113
Summary	p. 116
Prior Beliefs and Asset Pricing Models	p. 118
Prior Beliefs and Asset Pricing Models	p. 119
Preliminaries	p. 119
Quantifying the Belief About Pricing Model Validity	p. 121
Perturbed Model	p. 121
Likelihood Function	p. 122
Prior Distributions	p. 123
Posterior Distributions	p. 124
Predictive Distributions and Portfolio Selection	p. 126
Prior Parameter Elicitation	p. 127
Illustration: Incorporating Confidence about the Validity of an Asset Pricing Model	p. 128
Model Uncertainty	p. 129
Bayesian Model Averaging	p. 131
Illustration: Combining Inference from the CAPM and the Fama and French Three-Factor Model	p. 134
Summary	p. 135
Numerical Simulation of the Predictive Distribution	p. 135
Sampling from the Predictive Distribution	p. 136
Likelihood Function of a Candidate Model	p. 138
The Black-Litterman Portfolio Selection Framework	p. 141
Preliminaries	p. 142
Equilibrium Returns	p. 142
Investor Views	p. 144
Distributional Assumptions	p. 144
Combining Market Equilibrium and Investor Views	p. 146
The Choice of [tau] and [Omega]	p. 147
The Optimal Portfolio Allocation	p. 148
Illustration: Black-Litterman Optimal Allocation	p. 149
Incorporating Trading Strategies into the Black-Litterman Model	p. 153
Active Portfolio Management and the Black-Litterman Model	p. 154
Views on Alpha and the Black-Litterman Model	p. 157
Translating a Qualitative View into a Forecast for Alpha	p. 158
Covariance Matrix Estimation	p. 159
Summary	p. 161
Market Efficiency and Return Predictability	p. 162
Tests of Mean-Variance Efficiency	p. 164
Inefficiency Measures in Testing the CAPM	p. 167
Distributional Assumptions and Posterior Distributions	p. 168
Efficiency under Investment Constraints	p. 169
Illustration: The Inefficiency Measure, [Delta superscript R]	p. 170
Testing the APT	p. 171
Distributional Assumptions, Posterior and Predictive Distributions	p. 172
Certainty Equivalent Returns	p. 173
Return Predictability	p. 175
Posterior and Predictive Inference	p. 177
Solving the Portfolio Selection Problem	p. 180
Illustration: Predictability and the Investment Horizon	p. 182
Summary	p. 183
Vector Autoregressive Setup	p. 183
Volatility Models	p. 185
Garch Models of Volatility	p. 187
Stylized Facts about Returns	p. 188
Modeling the Conditional Mean	p. 189
Properties and Estimation of the GARCH(1,1) Process	p. 190
Stochastic Volatility Models	p. 194
Stylized Facts about Returns	p. 195
Estimation of the Simple SV Model	p. 195
Illustration: Forecasting Value-at-Risk	p. 198
An Arch-Type Model or a Stochastic Volatility Model?	p. 200
Where Do Bayesian Methods Fit?	p. 200
Bayesian Estimation of ARCH-Type Volatility Models	p. 202
Bayesian Estimation of the Simple GARCH(1,1) Model	p. 203
Distributional Setup	p. 204
Mixture of Normals Representation of the Student's t-Distribution	p. 206
GARCH(1,1) Estimation Using the Metropolis-Hastings Algorithm	p. 208
Illustration: Student's t GARCH(1,1) Model	p. 211
Markov Regime-switching GARCH Models	p. 214
Preliminaries	p. 215
Prior Distributional Assumptions	p. 217
Estimation of the MS GARCH(1,1) Model	p. 218
Sampling Algorithm for the Parameters of the MS GARCH(1,1) Model	p. 222
Illustration: Student's t MS GARCH(1,1) Model	p. 222
Summary	p. 225
Griddy Gibbs Sampler	p. 226
Drawing from the Conditional Posterior Distribution of [nu]	p. 227
Bayesian Estimation of Stochastic Volatility Models	p. 229
Preliminaries of SV Model Estimation	p. 230
Likelihood Function	p. 231
The Single-Move MCMC Algorithm for SV Model Estimation	p. 232
Prior and Posterior Distributions	p. 232
Conditional Distribution of the Unobserved Volatility	p. 233
Simulation of the Unobserved Volatility	p. 234
Illustration	p. 236
The Multimove MCMC Algorithm for SV Model Estimation	p. 237
Prior and Posterior Distributions	p. 237
Block Simulation of the Unobserved Volatility	p. 239
Sampling Scheme	p. 241
Illustration	p. 241
Jump Extension of the Simple SV Model	p. 241
Volatility Forecasting and Return Prediction	p. 243
Summary	p. 244
Kalman Filtering and Smoothing	p. 244
The Kalman Filter Algorithm	p. 244
The Smoothing Algorithm	p. 246
Advanced Techniques for Bayesian Portfolio Selection	p. 247
Distributional Return Assumptions Alternative to Normality	p. 248
Mixtures of Normal Distributions	p. 249
Asymmetric Student's t-Distributions	p. 250
Stable Distributions	p. 251
Extreme Value Distributions	p. 252
Skew-Normal Distributions	p. 253
The Joint Modeling of Returns	p. 254
Portfolio Selection in the Setting of Nonnormality: Preliminaries	p. 255
Maximization of Utility with Higher Moments	p. 256
Coskewness	p. 257
Utility with Higher Moments	p. 258
Distributional Assumptions and Moments	p. 259
Likelihood, Prior Assumptions, and Posterior Distributions	p. 259
Predictive Moments and Portfolio Selection	p. 262
Illustration: HLLM's Approach	p. 263
Extending The Black-Litterman Approach: Copula Opinion Pooling	p. 263
Market-Implied and Subjective Information	p. 264
Views and View Distributions	p. 265
Combining the Market and the Views: The Marginal Posterior View Distributions	p. 266
Views Dependence Structure: The Joint Posterior View Distribution	p. 267
Posterior Distribution of the Market Realizations	p. 267
Portfolio Construction	p. 268
Illustration: Meucci's Approach	p. 269
Extending The Black-Litterman Approach:Stable Distribution	p. 270
Equilibrium Returns Under Nonnormality	p. 270
Summary	p. 272
Some Risk Measures Employed in Portfolio Construction	p. 273
CVaR Optimization	p. 276
A Brief Overview of Copulas	p. 277
Multifactor Equity Risk Models	p. 280
Preliminaries	p. 281
Statistical Factor Models	p. 281
Macroeconomic Factor Models	p. 282
Fundamental Factor Models	p. 282
Risk Analysis Using a Multifactor Equity Model	p. 283
Covariance Matrix Estimation	p. 283
Risk Decomposition	p. 285
Return Scenario Generation	p. 287
Predicting the Factor and Stock-Specific Returns	p. 288
Risk Analysis in a Scenario-Based Setting	p. 288
Conditional Value-at-Risk Decomposition	p. 289
Bayesian Methods for Multifactor Models	p. 292
Cross-Sectional Regression Estimation	p. 293
Posterior Simulations	p. 293
Return Scenario Generation	p. 294
Illustration	p. 294
Summary	p. 295
References	p. 298
Index	p. 311
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