I have a portfolio consisting of bonds, stocks, fx and property. I'm using Monte Carlo to estimate the VaR of this portfolio. To generate the forward-looking interest rate risk factor scenarios I use the G2++ model which is calibrated to historical interest rate data. I have three term structures - EUR, USD and GBP.
I want to use a gaussian copula to correlate the risk factors. From what I understand the interest rate risk factor scenarios generated by the G2++ model are already correlated (due to the calibration) so the correlations between the different tenors for each term structure need to be set to zero. Correlations between different term structures tenors (e.g. EUR 2 year to USD 5 year) need to be estimated from historical data. So do the correlations between interest rates and the other risk factors (equity, fx and property).
After the correlation matrix is estimated I need to apply the Cholesky decomposition (gaussian copula) and multiply the result with the generated risk factor scenarios. The correlated scenarios can then be used to estimate VaR.
Here are the questions I have:
- Is this the correct approach? Are there any steps missing?
- The decomposed matrix (Cholesky) needs to be multiplied with the standard normal random vectors? Can it be also multiplied directly with the generated scenarios (after the standard normal random numbers have been used to generate the Monte Carlo scenarios)?
- If more risk factors (that are not part of the portfolio) are used in the calculation (e.g. if I include more interest rate tenors or more equities that are not represented by any instrument in the portfolio) should the results be different? The correlation matrix will change and the outcome of the Cholesky decomposition will be different. The standard normal random numbers will be different too. All these lead me to believe that a small deviation is expected but given a large number of scenarios should I expect material differences?