Not sure I am understanding spectral risk measures correctly.
Why is there an equal weighting scheme placed on the tail losses in expected shortfall.
Will that no bias the expected value of the loss towards the lower tail because the probability that the loss will occur is small compared to that which is closer to the p-value?
I don’t know if equal weights is a requirement of “spectral returns”…
But generally, this measure overweights certain bad outcomes because, 1) It’s more conservative to measure risk this way, and 2) This reflects diminishing marginal utility (lose $1 million is more than 10 times worse than lose $100k).
ES is just the expected value of a distribution given that it’s less than some quantile. So you’re still getting some probability weighting in there if you think about it from a distributional perspective. For instance, if you’re calculating ES from historical returns or Monte Carlo (i.e. rather than analytically), fewer extreme returns will occur, thus fewer of them will be included in this expected value.
Not sure where you’re going with the p-value point.
On risk neutrality, you can maximize a utility like mu-lambda*ES where lambda determines your risk appetite. You’re only risk neutral in that sense if lambda equals 0. Since you can add spectral risk measures and still get a spectral risk measure, you can actually take a weighted average of ESs at different confidence levels. So you could have a lambda for each, reflecting how much you care about different parts of the tail.