Bayesian Methods in Finance Book

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