Contents

Preface xv

About the Authors xvii

CHAPTER 1

Introduction 1

A Few Notes on Notation 3

Overview 4

CHAPTER 2

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

CHAPTER 3

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

Illustration: Posterior Trade-off and the Normal Mean

Parameter 35

Summary 37

Appendix: Definitions of Some Univariate and Multivariate

Statistical Distributions 38

The Univariate Normal Distribution 39

The Univariate Student’s t-Distribution 39

The Inverted χ2 Distribution 39

The Multivariate Normal Distribution 40

The Multivariate Student’s t-Distribution 40

The Wishart Distribution 41

The Inverted Wishart Distribution 41

CHAPTER 4

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

CHAPTER 5

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

CHAPTER 6

Bayesian Framework For Portfolio Allocation 92

Classical Portfolio Selection 94

Mean-Variance Efficient Frontier 97

Illustration: Mean-Variance Optimal Portfolio

with Portfolio Constraints 99

Bayesian Portfolio Selection 101

Prior Scenario 1: Mean and Covariance with Diffuse

(Improper) Priors 102

Prior Scenario 2: 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

CHAPTER 7

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

Appendix A: Numerical Simulation of the Predictive

Distribution 135

Sampling from the Predictive Distribution 136

CHAPTER 8

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 τ and _ 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

CHAPTER 9

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, _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