Marginal Tail Value-at-Risk (Marginal
TVaR) when underlying distribution is multivariate normal
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Suppose we have a set of risk factors
which we can characterise by an -dimensional
vector . Suppose that
the (active) exposures we have to these factors are characterised by another -dimensional
vector, . The aggregate
exposure is then .
The Value-at-Risk, , of the
portfolio of exposures at confidence
level , is defined as
the
The Marginal Tail Value-at-Risk, , is the
sensitivity of to a small change
in ’th exposure. It
is therefore:
In the case where the risk factors are multivariate normally
distributed with mean and covariance
matrix whose elements
are we have and hence . Hence .
Given the formula for the truncated first moments of a normal distribution
we have:
where , , is the
(standard) normal cumulative distribution function and is the
(standard) normal probability density function.
Hence:
The second of these terms can be expressed in terms of the
correlation between and in a manner
similar to Marginal
VaR when underlying distribution is multivariate normal.
As risks arising from individual positions interact there is
no universally agreed way of subdividing the overall risk into contributions
from individual positions. However, a commonly used way is to define the Contribution
to Tail Value-at-Risk, , of the ’th
position, to be as
follows:
Conveniently the then sum to the
overall VaR:
The property that the contributions to risk add to the total
risk is a generic feature of any risk measure that is (first-order)
homogeneous, a property that Tail Value-at-Risk exhibits.