How to calculate ES (P1.T4.EOC 1.17 and P1.T4.EOC 2.5)

[AUola2165]

There seems to be different ways to calculate the ES, depending, for example, whether the returns are continuous or discrete (I think).

In P1, T4, Chapter 1 EOC 1.17 the question goes "An investment has a uniform distribution where all outcomes between -40 and +60 are equally likely. What are the VaR and expected shortfall with a confidence level of 95%?" and the answer is "...Expected shortfall is 37.5. Conditional on the loss being greater than 35, the expected loss is halfway between 35 and 40, or 37.5."

Is this calculated as "median" (as it answer says "halfway"): 35,36,37,38,39,40 -> 37.5 is "halfway" or is this calculated as (35+36+37+38+49+40)/6=37.5?
Why does the answer say "halfway" when it should be the average?

In the next chapter EOC 2.5 question* goes:
"In the situation in Question 2.4, how is expected shortfall calculated?"
and the answer is "It is the average of the three worst losses."

Why is this calculated as the average of the three worst losses and not 4 worst losses as the "35" is included in the ES calculation in EOC 1.17 so should the 4th worst loss be included in the EOC 2.5 question.



(Question 2.4 being: "if there are 400 simulations on the loss (gain) from an investment, how is VaR with a 99% confidence level calculated"?)

 

[David Harper CFA FRM]

Hi @AUola2165

GARP is wrong on both 2.4 and 2.5. ES is a conditional average and has only one answer, ever (regardless of continuous or discrete). It has one answer for the same reason that there is always and only one sample mean for any given sample. In the case of 2.5, the 99% ES is the average of the conditional 1% loss so it is the average of the four worst losses. Always. Period.

In the case of 2.4, it is a discrete distribution and the 1% quantile (VaR is a quantile. That's it's definition) falls right "in between" the 4th and 5th losses such that--in this case--there are multiple 99.0% VaR. With equally-weighted n = 400, the 99.0% VaR can be either:

• 5th worst loss
• 4th worst loss
• Multiple values in between 5th and 4th depending on interpolation.

As usual, there setups lacks precision too (e.g., equally weighted presumption in the simulation), but to highlight this feature of GARP consider a tiny alteration (silly but for illustrative purposes only) in the question:

If there are 399 simulations on the loss (gain) from an investment, how is VaR with a 99% confidence level calculated? how is the expected shortfall calculated?

• There is still only one 99.0% ES but it is not quite as simple as the average of the worst four
• Now there is only one 99.0% VaR: the fourth worst loss.

Good catch on these GARP mistakes!