Value at risk methodology under soft conditions approach (fuzzy-stochastic approach) Zden ek Zme  skal Faculty of Economics, V  SB-Technical University of Ostrava, Sokolsk a 33, Ostrava 701 21, Czech Republic Accepted 25 August 2003 Available online 5 December 2003 Abstract The paper describes methodology of dealing with financial modelling under uncertainty with risk and vagueness aspects. An approach to modelling risks by the Value at Risk methodology under imprecise and soft conditions is solved. It is supposed that the input data and problem conditions is difficult to determine as real numbers or as some precise distribution function. Thus, vagueness is modelled through the fuzzy numbers of the linear T-number type. The combination of risk and vagueness is solved by fuzzy-stochastic methodology. Illustrative example is introduced. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Banking; Decision support systems; Finance; Fuzzy sets; Risk analysis; Uncertainty modelling 1. Introduction Typical feature of financial environment is uncertainty. This term is understood mostly as risk uncertainty (probability, stochastic) and is mod- elled by stochastic apparatus. However, the term uncertainty has the second aspect––the vagueness (sometimes called imprecision, non-preciseness, ambiguity) which is often neglected and could be modelled by fuzzy methodology. In this respect it is apparent that the general term uncertainty in- cludes two aspects: risk (stochastic) and vagueness (fuzzy) ones. These terms will be used in the paper. Distinguishing of risk and vagueness confirms a discussion in financial decision making for several years (see Keynes, 1937; Ellesberg, 1961; Olsen and Throuhton, 2000). What are the impacts of uncertainty on decision making? Interesting char- acteristics are described in Olsen and Throuhton (2000) here vagueness is called ambiguity. (1) Uncertainty influences selection. (2) Decision makers are ambiguity averse in general. (3) Ambiguity causes more weight to be placed on negative information. (4) Buyers pay lower prices for and insurers require higher premium on objects or hazards subject to greater difficulty in estima- tion of value or probability of outcome. (5) Risk aversion and ambiguity aversion have not been seen to be highly correlated. Recently, an application of fuzzy-stochastic methodology in financial modelling is being ex- tensively studied now. We can see four basic application areas. (1) Valuation of financial in- struments where basic assumptions of von Neu- mann and Morgenstern expected utility theory are E-mail address: zdenek.zmeskal@vsb.cz (Z. Zme  skal). 0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.08.048 European Journal of Operational Research 161 (2005) 337–347 www.elsevier.com/locate/dsw