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. [1]
Consider a portfolio (denoting the portfolio payoff). Then a spectral risk measure where is non-negative, non-increasing, right-continuous, integrable function defined on such that is defined by
where is the cumulative distribution function for X. [2] [3]
If there are equiprobable outcomes with the corresponding payoffs given by the order statistics . Let . The measure defined by is a spectral measure of risk if satisfies the conditions
Spectral risk measures are also coherent. Every spectral risk measure satisfies:
In some texts[ which?] the input X is interpreted as losses rather than payoff of a portfolio. In this case, the translation-invariance property would be given by , and the monotonicity property by instead of the above.
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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. [1]
Consider a portfolio (denoting the portfolio payoff). Then a spectral risk measure where is non-negative, non-increasing, right-continuous, integrable function defined on such that is defined by
where is the cumulative distribution function for X. [2] [3]
If there are equiprobable outcomes with the corresponding payoffs given by the order statistics . Let . The measure defined by is a spectral measure of risk if satisfies the conditions
Spectral risk measures are also coherent. Every spectral risk measure satisfies:
In some texts[ which?] the input X is interpreted as losses rather than payoff of a portfolio. In this case, the translation-invariance property would be given by , and the monotonicity property by instead of the above.
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cite journal}}
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help)