On risk measures and capital allocation for distributions depending on parameters with interval or fuzzy uncertainty

Since risks are regarded as emerging from uncertainty, they can be modeled using probabilistic, interval or fuzzy methods. The probabilistic literature on risk measures, though well developed, quantifies the risks by single values, which could seem restrictive for risk managers who would like to have more insight into the phenomena, like, e.g., an interval covering the single value. Therefore, in this paper, we study the VaR and TVaR risk measures for distributions with parameters of interval type, further extended to fuzzy numbers. In particular, we concentrate on the class of location and/or scale parameters, showing that in this case, the resulting risk measures are also in the form of intervals or, respectively, fuzzy numbers. Moreover, we apply the results to the capital allocation problem and detail the procedure for the normal, Pareto and Farlie-Gumbel-Morgenstern particular distributions. The formulas are numerically illustrated on interval and fuzzy parameters for some classical distributions; in this sense, some applications on real data sets are discussed.