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### Marginal Value-at-Risk (Marginal VaR) when underlying distribution is multivariate normal

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.