Relative Value-at-Risk (Relative VaR)
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Suppose we want to derive the (relative) portfolio
Value-at-Risk (relative VaR) when returns 
on
the 
exposures
are jointly Gaussian, assuming that the corresponding portfolio weights are 
and
corresponding benchmark weights are 
.
By jointly Gaussian we mean that the vector of returns 
is
distributed as a multivariate normal distribution 
,
where 
is
a vector of mean returns and 
is
a covariance matrix.
A property of any -dimensional Gaussian, i.e.
multivariate Normal, distribution that can be derived relatively simply from
the probability density function of such a distribution is that if 
and
if we have a constant vector 
then
is
univariate Normal 
for
some 
and
.
Specifically:
where the 
are
the elements of the covariance matrix .
By relative return we mean the return on the portfolio
relative to the return on the benchmark. For any given time period the return
on the portfolio is the weight the portfolio ascribes to the exposure times the
return on that exposure, i.e. is 
,
where 
is
a vector with components 
.
Likewise the return on the benchmark is is 
,
where 
is
a vector with components 
.
So the relative return* is 
.
Hence the relative return is distributed as a univariate Normal distribution
with 
and
.
The relative VaR with confidence level 
is
then 
with
these and 
, where 
is
the inverse Normal function.
* N.B. there is an assumption here that the returns over the
relevant time interval are small, otherwise there is an issue about whether to
use arithmetic relative or geometric relatives etc., see e.g. Relative Return
Computations).