Value-at-Risk
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The Value-at-Risk, VaR, of a portfolio 
is
the outcome (loss), 
, that will be exceeded on a
fraction 
of occasions. It requires a
time-scale (
) as well as a confidence level .
Where the support of the
distribution is continuous, the Value-at-Risk with confidence level is:
If the distribution is unbounded below then this means that
if the probability density function of the distribution is 
we
have:
If the lower bound is finite, e.g. 
then the
definition would replace 
with .
Note: sometimes and 
are
interchanged or losses are deemed positive rather than negative etc.
Visually the difference between VaR and Tail VaR (TVaR) may be
seen in either of the following charts:
VaR is not (in general) a coherent risk
measure, whilst TVaR is. VaR is arguably more shareholder focused and TVaR more
regulator/customer focused, see VaR versus TVaR mindsets.
If several different risk exposures are contributing to the
overall VaR then it often becomes important to identify the contribution each
is making to the total. This can be done using marginal Value-at-Risk.