Abstract
We propose a computational procedure to find the efficient frontier for the standard Markowitz mean-variance model with discrete variables. The integer constraints limit on the one hand the portfolio to contain a predetermined number of assets and, on the other hand, the proportion of the portfolio held in a given asset. We adapt the multiobjective algorithm NSGA for solving the problem. The algorithm ranks the solutions of each generation in layers based on Pareto non-domination. We have applied the procedure in sixty assets of ATHEX. We have also compared the algorithm with a single genetic algorithm. The computational results indicate that the procedure is promising for this class of problems.
Highlights
Every investor faces the problem of choice the appropriate assets in which he will invest his funds
We propose a computational procedure to find the efficient frontier for the standard Markowitz mean-variance model with discrete variables
This allows multiple Paretooptimal solutions to be found in a single simulation run. It appears that the first who tried to use genetic algorithms for finding the Pareto frontier in a multiobjective optimization problem was Schaffer [14]. His Vector Evaluated Genetic Algorithm (VEGA) gave encouraging results, it suffered from biasness towards some Pareto-optimal solutions
Summary
Every investor faces the problem of choice the appropriate assets in which he will invest his funds. Some other measures of risk have been used, e.g. Value-at-Risk [3,4]; and additional constraints were introduced in the standard model in order, for example, to avoid very small holdings, to restrict the total number of holdings and/or to take into consideration the roundlot of assets that can be bought or sold in a bunch [5] Since these additional constraints lead to sets of discrete variables and constraints, the resulting optimization problem becomes quite complex as it exhibits multiple local extrema and discontinuities [4,6,7,8].
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