Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint

Statistical modeling techniques—and factor models in particular—are extensively used in practice, especially in the insurance and finance industry, where many risks have to be accounted for. In risk management applications, it might be important to analyze the situation when fixing the value of a weighted sum of factors, for example to a given quantile. In this work, we derive the ( n − 1 ) -dimensional distribution corresponding to a n -dimensional i.i.d. standard Normal vector Z = ( Z 1 , Z 2 , … , Z n ) ′ subject to the weighted sum constraint w ′ Z = c , where w = ( w 1 , w 2 , … , w n ) ′ and w i ≠ 0 . This law is proven to be a Normal distribution, whose mean vector μ and covariance matrix Σ are explicitly derived as a function of ( w , c ) . The derivation of the density relies on the analytical inversion of a very specific positive definite matrix. We show that it does not correspond to naive sampling techniques one could think of. This result is then used to design algorithms for sampling Z under constraint that w ′ Z = c or w ′ Z ≤ c and is illustrated on two applications dealing with Value-at-Risk and Expected Shortfall.