Tsallis entropy of uncertain sets and its application to portfolio allocation

Hua Zhao; Hamed Ahmadzade; Mohammad GhasemiGol

doi:10.3934/jimo.2024032

Article Contents

2024, Volume 20, Issue 9: 2885-2905. Doi: 10.3934/jimo.2024032

This issue Previous Article Penalty method for the sparse portfolio optimization problem Next Article Product pricing, green effort decisions and coordination in a dynamic three-echelon green supply chain

Tsallis entropy of uncertain sets and its application to portfolio allocation

1.
School of Business Management Administration, Chongqing Technology and Business University, Chongqing 400067, China
2.
Department of Statistics, University of Sistan and Baluchestan, Zahedan, Iran
3.
School of Cybersecurity, Old Dominion University Norfolk, VA, US

^*Corresponding author: Hamed Ahmadzade
^*Corresponding author: Hamed Ahmadzade

Received: June 2023

Revised: December 2023

Early access: March 2024

Published: September 2024

Abstract / Introduction Full Text(HTML) Figure(2) / Table(7) Related Papers Cited by

Abstract

Tsallis entropy is a flexible device to measure indeterminacy of uncertain sets. A formula is obtained to calculate Tsallis entropy of uncertain sets via inversion of membership functions. Also, by considering Tsallis entropy as a risk measure, we optimize portfolio selection problems via mean-entropy models.

Keywords:

Mathematics Subject Classification: Primary: 91G10; Secondary: 91G70.

Citation:

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Figure 1. Function of $ T_{\beta}(s) $ for different values of $ \beta. $

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Figure 2. Comparative analysis of difference between performance ratio for same values of $ b $ in Models (7) and (8)

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Table 1. Securities

No	Uncertain Sets
1	$ \xi_1 = (0.02, 0.04, 0.06) $
2	$ \xi_2=(0.01, 0.04, 0.05, 0.07) $
3	$ \xi_3= (0.01, 0.03, 0.09) $
4	$ \xi_4= (-0.02, 0.04, 0.08) $
5	$ \xi_5=(-0.01, 0.05, 0.07, 0.09) $
6	$ \xi_6= (0.03, 0.07, 0.11) $

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Table 2. Expected value and Tsallis entropy

No	Expected value	Tsallis entropy
$ \xi_1 $	0.04	0.01
$ \xi_2 $	0.0425	0.0125
$ \xi_3 $	0.04	0.02
$ \xi_4 $	0.035	0.025
$ \xi_5 $	0.05	0.02
$ \xi_6 $	0.07	0.02

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Table 3. Optimal values in Model (3)

$ \lambda $	$ (y_1, y_2, y_3, y_4, y_5, y_6) $	Expected value
0.011	(0.9, 0, 0, 0, 0, 0.1)	0.043
0.012	(0.8, 0, 0, 0, 0, 0.2)	0.046
0.013	(0.7, 0, 0, 0, 0, 0.3)	0.049
0.015	(0.5, 0, 0, 0, 0, 0.5)	0.055
0.017	(0.3, 0, 0, 0, 0, 0.7)	0.061
0.019	(0.1, 0, 0, 0, 0, 0.9)	0.067

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Table 4. Optimal values in Model (4)

$ \delta $	$ (y_1, y_2, y_3, y_4, y_5, y_6) $	Tsallis entropy
0.045	(0.84, 0, 0, 0, 0, 0.16)	0.0117
0.048	(0.74, 0, 0, 0, , 0, 0.26)	0.0127
0.051	(0.63, 0, 0, 0, 0, 0.37)	0.0137
0.057	(0.43, 0, 0, 0, 0, 0.57)	0.0157
0.059	(0.37, 0, 0, 0, 0, 0.63)	0.0163
0.061	(0.3, 0, 0, 0, 0, 0.7)	0.017
0.064	(0.2, 0, 0, 0, 0, 0.8)	0.018
0.067	(0.1, 0, 0, 0, 0, 0.9)	0.019

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Table 5. Securities

No	Uncertain sets
1	$ \xi_1 = (0.03, 0.05, 0.07) $
2	$ \xi_2=(-0.01, 0.04, 0.09) $
3	$ \xi_3= (0.02, 0.22, 0.42) $
4	$ \xi_4= (0.01, 0.11, 0.21) $
5	$ \xi_5=(0.03, 0.07, 0.11) $
6	$ \xi_6= (0.02, 0.07, 0.12) $
7	$ \xi_7= (0.13, 0.23, 0.33) $
8	$ \xi_8= (0.2, 0.32, 0.44) $
9	$ \xi_9= (0.1, 0.22, 0.34) $
10	$ \xi_{10}= (0.25, 0.3, 0.35) $

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Table 6. Optimal values in Models (5) and (6)

Model	$ (y_1, y_2, y_3, y_4, y_5, y_6, y_7, y_8, y_9, y_{10}) $	Performance ratio
Mean-entropy	(0, 0, 0, 0, 0, 0, 0, 0, 0, 1)	12
Mean-variance	(0.73, 0, 0, 0, 0, 0, 0, 0, 0, 0.27)	8.36

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Table 7. Analysis of 3 indices in Models (7) and (8)

Pair	Asymp.Sig.(Right-tailed)
(CHIME-CHIMV)	0
(CRIME-CRIMV)	0
(CCCIME-CCCIMV)	0

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