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Â This session will be about the expected shortfall.

Â So, you can view the expected shortfall as an extension

Â 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 shape but it shouldn't necessarily be a bell shape.

Â And I look at the loss distribution and if you remember,

Â the value at risk is a level.

Â So that I have a one person probability to have a loss above that level.

Â So 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 short fall 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 shortfall because we have already, at our disposal,

Â another measure which is called the value at risk?

Â So in answer your question, what the advantages of the expected shortfall

Â with respect to the value at risk.

Â You have in fact two main advantages.

Â The first main advantage is that the expected shortfall is what is called

Â subadditive risk measure.

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.

Â So it mean 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 only

Â the port when you 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 will

Â be automatically lower than the sum of the two Expected Shortfall.

Â So, the two is measure computed on the individual portfolios.

Â So we have already 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 portfolios individually, the sum of the two risk when I consider them individually

Â should be above the risk when I consider them combined.

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

Â So intuitively, this is a nice notion subadditivity because it's

Â 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 that you will have always that

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

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

Â So this is the first advantage of the Expected Shortfall.

Â Now, what is the second advantage?

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

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

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

Â probability level you will have a loss above the value at risk.

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 a additional

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

Â So the Expected Shortfall give 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 so

Â the first I'm going to let you see is an expectation.

Â So you have the expectation of the loss return.

Â Knowing that the lost 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

Â I look at the average losses knowing that my loss is above the value at risk.

Â So this is a conditional expectation.

Â So I will not enter too much into the detail, but an application of

Â what is called the biased yuremwhich defines conditions and expectation.

Â Allows you to rewrite that conditional expectation, so

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

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

Â I can rewrite that as a standard expectation.

Â It will be the expectation of the loss return

Â multiplied by an indicator function.

Â And that indicator function tells 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

Â is above the value at risk.

Â So now if you remember the definition of the value at risk.

Â By construction, by definition, the value at risk is so that the probability,

Â 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 it will

Â be the average return multiplied by the integrate of function that

Â the loss return are above the value at risk divided by one minus five.

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Â