Improved estimation of covariance matrix for portfolio optimization - Priyanka Agarwal - Rez Chowdhury - Dzung Du - Nathan Mullen - Ka Ki Ng

Progress so far: 1.Simulations for 12 estimators presented last week 2.Implementation of Benninga’s two block estimator 3.Speed Optimization 4.Cleaning and documenting the code for the software 5.Portfolio with constraints 6. PCA estimator

Motivation behind shrinkage and portfolio of estimators 1.Covariance matrix not invertible. 2.Optimal portfolio with large short-sale positions. 3.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 1.Stocks in NYSE only (as in Disatnik/ Benninga) 2.First portfolio formed on Jan 1974; last portfolio formed for Total 360 monthly returns for each of the 6 simulations 4.In-sample period: 120 months and 60 months 5.Out-sample period: 12, 24 and 36 months 6.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 1.Simulations show consistency of our codes. 2.Performance improvement using shrinkage estimator and portfolio of estimators is within the same range. 3.Portfolio of estimators is simpler to use and implement. 4.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 1.Estimated covariance matrix with two blocks. 2.Each block has the sample variance on the diagonal. 3.Pair of stocks within the same block have the same covariance (  1 and  2 ). 4.Covariance between stocks from different blocks equals a third constant . 5.Disatnik and Benninga characterizes conditions on the covariances when unconstrained GMVP is positive. (refer to the paper)

Two Block Estimator – Arbitrary example 1.Each block with the same number of stocks divided based on permno. 2.  1 = (0.99) min(s i 2 ) 3.  2 = (0.99) min(s i 2 ) 4.  = (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 1.One block with stocks with positive beta and another with negative beta stocks. (positive  1 and  1 while negative  ) 2.More than two blocks * We may implement this if time permits (this was not tested by Disatnik, Benninga 2006)

Speed Optimization 1.Worst case scenario: ~ 75% faster Old code: ~3700 seconds Optimized code: ~900 seconds 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: 1.Fix covariance matrix estimation with constraints 2.Implement PCA and more than one factor industry models 3.Speed Optimization 4.Look into issue with shrinkage to constant correlation estimator 5.Fix memory issues with 60 months in-sample simulation 6.If time permits, implement a more financial oriented two-block estimator (this was not implemented by Disatnik, Benninga 2006)