Widely accepted methods for coming up with the co-variance matrix of assets?

Question

Question

What are the widely accepted ways for coming up with co-variance matrix of assets after the Markowitz's modern portfolio theory?

Question explained in more detail

1. After Modern portfolio theory was introduced, to my best knowledge, there are bunch of new theories of co-variance methods came out.

2. Exponential co-variance matrix and Ledoit-Wolf are two examples of those new methods.

3. Can somebody tell me the more recent advancement of the co-variance matrix to create a portfolio? If the paper has a github code written in Python, it would be more than welcome.

Answer 1

Multivariate volatility models for replacing the sample covariance matrix with in the mean-variance portfolio selection model:

1. RiskMetrics 1996 EWMA (Exponentially weighted moving average) covariance matrix

2. RiskMetrics 2006 EWMA covariance matrix

3. Multivariate DCC-GARCH covariance matrix

   • Jon Danielsson "Financial risk forecasting" has EWMA and GARCH for R and Matlab and looks like Python now too.
   • Kevin Sheppard's MFE toolbox for Matlab and Arch package for Python have EWMA and GARCH. RiskMetrics 2006 EWMA for Python is here.

4. Ledoit and Wolf (2003) covariance matrix based on the single factor index model:

   • Ledoit, O. and Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10:603-621.
   • Matlab code

5. Ledoit and Wolf (2004) covariance matrix based on the identity matrix:

   • Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88:365-411.
   • Matlab code
   • Python code (Sci-kit learn package)

See the second author's publication page if links change at https://www.econ.uzh.ch/en/people/faculty/wolf/publications.html

Answer 2

Ledoit and Wolf have a new paper ( November 2018 ) called "Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices" which has MATLAB code for the procedure at the end of the paper. The paper can be downloaded at SSRN.