How Asset Correlations Affect Simulated Portfolio VaR

Summary

The document compares simulating three assets independently with simulating them jointly through a multidimensional geometric Brownian motion. In the joint approach, correlated log returns are represented using a covariance structure and Cholesky decomposition. The author claims that mean returns and each asset’s individual VaR do not depend on which simulation procedure is used, while VaR for an equally weighted portfolio does depend on it. The portfolio return is described as the average across assets and simulation replications.

The key statistical point is that portfolio tail risk depends on co-movement: separate simulations do not preserve the assets’ joint return distribution, while a correlated simulation is intended to do so. The note poses a question and offers claims rather than a demonstration; it gives no parameter estimates, simulated outcomes, or validation. Results depend on the return model, estimated correlations, horizon, and precise VaR calculation. It is therefore a useful prompt about modeling dependence, not evidence that the stated claims hold under every setup.

Key ideas

• Joint simulations can represent dependence among asset returns through their covariance structure.
• Cholesky decomposition can be used to generate correlated log-return shocks.
• Portfolio VaR can reflect cross-asset co-movement, unlike VaR calculated separately for each asset.
• The document states claims about simulation outcomes but provides no numerical evidence or model validation.

Tags

# Does it make sense to simulate from the multidimensional GBM?


# Does it make sense to simulate from the multidimensional GBM?












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:

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

- 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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.