Expected shortfall (708.2. Dowd Question)

[kevolution]

I was doing the practice questions for this section and came across the answer but I'm wondering where the formula was derived.

Why is "($2.0 * 3.0%)" part of the equation and not "($2.0 * 8.0%)" because the f(X) corresponding to ($2.00) is 4a which should be 4 * (2.0%) = 8.0%, should it not?

 

[David Harper CFA FRM]

@kevolution the 95.0% ES is the conditional average of the 5.0% probability tail, so we are taking the average that includes the full density Pr[X = 3] = 2.0% but we only want 3.0% of the 8.0% of the Pr[X = 2] density. You might think of this as, if you imagine the vertical (pmf) bar that extends up to 8.0%, we are only "counting" the first 3.0% so that cumulatively we are counting the worst 5.0% only. To be consistent, we are then dividing by 5.0% to achieve a weighted average.

Notice, we could instead ask: what is the 90.0% ES? In which case the answer is given by: [($3.0 * 2%) + ($2 * 8%)] / 10% = $2.20. Are you an expired subscriber? Thanks,

append: source Q&A here https://forum.bionicturtle.com/threads/p2-t5-708-expected-shortfall-dowd-chapter-3.11166

 

[kevolution]

Thanks that clarifies the question. Yes I'm expired; I'll be taking the exam in November.