Properties
Spectral risk measures are also coherent. Every spectral risk measure satisfies:
- Positive Homogeneity: for every portfolio X and positive value, ;
- Translation-Invariance: for every portfolio X and, ;
- Monotonicity: for all portfolios X and Y such that, ;
- Sub-additivity: for all portfolios X and Y, ;
- Law-Invariance: for all portfolios X and Y with cumulative distribution functions and respectively, if then ;
- Comonotonic Additivity: for every comonotonic random variables X and Y, . Note that X and Y are comonotonic if for every .
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