spectral risk measure weighting

[andrnev]

In P1. T4.Chp1 section on Weighting and Spectral risk measures I do not follow the intuition behind this section at all. I understand the point where out of the two investments A and B that B would have a larger expected shortfall since the losses of B can be 10 times the VaR level even when the VaR level is the same for both portfilios. The matter following this point stumps me.

Question 1
"Expected shortfall assigned a probability based weight to all loss levels greater than VaR"
Consider the insurance contract example from the previous section, from one VaR confidence level two probabilities are shown i.e. 0.0064% and 0.9936% for ES calculation for the combined contracts.
1. Are these probabilities what is refered to as "probability based weight"?

Question 2
"If and only if the weights are a non decreasing function of the percentile of the loss distribution"
If I take the insurance contract example again from the previous section the probability based weight based on a 99% confidence percentile is 99.9936% - 99% = 0.9936% for the 21 mn loss but if we switch to a 99.5% confidence then the probability based weight drops to 99.9936% - 99.5% = 0.4946%.

1. Is this what is mean by weights being a function of the percentile?
2. How can you achieve a non decreasing weight?

Question 3

The figure 1.8 shows the weight for the expected shortfall measure at 10. How is this value computed?

 

[gsarm1987]

for your 2 questions, answers are yes and yes.
Note: ES is a spectral risk measure because it involves a weighted average of losses exceeding a certain percentile, effectively integrating the entire tail of the loss distribution beyond the VaR threshold. This means it considers not just the probability of extreme losses, but also their magnitude, which can be viewed as assigning a "spectrum" of weights to the tail losses.
VaR, on the other hand, is not a spectral risk measure. It is simply a threshold value such that the probability of a loss exceeding this value is a given percentage (e.g., 5% or 1%). VaR does not consider the size of the losses beyond this threshold, nor does it apply any weighting to losses exceeding the VaR.
from this graph: ES measure focuses on the average of losses that exceed the Value at Risk (VaR) at a certain percentile. In the graph, the weight for ES is uniform across all percentiles above the chosen VaR threshold until it reaches 100%, which reflects the proportion (1/(1-p) where p is the VaR percentile. The graph shows that the weight increases sharply as it approaches the 100th percentile, which means every point beyond the VaR percentile up to the 100th percentile is given equal importance when calculating ES