Not specifying a correlation matrix for the Monte Carlo Simulation's random returns is equivalent to assuming no correlation or a correlation coefficient of zero, which will seriously and adversely affect the results of the simulation. In the Quasi-Random Monte Carlo Simulated Asset Allocation methodology values are thus no longer drawn completely at random from a distribution, but rather are drawn purposefully according to a set of consistent asset class assumptions. [Quantitative Portfolio Optimization, Asset allocation and Risk management - Mikkel Rassmussen - 2003]
After having learned about "simple" Monte Carlo methodology in the context of mean variance portfolio optimization (Monte Carlo (resampling) in m.v. portfolio optimization) I understood that a major improvement would be considering the covariance/correlation matrix of the assets while drawing of random samples of returns.
All the papers and articles about Quasi Random Monte Carlo approach I have been able to find and read by now assume that returns of assets are normally distributed:
- One generates samples directly from the multivariate normal distribution (it is specified by a vector of means and usually by a covariance matrix).
- One generates samples for each asset from normal distribution (it is specified by mean and standard deviation) and then multiply them by a matrix $C$ such that $C \cdot C^T$ equals to the covariance or correlation matrix $\sum$ (thus introducing a correlated bias).
- $C$ can be generated according to Cholesky decomposition or from the eigenvalues and eigenvectors.
Considering all the above my questions are the following:
- Of course the assumption that returns are normally distributed is quite simplistic so given a sample of non-normally randomly generated returns (i.e. from fitted Gumbel distribution - via maximum likelihood estimation) is still correct multiply them by the matrix $C$ or this technique works only when returns are generated from normal distributions? If so which other techniques one should apply to non-normal randomly generated returns?
- In any case generating $C$ requires first the computation of covariance/correlation matrix from historical data. Is it preferred use correlation or covariance matrix as input of i.e. Cholesky decomposition (please specify advantages and disadvantages of both)?
Thank you all.