Hello, In the notes it says that ES is the probablity weighted average of the tail losses, yet the way we always calulate it assuming that all of the "slices" have equal weight. Is ES defined as equally weighting the slices or is this just a simplification that Dowd makes? Is there an assumption that the test makes? Same thing for Spectral risk measures. The example in the book gives higher weights to the "slices" that are more risky (contain bigger losses, which is expected) but nothing seems to be done with the probability of these slices, so I guess I need to ask the same question as above: does the test just assume that all of the slices will be equally weighted (from a probability standpoint)? Thanks! Shannon

Hi Shannon, Yes, ES is equal weight. The notes might be more helpful to read ES is "equally-weighted average of tail losses," or to just drop the "probability" and read "ES is a conditional average of tail losses." (although Dowd does refer to ES as a "probability-weighted average of tail losses.") Equal weights applied to the quantiles is not the same thing as equivalent probabilities: pdf(x) = f(x) is not constant, except for a uniform distribution. ES is an just an (equally) weighted average, just like the weighted average of any continuous/discrete/empirical distribution, except its concerns only the truncated tail region. (the "probability" is meant to acknowledge that, except for HS, not all the quantiles are equally likely). The spectral risk measure is a very general (i.e., flexible) formula, it applies weights to the quantiles, allowing for an infinite variety of weights. An analogy (just an analogy, nothing more!) would be simple historical volatility is a function of equally weighted squared returns (~ analogy to ES) EWMA applies greater weight to more recent returns (actually, the analogy to spectral would be ARCH(m), which allows for any weights, and includes the special case where the weights are equal) Here is an example of the difference: assume n = 400 P&L losses, where the worst 4 losses are 10, 20, 30, and 40. First, just note that our simple historical sort is uniquely easy for being a UNIFORM discrete distribution such that each outcome is equally likely: p(X=10) = 1/400, p(X=20) = 1/400, P(X=30) = 1/400, P(X=40) = 1/400. The 1% ES = 99% ES = average(10,20,30,40) = 25; i.e., ES is the equally-weighted average of (1 - 99%) tail losses. A spectral measure could be something like the following, to reflect risk-aversion: $10 * [p = 1/400] * 50 weight + $20 * [p = 1/400] * 75 weight + $30 * [p = 1/400] * 125 weight + $40 * [p=1/400] * 150 weight = $29.375 i.e., see how i assigned greater weights to worse outcomes? But this spectral measure is just a general case of the ES, where the weights are all the same (all weights = 1/[1-99%] = 100) $10 * [p = 1/400] * 100 + $20 * [p = 1/400] * 100 + $30 * [p = 1/400] * 100 + $40 * [p=1/400] * 100 = $25 = average(10,20,30,40) This is just the discrete application, same idea in the continuous distribution, really. I hope that is helpful,

That was incredibly helpful. The whole "probablity weighted" phrase irked me because he seemed to mention it a couple of times and then just assumed that each of the quantiles was equally likely by not putting any less weight on extreme events. Thank you!! Shannon