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Portfolio Opt - Practical Portfolio Optimization K V...

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Practical Portfolio Optimization K V Fernando NAG Ltd Wilkinson House Jordan Hill Oxford OX2 8DR United Kingdom email: [email protected]
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i Abstract NAG Libraries have many powerful and reliable optimizers which can be used to solve large portfolio optimization and selection problems in the financial industry. These versatile routines are also suitable for academic research and teaching. Key words Markowitz, mean-variance analysis, optimal portfolios, minimum variance portfolio, portfolio selection, portfolio allocation, portfolio diversification, portfolio optimization, efficient frontier, mean-variance frontier, MV efficiency
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ii Contents 1 Introduction 1 2 NAG Routines for Optimization 2 2.1 A Selection of Library Routines . . . . . . . . . . . . . . . . . 2 2.2 Quadratic Programming with Linear Constraints . . . . . . . 3 2.3 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . 3 2.4 Routines for Sparse Matrix Problems . . . . . . . . . . . . . . 3 2.5 Forward and Reverse Communication . . . . . . . . . . . . . 4 2.6 Hardware, Operating Systems and Environments . . . . . . . 4 3 Interfaces to Routines 4 3.1 Portfolio Weights . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Primary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 General Linear Constraints . . . . . . . . . . . . . . . . . . . 5 3.4 Nonlinear Constraints . . . . . . . . . . . . . . . . . . . . . . 6 3.5 Cold and Warm Starts . . . . . . . . . . . . . . . . . . . . . . 6 4 The Optimization Problems 6 5 Processing of Raw Data 7 5.1 The Covariance Matrix in Factored Form . . . . . . . . . . . 7 5.2 Determination of the Singular Values of the Cholesky Factors 9 5.3 If the Covariance Matrix Already Exists . . . . . . . . . . . . 9 5.4 Eigenvalues of the Covariance Matrix . . . . . . . . . . . . . . 10 5.5 Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Numerical Examples: Selection of Equities 10 7 Numerical Example: Asset Allocation 15 8 Transactions Costs 15 9 An Example Program 17 10 Acknowledgements 20
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1 1 Introduction The selection of assets or equities is not just a problem of finding attractive investments. Designing the correct portfolio of assets cannot be done by human intuition alone and requires modern, powerful and reliable mathe- matical programs called optimizers. The Numerical Algorithms Group Ltd (NAG) is world renowned for its work on numerical algorithms, and NAG routines for optimization are being used extensively in industry, commerce and academia. Many leading financial companies and institutions employ NAG optimizers to select, diversify and rebalance their portfolios. They are also used by business and management schools for teaching and research. Any investor would like to have the highest return possible from an in- vestment. However, this has to be counterbalanced by the amount of risk the investor is able or desires to take. The expected return and the risk mea- sured by the variance (or the standard deviation, which is the square-root of the variance) are the two main characteristics of a portfolio. Unfortunately, equities with high returns usually correlate with high risk. The behaviour of a portfolio can be quite different from the behaviour of individual components of the portfolio. The risk of a properly constructed portfolio from equities in leading markets could be half the sum of the risks of individual assets in the portfolio. This is due to complex correlation patterns between individual assets or equities. A good optimizer can exploit the
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