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I definitely struggle to understand the following interpretation of VAR (value at risk) provided by Jorion

$$VAR(c)=E[X]−Q(X,c)$$

where $X$ is a random variable, $E[X]$ its expected value, $Q(X,c)$ the quantile of its distribution such that the associated probability is c.

Isn't VAR the quantile itself? Why should it be viewed as a deviation from the mean?

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  • $\begingroup$ Two reasons: 1) "Risk" is interpreted as deviation from what you would expect. 2) If VAR were equal to the quantile you would not need a new word for it. $\endgroup$
    – g g
    Nov 18, 2020 at 15:20

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VaR is the quantile of the loss distribution but $X$ in your post no doubt denotes the the future value of the portfolio. If $E[X]=10$ (expected future value of portfolio) and the quantile $Q(X,c)=4$ for some $c$, then your VaR, that is the lost that represents the outcome at the quantile, is $$VaR(c) = E[X] - Q(X,c) = 10 - 4 = 6$$

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