[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. The other two parts are the sum of the Expected Shortfall computed with only the port when you 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 Shortfall. So, the two is measure computed on the individual portfolios. So we have already heard about the type of notion which is called diversification. So if I look at two portfolio, intuitively when I look at the two portfolios individually, the sum of the two risk when I consider them individually should be above the risk when I consider them combined. So when I look at the joint position made up 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 that you will have always that the measure of risk on the sum of two portfolios will be always lower than the sum of the risk measure computed on the two individual portfolios. So this is the first advantage of the Expected Shortfall. Now, what is the second advantage? And I talked about that a little bit earlier in the video is that the value at risk just gives you a single point in the PNL distribution. So the only information that you have is that was a one person probability level you will have a loss above the value at risk. 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 a additional information, which is the average loss, when you have a loss above $1 million. So the Expected Shortfall give you an additional information. 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 I'm going to let you see is an expectation. So you have the expectation of the loss return. Knowing that the lost 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 biased yuremwhich defines conditions and 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 return is above value at risk. I can rewrite that as a standard expectation. It will be the expectation of the loss return multiplied by an indicator function. And that indicator function tells 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 probability, 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 integrate of function that the loss return are above the value at risk divided by one minus five. [MUSIC]
