CH4 Page 17 Example

[Brett Kim]

Taking just investment Rx here - I can understand why the VaR(95%) is $34.91m. But when we go on to look at the conditional TVaR the integral is different to what has been shown in the previous examples or the summary?

We are given that the expected shortfall for a continuous variable is int^L_-inf [(L - x) f(x) dx] so I would have expected to set L = -34.91m with the bottom limit of the integral at -inf.

Why is it for this example we are setting the limits of the integral to 59.91465 and inf instead?
And why are is it doing what looks to be (x - L) inside the integral instead of (L - x)?

 

[John Potter]

Hi Brett,
A nice way to think about VaR (5%) is that it tells us how much we lose if the 5th worst outcome out of 100 outcomes occurs. Both investments are 25 - R, so the 5th worst outcome is when F(R) = 0.95. This is when R = 59.91465 (or 55.56 for the other investment, I made these examples in R deliberately to get the VaRs to be similar).
Now conditional TVaR is the extra amount we would expect to lose on top of what we have already lost. So, we need to go back to the PDF and integrate from the F(0.95) point. The signs might appear the wrong way round, perhaps because we are talking about 25 - R, ie the higher R is, the more we lose.
It's good to think about both VaR and TVaR in an intuitive way rather than concentrating on formulae that dictate what the lower limit of an integral should be. This is going to be especially true now we are in the open-book exam era, where I predict there will be fewer questions that allow us to just blindly apply formulae,
Good luck!
John

 

[Brett Kim]

Hi John, thanks very much for your explanation I understand now exactly what the example is doing. I think my biggest confusion was not realizing that the larger the R the larger the loss, hence why integrating up from 59.91465 (the 95th percentile cut off) gives the 5% worst outcomes in the tail.