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

Introduction & Background  The efficient frontier

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

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

Original problem : extension of Markowitz portfolio optimization Diversity Extension

Adding The L-2 norm constraint

L-1 norm constraint:

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

Robust Model  The robust model  E is an ellipsoid

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

Robust Optimization (cont’d)

Robust Optimization Ellipsoids  Ellipsoids  Fact iff

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

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

Random Matrix Confidence Interval  Confidence interval

Random Matrix Formulation  Problem to solve

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

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

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

Contribution to Variance Model

QQP Formulation  Add artificial :

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