 # Spectral Measures

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Analyses of investments, insurance, and economic policies often depend on an ability to collapse a large set of probability-weighted outcomes to just a few numbers. Examples of this process are computations of "expected utility", (or its cousin, "certainty equivalent wealth"), means and standard deviations of the outcomes, and "value at risk". Each of these measures has strengths as well as deficiencies, the latter including frequent difficulty in coping with negative outcomes in expected utility theory, the theoretical nonexistence of standard deviations for outcomes drawn from certain distributions, and the lack of sensitivity of value at risk to the magnitude of the worst outcomes.

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In recent years various spectral measures have developed as an attractive alternative method in evaluating probabilistic outcomes. A spectral measure is one in which the magnitude of each event is weighted according to the position of that event in a sorted list of events, the so-called order statistic of the event. A set of weightings is said to be spectral and satisfies various desirable properties such as subadditivity if it is nonincreasing over the order statistics, always non-negative, and sums to one. Thus, if there were five events, would be a spectral risk aversion function, as would and ; but would violate the nonincreasing requirement. Thus, the spectral measure of equal probability materializations using a spectrum of is because .

In this Demonstration, you choose a probability distribution that generates an independent and identically distributed collection of random variables. You may also specify the length of the collection. You then choose a family of risk aversion functions from which to generate weights as well as the parameters that instantiate a particular risk aversion function from that family.

In "BubbleChart" view, the Demonstration responds by generating a bubble chart in which the value of each bubble is the position of a collection item in the sorted version of the collection, the value is the value of the collection item, and for which the area of each bubble is proportional to the value of the spectrum at that position. Thus, in the dataset example given, there would be a bubble at whose area would be proportional to . Two lines decorate the graphic. A gold line shows the mean value of the collection and a green line shows the spectral measure of the collection.

In "Spectrum" view, the Demonstration responds by generating a plot in which the value of each point is the position of a collection item in the sorted version of the collection and the value is the value of the risk aversion function for that position.

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Contributed by: Seth J. Chandler and Roger J. Brown (October 2010)
Open content licensed under CC BY-NC-SA

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The "threshold" family of spectra has only two values: a constant value up to some position in the spectrum and zero thereafter. The risk measure "expected shortfall", which is growing in popularity and is also known as "conditional value at risk (cvar)", can be generated using a member from this family.

Pioneering work by Carlo Acerbi has been done in the field. A brief discussion of spectral measures may be found at Spectral risk measure.

The set of permissible spectral weights is the set of nonincreasing probability density functions (sometimes also known as probability mass functions for discrete distributions).

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