[MUSIC] Hi I'm Olivier 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 a value at risk or how can we define the value at risk? And the second question that we will address is how can we compute that value at risk in practice. So, what is the value at risk? So you have a very informal definition which so in deed a quantitive and a very synthetic risk measure and the value at risk try to address 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 corresponds to the level so that I have a 1% probability level to be above that value at risk through that level. So to be concrete, if my value at risk is equal to $1 million loss, it means that I will have a 1% probability level to have a loss above that $1 million. Okay, 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 a probability level and I can see that the poverty level is 60 equal to 1 minus alpha. So is alpha is equal to 99%, 1 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 denoted by RT and I will add something, and that something is called the value at risk. So that when I consider the return on my portfolio plus my value at risk, the sum of the two have a probability that this sum is negative will be equal to 1 minus alpha. For example, equal to one person. So if I look at the definition, the value at risk is in fact, can be viewed as a capital as a safety caution. And this is why the value at risk by convention and this is what you will find in the recommendation of the Bazaar committee, the value at risk is defined with a positive sign. Okay, do in fact it correspond to. An actuarial standard, so for an actuary, when I look at a loss, the loss has a positive side. I will lose $1 million with a positive side. If I'm a finals guy, so finals guys are used to use a minus, so for a finals 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 the return on your portfolio is above the value at risk is equal to 1 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 point equal to one person 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 define as a sum of the allocation, here the allocation already noted by a, but if you look at other presentation it might be for example define as W for weights. So 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 asset and I sum everything, I will have the return on my portfolio. So this is the definition of a portfolio return. [MUSIC]