Hi Tom,
If we continue to assume your losses are a simple historical simulation (i.e., sorted and each has the same weight of 1.0%), then you can't use 96*1.5%; unless that reflects the parent distribution, in which case it would have to reflect 99, 98, 97, 96, 96. Then your formula works, but only b/c the 5th worst is also 96. To get the 95.5 ES, you need the 4.5% tail, which means that you need the worst four losses plus 0.5%*the 5th worst. (so, you'd never get this question). Your numerator is retrieving the actual "parent" x*f(x) values; or in continuous terms, x*f(x)dx, so something should feel wrong about 96*1.5%.
While we are here, this does just happen to be a sorted list were each loss is given the same weight of 1/n; i.e., simple historical simulation. In the (Dowd) variations, the 1% weights can vary according to some rule (e.g., EWMA), but still if we want the X% ES, we do need to retrieve the parent's (1-x)% probabilities "as they actually are." Thanks,
If we continue to assume your losses are a simple historical simulation (i.e., sorted and each has the same weight of 1.0%), then you can't use 96*1.5%; unless that reflects the parent distribution, in which case it would have to reflect 99, 98, 97, 96, 96. Then your formula works, but only b/c the 5th worst is also 96. To get the 95.5 ES, you need the 4.5% tail, which means that you need the worst four losses plus 0.5%*the 5th worst. (so, you'd never get this question). Your numerator is retrieving the actual "parent" x*f(x) values; or in continuous terms, x*f(x)dx, so something should feel wrong about 96*1.5%.
While we are here, this does just happen to be a sorted list were each loss is given the same weight of 1/n; i.e., simple historical simulation. In the (Dowd) variations, the 1% weights can vary according to some rule (e.g., EWMA), but still if we want the X% ES, we do need to retrieve the parent's (1-x)% probabilities "as they actually are." Thanks,