Krzysztof Piasecki , Joanna Siwek

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The main purpose of this article is to present characteristics of a two-asset portfolio in case of present values of assets being given by a triangular fuzzy number. Fuzzy expected discounting factor for a portfolio and imprecision risk assessments for the imprecision burdening a portfolio were appointed throughout the paper. Thanks to this, the influence of portfolio diversification on an imprecision risk was analyzed and some interesting conclusions were stated.


two-asset portfolio, present value, triangular fuzzy number, discounting factor


Buckley I. J., (1987), The Fuzzy Mathematics of Finance, Fuzzy Sets and Systems, 21, 257–273.

Caplan B., (2001), Probability, Common Sense, and Realism: a Reply to Hulsmann and Block, The Quarterly Journal of Austrian Economics, 4 (2), 69–86.

Chiu, C. Y., Park, C. S., (1994), Fuzzy Cash Flow Analysis Using Present Worth Criterion, The Engineering Economist, 39 (2), 113–138.

Czerwiński Z., (1960), Enumerative Induction and the Theory of Games, Studia Logica, 10, 29–38.

Czerwiński Z., (1969), Matematyka na usługach ekonomii, PWN, Warszawa.

Duan L., Stahlecker P., (2011), A Portfolio Selection Model Using Fuzzy Returns, Fuzzy Optimization and Decision Making, 10 (2), 167–191.

Dubois D., Prade H., (1978), Operations on Fuzzy Numbers, International Journal of Systems Science 9, 613–626.

Dubois D., Prade H., (1979), Fuzzy Real Algebra: Some Results, Fuzzy Sets and Systems, 2, 327–348.

Fang Y., Lai K. K., Wang S., (2008), Fuzzy Portfolio Optimization. Theory and Methods, Lecture Notes in Economics and Mathematical Systems, 609, Springer, Berlin.

Greenhut J. G., Norman G., Temponi C. T., (1995), Towards a Fuzzy Theory of Oligopolistic Competition, IEEE Proceedings of ISUMA-NAFIPS, 286–291.

Guo S., Yu L., Li X., Kar S., (2016), Fuzzy Multi-Period Portfolio Selection with Different Investment Horizons, European Journal of Operational Research, 254 (3), 1026–1035.

Gupta P., Mehlawat M. K., Inuiguchi M., Chandra S., (2014), Fuzzy Portfolio Optimization. Advances in Hybrid Multi-criteria Methodologies, Studies in Fuzziness and Soft Computing, 316, Springer, Berlin.

Gutierrez I., (1989), Fuzzy Numbers and Net Present Value, Scandinavian Journal of Management, 5 (2), 149–159.

Hiroto K., (1981), Concepts of Probabilistic Sets, Fuzzy Sets and Systems, 5, 31–46.

Huang X., (2007a), Two New Models for Portfolio Selection with Stochastic Returns Taking Fuzzy Information, European Journal of Operational Research, 180 (1), 396–405.

Huang X., (2007b), Portfolio Selection with Fuzzy Return, Journal of Intelligent & Fuzzy Systems, 18 (4), 383–390.

Khalili S., (1979), Fuzzy Measures and Mappings, Journal of Mathematical Analysis and Applications, 68, 92–99.

Klir G. J., (1993), Developments In Uncertainty-Based Information, Advances in Computers, 36, 255–332.

Knight F. H., (1921), Risk, Uncertainty, and Profit, Hart, Schaffner & Marx, Houghton Mifflin Company, Boston, MA.

Kolmogorov A. N., (1933), Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin.

Kolmogorov A. N., (1956), Foundations of the Theory of Probability, Chelsea Publishing Company, New York.

Kosko B., (1986), Fuzzy Entropy and Conditioning, Information Sciences, 40, 165–174.

Kosko B., (1990), Fuzziness vs Probability, International Journal of General Systems, 17 (2/3), 211–240.

Kuchta D., (2000), Fuzzy Capital Budgeting, Fuzzy Sets and Systems, 111, 367–385.

Lambalgen M. von, (1996), Randomness and Foundations of Probability: Von Mises’ Axiomatization of Random Sequences, Institute of Mathematical Statistics Lecture Notes – Monograph Series 30, 347–367.

Lesage C., (2001), Discounted Cash-Flows Analysis. An Interactive Fuzzy Arithmetic Approach, European Journal of Economic and Social Systems, 15 (2), 49–68.

Li Ch., Jin J., (2011), Fuzzy Portfolio Optimization Model with Fuzzy Numbers, w: Li S., Wang X., Okazaki Y., Kawabe J., Murofushi T., Guan L., (red.), Nonlinear Mathematics for Uncertainty and its Applications, Advances in Intelligent and Soft Computing, 100, 557–565.

Liu Y.-J., Zhang W.-G., (2013), Fuzzy Portfolio Optimization Model Under Real Constraints, Insurance: Mathematics and Economics, 53 (3), 704–711.

de Luca A., Termini S., (1972), A Definition of a Non-Probabilistic Entropy in The Settings of Fuzzy Set Theory, Information and Control, 20, 301–313.

de Luca A., Termini S., (1979), Entropy And Energy Measures Of Fuzzy Sets, w: Gupta M. M., Ragade R. K., Yager R. R., (red.), Advances in Fuzzy Set Theory and Applications, 321–338.

Markowitz H. S. M., (1952), Portfolio Selection, Journal of Finance, 7 (1), 77–91.

Mehlawat M. K., (2016), Credibilistic Mean-Entropy Models for Multi-Period Portfolio Selection with Multi-Choice Aspiration Levels, Information Science, 345, 9–26.

Mises R. von, (1957), Probability, Statistics And Truth, The Macmillan Company, New York.

Mises L. von, (1962), The Ultimate Foundation of Economic Science an Essay on Method, D. Van Nostrand Company, Inc., Princeton.

Piasecki K., (2011a), Rozmyte zbiory probabilistyczne, jako narzędzie finansów behawioralnych, Wyd. UE, Poznań.

Piasecki K., (2011b), Effectiveness of Securities with Fuzzy Probabilistic Return, Operations Research and Decisions, 21 (2), 65–78.

Piasecki K., (2011c), Behavioural Present Value, SSRN Electronic Journal, DOI:10.2139/ssrn.1729351.

Piasecki K., (2014b), Behawioralna wartość bieżąca – nowe podejście, Optimum Studia Ekonomiczne 67, 36–45.

Piasecki K., Siwek J., (2015), Behavioural Present Value Defined as Fuzzy Number – a New Approach, Folia Oeconomica Stetinensia, 15 (2), 27–41.

Saborido R., Ruiz A. B., Bermúdez J. D., Vercher E., Luque M., (2016), Evolutionary Multi-Objective Optimization Algorithms for Fuzzy Portfolio Selection, Applied Soft Computing, 39, 48–63.

Sadowski W., (1977), Decyzje i prognozy, PWN, Warszawa.

Sadowski W., (1980), Forecasting and Decision Making, Quantitative Wirtschafts- und Unternehmensforschung, Springer-Verlag, Berlin Heidelberg.

Sheen J. N., (2005), Fuzzy Financial Profitability Analyses Of Demand Side Management Alternatives From Participant Perspective, Information Sciences, 169, 329–364.

Siwek J., (2015), Portfel dwuskładnikowy – studium przypadku dla wartości bieżącej danej jako trójkątna liczba rozmyta, Studia Ekonomiczne. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach 241, 140–150.

Tsao C.-T., (2005), Assessing the Probabilistic Fuzzy Net Present Value For a Capital, Investment Choice Using Fuzzy Arithmetic, Journal of Chinese Insitute of Industrial Engineers, 22 (2), 106–118.

Ward T. L., (1985), Discounted Fuzzy Cash Flow Analysis, 1985 Fall Industrial Engineering Conference Proceedings, 476–481.

Wu X.-L., Liu Y. K., (2012), Optimizing Fuzzy Portfolio Selection Problems by Parametric Quadratic Programming, Fuzzy Optimization and Decision Making, 11 (4), 411–449.

Zadeh L., (1965), Fuzzy Sets, Information and Control, 8, 338–353.

Zhang X., Zhang W.-G., Xiao W., (2013), Multi-Period Portfolio Optimization under Possibility Measures, Economic Modelling, 35, 401–408.

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