Improved estimation of covariance matrix for portfolio optimization
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Improved estimation of covariance matrix for portfolio optimization. Priyanka Agarwal Rez Chowdhury Dzung Du Nathan Mullen Ka Ki Ng. Progress so far:. Simulations for 12 estimators presented last week Implementation of Benninga’s two block estimator Speed Optimization

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Improved estimation of covariance matrix for portfolio optimization

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Improved estimation of covariance matrix for portfolio optimization

  • Priyanka Agarwal

  • Rez Chowdhury

  • Dzung Du

  • Nathan Mullen

  • Ka Ki Ng


Progress so far:

  • Simulations for 12 estimators presented last week

  • Implementation of Benninga’s two block estimator

  • Speed Optimization

  • Cleaning and documenting the code for the software

  • 5.Portfolio with constraints

  • 6. PCA estimator


Motivation behind shrinkage and portfolio of estimators

  • Covariance matrix not invertible.

  • Optimal portfolio with large short-sale positions.

  • Large N (number of assets) as compared to T (time-series) - This makes the matrix ill-conditioned with large off-diagonal elements amplifying the estimation error.


Simulations

  • Stocks in NYSE only (as in Disatnik/ Benninga)

  • First portfolio formed on Jan 1974; last portfolio formed for 2003

  • Total 360 monthly returns for each of the 6 simulations

  • In-sample period: 120 months and 60 months

  • Out-sample period: 12, 24 and 36 months

  • Compare the performance of shrinkage estimators vs. portfolio of estimators


Table 1.a


Table 1.b


Table 1.c


Table 2.a


Table 2.b


Table 2.c


Conclusion

  • Simulations show consistency of our codes.

  • Performance improvement using shrinkage estimator and portfolio of estimators is within the same range.

  • Portfolio of estimators is simpler to use and implement.

  • Shrinkage estimator gives rise to a new type of error.


Two Block Estimator

  • Motivation: To overcome the drawbacks of the short sale constraint

  • Discontinuity imposed on the relation between asset statistics and optimal asset weights

  • Solution obtained is numerical and not analytical

  • Result: Produces a positive GMVP in an unconstrained optimization


Two Block Estimator - Methodology

  • Estimated covariance matrix with two blocks.

  • Each block has the sample variance on the diagonal.

  • Pair of stocks within the same block have the same covariance (1 and 2 ).

  • Covariance between stocks from different blocks equals a third constant .

  • Disatnik and Benninga characterizes conditions on the covariances when unconstrained GMVP is positive. (refer to the paper)


Two Block Estimator – Arbitrary example

  • Each block with the same number of stocks divided based on permno.

  • 1 = (0.99) min(si2)

  • 2 = (0.99) min(si2)

  •  = (0.99) min(1, 2 )


Two Block Estimator – Results

Portfolio with two block estimator performs the best amongst all with no constraints.


Two Block Estimator – Improvements

  • One block with stocks with positive beta and another with negative beta stocks. (positive 1 and 1 while negative )

  • More than two blocks

* We may implement this if time permits (this was not tested by Disatnik, Benninga 2006)


Speed Optimization

  • Worst case scenario: ~ 75% faster

  • Old code: ~3700 seconds

  • Optimized code: ~900 seconds

  • 2. 60 months in-sample takes longer to run than 120 months in-sample.

  • 3. ‘Out of memory’ issues with 60 months in-sample simulations


Backup


Estimate of shrinkage constant (shrinkage to market)


Future actions:

  • Fix covariance matrix estimation with constraints

  • Implement PCA and more than one factor industry models

  • Speed Optimization

  • Look into issue with shrinkage to constant correlation estimator

  • Fix memory issues with 60 months in-sample simulation

  • If time permits, implement a more financial oriented two-block estimator (this was not implemented by Disatnik, Benninga 2006)


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