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This session will be about the expected shortfall, so

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

So again I will look at two questions.

So the first question is what is the expected shortfall?

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

So again the expected shortfall is a quantitative and

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

So what is the average loss when I

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

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

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

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

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

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

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

When I look at the expected shortfall,

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

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

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

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

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

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

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

because we have already at our disposal another measure which

is called the value-at-risk?

So an actual question what are the advantages of the expected

shortfall with respect to the value-at-risk?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So intuitively this is a nice notion.

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

So the expected shortfall is always subadditive.

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

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

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

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

So this is a first advantage of the expected shortfall.

Now what is a second advantage, and

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

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

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

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

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

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

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

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

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

how can we define formally the expected shortfall.

So, this is again some formula.

So the first one that you see is an expectation.

So you have the expectation of the loss return.

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

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

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

my loss is above the value-at-risk.

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

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

conditional expectation, allows you to rewrite that conditional expectation.

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

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

At least I can rewrite that as standard expectation.

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

And that indicator function takes the value one,

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

So this is the expectation, and

you will divide by the probability of the conditioning event.

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

the value-at-risk.

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

by definition the value-at-risk is so

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

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

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

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

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