Examples of Spectral Risk Measures

Let's take the usual definition of a spectral risk measure.

If we look at the integral we see that spectral risk measures have the property that the risk measure of a random variable $X$ can be represented by a combination of the quantiles of $X$.

Since the quantile function is rather friendly one gets that every spectral risk measure is also a coherent risk measure.

Examples are the expected value and the expected shortfall (CVaR). In those cases, the spectral representation yields a very convenient way to approximate the measure by simply weighing the quantiles of our dataset. That yields the following questions:

Are there any other known measures that have a spectral representation? If we relax the assumptions on the spectrum $\phi$, can we obtain (approximative sequences of) other (possibly non-coherent) risk measures?

EDIT: In reaction to the comment by @Joshua Ulrich I want to provide an example of what I want to achieve and some more details.

• Example: The Conditional Value at Risk. We have the following formula: $\text{CVaR}_\alpha(X) = -\frac{1}{\alpha}\int_0^{\alpha}F^{-1}_X(p)dp$. From sample $X_i$, $i=1,\ldots,N$, we can calculate the CVaR by taking the order statistics that are in the $\alpha$-tail of the sample, average, and divide by $\alpha$. We can see that this is measure has a spectral representation with $\phi(p) = \frac{1}{\alpha}$ for $p \in [0,\alpha]$ and $\phi(p) = 0$ for $p \in (\alpha, 1]$. So its easy to check: The CVaR is a spectral measure.

Obviously, the "order statistics + weighted average" procedure does not only work for the CVaR, it works for all spectral measures: From the definition of spectral measure we see that, after discretizing the integral, we have an approximation of the measure that is a linear combination of quantiles which is very easy to compute.

In fact its so easy that I would like to compute as many risk measures as possible this way (very easy if you do monte carlo or scenarios for example). For the computation only, I dont need all the assumptions about $\phi$ so lets forget about them for a moment and see what else we can calculate this way.

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This is a bit broad, and could lead to list-like answers. Could you provide more details on what you're actually trying to accomplish? –  Joshua Ulrich Oct 20 '13 at 12:32
Well, I am not sure I understand what you want but what is clear is that $CVAR_{\alpha}$ works as $\Phi(p)=\frac{1}{\alpha} \cdot 1_{p \in [0,\alpha]}$ which is a density of probability. So you can build any other risk measure chosing for $\Phi$ any density on the compact set $[0,1]$, and by scaling on a compact set. I would try a "hat function" first, then sinusoids... –  statquant Oct 21 '13 at 12:20
Quite a few risk measures seem unlikely to have a representation like this, since their units differ. For example, annualized volatility. Some risk measures, such as scenario playbacks of the '97 Asian crisis, will trivially have a spectral representation but not in any computationally useful way. –  Brian B Oct 21 '13 at 13:38
@statquant Well, the aim is not to try a different spectrum $\phi$ but rather to identify the spectra of other well known risk measures. Brian B: I dont understand your comment about units. The units are irrelevant for a risk measure as far es the spectral property is concerned, right? –  vanguard2k Oct 23 '13 at 6:47