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Suppose I have times series data on 3 assets and I do $N$ simulations (GBM) first for each of assets individually and then from a multidimensional GBM since their log-returns are correlated (I use Cholesky decomposition). My claims:

  1. Mean returns and individual VaRs are independent of the simulation procedures mentioned above.

  2. VaR of an equally weighted portfolio is dependent on the simulation procedure. I calculate expected return of a portfolio as $N^{-1}\sum_i\sum_kx_{ik}/3$, where $k = 1,2,3$ indexes an asset and $i = 1,2,\dots,N$ indexes a replication and $x$ is an overall return of an asset.

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  • $\begingroup$ When you say "independent of the simulation procedures mentioned above", what do you mean? That the results do not depend whether you use the first (individual) simulations or the second (multidimensional) simulation? $\endgroup$ – Alex C Jun 20 '18 at 1:23
  • $\begingroup$ @AlexC yes, exactly. $\endgroup$ – tosik Jun 20 '18 at 12:00

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