[MUSIC] So now let me look at more concretely how we can compute the Value-at-Risk in practice. So if you look at the way that we compute it in practice, there are two methods. So the first method is called a variance-covariance method, and the second method is called the historical method. The first method, the variance-covariance method, implicitly assumes that the return can be approximated by a nice Gaussian distribution, so a bell- shaped distribution which is symmetric. So we have seen that this is sometimes, indeed, a correct assumption, for example, when I'm along the S & P of 500, but in other cases when the profit and loss distribution is asymmetric, an assumption of a symmetry distribution is probably incorrect. So the main message here is that you should avoid to use the variance-covariance approach when you have asymmetric distribution. So now let me look a little bit more into detail to the formula, okay. So if I look at the Value-at-Risk and the variance-covariance approach, we can see that the Value-at-Risk, of course, will depend on the allocation, so the weight. Okay, so you have the allocation in asset 1, a1. You will have the allocation in asset n, an. And, of course, it will also depend on the probability level, alpha, for example, 99%. So how can I compute that? It's very simple. So what you basically have to do is to compute the expected return, or the average return, of your portfolio, and you have to compute the volatility or the standard deviation of your portfolio. And then you will multiply that by the quintile of Gaussian distribution, which at the 99% level is equal to 2.3 and something. Okay, so again, this is very simple to compute because you only have to compute a mean and a volatility. And here the mean is denoted by mu and the volatility is simply denoted by sigma. Now, if I want to compute that from the characteristics of the individual assets, and the characteristics of the individual assets will be the individual expected return, and the individual volatility and also the covariance, you can see that I have, indeed, an explicit formula to compute the mu, so the expected return of my portfolio, and the variance, which is the square volatility of the return of my portfolio. So how do you compute my mean? Very simple, I take all my weights, all my allocation, I multiply by the expected return of all the individual assets and I sum. And this will give me the expected return on my portfolio. For the variance, you have a slightly more complicated formula where you can see that you have a double sum. And the double sum will involve the allocation of the weight in each of the individual assets, multiplied by the covariance and the variance of your individual asset, okay. So this is how you compute under the assumption of a Gaussian distribution. So, now, what can we do if we have an asymmetric profit and loss distribution? In that case, what I advise is simply to use what is called the historical approach. So again this is very simple to compute and to implement because what you have to do is just to collect what we call the history of return on your portfolio. And then what you will do is simply compute what is called the empirical quantile of the loss distribution, okay. So what you do is, in fact, very simple. And so you look at the return of your portfolio, so you look at the history of the portfolio return, you put a minus in front of all these returns and you rank them. And the empirical quantile will be simply the level so that you have 99% of the data which are below and 1%of the data above. So a percentile is very similar to something that you have probably seen in your statistical courses, which is the median. And the median is the percentile at a 50% probability level. So in that case, you have 50% of your observations which are below, and 50% which are above. So, here, instead of having 50% below and 50% above, I will have 99% below and 1% above. So, again, this is very simple to compute. So, what are the learning outcomes of the session? So, what we learn is what is the Value-at-Risk? And we saw that the Value-at-Risk is a quantitative and very significant measure of risk. It's a quantile of the loss distribution, and we saw that it's very simple to compute under either the variance-covariance approach or the historical approach. [MUSIC]