Using medians in portfolio optimization

Abstract

Some new portfolio optimization models are formulated by adopting the sample median instead of the sample mean as the investment efficiency measure. The median is a robust statistic, which is less affected by outliers than the mean, and in portfolio models this is particularly relevant as data are often characterized by attributes such as skewness, fat tails and jumps, which may strongly bias the mean estimate. As in mean/variance optimization, the portfolio problems are formulated as finding the optimal weights, for example, wealth allocation, which maximize the portfolio median, with risk constrained by some risk measure, respectively, the Value-at-Risk, the Conditional Value-at-Risk, the Mean Absolute Deviation and the Maximum Loss, for a whole of four different models. All these models are formulated as mixed integer linear programming problems, which, at least for moderate sized problems, are efficiently solved by standard software. Models are tested on real financial data, compared to some benchmark portfolios, and found to give good results in terms of realized profits. An important feature is greater portfolio diversification than that obtained with other portfolio models.

Keywords:

• finance
• risk management
• portfolio models
• mixed-integer linear programming
• robust statistics

Acknowledgements

Three anonymous referees are acknowledged for their comments. This paper was partly developed while the author was working as visiting professor at the Instituto Matematicas Antonio de Castro Brzezicki of the Universidad de Sevilla. All institute personnel are kindly acknowledged for their support.