[MUSIC] Hi I'm Olivier so in this session I will talk about the value at risk and how we can use that value at risk. So, again, we will address two questions. The first question is, what is value at risk, or how can we define the value at risk? And the second question that we'll address is how can we compute the value at risk in practice? So what is the value at risk? So you have a very informal definition. So it's indeed a quantitative and very synthetic risk measure and the value at risk tries to address a very simple question. So, what is the minimum loss level so that you can have for a given probability. Okay, so let's look at the graph so we see that we have the loss distribution and here the loss distribution is symmetric but it doesn't need to be symmetric. So when you look at the value at risk here, you focus on the right tail of the loss distribution. And as you can see in the right tail, I have a level which is called the value at risk, and that level correspond to the level so that I have a 1% probability level to be above that value at risk of that level. So to be concrete, if my value at risk is equal to $1 million loss, it mean that I will have a 1% probability level to have loss above that $1 million. OkY, so this is in the definition of the value at risk. So let me now be a little bit more formal in terms of the definition and look at the formula. So the first formula correspond to probability level, and I can see that the probability level is fixed here equal to one minus alpha. So alpha is equal to 99%, one minus alpha will be equal to 1%. And I look at the 1% probability and I look at the probability of which type of event. When the event is very simple I look at the return of my portfolio, which is here denoted by Rt, and I will add something, and that something is called the value at risk. So when I consider the return on my portfolio plus my value at risk, the sum of the two, I have a probability that this sum is negative. Will be equal to one minus alpha, for example, equal to 1%. So if I look at the definition, the value at risk is in fact can be viewed as a capital, as a safety cushion. And this is why the value at risk by convention, and this is what you will find in the recommendation of the Basel committee. The value at risk is defined with a positive sign, okay? So in fact it corresponds to an actuarial standard. So for an actuary when I look at a loss, the loss has a positive sign. We lose $1 million with a positive sign. If I'm a finance guy, so, finance guys are used to use minus. So for a finance guy, you don't lose $1 million, but you will lose minus $1 million. So this is a subtle difference when you look at the definition of the value at risk. So the value at risk is defined with a positive sign. So now you can see that this first definition can be written as a probability that minus that return on your portfolio is above the value at risk is equal to one minus alpha. And this is exactly the definition that I have used when I was looking at the distribution of my losses. When I was looking at the level so that I have a probability equal to 1% to be above that level so above $1 million for example. Now if you look at the definition of the return of the portfolio, so the return of portfolio is simply defined as a sum of the allocation. Here the allocation are denoted by a, but if you look at other representation, it might be, for example, defined as w for weights. Also, for example, if I have 50% of my investment in IBM, this a will be equal to 50%. If I have 20% in Dell, the a will be equal to 20%. So if I multiply my weights or my allocation by my return, and I do that for all of my assets and I sum everything, I will have the return on my portfolio, so this is the definition of a portfolio. [MUSIC]