Fuzzy Numbers and MCDM Methods for Portfolio Optimization
A new deployment of the multiple criteria decision
making (MCDM) techniques: the Simple Additive Weighting
(SAW), and the Technique for Order Preference by Similarity to
Ideal Solution (TOPSIS) for portfolio allocation, is demonstrated in
this paper. Rather than exclusive reference to mean and variance as in
the traditional mean-variance method, the criteria used in this
demonstration are the first four moments of the portfolio distribution.
Each asset is evaluated based on its marginal impacts to portfolio
higher moments that are characterized by trapezoidal fuzzy numbers.
Then centroid-based defuzzification is applied to convert fuzzy
numbers to the crisp numbers by which SAW and TOPSIS can be
deployed. Experimental results suggest the similar efficiency of these
MCDM approaches to selecting dominant assets for an optimal
portfolio under higher moments. The proposed approaches allow
investors flexibly adjust their risk preferences regarding higher
moments via different schemes adapting to various (from
conservative to risky) kinds of investors. The other significant
advantage is that, compared to the mean-variance analysis, the
portfolio weights obtained by SAW and TOPSIS are consistently
well-diversified.
[1] H. M. Markowitz, "Portfolio selection," Journal of Finance, vol. 7, pp.
77-91, 1952.
[2] C. L. Hwang and K. Yoon, Multiple attribute decision making: Methods
and applications. Heidelberg: Springer-Verlag, 1981.
[3] S. H. Zanakis, A. Solomon, N. Wishart, and S. Dublish, "Multi-attribute
decision making: A simulation comparison of select methods,"
European Journal of Operational Research, vol. 107, pp. 507-529,
1998.
[4] Y. H. Chang and C. H. Yeh, "Evaluating airline competitiveness using
multiattribute decision making," Omega , vol. 29, pp. 405-415, 2001.
[5] M. Modarres and S. Sadi-Nezhad, "Fuzzy simple additive weighting
method by preference ratio" Intelligent Automation and Soft Computing,
vol. 11, pp. 235-244, 2005.
[6] S. Y. Chou, Y. H. Chang, and C. Y. Shen, "A fuzzy simple additive
weighting system under group decision-making for facility location
selection with objective/subjective attributes," European Journal of
Operational Research, vol. 189, pp. 132-145, 2008.
[7] E. Bottani and A. Rizzi, "A fuzzy TOPSIS methodology to support
outsourcing of logistics services," Supply Chain Management, vol. 11,
pp. 294-308, 2006.
[8] H. S. Shih, H. J. Shyur, and E. S. Lee, "An extension of TOPSIS for
group decision making," Mathematical and Computer Modelling, vol.
45, pp. 801-813, 2007.
[9] M. A. Abo-Sinna and T. H. Abou-El-Enien, "An interactive algorithm
for large scale multiple objective programming problems with fuzzy
parameters through TOPSIS approach," Applied Mathematics and
Computation, vol. 177, pp. 515-527, 2006.
[10] G. R. Jahanshahloo, F. H. Lotfi, and R. A. Davoodi, "Extension of
TOPSIS for decision-making problems with interval data: Interval
efficiency," Mathematical and Computer Modelling, vol. 49, pp. 1137-
1142, 2009.
[11] Y. Chen, D. M. Kilgour, and K. W. Hipel, "An extreme-distance
approach to multiple criteria ranking," Mathematical and Computer
Modelling, vol. 53, pp. 646-658, 2011.
[12] A. Awasthi, S. S. Chauhan, and S. K. Goyal, "A multi-criteria decision
making approach for location planning for urban distribution centers
under uncertainty," Mathematical and Computer Modelling, vol. 53, pp.
98-109, 2011.
[13] E. Jondeau and M. Rockinger, "Optimal portfolio allocation under
higher moments," European Financial Mangement, vol. 12, pp. 29-55,
2006.
[14] L. Martellini and V. Ziemann, "Improved estimates of higher-order
comoments and implications for portfolio selection," Review of
Financial Studies, vol. 23, pp. 1467-1502, 2010.
[15] L. A. Zadeh, "Fuzzy sets," Information and Control, vol. 8, pp. 338-353,
1965.
[16] R. C. Scott and P. A. Horvath, "On the direction of preference for
moments of higher order than the variance," Journal of Finance, vol. 35,
pp. 915-919, 1980.
[17] T. T. Nguyen and L. N. Gordon-Brown, "Constrained fuzzy hierarchical
analysis for portfolio selection under higher moments," IEEE
Transactions on Fuzzy Systems, vol. 20, no. 4, pp. 666-682, 2012.
[18] Y. M. Wang, J. B.Yang, D. L. Xu, and K. S. Chin, "On the centroids of
fuzzy numbers," Fuzzy Sets and Systems, vol. 157, pp. 919-926, 2006.
[19] Y. J. Wang and H. S. Lee, "The revised method of ranking fuzzy
numbers with an area between the centroid and original points,"
Computers and Mathematics with Applications, vol. 55, pp. 2033-2042,
2008.
[20] H. M. Markowitz, Portfolio selection: Efficient diversification of
investments, New York: Wiley, 1959
[1] H. M. Markowitz, "Portfolio selection," Journal of Finance, vol. 7, pp.
77-91, 1952.
[2] C. L. Hwang and K. Yoon, Multiple attribute decision making: Methods
and applications. Heidelberg: Springer-Verlag, 1981.
[3] S. H. Zanakis, A. Solomon, N. Wishart, and S. Dublish, "Multi-attribute
decision making: A simulation comparison of select methods,"
European Journal of Operational Research, vol. 107, pp. 507-529,
1998.
[4] Y. H. Chang and C. H. Yeh, "Evaluating airline competitiveness using
multiattribute decision making," Omega , vol. 29, pp. 405-415, 2001.
[5] M. Modarres and S. Sadi-Nezhad, "Fuzzy simple additive weighting
method by preference ratio" Intelligent Automation and Soft Computing,
vol. 11, pp. 235-244, 2005.
[6] S. Y. Chou, Y. H. Chang, and C. Y. Shen, "A fuzzy simple additive
weighting system under group decision-making for facility location
selection with objective/subjective attributes," European Journal of
Operational Research, vol. 189, pp. 132-145, 2008.
[7] E. Bottani and A. Rizzi, "A fuzzy TOPSIS methodology to support
outsourcing of logistics services," Supply Chain Management, vol. 11,
pp. 294-308, 2006.
[8] H. S. Shih, H. J. Shyur, and E. S. Lee, "An extension of TOPSIS for
group decision making," Mathematical and Computer Modelling, vol.
45, pp. 801-813, 2007.
[9] M. A. Abo-Sinna and T. H. Abou-El-Enien, "An interactive algorithm
for large scale multiple objective programming problems with fuzzy
parameters through TOPSIS approach," Applied Mathematics and
Computation, vol. 177, pp. 515-527, 2006.
[10] G. R. Jahanshahloo, F. H. Lotfi, and R. A. Davoodi, "Extension of
TOPSIS for decision-making problems with interval data: Interval
efficiency," Mathematical and Computer Modelling, vol. 49, pp. 1137-
1142, 2009.
[11] Y. Chen, D. M. Kilgour, and K. W. Hipel, "An extreme-distance
approach to multiple criteria ranking," Mathematical and Computer
Modelling, vol. 53, pp. 646-658, 2011.
[12] A. Awasthi, S. S. Chauhan, and S. K. Goyal, "A multi-criteria decision
making approach for location planning for urban distribution centers
under uncertainty," Mathematical and Computer Modelling, vol. 53, pp.
98-109, 2011.
[13] E. Jondeau and M. Rockinger, "Optimal portfolio allocation under
higher moments," European Financial Mangement, vol. 12, pp. 29-55,
2006.
[14] L. Martellini and V. Ziemann, "Improved estimates of higher-order
comoments and implications for portfolio selection," Review of
Financial Studies, vol. 23, pp. 1467-1502, 2010.
[15] L. A. Zadeh, "Fuzzy sets," Information and Control, vol. 8, pp. 338-353,
1965.
[16] R. C. Scott and P. A. Horvath, "On the direction of preference for
moments of higher order than the variance," Journal of Finance, vol. 35,
pp. 915-919, 1980.
[17] T. T. Nguyen and L. N. Gordon-Brown, "Constrained fuzzy hierarchical
analysis for portfolio selection under higher moments," IEEE
Transactions on Fuzzy Systems, vol. 20, no. 4, pp. 666-682, 2012.
[18] Y. M. Wang, J. B.Yang, D. L. Xu, and K. S. Chin, "On the centroids of
fuzzy numbers," Fuzzy Sets and Systems, vol. 157, pp. 919-926, 2006.
[19] Y. J. Wang and H. S. Lee, "The revised method of ranking fuzzy
numbers with an area between the centroid and original points,"
Computers and Mathematics with Applications, vol. 55, pp. 2033-2042,
2008.
[20] H. M. Markowitz, Portfolio selection: Efficient diversification of
investments, New York: Wiley, 1959
@article{"International Journal of Information, Control and Computer Sciences:56794", author = "Thi T. Nguyen and Lee N. Gordon-Brown", title = "Fuzzy Numbers and MCDM Methods for Portfolio Optimization", abstract = "A new deployment of the multiple criteria decision
making (MCDM) techniques: the Simple Additive Weighting
(SAW), and the Technique for Order Preference by Similarity to
Ideal Solution (TOPSIS) for portfolio allocation, is demonstrated in
this paper. Rather than exclusive reference to mean and variance as in
the traditional mean-variance method, the criteria used in this
demonstration are the first four moments of the portfolio distribution.
Each asset is evaluated based on its marginal impacts to portfolio
higher moments that are characterized by trapezoidal fuzzy numbers.
Then centroid-based defuzzification is applied to convert fuzzy
numbers to the crisp numbers by which SAW and TOPSIS can be
deployed. Experimental results suggest the similar efficiency of these
MCDM approaches to selecting dominant assets for an optimal
portfolio under higher moments. The proposed approaches allow
investors flexibly adjust their risk preferences regarding higher
moments via different schemes adapting to various (from
conservative to risky) kinds of investors. The other significant
advantage is that, compared to the mean-variance analysis, the
portfolio weights obtained by SAW and TOPSIS are consistently
well-diversified.", keywords = "Fuzzy numbers, SAW, TOPSIS, portfolio
optimization, higher moments, risk management.", volume = "6", number = "12", pages = "1672-13", }
