# Quasi Random Monte Carlo in m.v. portfolio optimization

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:

1. 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?
2. 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.

...this technique works only when returns are generated from normal distributions?

Yes and no. Multiplying them by $$C$$ will produce the correlation that you wanted, but it won't preserve the distribution in general. Remember that when we apply $$C$$ to a vector of i.i.d. random variables $$\boldsymbol{x}$$ that the resultant vector element is $$\sum_j C_{ij}x_j$$, which is a weighted sum of the i.i.d. random variables. In general the sum of two (or more) random variables from some distribution need not follow the same distribution as its constituents. For example this doesn't hold for the uniform distribution, but it does hold for the normal distribution. (Interesting it also holds for the Cauchy distribution!). Some of the distributions this works for:

• Binomial
• Negative binomial
• Poisson
• Normal
• Cauchy
• Gamma
• $$\chi^2$$

The fact that the variance holds comes from $$\mathbb{V}(C\boldsymbol{x}) = CC^T\mathbb{V}(x) = CC^TI = \Sigma$$ where for i.i.d. standardised random variables $$\boldsymbol{x}$$ we have $$\mathbb{V}(x) = I$$.

Is it preferred use correlation or covariance matrix

The method requires the covariance matrix. If you are trying to estimate this from data then we need to recover a positive definite matrix, which has it's own challenges. There are issues with numeric stability and computer performance (cf. PCA as an alternative to Cholesky), as well as ensuring the positive definite requirement, which are addressed in this question: Principle Component Analysis vs. Cholesky Decomposition for MonteCarlo

how one should deal with this?

Again, several of these issues are dealt with in the answer to the previous point.

• Thank you very much for the detailed explanation. I will try do more tests. – Nipper Jul 8 at 21:07
• Having some spare time I re-evaluate the whole topic: so Cholesky decomposition applied on a samples of randomly generated returns from a Gumbel distribution will not preserve the distribution. All things considered is it better using Archimedean Copulas? – Nipper Jul 10 at 0:18
• @Nipper sounds like a whole new question. – oliversm Jul 10 at 0:20