Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint
Louvain Finance Center & CORE, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
Received: 30 May 2018 / Revised: 21 June 2018 / Accepted: 22 June 2018 / Published: 25 June 2018
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
-dimensional distribution corresponding to a n
-dimensional i.i.d. standard Normal vector
subject to the weighted sum constraint
. This law is proven to be a Normal distribution, whose mean vector
and covariance matrix
are explicitly derived as a function of
. 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
under constraint that
and is illustrated on two applications dealing with Value-at-Risk and Expected Shortfall.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
Share & Cite This Article
MDPI and ACS Style
Vrins, F. Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint. Risks 2018, 6, 64.
Vrins F. Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint. Risks. 2018; 6(3):64.
Vrins, Frédéric. 2018. "Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint." Risks 6, no. 3: 64.
Show more citation formats
Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
[Return to top]
For more information on the journal statistics, click here
Multiple requests from the same IP address are counted as one view.