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)?
- I already made an attempt to apply Quasi Random Montecarlo assuming that returns are normally distributed. I noticed that using Cholesky decomposition to introduce a correlation bias is very unstable. Specifically computing differently the covariance matrix of historical returns (I'm not referring to the technique itself i.e. Empirical covariance v.s. Shrinked covariance but rather how one annualize the matrix i.e. 1 year, 20 years, 100 years etc.) leads to great differences in allocation results. So I was wondering, how one should deal with this?
Thank you all.