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The Tail Value-at-Risk, TVaR, of a portfolio is
defined as the expected outcome (loss), conditional on the loss exceeding the Value-at-Risk (VaR), of
Where the support of the
distribution is continuous the VaR with confidence level is usually
defined as follows:
The corresponding Tail Value-at-Risk would then be defined
Visually the difference between VaR and Tail VaR 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.
Writers use Tail VaR (TVaR) and Conditional VaR
(CVaR) largely interchangeably, usually with the same loss trigger as the
quantile level that would otherwise be applicable if the focus was on VaR.
Occasionally, TVaR and/or CVaR are differentiated, with one being expressed in
terms of the loss beyond the VaR rather than below zero. However, such a
definition inherits some technical weaknesses attributable to VaR (i.e. that it
no longer exhibits diversification properties we might ‘expect’ a risk measure
Another term that means much the same thing is Conditional
Tail Expectation (CTE), although perhaps this is more likely to refer to
the right tail of a distribution rather than the left tail, i.e. it might focus
on upside rather than downside, and the bound beyond which it is calculated may
not be expressed in a VaR-like form.
Shortfall has a similar meaning, but might use a trigger level set more
generically, e.g. it might include all returns below some level (e.g. zero) and
more commonly might no longer contain the multiplier
included in the definition of TVaR a above.
If several different risk exposures are contributing to the
overall TVaR then it often becomes important to identify the contribution each
is making to the total. This can be done using marginal Tail Value-at-Risk
(or marginal TVaR).