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 vector . 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 then  can 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.




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