The expected **equity premium** is an important parameter in many financial models, especially within portfolio optimization. A good forecast of the future equity premium is therefore of great interest. In this thesis we seek to forecast the equity premium, use it in portfolio optimization and then give evidence on how sensitive the results are to estimation errors and how the impact of these can be minimized.Linear prediction models are commonly used by practitioners to forecast the expected equity premium, this with mixed results. To only choose the model that performs the best in-sample for forecasting, does not take model uncertainty into account. Our approach is to still use linear prediction models, but also taking model uncertainty into consideration by applying Bayesian model averaging…

*Contents*

1 Introduction

1.1 Objectives

1.2 Problem deﬁnition

1.3 Limitations

1.4 Contributions

1.5 Outline

I Equity Premium Forecasting using Bayesian Statistics

2 The Equity Premium

2.1 What is the *equity premium*?

2.2 Historical models

2.3 Implied models

2.4 Conditional models

2.5 Multi factor models

2.6 A short summary of the models

2.7 What is a good model?

2.8 Chosen model

3 Linear Regression Models

3.1 Basic deﬁnitions

3.2 The classical regression assumptions

3.3 Robustness of OLS estimates

3.4 Testing the regression assumptions

4 **Bayesian Statistics**

4.1 Basic deﬁnitions

4.2 Suﬃcient statistics

4.3 Choice of prior

4.4 Marginalization

4.5 Bayesian model averaging

4.6 Using BMA on linear regression models

5 The Data Set and Linear Prediction

5.1 Chosen series

5.2 The historical equity premium

5.3 Factors explaining the equity premium

5.4 Testing the assumptions of linear regression

5.5 Forecasting by linear regression

6 Implementation

6.1 Overview

6.2 Linear prediction

6.3 Bayesian model averaging

6.4 Backtesting

7 Results

7.1 Univariate forecasting

7.2 **Multivariate forecasting**

7.3 Results from the backtest

8 Discussion of the Forecasting

II Using the Equity Premium in Asset Allocation

9 Portfolio Optimization

9.1 Solution of the Markowitz problem

9.2 Estimation error in Markowitz portfolios

9.3 The method of portfolio resampling

9.4 An example of portfolio resampling

9.5 Discussion of portfolio resampling

10 Backtesting Portfolio Performance

10.1 Backtesting setup and results

11 Conclusions

Bibliography

A Mathematical Preliminaries

A.1 Statistical deﬁnitions

A.2 Statistical distributions

B Code

B.1 Univariate predictions

B.2 Multivariate predictions

B.3 Merge time series

B.4 Load data into Matlab from Excel

B.5 Permutations

B.6 Removal of outliers and linear prediction

B.7 setSubColumn

B.8 Portfolio resampling

B.9 Quadratic optimization

Author: Bjurgert, Johan

Source: Linköping University

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