A benchmark approach to asset management
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Eckhard Platen is a Professor of Quantitative Finance at the University of Technology, Sydney. Prior to this appointment, he was Head of the Centre for Financial Mathematics in the Institute of Advanced Studies at the Australian National University and is Adjunct Professor of this University. He has a PhD in Mathematics from the Technical University in Dresden and obtained his Dr Sc from the Academy of Sciences in Berlin. He is co-author of two well-known books on numerical methods for stochastic differential equations, published by Springer, and has authored more than a hundred papers in finance and applied mathematics. He serves on the editorial boards of five international journals in finance and applicable mathematics. His current research interests cover areas ranging from financial market modelling, portfolio optimisation, derivative pricing and dynamic risk analysis to numerical methods in finance. He initiated and has been co-organising the annual Quantitative Methods in Finance conference series. University of Technology Sydney, School of Finance & Economics and Department of Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia. Tel: +61 2 9514 7759; Fax: +61 2 9514 7711; E-mail: [email protected]
Abstract This paper aims to discuss the optimal selection of investments for the short and long run in a continuous time financial market setting. First, it documents the almost sure pathwise long-run outperformance of all positive portfolios by the growth optimal portfolio. Secondly, it assumes that every investor prefers more rather than less wealth and keeps the freedom to adjust his or her risk aversion at any time. In a general continuous market, a two fund separation result is derived which yields optimal portfolios located on the Markowitz efficient frontier. An optimal portfolio is shown to have a fraction of its wealth invested in the growth optimal portfolio and the remaining fraction in the savings account. The risk aversion of the investor at a given time determines the volatility of her/his optimal portfolio. It is pointed out that it is usually not rational to reduce risk aversion further than is necessary to achieve the maximum growth rate. Assuming an optimal dynamics for a global market, the market portfolio turns out to be growth optimal. The discounted market portfolio is shown to follow a particular time transformed diffusion process with explicitly known transition density. Assuming that the drift of the discounted market portfolio grows exponentially, a parsimonious and realistic model for its dynamics results. It allows for efficient portfolio optimisation and derivative pricing. Keywords: growth optimal portfolio, portfolio selection, two fund separation, risk aversion, minimal market model
Introduction Throughout recent decades, there has been an ongoing debate on the following important question: If an investor invests for the ‘long run’, this means with still
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Journal of Asset Management
several decades to go, is the growth optimal portfolio an appropriate c
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