Paper:

# Risk Analysis of Portfolios Under Uncertainty: Minimizing Average Rates of Falling

## Yuji Yoshida

Faculty of Economics and Business Administration, University of Kitakyushu, 4-2-1 Kitagata, Kokuraminami, Kitakyushu 802-8577, Japan

A portfolio model to minimize the risk of falling under uncertainty is discussed. The risk of falling is represented by the value-at-risk of rate of return. Introducing the perception-based extension of the average value-at-risk, this paper formulates a portfolio problem to minimize the risk of falling with fuzzy random variables. In the proposed model, randomness and fuzziness are evaluated respectively by the probabilistic expectation and the mean with evaluation weights and λ-mean functions. The analytical solutions of the portfolio problem regarding the risk of falling are given. This paper gives formulae to show the explicit relations among the following important parameters in portfolio: the expected rate of return, the risk probability of falling and bankruptcy, and the average rate of falling regarding the asset prices. A numerical example is given to explain how to obtain the optimal portfolio and these parameters from the asset prices in the stock market.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.15, No.1, pp. 56-62, 2011.

- [1] H. Markowitz, “Mean-Variance Analysis in Portfolio Choice and Capital Markets,” Blackwell, Oxford, 1990.
- [2] M. C. Steinbach, “Markowitz revisited: Mean-variance model in financial portfolio analysis,” SIAM Review, Vol.43, pp. 31-85, 2001.
- [3] Z. Zmeškal, “Value at risk methodology of international index portfolio under soft conditions (fuzzy-stochastic approach),” Int. Review of Financial Analysis, Vol.14, pp. 263-275, 2005.
- [4] Y. Yoshida, “Fuzzy extension of estimations with randomness: The perception-based approach,” in: V. Torra, Y. Narukawa and Y. Yoshida, eds., ‘Modeling Decisions for Artificial Intelligence – MDAI 2007,’ Lecture Notes in Artificial Intelligence, Vol.4617 (Springer, August, 2007), pp. 295-306, 2007.
- [5] Y. Yoshida, “Mean values, measurement of fuzziness and variance of fuzzy random variables for fuzzy optimization,” Proc. of SCIS & ISIS 2006, (Joint 3rd Intern. Conf. on Soft Comp. and Intell. Sys. and 7th Intern. Symp. on Advanced Intell. Sys.), Tokyo, (Sept., 2006) pp. 2277-2282, 2006.
- [6] Y. Yoshida, “A perception-based portfolio under uncertainty: Minimization of average rates of falling,” in: V. Torra, Y. Narukawa and M. Inuiguchi, eds., ‘Modeling Decisions for Artificial Intelligence – MDAI 2009,’ Lecture Notes in Artificial Intelligence, Vol.5861 (Springer, November, 2009), pp. 149-160, 2009.
- [7] L. El Chaoui, M. Oks, and F. Oustry, “Worst-case value at risk and robust portfolio optimization: A conic programming approach,” Operations Research, Vol.51, pp. 543-556, 2003.
- [8] A. Meucci, “Risk and Asset Allocation,” Springer-Verlag, Heidelberg, 2005.
- [9] Z. Zmeškal, “Value at risk methodology under soft conditions approach (fuzzy-stochastic approach),” European J. Oper. Res., Vol.161, pp. 337-347, 2005.
- [10] L. A. Zadeh, “Fuzzy sets,” Inform. and Control, Vol.8, pp. 338-353, 1965.
- [11] H. Kwakernaak, “Fuzzy random variables – I. Definitions and theorem,” Inform. Sci., Vol.15, pp. 1-29, 1978.
- [12] M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” J. Math. Anal. Appl., Vol.114, pp. 409-422, 1986.
- [13] R. Kruse and K. D. Meyer, “Statistics with Vague Data,” Riedel Publ. Co., Dortrecht, 1987.
- [14] R. R. Yager, “A procedure for ordering fuzzy subsets of the unit interval.,” Inform. Sciences, Vol.24, pp. 143-161, 1981.
- [15] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent measures of risk,” Mathematical Finance, Vol.9, pp. 203-228, 1999.
- [16] L. Campos and A.Munoz, “A subjective approach for ranking fuzzy numbers,” Fuzzy Sets and Systems, Vol.29, pp. 145-153, 1989.
- [17] C. Carlsson and R. Fullér, “On possibilistic mean value and variance of fuzzy numbers,” Fuzzy Sets and Systems, Vol.122, pp. 315-326, 2001.
- [18] P. Fortemps and M. Roubens, “Ranking and defuzzification methods based on area compensation,” Fuzzy Sets and Systems, Vol.82, pp. 319-330, 1996.
- [19] M. López-Díaz and M. A. Gil, “The λ-average value and the fuzzy expectation of a fuzzy random variable,” Fuzzy Sets and Systems, Vol.99, pp. 347-352, 1998.
- [20] Y. Yoshida, “A mean estimation of fuzzy numbers by evaluation measures,” in: M. Ch. Negoita et al. eds., ‘Knowledge-Based Intelligent Information and Engineering Systems, Part II,’ Lecture Notes in Artificial Intelligence, Vol.3214 pp. 1222-1229. Springer, September, 2004.
- [21] Y. Yoshida, M. Yasuda, J. Nakagami, and M. Kurano, “A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty,” Fuzzy Sets and Systems, Vol.157, pp. 2614-2626, 2006.
- [22] M. Inuiguchi and T. Tanino, “Portfolio selection under independent possibilistic information,” Fuzzy Sets and Systems, Vol.115, pp. 83-92, 2000.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 International License.