Kaczmarek Krzysztof, Dymova Ludmila, Sevastjanov Pavel
Department of Computer Science, Czestochowa University of Technology, Dabrowskiego 73, 42-201 Czestochowa, Poland.
Entropy (Basel). 2020 Aug 25;22(9):932. doi: 10.3390/e22090932.
In this paper, first we show that the variance used in the Markowitz's mean-variance model for the portfolio selection with its numerous modifications often does not properly present the risk of portfolio. Therefore, we propose another treating of portfolio risk as the measure of possibility to earn unacceptable low profits of portfolio and a simple mathematical formalization of this measure. In a similar way, we treat the criterion of portfolio's return maximization as the measure of possibility to get a maximal profit. As the result, we formulate the portfolio selection problem as a bicriteria optimization task. Then, we study the properties of the developed approach using critical examples of portfolios with interval and fuzzy valued returns. The α-cuts representation of fuzzy returns was used. To validate the proposed method, we compare the results we got using it with those obtained with the use of fuzzy versions of seven widely reputed methods for portfolio selection. As in our approach we deal with the bicriteria task, the three most popular methods for local criteria aggregation are compared using the known example of fuzzy portfolio consist of five assets. It is shown that the results we got using our approach to the interval and fuzzy portfolio selection reflect better the essence of this task than those obtained by widely reputed traditional methods for portfolio selection in the fuzzy setting.
在本文中，首先我们表明，马科维茨均值 - 方差模型中用于投资组合选择的方差及其众多修正形式，往往不能恰当地体现投资组合的风险。因此，我们提出将投资组合风险视为投资组合获得不可接受的低利润可能性的一种度量，并对该度量进行简单的数学形式化。同样地，我们将投资组合回报最大化准则视为获得最大利润可能性的一种度量。结果，我们将投资组合选择问题表述为一个双准则优化任务。然后，我们使用具有区间值和模糊值回报的投资组合的关键示例来研究所提出方法的性质。使用了模糊回报的α - 截集表示。为了验证所提出的方法，我们将使用该方法得到的结果与使用七种广泛认可的投资组合选择模糊方法得到的结果进行比较。由于在我们的方法中处理的是双准则任务，因此使用由五种资产组成的模糊投资组合的已知示例，比较了三种最流行的局部准则聚合方法。结果表明，我们用于区间和模糊投资组合选择的方法所得到的结果，比在模糊环境下广泛认可的传统投资组合选择方法所得到的结果，能更好地反映该任务的本质。
