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Â This session will be about the expected shortfall, so

Â you can view the expected shortfall as an extension of the concept of value-at-risk.

Â So again I will look at two questions.

Â So the first question is what is the expected shortfall?

Â And the second question is how can I compute the expected shortfall?

Â So again the expected shortfall is a quantitative and

Â synthetic measure of risk and it answer a very simple question.

Â So what is the average loss when I

Â know that my loss will be above the value-at-risk?

Â And if you remember, the value-at-risk is a quantile of a loss distribution.

Â So how can I define that informally looking at the graph?

Â So if you remember, the graph of the loss distribution, so

Â here we have a nice bell shaped, but it shouldn't necessarily be a bell shape.

Â And I look at the loss distribution, and if you remember, the value-at-risk is

Â a level, so that I have a 1% probability to have a loss above that level.

Â When I look at the expected shortfall,

Â what I will do is simply look at the averages of the losses

Â when I know that I will have a loss above the value-at-risk.

Â So for example if the value-at-risk is equal to $1 million,

Â the expected shortfall will tell me whether in average the loss when I

Â have a loss above $1 million will be equal for example to $50 million or $1 million.

Â So it will be the average loss when I know that my loss is above my value-at-risk.

Â So of course the question that you should ask me is why should we use the expected

Â because we have already at our disposal another measure which

Â is called the value-at-risk?

Â So an actual question what are the advantages of the expected

Â shortfall with respect to the value-at-risk?

Â You have in fact two main advantages, the first main advantage is

Â that the expected shortfall is what is called a subadditive is measured.

Â So let us look at the formula defined in subadditivity of the expected shortfall.

Â As you can see, we have the expected shortfall computed on a1 plus a2 so

Â that means that I will have one portfolio made of a1 and

Â one portfolio made of a2, and I look at the sum.

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The other two parts are the sum of the expected shortfall computed with

Â only the portfolio made of a1, and only made of the portfolio made of a2.

Â And as you can see the expected shortfall computed on a1 plus a2

Â will be automatically lower than the sum of the two expected shortfalls.

Â So the two it is a measure computed on the individual portfolios.

Â So you have probably heard about the type of notion which is called diversification.

Â So if I look at two portfolio, intuitively, when I look at the two

Â portfolio individually, the sum of the two risk when I consider them

Â individually should be or both the case when I consider them combined.

Â So when I look at the joint position make of a1 and a2.

Â So intuitively this is a nice notion.

Â Subadditivity because it's the translation of the notion of diversification.

Â So the expected shortfall is always subadditive.

Â This is not necessarily the case for the value-at-risk.

Â So the value-at-risk doesn't ensure you that you will have always at

Â the measure of risk on the sum of two portfolio will be always

Â lower than the sum of the risk measure computed on the two individual portfolios.

Â So this is a first advantage of the expected shortfall.

Â Now what is a second advantage, and

Â I talked about that a little bit earlier in the video is that

Â the value of risk just gives you a single point in the PNL distribution.

Â So the only information that you have is that is the one person

Â probability level you will have a loss above the value of three.

Â So for example above $1 million, but you have no clue about whether

Â that loss will be $1 million, $2 million or $10 million.

Â So when you look at the expected shortfall, you have an additional

Â information, which is the average loss when you have a loss above $1 million.

Â So the expected shortfall gives you an additional information.

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So now let us look at how we can compute the expected shortfall, and

Â how can we define formally the expected shortfall.

Â So, this is again some formula.

Â So the first one that you see is an expectation.

Â So you have the expectation of the loss return.

Â Knowing that the loss return is above the value-at-risk.

Â So it is exactly the notion that I show you on the graph.

Â I look at the value-at-risk and I look at the average losses knowing that

Â my loss is above the value-at-risk.

Â So this is a conditional expectation, so I will not enter too much into

Â the detail but an application of what is called the Bayes' theorem which defines

Â conditional expectation, allows you to rewrite that conditional expectation.

Â So the expectation of the loss return, knowing so it's a conditional expectation.

Â So knowing something which is knowing that the loss of return is above value.

Â At least I can rewrite that as standard expectation.

Â It will be an expectation of velocity return, multiply by an indicator function.

Â And that indicator function takes the value one,

Â if indeed you are above the value-at-risk, and zero otherwise.

Â So this is the expectation, and

Â you will divide by the probability of the conditioning event.

Â And here the conditioning event is simply that the loss return is above

Â the value-at-risk.

Â So now if you remember the definition of the value-at-risk by construction,

Â by definition the value-at-risk is so

Â that the that the loss return is above the value-at-risk is equal to 1 minus alpha.

Â And this gives you the final formula for the expected shortfall.

Â It will be the average return multiplied by the indicator function that the rate

Â the loss return above the value-at-risk, divided by 1 minus alpha.

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Â