| Περίληψη : | For both investment firms and private investors, portfolio management can be a very hard and time-consuming process. Unfortunately, there is no standard method to construct an optimal portfolio, applicable in every case. Portfolio optimization is a multi-objective problem involving simultaneously return maximization and risk minimisation objectives. Thus, typical optimization approaches aim to eliminate suboptimal solutions, i.e. portfolios for which expected return can increase without increasing risk, or vice versa. A family of solutions is often derived defining the so-called efficient frontier in return-risk space. Selecting an optimal portfolio located on the efficient frontier depends of subjective factor, i.e. the investor's attitude to risk.The first part of the report presents the fundamental theory of portfolio optimization and different formulations of optimization problems whose solution defines the efficient frontier. The starting point of the chapter is the Markowicz theory, in which risk is represented by the variance of the portfolio's return, so that the two conflicting objectives which need to be balanced are expected return and portfolio variance. The model can be extended by incorporating portfolio diversification constraints and transaction costs. Alternative models are also considered in which risk is represented by different metrics, such as the portfolios Value at Risk (VaR) and Conditional Value at Risk (CVaR), sometimes called expected shortfall. The resulting optimization problem in this case is Integer Linear Programming or standard linear Programming, respectively. In this formulation probabilistic constraints are translated into standard linear constraints on the portfolio's parameters by sampling the random joint distribution of the portfolio's assets if this is known, or by considering historic data of the assets' variations. Alternative methods are also reviewed and the corresponding optimization problems are formulated and solved. These include portfolios with risk-free assets, Sharp-ratio methods, methods based on Utility functions, etc.The second part of the report applies the theory summarised in the first part of the report to real financial data. Two sets of data are considered. The first consists of three asset indexes, specifically stocks (based on the S&P 500 index), bonds (based on the 10-year US Treasury bond index) and cash invested in the money market whose return is assumed to correspond to the 1-day Federal Fund rate. The time-series for these three assets is constructed from annual prices between years 1960 and 2003. The model in this case is relatively simple and its main purpose is to illustrate the theory presented in the first part 1 of the report. In the second case study the dataset consists of the closing prices of 30 assets selected from the Dow Jones listed companies for 1000 consecutive trading days between 22 May 2017 and 13 May 2021. The various optimization algorithms have been implemented in Matlab, which is also applied to data analysis in both case studies to investigate the composition of the portfolios lying on the efficient frontier, the portfolios' sensitivity to estimation errors in the elements of the covariance matrix of the assets' return, the effect of transaction costs, etc. The conclusions of the report are summarised in Section 3, while the References and Matlab code are included in Sections 4 and 5, respectively.
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