Why it's named Expected Shortfall?

[Steve Jobs]

I'm reviewing the materials not to pass exams anymore but to say something simple in any job interview and sometimes I come across such questions which I apologize in case I'm wasting your time.

Why it's named Expected Shortfall?
Is it because most banks use VaR and since the ES is the avergae of loss qouantiles beyond VaR, so ES - VaR = the amount that banks will be short?

 

[David Harper CFA FRM]

Hi @Steve Jobs

By all means, please, it's a great question that I never thought to ask. I will just say that, historically the FRM always used ES and treated as synonyms, though rarely invoked, conditional VaR and expected tail loss. To me, personally, "expected tail loss" is the most accurately descriptive, as ES is average in the tail. My only problem with conditional VaR is that it tends to encourage the mistaken notion that the ES metric depends directly on the VaR metric. Because I don't know the answer to your question, I looked to my two favorite reference on the matter:

Assigned Dowd (which maybe we should note is now almost ten years old, yikes!), Chapter 2 footnote 16, which does not answer your question but merely introduces some distinctions:

"The ES is one of a family of closely related risk measures, members of which have been variously called the expected tail loss, tail conditional expectation (TCE), tail VaR, conditional VaR, tail conditional VaR and worst conditional expectation, as well as expected shortfall (ES). Different writers have used these terms in inconsistent ways, and there is an urgent need to cut through the confusion created by all this inconsistent terminology and agree on some consensus nomenclature. This said, the substantive point is that this family of risk measures has two significant substantially distinct members. The first is the measure we have labelled the ES, as defined in Equation (2.4); this is defined in terms of a probability threshold. The other is its quantile-delimited cousin, most often labelled as the TCE, which is the average of losses exceeding VaR, i.e.,TC Eα=−E[X|X>qα(X)]. The ES and TCE will always coincide when the loss distribution is continuous, but the TCE can be ambiguous when the distribution is discrete, whereas the ES is always uniquely defined (see Acerbi (2004, p. 158)). We therefore ignore the TCE in what follows, because it is not an interesting statistic except where it coincides with the ES"

And Carol Alexander, MRA volume 4, (if anybody knows, I would think she would know) which appears to hint toward and answer (emphasis mine):

"16. Unless VaR is measured using a simple model, such as the normal linear model, it is not sub-additive. That is, the sum of the stand-alone component VaRs may be greater than the total VaR. In this case the whole concept of risk budgeting flies out of the window. Traders could keep within risk limits for each portfolio but the total limit for the desk could be exceeded. Desk managers could adhere to strict limits, but the total risk budget for the organization as a whole could still be exceeded. Hence, for risk budgeting purposes most large economic capital driven organizations use a risk metric that is associated with VaR, and which is sub-additive. This is a conditional VaR metric that we call expected tail loss or, if measured relative to a benchmark, expected shortfall. Conditional VaR satisfies all the properties for being a coherent risk metric, in a sense that will presently be made precise."

 

[Steve Jobs]

My favorite part of the quotes is for sure the below, although it's not in the main text, but at least as footnote!

Dowd: Different writers have used these terms in inconsistent ways, and there is an urgent need to cut through the confusion created by all this inconsistent terminology and agree on some consensus nomenclature.

Yeah I liked the term " Expected Tail Loss". The word tail is visual and reminds us that it's a statistical concept and not something to be used out of the statistical context to give any sense of false assurance.

Just one more clarification, If we say Average Tail Loss then this implies that the weight given to each quantile was the same, right? but if we sat Expected instead of Average then it means the weights might not necessary the same.

However, still no idea why it's called Expected Shortfall!, Thanks David,

 

[Alex_1]

Hi, I usually like to break down concepts consisting of several words into the single parts.

In this case the "expected" part should be clear as it is an "average in the tail" (as David also indicates).

Now for the "shortfall" part: I find the quote from Carol Alexander helpful -> "This is a conditional VaR metric that we call expected tail loss or, if measured relative to a benchmark, expected shortfall."
Starting from this quote I would interpret the ES as the return which falls short of a benchmark, in other words what would be the expected return in the worst q% of the cases (q being the quantile applied to the ES measure) which then represents the unreached return of the benchmark (hence a return which failed to meet some pre-set goal and "fell short" of this goal). I think this is probably easier to understand for a native speaker (which I am not) due to the fact that "to fall short of something" is an expression used in English.

I am not sure this helps for a better understanding of the terminology...

 

[David Harper CFA FRM]

I like your interpretation @Alex_1

@Steve Jobs Your clarification, to me, runs the risk of misintepreting ES. I would say it is always an expected value. Similar to when we refer to the mean (or average) of a distribution: we almost always mean a (probability) weighted average. The ES is similarly a probability-weighted average, but of the mass in the tail (which can be called a child distribution) rather than the entire body of the distribution (which can be called a parent).

You might be thinking of Dowd's method for approximating (estimating) the ES by averaging the tail VaRs (i.e., tail quantiles), which does give each computed tail VaR equal weight, but this is a means to estimating ES which doesn't really give all tail values equal weight as the VaRs are already probability-adjusted. This method (which often clouds an understanding of the ES definition, but must be understood as a mere shortcut) can be confusing, but shouldn't distract from the meaning of ES as a conditional, probability-weighted mean. Thanks!