Marginal Value-at-Risk (Marginal VaR)
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Suppose we have a set of risk
factors which we can characterise by an -dimensional
Suppose that the (active) exposures we have to these factors are characterised
by another -dimensional vector, .
Then the aggregate exposure is .
The Value-at-Risk of the portfolio of exposures at
confidence level , ,
is usually defined to be the value such that .
The Marginal Value-at-Risk, ,
is the sensitivity of to
a small change in ’th exposure. It is therefore:
Because Value-at-Risk is (first-order) homogeneous (for a
continuous probability distribution) it satisfies the Euler capital allocation
principle and hence:
If the risk factors are multivariate normally distributed
be expressed using a relatively simple formula, see here.
The Marginal Value-at-Risk is sometimes called the
incremental value at risk (perhaps because a leading software vendor uses the
latter terminology). More usually incremental value at risk is defined as
follows, where is
the same as except
that it has 0 for its ’th entry.