Portfolio optimisation and diversification
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David King holds a BSc in Mathematics (1986) and an MSc in Mathematics (1987), both from the University of Warwick. His investment career commenced in 1989 and he has worked in a variety of roles covering both quantitative analysis and risk analysis and has also managed both active and passive portfolios. He is currently Head of Investment Risk at Schroder Investment Management. Investment Risk Group, Schroder Investment Management, 31 Gresham Street, London EC2V 7QA, UK. Tel: þ 44 020 7658 7675; Fax þ 44 020 7658 2836; E-mail: [email protected]
Abstract Portfolio optimisation can produce overly concentrated portfolios that both practitioners and clients may find difficult to accept. This paper shows, through adjustments to the objective function, how to alter levels of portfolio diversification using the same quadratic programming methodology used in the standard portfolio optimisation process. The ideas can be extended to target differential levels of diversification at multiple levels of asset categorisation and can thus be used in a variety of settings. Journal of Asset Management (2007) 8, 296–307. doi:10.1057/palgrave.jam.2250082 Keywords: portfolio optimisation, diversification, quadratic programming
Introduction It is well documented (see eg Michaud, 1989) that mean–variance optimisers, if left to their own devices, can sometimes lead to unintuitive portfolios with extreme positions in asset classes that are unpalatable to investors and clients alike. These characteristics can prove to be major stumbling blocks to the acceptance of optimisers by portfolio managers as mainstream tools for portfolio construction. Constraints on the weights of asset classes in optimal portfolios can be used to mitigate the effect of extreme positions in optimised portfolios. In many cases, however, the process of imposing lower and upper bounds upon the weights of assets in portfolios can result in prior views on the composition of the desired portfolio being imposed on the process. The optimiser may effectively end up being ‘pointed in the right direction’ by asset class constraints. Over the years, techniques for attempting to improve the portfolio optimisation process have been developed, in the main focusing upon modifications to the input parameters — see, for example, Black and Litterman
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(1992), Ledoit and Wolf (2004) and Fabozzi et al. (2007). Scherer (2004) and Michaud (1998) provide comprehensive discussions of the various approaches adopted. In this paper, we present a nonparametric approach to creating diversified portfolios that takes the form of an adjunct to the standard mean– variance optimisation objective function. Levels of diversification can be varied directly via the use of scalars that are akin to risk aversion parameters. The method also allows diversification at various hierarchical levels to be targeted and thus allows the creation of diversified portfolios at different levels of asset class and classification. The ideas described can be used in conjunction with any methods that target b
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