Mathematical Definition
Given a confidence level, the VaR of the portfolio at the confidence level is given by the smallest number such that the probability that the loss exceeds is at most . Mathematically, if is the loss of a portfolio, then is the level -quantile, i.e.
The left equality is a definition of VaR. The right equality assumes an underlying probability distribution, which makes it true only for parametric VaR. Risk managers typically assume that some fraction of the bad events will have undefined losses, either because markets are closed or illiquid, or because the entity bearing the loss breaks apart or loses the ability to compute accounts. Therefore, they do not accept results based on the assumption of a well-defined probability distribution. Nassim Taleb has labeled this assumption, "charlatanism." On the other hand, many academics prefer to assume a well-defined distribution, albeit usually one with fat tails. This point has probably caused more contention among VaR theorists than any other.
VaR can also be written as a distortion risk measure given by the distortion function
Read more about this topic: Value At Risk
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