1 / 27

Portfolio Diversity and Robustness

Portfolio Diversity and Robustness. TOC. Markowitz Model Diversification Robustness Random returns Random covariance Extensions Conclusion. Introduction & Background. The classic model S - Covariance matrix (deterministic) r – Return vector (deterministic) Solution via KKT conditions.

arin
Download Presentation

Portfolio Diversity and Robustness

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Portfolio Diversity and Robustness

  2. TOC • Markowitz Model • Diversification • Robustness • Random returns • Random covariance • Extensions • Conclusion

  3. Introduction & Background The classic model S - Covariance matrix (deterministic) r – Return vector (deterministic) Solution via KKT conditions

  4. Introduction & Background The efficient frontier

  5. Problems and Concerns Number of assets vs. time period Empirical estimate of Covariance matrix is noisy Slight changes in Covariance matrix can significantly change the optimal allocations Sparse solution vectors Without diversity constraints the optimal solution allows for large idiosyncratic exposure

  6. Outline Diversity Constraints L1/L2-norms Robust optimization via variation in returns vector Variation in Covariance Estimators via Random Matrix theory Results Further developments

  7. Diversity Extension Original problem : extension of Markowitz portfolio optimization

  8. Adding The L-2 norm constraint

  9. L-1 norm constraint:

  10. Robust optimization The classic model Robust: letting r vary i.e. adding infinitely many constraints

  11. Robust Model The robust model E is an ellipsoid

  12. Robust Model (cont’d) Family of constraints: it can be shown that The new Robust Model:

  13. Robust Optimization (cont’d)

  14. Robust Optimization Ellipsoids Ellipsoids Fact iff

  15. Random Matrix Theory • Covariance Matrix is estimated rather than deterministic • The Eigenvalue/Eigenvector combinations represent the effect of factors on the variation of the matrix • The largest eigenvalue is interpreted as the broad market effect on the estimated Covariance Matrix

  16. Random Matrix Implementation compute the covariance and eigenvalues of the empirical covariance matrices Estimate the eigenvalue series for the decomposed historical covariance matrices Calculate the parameters of the eigenvalue distribution Perturb the eigenvalue estimate according to the variability of the estimator

  17. Random Matrix Confidence Interval Confidence interval

  18. Random Matrix Formulation Problem to solve

  19. Markowitz and Robust Portfolio Return is assumed to be random r~N(m,S) Robust portfolio also lies on efficient frontier

  20. Efficient Frontier Perturbed Covariance The worst case perturbed Covariance matrix shifts the entire efficient frontier

  21. Further Extensions • Contribution to variance constraints • Multi-Moment Models • Extreme Tail Loss (ETL) • Shortfall Optimization

  22. Contribution to Variance Model

  23. QQP Formulation • Add artificial :

  24. We’d Like To Thank

More Related