Value at risk methodology of international index portfolio under soft conditions (fuzzy-stochastic approach)

The approach to modelling uncertainty of the international index portfolio by the value at risk (VAR) methodology under soft conditions by fuzzy-stochastic methodology is described in the paper. The generalised term uncertainty is understood to have two aspects: risk modelled by probability (stochastic methodology) and vagueness sometimes called impreciseness, ambiguity, softness is modelled by fuzzy methodology. Thus, hybrid model is called fuzzy-stochastic model. Input data for a stochastic model are unique distribution functions and crisp (real) data. Input data for fuzzy model are fuzzy numbers and crisp (real) data. Input data for hybrid model are fuzzy probability distribution functions, unique distribution functions, and crisp (real) data. Softly defined VAR model is constructed as hybrid model because it is supposed that the input data are difficult to determine as crisp numbers or as some unique distribution functions. Risk is modelled by stochastic methodology on the VAR basis and vagueness is modelled through the fuzzy numbers. The analytical delta normal VAR methodology for international index portfolio under soft conditions is described including illustrative example. It is shown, that methodology described could be considered to be generalised sensitivity analysis.