[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 assume that the return can be approximated by a nice Gaussian distribution. So a bell shape distribution which is symmetric. So we have seen that this is sometimes indeed a correct assumption, for example, when I'm long the S&P 500. But in other cases, when the profit and loss distribution is asymmetric, an assumption of 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 under the variance-covariance approach, we can see that the value-at-risk of course will depend on the allocation, so the weights, 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 quantile of Gaussian distribution which is at a 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 asset will be the individual expected return and the individual volatility and also the covariance, you can see that I have indeed explicit formula to compute the mu. So the expected return on my portfolio and the variance which is the squared volatility of the return of my portfolio. So how do compute my mean? Very simple, I take all my weights, all my allocation, I multiply by the expected return of all the individual asset 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 or the weight in each of the individual asset 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 this return and you rank them. And the empirical quintile will be simply the level so that you have 99% of the data which are below and 1% of the data which will be 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 simply the percentile at a 50% probability level, so in that case you have 50% of your observation 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 this session? So what we learned is, what is the value-at-risk, and we saw that the value-at-risk is quantitative and very scientific measure of risk. It's a quantile of the loss distribution, and we saw that it's very simple to compute, and either the variance-covariance approach or the historical approach. [MUSIC]