Introduction and motivation
Modern portfolio theory is based on Harry Markowitz’s work on mean-variance portfolios , which stated that a rational investor should either maximise their expected return for a given level of risk, or minimise their risk for a given return. These two principles lead to an efficient frontier of portfolios, among which the investor is free to choose.
More than sixty years on, there is no widely accepted practical implementation of the mean-variance portfolio theory. Typical optimisers rely on quadratic programming and deterministic algorithms to find "optimal" portfolios. However, when practical limitations of the financial markets, such as transaction costs and minimal transaction lots, are included in the model it becomes an NP-hard problem and a global optimal solution cannot be obtained by traditional mathematics programming techniques.
In this paper an evolutionary algorithm (EA), which allows for much more freedom in the functional form, is used. These types of algorithms are heuristic and stochastic search methods, and are often well-suited to find good solutions to optimisation problems where the search space has many local minima and/or there are no known well-performing deterministic search methods. The proposed solution with no short-selling constraint has been extensively tested on several portfolios ranging in size from a few to hundreds of stocks. Please note that if the no short-selling constraint was to be relaxed, the EA would be more complex, since additional constraints would need to be introduced, such as maximum leverage level, maximum short position size, maximum proportion of assets to be held short and etc. Please see  for the detailed discussion of the impact of the short-selling constraint on the portfolio optimisation.
The rest of the paper is organised as follows. The next section sets out the description of the problem to be solved. Section 3 illustrates the EA which has been implemented, together with the chromosomes’ representation, fitness functions and genetic operators used. Section 4 provides details of the parameter settings for the various experiments carried out with the proposed method. Section 5 discusses the experimental results of the proposed method. Conclusions are drawn in Section 6.
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