Marginal Value-at-Risk (Marginal VaR)
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, . Then the aggregate
exposure is .
The Tail Value-at-Risk, , of the
portfolio of exposures at confidence
level , is defined as
the value such that . The Marginal
Value-at-Risk, , is the
sensitivity of to a small change
in ’th exposure,
i.e.:
In the case where the risk factors are multivariate normally
distributed with mean and covariance
matrix whose elements
are we have the
following.
As we have where is the standard
deviation of the volatility of the (active) portfolio return, otherwise known
if we are focusing on active exposures as the (ex-ante) tracking error.
The last part of this equation can be expressed in terms of
the correlation between and as follows.
Suppose we view the as corresponding
to time series with elements
(which without loss of generality can be assumed to be de-meaned, i.e. to have
their means set to zero) and as corresponding
to a time series . Then the
correlation between and would be
(ignoring any small sample adjustment):
We would also have:
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 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 Value-at-Risk exhibits.