Here are the following steps to calculate Monte Carlo VaR. I am learning how to proceed with each steps and I would need somebody who can explain. Do I have to create only 1 vector in step 4 (even if i have multi asset portfolio) ? In which case I don't understand which mu and sigma i use to created my standard normal variates since I want to mimic estimators from my assets log returns.
Here are the steps I have managed to pickup using different sources:
Estimate the portfolio's current value $P_0$.
Build the portfolio's covariance matrix using stock historical data.
Create the Cholesky decomposition of the covariance matrix.
Generate a vector of n independent standard normal variates
multiply the matrix resulting from the Cholesky decomposition with the vector of standard normal variates in order to get a vector of correlated variates.
Calculate the assets' terminal prices using geometric brownian motion. $S_i(T) = S_i(0) \cdot e^{((\mu-\frac{\sigma^2}{2})T + \sigma \sqrt{T} \epsilon_i})$, where $\epsilon_i$ corresponds to the correlated random variate for asset $i$ obtained from the vector of correlated variates.
reevaluate the portfolio's value at time $T$, $P_T$, using the stock prices generated in the previous step.
Calculate the portfolio return using $R_T=\frac{P_T−P_0}{P_0}$
Repeat steps 4-8 many times (for example $n=10,000$ simulations).
Sort the returns in ascending order.