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Â So now let me look at more concretely how we can compute

Â the value-at-risk in practice.

Â So if you look at the way that we compute it in practice, there are two methods.

Â So the first method is called a variance-covariance method and

Â the second method is called the historical method.

Â The first method, the variance-covariance method, implicitly assume that the return

Â can be approximated by a nice Gaussian distribution.

Â So a bell shape distribution which is symmetric.

Â So we have seen that this is sometimes indeed a correct assumption, for example,

Â when I'm long the S&P 500.

Â But in other cases, when the profit and loss distribution is asymmetric,

Â an assumption of symmetry distribution is probably incorrect.

Â So the main message here is that you should avoid to use

Â the variance-covariance approach when you have asymmetric distribution.

Â So now let me look a little bit more into detail to the formula, okay?

Â So if I look at the value-at-risk under the variance-covariance approach,

Â we can see that the value-at-risk of course will depend on the allocation, so

Â the weights, okay?

Â So you have the allocation in asset 1 a1,

Â you will have the allocation in asset n an.

Â And, of course, it will also depend on the probability level alpha, for example, 99%.

Â So, how can I compute that?

Â It's very simple.

Â So what you basically have to do is to compute the expected return or

Â the average return of your portfolio.

Â And you have to compute the volatility or the standard deviation of your portfolio.

Â And then you will multiply that by the quantile of Gaussian

Â distribution which is at a 99% level is equal to 2.3 and something.

Â Okay, so again, this is very simple to compute because you only have to compute

Â a mean and a volatility.

Â And here the mean is denoted by mu and the volatility is simply denoted by sigma.

Â Now if I want to compute that from the characteristics of the individual assets,

Â and the characteristics of the individual asset will be the individual

Â expected return and the individual volatility and also the covariance,

Â you can see that I have indeed explicit formula to compute the mu.

Â So the expected return on my portfolio and

Â the variance which is the squared volatility of the return of my portfolio.

Â So how do compute my mean?

Â Very simple, I take all my weights, all my allocation,

Â I multiply by the expected return of all the individual asset and I sum.

Â And this will give me the expected return on my portfolio.

Â For the variance, you have a slightly more complicated formula

Â where you can see that you have a double sum.

Â And the double sum will involve the allocation or

Â the weight in each of the individual asset multiplied by the covariance and

Â the variance of your individual asset, okay?

Â So this is how you compute under the assumption of a Gaussian distribution.

Â So now, what can we do if we have an asymmetric profit and loss distribution?

Â In that case,

Â what I advise is simply to use what is called the historical approach.

Â So again, this is very simple to compute and to implement because what you have

Â to do is just to collect what we call the history of return on your portfolio.

Â And then, what you will do is simply compute what is called

Â the empirical quantile of the loss distribution, okay?

Â So what you do is in fact very simple and

Â so, you look at the return of your portfolio.

Â So you look at the history of the portfolio return,

Â you put a minus in front of all this return and you rank them.

Â And the empirical quintile will be simply the level so that you have 99%

Â of the data which are below and 1% of the data which will be above.

Â So a percentile is very similar to something that you have

Â probably seen in your statistical courses which is the median.

Â And the median is simply the percentile at a 50% probability level, so in that

Â case you have 50% of your observation which are below and 50% which are above.

Â So here, instead of having 50% below and 50% above,

Â I will have 99% below and 1% above.

Â So again, this is very simple to compute, so

Â what are the learning outcomes of this session?

Â So what we learned is, what is the value-at-risk, and we saw that

Â the value-at-risk is quantitative and very scientific measure of risk.

Â It's a quantile of the loss distribution, and

Â we saw that it's very simple to compute, and

Â either the variance-covariance approach or the historical approach.

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Â