Allocating risk properly to subunits is crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using coherent measures of risk it is impossible to allocate risk satisfying simultaneously the natural game theoretical requirements of Core Compatibility and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games as being the only risk allocation method satisfying Strong Monotonicity, Equal Treatment Property and Efficiency. Moreover, we clarify and interpret the related game theoretical requirements that have appeared in the literature so far and have been applied to risk allocation.
The water balance of Zala County was analyzed using remote-sensing based actual evapotranspiration (ETA) (Kovacs 2011) and runoff (R) in the context of land cover types. The highest mean ETA rates were determined for water bodies (658 mm/year) and wetlands (622 mm/year). Forests have higher values than agricultural areas, and the lowest rates belong to artificial surfaces. Mean annual runoff is the largest on artificial surfaces (89 mm/year), and it decreases on the other land cover types. For climate change impact analysis a Budyko-model was used in spatially-distributed mode. The parameter of the Budyko-model (α) was calculated for pixels without surplus water. For the extra-water affected pixels a linear model with β-parameter (actual evapotranspiration / panevapotranspiration) was used. These parameters (α and β) can be used for evaluating future ETA and R in spatially-distributed mode. According to the predictions, the mean annual evapotranspiration may increase about 27 mm while the runoff may decrease to the one third to the present amount by end of the century