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Improved estimation of covariance matrix for portfolio 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)