[MUSIC] >> Now that you have defined formally what is the expected shortfall, let me explain you how we can compute the expected shortfall in practice. So, as we saw in the value at risk, we'll have two main approaches. The variance-covariance approach, which is based on the assumption of Gaussian return. And the historical approach. So, let me first focus on the variance-covariance approach. So, you see that you will have something which is quite similar to the formula that we saw in the value at risk. So, you recognize the mu. So the mu correspond to the expected return of the portfolio. You recognize a sigma, which is the volatility of the portfolio return. You recognize z alpha, which is the quintile of the Gaussian distribution which will be equal to 2.3 something if you use a priority level equal to 99%. And here you see that you have an additional function which is the phi. And the phi is simply the density of a Gaussian distribution. This might look a little bit complicated, but you can see that the only that you need to compute is an average of the portfolio return and the standard deviation of the volatility of the portfolio return. And then you simply plug you replace in the formula that you have on the slides. So now that we have seen how to compute the expected shortfall using the variance-covariance approach, let me focus on the historical approach. So this approach will be very similar to what we saw for the value at risk. So, what I will do is simply look at the history of return, and so I will collect data, for example, on the S&P 500. So return on the S&P 500, and I will take a minus. Then what I will do is only focus on the loss return which are above the value at risk, and I will compute the empirical average. This is what you see in the formula. You see that I have a sum. Okay? I have a sum on the return of the S&P 500 multiplied by the indicator function. So which takes value 1 if you are above the value at risk, and zero if you are below. I have a sum divided by T, so this correspond to an empirical average, but only for the return which are above the value at risk. Now, of course, I would also have to divide by 1 minus alpha, which correspond to the 1% if I use alpha is equal to 99%. So what are the learning outcomes of this session? So, again, we have three learning outcomes. The first one is that the expected shortfall is indeed a simple synthetic and quantitative risk measure. It correspond to the average loss return when my loss return is above the value at risk, and the value at risk correspond to the quantile of a loss distribution. And the third outcome that we saw is that it's very, very simple to compute. Under the variance-covariance approach, you compute an empirical mean. You compute an empirical standard deviation to the volatility, and you replace in the formula. And for the historical approach, the only study that you have to do is to compute an empirical average of the loss return which are above the value at risk. So this is very simple to implement. [MUSIC]