مسائل بهینه سازی نمونه کارها در اقدامات خطر مختلف با استفاده از الگوریتم ژنتیک

This paper introduces a heuristic approach to portfolio optimization problems in different risk measures by employing genetic algorithm (GA) and compares its performance to mean–variance model in cardinality constrained efficient frontier. To achieve this objective, we collected three different risk measures based upon mean–variance by Markowitz; semi-variance, mean absolute deviation and variance with skewness. We show that these portfolio optimization problems can now be solved by genetic algorithm if mean–variance, semi-variance, mean absolute deviation and variance with skewness are used as the measures of risk. The robustness of our heuristic method is verified by three data sets collected from main financial markets. The empirical results also show that the investors should include only one third of total assets into the portfolio which outperforms than those contained more assets.

Expect return and risk are the most important parameters with regard to portfolio optimization problems. One of the main contributions on this problem is by Markowitz, 1952 and Markowtitz, 1991 who introduced mean–variance model, but the standard mean–variance model is based on assumption that investors are risk averse and the return of assets are normally distributed. Jia and Dyer (1996) noted that these conditions are rarely satisfied in practice. The mean–variance objective function may not be the best choice available to investors in terms of an appropriate risk measure. Furthermore, other risk measures may be more appropriate. From a practical point of view, real world investors have to face a lot of constraints in risk models: trading limitation, size of portfolio, etc. Such as constraints may be formed in a nonlinear mixed integer programming problem which is considerably more difficult to solve than the original model. Several researchers have attempted to find this problem by a variety of techniques, but exact solution methods fail to solve large-scale instances of the problem. Therefore, several researchers try to improve algorithms by using the state-of-the-art mathematical programming methodology to solving portfolio problems. The purpose of this paper is to show that portfolio optimization problems containing cardinality constrained efficient frontier can be successfully solved by the state-of-the-art genetic algorithms if we use the different risk measures such as mean–variance, semi-variance, mean absolute deviation and variance with skewness. We also show that practical portfolio optimization problems consisting of different numbers of assets drawn from three main markets stock indices can be solved by a genetic algorithm within a practical amount of time. The remainder of this paper is organized as follows. Section 2 describes the portfolio optimization in the risk measures which we want to solve. In Section 3 investigates basic structure of genetic algorithm. Section 4, our proposed algorithm was introduced. Section 5 provides our computational results using C++ programming. It shows that cardinality constrained portfolio optimization problems can be solved in different risk measures without difficulty. Conclusion is given in Section 6.

Genetic algorithm is robust to solve mixed nonlinear and integer programming problems and effective for solving the portfolio optimization problems in different risk measures. GA prominent advantage over other exact search methods is its flexibility and its ability to easily obtain a good solution to a problem where the other deterministic methods cannot achieve optimality in an easy manner. The main objective of this paper was to investigate genetic algorithm for solving difficult portfolio optimization problems with different risk models. Specifically, a number of portfolio optimization problems including cardinality constraint can be solved by the state-of-the-art GA in a practical amount of time if we use mean–variance, semi-variance and variance with skewness as the measures of risk. The application of our GA in the proposed portfolio optimization problem is attractive because they are able to deal with a class of objective functions which are difficult to solve by other exact search algorithms found in literature. The contribution of this paper showed the efficiency using GA to solve these portfolio optimization problems in different risk measures. It also verified that investors should not consider K values above one third of total assets since they are obviously dominated by those with relatively less K values. The GA method developed in the paper can provide an efficient and convenient tool for investors. With different risk tendencies, investors are able to find efficient frontier based on a fixed amount of assets, as well as a lower bound of each asset to avoid minor investment which might increase transaction costs. In terms of number of assets hold in the portfolio, our research provides a clear fact that a small size of portfolio could have a better performance than those of bigger one. The future study is to extend a class of problems approach such as: (a) Investigating other algorithms and their performance in different risk measures, such as simulated annealing, tabu search and neural network. (b) Applying other algorithms to portfolio optimization problems and comparing the results to those obtained with previous heuristic and mathematical programming algorithms. (c) Analyzing other algorithms that are more appropriate and efficient for specific risk measures.