SPECTRAL RISK MEASURE

Spectral risk measure is one of the risk measures that are used a weighted average of outcomes associated with bad outcomes, customarily, included with larger weights. Spectral risk measures enable us to associate the risk measure with the user’s attitude toward risk. SRMs can be applied to a number of problems like setting capital requirements, setting margin requirements for futures clearinghouses, obtaining optimal risk-expected return tradeoffs and so on.

What Is Spectral Risk Measure?

Spectral risk measure or SRM is a risk measure that is calculated as a weighted average of outcomes, the weights of which depend on the user’s attitude towards the risk. This is a function that measures portfolio returns and the number of outputs of the numeraire (typically a currency) to be kept in reserve. A spectral risk measure is always a coherent risk measure, but it is true that coherent risk measure does not always come as spectral risk measure.

One of the advantages that Spectral risk measure carries, it possesses an accord with risk aversion, particularly to a utility function, by means of weights given to the possible portfolio returns to the expected possible returns.

The theory of spectral risk measure is considered as one of the most potential and interesting development in financial risk areas shares common property sub-additivity with coherent risk measures.

How Spectral Risk Measure Works?

As a matter of fact, spectral risk measures (SRMs) can be related to the coherent risk measures proposed by Artzner et al (1997, 1999). That's because both these phenomena have a property called "subadditivity" in common. Subadditivity can be explained as follows:

Suppose ρ(.) is a measure of risk, and if A and B are any two positions. In this case, subadditivity implies that it will always be the case that:

ρ(A + B) ≤ ρ(A) + ρ( B).

Subadditivity suggests that individual risks typically diversify (or, at worst, stop increasing) when all the risky positions are put together. Hence, the spectral risk measures enable us to relate the risk measure to a user's perception and sensitivity towards risk.

In other words, it can be expected that if a user is more risk-averse, other things being equal, then that user should face a higher risk, as given by the value of the SRM.

Examples Of Spectral Risk Measure

Expected shortfall is one of the best examples of Spectral risk measure.

In order to understand Expected Shortfall, we first need to know about the Value at Risk (VaR), which is a statistic used to quantify the risk of a portfolio. VaR indicates the maximum loss that can be expected with a certain confidence level and is used to answer the following question:

What value of a given portfolio is at risk?

Now, coming to the Expected Shortfall (ES) or Conditional VaR (CVaR), which is a function of two parameters:

• N (the time horizon in days)
• X% (the confidence level)

Conditional VaR or CVaR represents the expected loss in an N-day period, assuming that the loss exceeds the Xth percentile of the loss distribution. Mathematically explaining the phenomenon:

If N = 88

   X= 12

then as per the definition, the Expected Shortfall in this case, is the amount lost over a 12-day period, assuming that the loss exceeds the 88th percentile of the loss distribution. 

So, basically, the Expected Shortfall is readily used as a statistic to determine how risky a portfolio can prove to be. Given a certain confidence level, CVaR represents the expected loss when it is greater than the Value at Risk (VaR) calculated with that confidence level. In other words, this measure can be used to answer the question:

If things go bad, how much loss should be expected?

Spectral Risk Measure vs Distortion Risk Measure

As mentioned earlier, SRM is a risk measure that is calculated as a weighted average of outcomes, the weights of which depend on the user’s attitude towards the risk.

Mathematically, SRM can be defined as (M_phi), a weighted average of the quantiles, q, of the loss distribution

M_phi = int_{_0^1} phi(p) q_p dp

for some weighting function phi(p) that reflects the user’s risk aversion.

On the other hand, distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio. These measures can be defined as follows:

A distortion function is a non-decreasing function with g(0) = 0 and g(1) = 1, and g : [0, 1] →

[0, 1] .

A distortion risk measure associated with distortion function g, for a random loss X with

decumulative distribution function S(x) is

ρg(X) = Z ∞

0

g(S(x))dx

The main difference between spectral and distortion risk measures is that distortion risk measures are law-invariant and monotone, but not coherent while spectral risk measures are fully coherent.

So, these are the basic concepts about spectral risk measure that students pursuing a higher degree in Finance should know. Students can either rely on college lectures or books like:

• Risk Management: Foundations For a Changing Financial World By Walter V. "Bud" Haslett, Jr. (published by John Wiley & Sons, 2010)
• A Probability Metrics Approach to Financial Risk Measures By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi (published by John Wiley & Sons, 2011)

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References:

• https://www.risk.net/risk-magazine/technical-paper/1506669/var-versus-expected-shortfall
• https://arxiv.org/ftp/arxiv/papers/1103/1103.5674.pdf
• https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.488.4957&rep=rep1&type=pdf
• https://quant.stackexchange.com/questions/23016/spectral-and-distortion-risk-measures
• https://blogs.cfainstitute.org/investor/2014/02/26/are-spectral-risk-measures-respectable-enough/