rapport_claes - Semester Project Portfolio Optimization Claes Osterlof Supervisors Dr Olivier Leveque Professor Rudiger Urbanke Summer semester 2005

rapport_claes - Semester Project Portfolio Optimization...

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Semester Project Portfolio Optimization Claes Osterlof Supervisors: Dr. Olivier L´ evˆ eque, Professor Rudiger Urbanke Summer semester 2005 Communications Theory Laboratory I&C 1
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Contents 1 Introduction 5 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Basic settings in portfolio optimization . . . . . . . . . . . . . . . . . . . . 6 1.3 Constantly rebalanced portfolio . . . . . . . . . . . . . . . . . . . . . . . . 7 2 The i.i.d. model 8 2.1 The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Windowing for the i.i.d. model . . . . . . . . . . . . . . . . . . . . 9 2.1.2 The forgetting factor for the i.i.d. model . . . . . . . . . . . . . . . 9 3 The Markovian approach 10 3.1 Markov model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Windowing for the Markov model . . . . . . . . . . . . . . . . . . . 11 3.1.2 Forgetting factor for the Markov model . . . . . . . . . . . . . . . . 11 4 Implementations and results 12 4.1 Dealing with the splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 I.i.d. stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2.1 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2.2 Forgetting factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Markov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Markov modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.1 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.2 Forgetting factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.5 Smoothening of the transition matrix . . . . . . . . . . . . . . . . . . . . . 21 4.6 Windowing VS Forgetting factor . . . . . . . . . . . . . . . . . . . . . . . . 22 4.7 I.i.d. VS Markov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Future work 24 6 Conclusion 24 7 Personal notes 25 A Notations 26 B References 26 2
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C Matlab codes 27 3
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Abstract Portfolio optimization can be defined as finding a portfolio to improve the risk/return trade-off of the portfolio. The process seeks to create a portfolio with either the highest potential return at a specific historical risk, or the lowest his- torical risk for a specific potential return. It is also the process of analyzing one’s portfolio, maximizing expected returns for given risks, and rebalancing when nec- essary, to ensure optimal risk-adjusted returns. In this semester project, we revisit mainly two methods of portfolio optimization, one is the i.i.d. case and the other is a Markov model approach. In the i.i.d. approach, we consider that the changes from day to day of the returns are i.i.d.. If the returns really were i.i.d., this would lead to best performance of wealth for our methods. The Markov model builds a Markov chain from the stocks and maximizes the expected value of wealth. Based on these two approaches, some extensions have been considered, such as window- ing and forgetting factors. We start the report with a short historical overview of portfolio optimization as an introduction to our work, then the theoretical aspects behind our two models are developed and finally the implementation and results are discussed as this was the main work during this project. 4
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1 Introduction A portfolio optimization problem is considered as follows: an investor has an initial wealth S 0 at time 0, and wants to maximize his wealth at time T . In order to do this, the investor has to choose how much to invest in each of the considered stocks. In other words, he needs to determine how many shares, in each stock, should be held at each time t, t = 0 , ..., T , in order to maximize his expected wealth at time T . The main objective of portfolio problems is to give a solution to the basic problem mentioned above, which is maximizing wealth.
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