EconPapers    
Economics at your fingertips  
 

Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint

Frédéric Vrins ()
Additional contact information
Frédéric Vrins: Louvain Finance Center & CORE, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

Risks, 2018, vol. 6, issue 3, 1-13

Abstract: 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.

Keywords: conditional distribution; conditional sampling; multi-factor models (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 M2 M4 K2 (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations Track citations by RSS feed

Downloads: (external link)
https://www.mdpi.com/2227-9091/6/3/64/pdf (application/pdf)
https://www.mdpi.com/2227-9091/6/3/64/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jrisks:v:6:y:2018:i:3:p:64-:d:154355

Access Statistics for this article

Risks is currently edited by Prof. Dr. J. David Cummins

More articles in Risks from MDPI, Open Access Journal
Bibliographic data for series maintained by XML Conversion Team ().

 
Page updated 2018-10-02
Handle: RePEc:gam:jrisks:v:6:y:2018:i:3:p:64-:d:154355