[MUSIC]

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 shape 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 one person probability to have a loss above that level.

So 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 short fall 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

shortfall because we have already, at our disposal,

another measure which is called the value at risk?

So in answer your question, what 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

subadditive risk measure.

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 it mean that I will have one portfolio made of a1 and

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