Expected Shortfall

[ckyeh]

Dear David:

On your spreadsheet「5[1].d.1.ES_subadditive」, 「VaR not Subadditive」:
# Bonds ES @ 1 -
in Port 5.00%
1 0.4000
2 0.8000
3 1.0238

I conclude:
If the portfolio is 1 bond, ES is 0.4 bond will default, so ES is 0.4*100=﹩40.
If the portfolio is 2 bond, ES is 0.8 bond will default, so ES is 0.8*100=﹩80.
If the portfolio is 3 bond, ES is 1.0238 bond will default, so ES is 1.0238*100=﹩102.38.

Is it correct?

On your spreadsheet「5[1].d.1.ES_subadditive」, 「Normal_ES」:
0.798=0.3989/50%
ES=PDF probability density function/alpha?
But I see 2010-5-d-Market-Risk, page 10:
ES= cumulative distribution function/alpha, right?
Which one should be used to calculate ES, probability density function or cumulative distribution function?

Thanks

 

[David Harper CFA FRM]

Hi ckyeh,

In regard to the 3-bond example, your numbers are exactly correct because the (simplifying) assumption is that LGD =100% (recovery = 0%) such that (eg), for the 3-bond portfolio, the average number of bonds defaulting for the 5% worst (conditional on 5% worst) is 1.038, which is $102.38.

In regard to ES, it is hard for me to characterize along those lines. Both are using the PDF. The difference arises from the continuous versus discrete distribution (I suppose the continuous is the general form…). My normal XLS may just be an approximation, it matched Dowd's but I am not certain that is equal to the true integral. In either case, it is the same issue encountered when taking the mean (expected value) of a probability (b/c ES is a mean ... the easiest way to access the ES is to see that is is really just a conditional mean!): if the var is discrete, you have a straightforward sum of products. But if the var is continuous, you must integrate so it's not quite a CDF but "area under the curve."

David