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# Expected Shortfall

#### caramel

##### Member
HI David
I was a little confused and after reading thr was another thread , I was wondering if this is another Garp error

From the handbook FRM exam 2003 question 5
Given the following 30 ordered percentage returns of an asset, calculate the VAR and expected shortfall at a 90% confidence interval
-16, -14,-10, -7, -7,-5, -4, -4, -4, -3, -1, -1, 0,0, 0, .........................
Ans was given as VAR 10 and ES 15
the 10% lower cutoff point is the third lowest observation which is VAR =10 , the ES is then the avg the of the observations in the tails which is 15

this is different from the below thread.

#### caramel

##### Member
So David should I stick to the solution in the thread and not go with GARP on this one

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi caramel

Yes absolutely, it's an embarrassing error in the handbook. (Although I've submitted this to GARP before in another context, given the nature of so fundamental an error, I just submitted this thread here to GARP). You might infer that the thread, to which you link, is correct since we've discussed the issue there. In the above question:
• 90% VaR = 10, the 3rd worst loss, is acceptable. I would prefer GARP followed the assignments and use the 4th worst, but under a discrete (HS) distribution, lacking a standard, there are several correct answers: 3rd worst, 4th worst, or interpolation between 3rd and 4th. Put another way, lacking a definitinal standard of VaR in the discrete distribution, VaR due the fact that it is a quantile does suffer ambiguity in the discrete case
• But 90% ES is not ambiguous. The Q&A is incorrect. The 90% ES is the (conditional) average of the 10% loss tail, which is the average of the worst 3 (out of 30), not the worst 2. ES does not depend on the discrete calibration of VaR. The answer given is the 93.33% ES; i.e., average of the worst 2/30. The only correct answer for 90% ES is average(16,14,10) = 13.33. Thanks,

#### cqbzxk

##### Member
Hi caramel

Yes absolutely, it's an embarrassing error in the handbook. (Although I've submitted this to GARP before in another context, given the nature of so fundamental an error, I just submitted this thread here to GARP). You might infer that the thread, to which you link, is correct since we've discussed the issue there. In the above question:
• 90% VaR = 10, the 3rd worst loss, is acceptable. I would prefer GARP followed the assignments and use the 4th worst, but under a discrete (HS) distribution, lacking a standard, there are several correct answers: 3rd worst, 4th worst, or interpolation between 3rd and 4th. Put another way, lacking a definitinal standard of VaR in the discrete distribution, VaR due the fact that it is a quantile does suffer ambiguity in the discrete case
• But 90% ES is not ambiguous. The Q&A is incorrect. The 90% ES is the (conditional) average of the 10% loss tail, which is the average of the worst 3 (out of 30), not the worst 2. ES does not depend on the discrete calibration of VaR. The answer given is the 93.33% ES; i.e., average of the worst 2/30. The only correct answer for 90% ES is average(16,14,10) = 13.33. Thanks,

Hi David, I get confused about ES, I met lot of practices before, for example, there is typical question:
Confidence level -----Tail VaR
95%--------------------- 3
96% --------------------3.25
97% --------------------3.6
98% ---------------------4
99% -------------------4.75
What is ES at 95% level?
3.25+3.6+4+4.75/4 or 3+3.25+3.6+4+4.75/5 which one is correct? thanks !

#### cqbzxk

##### Member
the notes also said, the tail mass is divided into n equal slices and the corresponding n-1 VaR are computed..

#### cqbzxk

##### Member
Hi cqbzxk, we just discussed this question here at http://www.bionicturtle.com/forum/t...at-your-remember-here.5923/page-14#post-24224
... i don't have the source question, so i feel like it's unproductive to keep trying to figure it out

where is this question exactly?

Hi David, the question I post above was not from exam, it is from Notes "Estimating Market Risk Measures" Topic 1 my version is 2011, it seems that the notes only use (3.25+3.6+4+4.75)/4 = 2.003 to explain ES at 95%, this made me a little bit confuse,

#### cqbzxk

##### Member
my confusion is that where is the 1/1-confidence ? suppose 1/5% * (x*f(x) + .....)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
that method is not computing an accurate ES. I guess, I don't have the source, it is using an approximation method: you can approximate the 95% VaR by taking the average of the any (N) VaRs in the tail, as but at 3 or 4 it's totally rough estimate. Averaging the (N) tail VaRs is an approximation shortcut (very inaccurate for low N) that does not bother with the effort to retrieve the accurate 1/signficance * sum of: (loss)*f(x). Thanks,