Spectral risk measure

A Spectral risk measure is a risk measure given as a weighted average of outcomes where bad outcomes are, typically, included with larger weights. A spectral risk measure is a function of portfolio returns and outputs the amount of the numeraire (typically a currency) to be kept in reserve. A spectral risk measure is always a coherent risk measure, but the converse does not always hold. An advantage of spectral measures is the way in which they can be related to risk aversion, and particularly to a utility function, through the weights given to the possible portfolio returns. A Spectral risk measure is a risk measure given as a weighted average of outcomes where bad outcomes are, typically, included with larger weights. A spectral risk measure is a function of portfolio returns and outputs the amount of the numeraire (typically a currency) to be kept in reserve. A spectral risk measure is always a coherent risk measure, but the converse does not always hold. An advantage of spectral measures is the way in which they can be related to risk aversion, and particularly to a utility function, through the weights given to the possible portfolio returns. Consider a portfolio X {displaystyle X} (denoting the portfolio payoff). Then a spectral risk measure M ϕ : L → R {displaystyle M_{phi }:{mathcal {L}} o mathbb {R} } where ϕ {displaystyle phi } is non-negative, non-increasing, right-continuous, integrable function defined on [ 0 , 1 ] {displaystyle } such that ∫ 0 1 ϕ ( p ) d p = 1 {displaystyle int _{0}^{1}phi (p)dp=1} is defined by where F X {displaystyle F_{X}} is the cumulative distribution function for X. If there are S {displaystyle S} equiprobable outcomes with the corresponding payoffs given by the order statistics X 1 : S , . . . X S : S {displaystyle X_{1:S},...X_{S:S}} . Let ϕ ∈ R S {displaystyle phi in mathbb {R} ^{S}} . The measure M ϕ : R S → R {displaystyle M_{phi }:mathbb {R} ^{S} ightarrow mathbb {R} } defined by M ϕ ( X ) = − δ ∑ s = 1 S ϕ s X s : S {displaystyle M_{phi }(X)=-delta sum _{s=1}^{S}phi _{s}X_{s:S}} is a spectral measure of risk if ϕ ∈ R S {displaystyle phi in mathbb {R} ^{S}} satisfies the conditions Spectral risk measures are also coherent. Every spectral risk measure ρ : L → R {displaystyle ho :{mathcal {L}} o mathbb {R} } satisfies: 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 {displaystyle ho (X+a)= ho (X)+a} instead of the above.