Estimating Expected Shortfall

[brian.field]

Hull states that "...slice the tail into a large number of n slices, each of which has the same probability mass ..."

However, his examples do not actually do this. F0r instance, splitting the 95% to 100% tail into 9 slices, i.e., 95.0% to 95.5%, 95.5% to 96.0%, 96.0% to 96.5%, .....99.0% to 99.5% creates 9 slices of size 0.50%.

Clearly, the probability density or mass in the 95.0% to 95.5% slice is greater than the probability density or mass in the 99.0% to 99.5% slice, for instance, although they are "close" when considering a normal distribution.

Have you any thoughts in this @David Harper CFA FRM?

 

[ami44]

Hi Brian,

mass and density are not the same. All the slices have the same mass by definition, which is 0,5% probability, but the probability density in each slice might very.

I made this little picture to illustrate, which shows the density of a standard normal distribution.



I marked the two slices one for 99%-99.5% and one for 95%-95.5%.
The probability mass (or just probability) is the area under the curve, which is for both slices 0.5%. The probability density is the blue line and varies in each slice.
Did that address your point?

 

[brian.field]

Thanks @ami44 - I really consider you to be extremely knowledgeable, so I need to think about what you said to convince myself that it it true. One thing that I found a bit unclear on this question is the fact that the question used the term "mass". I have always associated probability mass as the area under the curve for a discrete distribution whereas density is the area under the curve for a continuous distribution. I think the fact that Hull uses mass is a bit confusing to me. It is not clear to me whether you are saying that the areas under each slice are the same - I don't think this is true but I need to confirm my intuition here. As for Hull's use of slices of the same mass, I think he should have said slices of the same size in terms distances between quantiles....

 

[ami44]

You just seem to have a slight problem with the terms:

- for continuous distribution the density is the curve (e.g. The blue curve in my picture. The density does not denote probability directly, but probability density. That means the area under the curve for a given interval is the probability, that the value of the random variable will fall into that interval.
- for discrete distributions you normally do not use densities, because they would show infinitly high peaks (delta functions), but cumulative distribution functions (cdf) can be used for both cases. If a density exists, the cdf is the integral of that density.
The value of the cdf at point x gives you the probability for the random variable to have a value equal or lower than x.

- probability mass is just a fancy word for probability.

If the areas below the two slices in my picture should not be equal, than only because I made an error. I explicitly constructed them to be both 0.005.

If you need more explanation, feel free to ask.

 

[brian.field]

Ahhh I see. Thank you @ami44. The values of the integrals are the same in each case, i.e., that probabilities are the same, i.e., Hull is slicing up the probabilities, in essence. This wasn't entirely clear to me - not sure why - just wasn't clicking.

 

[brian.field]

This is clear to me now. Thanks again.

 

[brian.field]