FRM handbook Example 15.10: FRM EXAM 2008 - Question 5-8

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hello,

The question ask about the expected shortfall and the solution said the VAR is 1428 while the ESF is 1861.

The followings are the sorted 15 worst P&L from 1200 past days data.
-2833, -2333, -2228, -2084, -1960, -1751, -1679, -1558, -1542, -1484, -1450, -1428, 1368, -1346, -1319

According to the handbook definition, ESF = E[-X|X<-VAR]. According to my understanding, If the VAR is 1428, the ESF be computed as [2833+2333+2228+2804+1960+1751+1679+1558+1542+1484+1450]/11 = 1900.181818 which is not the same as the solution suggested.

Another question is that, the VAR is inf{x: P(Loss > x) <= 1-C}, so in the above question, P(Loss > 1368) = 12/1200 = 0.01. I don't understand why the VAR is 1428 instead of 1368.

Many Thanks

 

[David Harper CFA FRM]

Hi wingsiu,

The ES formula is as you suggest, but in this case it needs to be the average of the worst 12 losses, not 11, as 12 = 1% * 1,200 observations (this does not depend on the VaR. Rather, it is the average of the lowest 1%, which in this discrete case, must include 12 observations). By my calculations, answer (c) is exactly correct: I get 1,861 as the average of the worst 12 and therefore the 99% ES.
(the explain, i think, is just a means to quickly approximate. The ES does not take a median)

In regard to VaR, technically (i.e., following the assigned Dowd), you are absolutely correct. The best answer for a 99% VaR, in this discrete case, is the 13th worst loss (i.e., 13 = 1%*1200 + 1). However, in this discrete case, the 12th worst is also acceptable, as is an interpolation (e.g., PERCENTILE) between 1368 and 1428.

Hope that helps, David