Examples of Spectral Risk Measures

Question

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.

Answer 1

I believe that Prospect Theory (as defined by Kahneman, Amos, and Tversky) implicitly makes use spectral risk measures. Though I am not able to find any literature linking the two, I think there is clear link between the intuitions regarding loss aversion. The key difference is that spectral risk measures are normative; we assume that the utility function is known. Prospect Theory, on the other hand, is inherently descriptive (i.e., reflects observed behaviors). Also, I am aware that spectral risk measures are extended to portfolio risk, while Prospect Theoretic measures deal with generic utility.

source: Wikipedia. Prospect Theory

Again, while I haven't seen any literature on the topic, it would be interesting if someone were to show that Prospect Theoretic risk measures (which are typified by Exhibit A) meet the coherence standards for a spectral risk measure given by:

${\displaystyle \rho :{\mathcal {L}}\to \mathbb {R} }$ satisfies:

• Positive Homogeneity: for every portfolio X and positive value ${\displaystyle \lambda >0} \lambda >0$, ${\displaystyle \rho (\lambda > X)=\lambda \rho (X)}$;
• Translation-Invariance: for every portfolio X and $\alpha \in \mathbb {R}$, ${\displaystyle \rho (X+a)=\rho (X)-a}$;
• Monotonicity: for all portfolios X and Y such that ${\displaystyle X\geq Y}$ , ${\displaystyle \rho (X)\leq \rho (Y)}$;
• Sub-additivity: for all portfolios X and Y, ${\displaystyle \rho (X+Y)\leq \rho (X)+\rho (Y)}$;
• Law-Invariance: for all portfolios X and Y with cumulative distribution functions ${\displaystyle F_{X}}$ and ${\displaystyle > F_{Y}}$ respectively, if ${\displaystyle F_{X}=F_{Y}}$ then ${\displaystyle \rho (X)=\rho (Y)}$;
• Comonotonic Additivity: for every comonotonic random variables X and Y, ${\displaystyle \rho (X+Y)=\rho (X)+\rho (Y)}$. Note that X and Y are comonotonic if for every ${\displaystyle \omega _{1},\omega > _{2}\in \Omega :\;(X(\omega _{2})-X(\omega _{1}))(Y(\omega _{2})-Y(\omega _{1}))\geq 0}$

Answer 2

As I know, I think that spectral risk measure is a new kind of measure developed from the CVaR (weighted average value of VaR) and in the framework of coherent risk measures.

If you can prove that a risk measure is coherent then you can add any types of weighted function $\phi$ to make it a spectral risk measure. The underlying idea is that the sum of any number of coherent measures is also a coherent risk measure.