Online Algorithms for the Portfolio Selection Problem 1st Edition by Robert Dochow 9783658135270 3658135271

1 Introduction

1.1 Preliminaries

1.2 Motivation and Research Questions

1.3 Structure of the Thesis

2 Portfolio Selection Problems

2.1 Preliminaries

2.1.1 Online and Offline Algorithms

2.1.2 Mathematical Programming

2.1.3 Asset Prices, Conversion Rates and Return Factors

2.2 Selected Portfolio Selection Problems

2.2.1 General Portfolio Selection Problem

2.2.2 Constant Rebalancing Problem

2.2.3 Semi-Portfolio Selection Problem

2.2.4 Semi-Constant Rebalancing Problem

2.2.5 Buy-and-Hold Problem

2.2.6 Conversion Problem

2.3 Standard Working Models

2.3.1 Portfolio Selection Problem

2.3.2 Conversion Problem

2.4 Conclusions

3 Performance Evaluation

3.1 Preliminaries

3.1.1 Problem Statement

3.1.2 Efficient Markets Hypothesis

3.1.3 Time Complexity

3.2 Selected Performance Measures

3.2.1 Measures of Return on Investment

3.2.2 Measures of Risk

3.2.3 Measures of Risk-adjusted Performance

3.3 Selected Benchmarks

3.3.1 Offline Benchmarks: Buy-and-Hold

3.3.2 Offline Benchmarks: Constant Rebalancing

3.3.3 Online Benchmarks

3.4 Statistical Analysis

3.4.1 Selected Statistical Measures

3.4.2 Hypothesis Testing

3.4.3 Selected Sampling Techniques

3.5 Competitive Analysis

3.5.1 Competitive Ratio

3.5.2 Performance Ratio

3.5.3 Comparative Ratio

3.5.4 Average-Case Competitive Ratio

3.5.5 Concept of Universality

3.5.6 Competitive Ratio as Performance Measure

3.6 Conclusions

4 Selected Algorithms from the Literature

4.1 Preliminaries

4.1.1 Virtual Market

4.1.2 Projection onto a Simplex

4.1.3 Information and Algorithms

4.2 Follow-the-Winner Algorithms

4.2.1 Successive Constant Rebalanced Algorithm

4.2.2 Universal Portfolio Algorithm

4.2.3 Exponential Gradient Algorithm

4.2.4 Online Newton Step Algorithm

4.3 Follow-the-Loser Algorithms

4.3.1 Anti Correlation Algorithm

4.3.2 Passive Aggressive Mean Reversion Algorithm

4.3.3 Confidence Weighted Mean Reversion Algorithm

4.3.4 Online Moving Average Mean Reversion Algorithm

4.3.5 Robust Median Reversion Algorithm

4.4 Conclusions

5 Proposed Algorithms with Risk Management

5.1 Preliminaries

5.1.1 Worst-Case Logarithmic Wealth Ratio

5.1.2 Universal Portfolio Algorithm

5.1.3 Risk-adjusted Portfolio Selection Algorithm

5.1.4 Combined Risk-adjusted Portfolio Selection Algorithm

5.2 Comparison of Competitiveness

5.3 Numerical Results

5.4 Conclusions

6 Empirical Testing of Algorithms

6.1 Preliminaries

6.1.1 Algorithms and Parameters

6.1.2 Related Work

6.1.3 Dataset and Description

6.2 Test Design

6.3 Numerical Results: Expected Performance

6.4 Numerical Results: Beating the Benchmark

6.5 Conclusions

7 A Software Tool for Testing Algorithms

7.1 Preliminaries

7.2 Primary Functions

7.2.1 Executing Sampling

7.2.2 Running Algorithms

7.2.3 Measuring Performance

7.3 Conclusions

8 Conclusions and Future Work

8.1 Portfolio Selection Problems

8.2 Online Algorithms with Risk Management

8.3 Empirical Testing

8.4 Concluding Remarks

A Proofs

A.1 Bounds on the Number of Allocations

A.2 Asymptotic Behavior of the Number of Allocations

B Numerical Results

B.1 Numerical Results: Expected Performance

B.2 Numerical Results: Beating the Benchmark