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<p>Portfolio selection problems, considering returns of the securities as uncertain variables, are an important area of contemporary research. In this line, linear and zigzag uncertainty distributions are popularly being used. These distributions contain two and three parameters, <i>a</i>,&#xa0;<i>b</i> and <i>a</i>,&#xa0;<i>b</i>,&#xa0;<i>c</i>, respectively. In this paper, two families of uncertainty distributions containing one arbitrary constant each have been introduced, and the properties are studied. Linear and zigzag uncertainty distributions then become a particular member of the respective family. This is achieved by introducing the arbitrary constants <i>h</i> and <i>k</i>, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10700_2025_9452_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="156" /> </InlineMediaObject> <EquationSource Format="TEX">\(0 \le h&lt; 1,~0&lt;k&lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>h</mi> <mo>&lt;</mo> <mn>1</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mn>0</mn> <mo>&lt;</mo> <mi>k</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, representing the values of the cumulative uncertainty level at <i>a</i> (for linear) and <i>b</i> (for zigzag). Then, by varying the values of <i>h</i> and <i>k</i>, that is, by changing the slope(s) of the line segment(s), one can fit different distributions having one or two line segments. The newly introduced families have been successfully applied to model and solve portfolio selection problems in an uncertain environment. The solution that fulfills the investor’s requirements may then be chosen. The proposed method of solution has been illustrated by a numerical example. In constructing portfolio selection problems, we considered the adjustment of securities and the transaction costs involved. The solutions obtained for optimal returns and risks for different values of the arbitrary constants are compared.</p>

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Generalized mean semi-absolute deviation model of portfolio selection based on uncertainty theory

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Sanjoy Chhatri,
  • Debasish Bhattacharya,
  • Birojit Das

摘要

Portfolio selection problems, considering returns of the securities as uncertain variables, are an important area of contemporary research. In this line, linear and zigzag uncertainty distributions are popularly being used. These distributions contain two and three parameters, a, b and a, b, c, respectively. In this paper, two families of uncertainty distributions containing one arbitrary constant each have been introduced, and the properties are studied. Linear and zigzag uncertainty distributions then become a particular member of the respective family. This is achieved by introducing the arbitrary constants h and k, \(0 \le h< 1,~0<k<1\) 0 ≤ h < 1 , 0 < k < 1 , representing the values of the cumulative uncertainty level at a (for linear) and b (for zigzag). Then, by varying the values of h and k, that is, by changing the slope(s) of the line segment(s), one can fit different distributions having one or two line segments. The newly introduced families have been successfully applied to model and solve portfolio selection problems in an uncertain environment. The solution that fulfills the investor’s requirements may then be chosen. The proposed method of solution has been illustrated by a numerical example. In constructing portfolio selection problems, we considered the adjustment of securities and the transaction costs involved. The solutions obtained for optimal returns and risks for different values of the arbitrary constants are compared.