Allocation Processes for Worst Scenarios in Fuzzy Asset Management Using Weighted Average Value-at-Risks

By Yuji Yoshida


A dynamic portfolio allocation is discussed in asset management with fuzziness. By perception-based extension for fuzzy random variables, a dynamic portfolio model for weighted average value-at-risks of fuzzy random variables is introduced. By dynamic programming and mathematical programming, this paper derives analytical solutions of the optimization problem for dynamic worst scenarios. A few numerical examples are given to discuss the results. We find the weighted average value-at-risk is a more reasonable criterion than value-at-risk.

Full Text:



P. Artzner, F. Delbaen, J.-M. Eber and D .Heath, “Coherent measures of risk,” Mathematical Finance, vol.9, pp.203-228, 1999.

R. E. Bellman and L. A. Zadeh, “Decision-making in a fuzzy environment,” Manag. Sci., Ser. B., vol.17, pp.141-164, 1970.

P. Fortemps and M. Roubens, “Ranking and defuzzification methods based on area compensation,” Fuzzy Sets and Systems, vol.82, pp.319-330, 1996.

H. Katagiri, M. Sakawa, K. Kato and I. Nishizaki, “Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability,” European J Oper. Res., vol.188, pp.530-539, 2008.

R. Kruse and K. D. Meyer, Statistics with Vague Data, Riedel Publ. Co., Dortrecht, 1987.

S. Kusuoka, “On law-invariant coherent risk measures,” Advances in Mathematical Economics, vol.3, pp.83-95, 2001.

H. Kwakernaak, “Fuzzy random variables -- I. Definitions and theorem,”

Inform. Sci., vol.15, pp.1-29, 1978.

H. Markowitz, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Blackwell, Oxford, 1990.

S. R. Pliska, Introduction to Mathematical Finance: Discrete-Time Models, Blackwell Publ., New York, 1997.

R. T. Rockafellar and S. Uryasev, “Optimization of conditional value-at-risk,” J. of Risk, vol.2, pp.21-41, 2000.

R. T. Rockafellar and S. Uryasev, “Conditional value-at-risk for general loss distribution functions,” J. of Banking and Finance, vol.26, pp.1443-1471, 2002.

S. M. Ross, An Introduction to Mathematical Finance, Cambridge Univ. Press, Cambridge, 1999.

S. Sarykalin, G. Serraino and S .Uryasev, “Value-at-risk vs. conditional value-at-risk in risk management and optimization," Tutorials in Informs Research, pp.270-294, 2008.

M. C. Steinbach, “Markowitz revisited: Mean-variance model in financial portfolio analysis," SIAM Review, vol.43, pp.31-85, 2001.

H. Tanaka, P, Guo, “Portfolio selection based on upper and lower exponential possibility distributions,” European J Oper. Res., vol.114, pp.115-126, 1999.

H. Tanaka, P. Guo and I. B. Turksen, “Portfolio selection based on fuzzy probabilities and possibility distributions,” Fuzzy Sets and Systems, vol.111, pp.387-397, 2000.

D. Tasche, “Expected shortfall and beyond,” J. of Banking and Finance, vol.26, pp.1519-1533, 2002.J.

Watada, “Fuzzy portfolio model for decision making in investment,” Y. Yoshida (ed): Dynamical aspects in fuzzy decision making, Physica-Verlag, Heidelberg, pp.141-162, 2001.

Y. Yoshida, “Mean values, measurement of fuzziness and variance of fuzzy random variables for fuzzy optimization,” Proceedings of SCIS & ISIS 2006, pp.2277-2282, Sept. 2006.

Y. Yoshida, “Fuzzy extension of estimations with randomness: The perception-based approach,” Proceedings of MDAI 2007, LNAI 4529, Springer, pp.295-306, 2007.

Y. Yoshida, “Perception-based estimations of fuzzy random variables: Linearity and convexity,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol.16, suppl., pp.71-87, 2008.

Y. Yoshida, “An estimation model of value-at-risk portfolio under uncertainty,” Fuzzy Sets and Systems, vol.160, pp.3250-3262, 2009.

Y. Yoshida, “Risk analysis of portfolios under uncertainty: Minimizing average rates of falling,” Journal of Advanced Computational Intelligence and Intelligent Informatics, vol. 15, pp.56-62, 2011.

Y. Yoshida, “Dynamic fuzzy asset management for worst scenarios with average value-at-risks,” Proceedings of CiSt 2018 (2nd IEEE OMCS), IEEE press, Marakech in Morocco, pp.437-442, Oct. 2018.

L. A. Zadeh, “Fuzzy sets,” Inform. and Control, vol.8, pp.338-353, 1965.