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. Original problem : extension of Markowitz portfolio optimization Diversity Extension

8. Adding The L-2 norm constraint

11. L-1 norm constraint:

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

14. Robust Model  The robust model  E is an ellipsoid

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

16. Robust Optimization (cont’d)

17. Robust Optimization Ellipsoids  Ellipsoids  Fact iff

18. 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

19. 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

20. Random Matrix Confidence Interval  Confidence interval

21. Random Matrix Formulation  Problem to solve

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

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

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

25. Contribution to Variance Model

26. QQP Formulation  Add artificial :

27. We’d Like To Thank