Abstract
Markowitz's classical model and other models derived from it have raised the portfolio problem using statistical instruments that assume a regular and efficient market. In this paper, the authors propose an alternative method using fuzzy numbers to represent the uncertainty of the future return on assets. Subsequently, we define measures for risk and for excess return on a portfolio. The resulting problem is a nonlinear multi-objective program with fuzzy parameters. Finally, we introduce one method to solve the resulting problem and an example.
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Notes
For example, Tobin (1958) and Sharpe (1970), in order to simplify the problem, assume that R p =R f , but this is not a very realistic case. It is one of the hypotheses of an efficient market.
In our study, we do not consider the possibility that x k <0, that is, short sales are not allowed. Black's portfolio model (see Black, 1972) includes this possibility.
Markowitz's Mean-Semivariance Model (1989), for example, takes this consideration into account.
References
Black, F. (1972) ‘Capital Market Equilibrium with Restricted Borrowing’, Journal of Business, 45 (3), 444–455.
Elton, E. J. and Gruber, M. J. (1995) Modern Portfolio Theory and Investment Analysis, John Wiley & Sons, New York.
Francis, J. C. and Archer, S. H. (1979) Portfolio Analysis, Prentice-Hall, New Jersey.
Gil Aluja, J. (1997) ‘Towards a New Paradigm of Investment Selection in Uncertainty’, Fuzzy Sets and Systems, 84, 187–197.
Isaac, R. M. and James, D. (2000) ‘Just Who Are You Calling Risk Averse’, Journal of Risk and Uncertainty, 20 (2), 177–187.
Markowitz, H. M. (1952) ‘Portfolio Selection’, Journal of Finance, 7, 77–91.
Markowitz, H. M. (1959) Portfolio Selection, John Wiley & Sons, New York.
Markowitz, H. M. (1989) Mean-Variance Analysis in Portfolio Choice and Capital Markets, Basil Blackwell, Oxford.
Ortí, F. (1999) ‘Optimización Borrosa. Aplicación al problema de Análisis y Selección de Carteras de Valores’, Doctoral Thesis, Universitat Rovira i Virgili, Reus (Spain).
Ortí, F., Sáez, J. and Terceño, A. (1999) ‘Valoración de la eficiencia de títulos de renta variable. Aplicación al mercado español’, Actas del VI Congreso de S.I.G.E.F., Morelia (México), pp. 132–140.
Sakawa, M. (1993) Fuzzy Sets and Interactive Multi-objective Optimization, Plenum Press, New York.
Sakawa, M. and Yano, H. (1989) ‘Interactive Decision Making for Multi-objective Nonlinear Programming Problems with Fuzzy Parameters’, Fuzzy Sets and Systems, 29, 315–326.
Sharpe, W. F. (1970) Portfolio Theory and Capital Markets, McGraw-Hill, New York.
Tanaka, H. and Guo, P. (1999) ‘Portfolio Selection Based on Possibility Theory’, in R. A. Ribeiro, H. J. Zimmermann, R. R. Yager and J. Kacprzyk, (eds.), Soft Computing in Financial Engineering. Studies in Fuzziness and Soft Computing, Vol. 28, Physica-Verlag, Heidelberg, 159–185.
Tanaka, H., Guo, P. and Türksen, I. B. (2000) ‘Portfolio Selection Based on Fuzzy Probabilities and Possibility Distributions’, Fuzzy Sets and Systems, 111, 387–397.
Terceño, A., Sáez, J. and Ortí, F. (1996) ‘Tratamiento de la Incertidumbre en la Modelización del Problema de Constitución de Carteras Óptimas’, Actas del III Congreso de S.I.G.E.F., Vol. III, Buenos Aires (Argentina), Paper 2.59.
Tobin, J. (1958) ‘Liquidity Preference as Behavior Toward Risk’, Review of Economics Studies, 25, 65–86.
Turner, A. L. and Weigel, E. J. (1990) ‘An Analysis of Stock Market Volatility’, in Russell Research Commentaries, Frank Russell Co, (ed.) Tacoma, WA.
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Ortí, F., Sáez, J. Portfolio optimisation: A fuzzy multi-objective approach. J Asset Manag 9, 138–148 (2008). https://doi.org/10.1057/jam.2008.16
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DOI: https://doi.org/10.1057/jam.2008.16