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Published byGertrude Clarke Modified over 3 years ago

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Portfolio Diversity and Robustness

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TOC Markowitz Model Diversification Robustness Random returns Random covariance Extensions Conclusion

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

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Introduction & Background The efficient frontier

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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

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

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Original problem : extension of Markowitz portfolio optimization Diversity Extension

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Adding The L-2 norm constraint

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L-1 norm constraint:

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

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Robust Model The robust model E is an ellipsoid

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

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Robust Optimization (cont’d)

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Robust Optimization Ellipsoids Ellipsoids Fact iff

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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

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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

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Random Matrix Confidence Interval Confidence interval

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Random Matrix Formulation Problem to solve

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

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

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

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Contribution to Variance Model

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QQP Formulation Add artificial :

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