Spectral risk measure

Definition

Consider a portfolio X (denoting the portfolio payoff). Then a spectral risk measure M ϕ : L → R where ϕ is non-negative, non-increasing, right-continuous, integrable function defined on [ 0 , 1 ] such that ∫ 0 1 ϕ ( p ) d p = 1 is defined by

where F X is the cumulative distribution function for X.

If there are S equiprobable outcomes with the corresponding payoffs given by the order statistics X 1 : S , . . . X S : S . Let ϕ ∈ R S . The measure M ϕ : R S → R defined by M ϕ ( X ) = − δ ∑ s = 1 S ϕ s X s : S is a spectral measure of risk if ϕ ∈ R S satisfies the conditions

1. Nonnegativity: ϕ s ≥ 0 for all s = 1 , … , S ,
2. Normalization: ∑ s = 1 S ϕ s = 1 ,
3. Monotonicity : ϕ s is non-increasing, that is ϕ s 1 ≥ ϕ s 2 if s 1 < s 2 and s 1 , s 2 ∈ { 1 , … , S } .

Properties

Spectral risk measures are also coherent. Every spectral risk measure ρ : L → R satisfies:

1. Positive Homogeneity: for every portfolio X and positive value λ > 0 , ρ ( λ X ) = λ ρ ( X ) ;
2. Translation-Invariance: for every portfolio X and α ∈ R , ρ ( X + a ) = ρ ( X ) − a ;
3. Monotonicity: for all portfolios X and Y such that X ≥ Y , ρ ( X ) ≤ ρ ( Y ) ;
4. Sub-additivity: for all portfolios X and Y, ρ ( X + Y ) ≤ ρ ( X ) + ρ ( Y ) ;
5. Law-Invariance: for all portfolios X and Y with cumulative distribution functions F X and F Y respectively, if F X = F Y then ρ ( X ) = ρ ( Y ) ;
6. Comonotonic Additivity: for every comonotonic random variables X and Y, ρ ( X + Y ) = ρ ( X ) + ρ ( Y ) . Note that X and Y are comonotonic if for every ω 1 , ω 2 ∈ Ω : ( X ( ω 2 ) − X ( ω 1 ) ) ( Y ( ω 2 ) − Y ( ω 1 ) ) ≥ 0 .

In some texts the input X is interpreted as losses rather than payoff of a portfolio. In this case the translation-invariance property would be given by ρ ( X + a ) = ρ ( X ) + a instead of the above.

Examples