The instability and high sensitivity of optimisation results can be augmented by adding another layer of quantitative methodology in the form of Monte Carlo Simulation. The name Monte Carlo alludes to the nature of the simulation procedure, which, in essence, involves drawing random numbers from a distribution, and then using the random numbers as inputs for a mathematical process, in this case portfolio optimisation. [Quantitative Portfolio Optimization, Asset allocation and Risk management - Mikkel Rassmussen - 2003]
I'm currently trying to apply Monte Carlo techniques in the context of mean variance portfolio optimization.
According to what I have learned until now the most basic and simple model is "Resampling" and it consists in the following steps:
- For each asset fit the historical returns (daily, weekly or monthly data) with a distribution of the parametric family (normal, Student's t, etc.) and obtain the specific parameters (mean, variance).
- For each asset generate a random returns from its specific probabilistic distribution.
- Performing mean-variance optimization (tangency portfolio which implies Sharpe-Ratio maximization) using the generated random returns to compute expected returns and covariance matrix.
- Repeat point 2. and 3. for n times.
- Average the weights of all portfolios.
My questions are the following:
- How one should compute correctly statistics (expected return, expected volatility) of the final averaged optimized portfolio?
- Is not very clear to me if one should average the weights of all portfolios (point 5.) according to some techniques or just computing the simple mean. If the first, which are these techniques?
- Are there ways to improve the "Resampling" other than trying different probability distributions (i.e. generate expected returns not directly from a probability distribution but applying i.e. Single Index Model - $R_{it}=\alpha_i+\beta_i \cdot R_{mt} + \epsilon_{it}$ - the random component in that case would be noise $\epsilon_{it}$?
- Does makes sense generate random return with a multivariate probability distribution (mean is the mean of each asset and variance is the covariance matrix)? Doing so I noticed that all assets are always in the portfolio.