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Marginal Tail Value-at-Risk (Marginal TVaR) 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,  . 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.