Improved estimation of covariance matrix for portfolio optimization

- Priyanka Agarwal- Rez Chowdhury- Dzung Du- Nathan Mullen- Ka Ki Ng

Progress since the last presentation:

1. Progress in implementation of Ledoit’s Industry Factors estimator

2. Standard error calculations (assuming iid Gaussian distribution)

3. Calculations for annual returns for estimators

4. Other work in progress:

--The option to not look at the out-of-sample data when selecting portfolio.

--Writing portfolio data out to text file (to facilitate analysis)

--Started standard error calculation using bootstrapping method

Progress in implementation of Ledoit’s Industry Factors estimator

Sharpe’s Single-index model (p.5 of Ledoit):

Industry Factors model (p.14 of Ledoit):

ikc is a “dummy variable” equal to 1 if the stock “i” corresponds to the industry “k”. It is equal to zero otherwise.

ktz is equal to the industry “k”s average return at time “t”.

Progress in implementation of Ledoit’s Industry Factors estimator (cont.)

Ledoit uses the same industry grouping as Fama and French’s 1997 “Industry Costs of Equity” in Appendix A:

-Uses ranges of Standard Industrial Classification (SIC) codes to group stocks of similar industry into the same group.

-SIC code is a 4 digit code ranging from 0100-9999.-The 2 first numbers (from left to right) refer to a “major

group”, whereas the first 3 numbers refer to an “industry group”, and all four refer to the specific industry (CRSP website)

-Fama and French break up different industries into “industry” groups different from the one’s indicated by the SIC code.

Progress in implementation of Ledoit’s Industry Factors estimator (cont.)

• For example (partial screen-shot from Fama and Frech Appendix A):

(see http://www.ehso.com/siccodes.php for a nice list of SIC codes and meanings)

Progress in implementation of Ledoit’s Industry Factors estimator (cont.)

• We put all of these mappings into a big chain of if-elseif statements

• Our testing revealed a small mistake in Fama and French’s industry grouping:– SICs 3210-3219 are included in two different industry

groupings:

Group “321x” refers to “Flat Glass”, which is likely best grouped into Construction Materials

Progress in implementation of Ledoit’s Industry Factors estimator (cont.)

• 32 SIC codes exists in the 1960-2009 data that don’t map to Fama and French Industry groups– We will manually place these in the groups where

we think they fit best.• At this point we have all of the code that

calculates the industry averages.• Need to use these to calculate Betas, and then

the covariance matrix (see page 245 of text)

Calculated Standard Deviation and Standard Error

Estimator SD-Led SE-Led SD-Ours SE-Ours(annualized) SE-Ours(monthly)

Identity 17.75 0.44 18.42 0.78 0.23

Constant correlation 14.27 0.19 13.22 0.56 0.16

Pseudo-inverse 12.37 0.23 12.1 0.51 0.15

Market model 12 0.16 11.15 0.47 0.14

Principal components 10.31 0.16 9.51 0.4 0.12

Shrinkage to identity 10.21 0.17 9.87 0.42 0.12

Shrinkage to market 9.55 0.15 8.95 0.38 0.11

T = 23 * 12 = 276

(standard error formula taken from Note)

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Future Work

• Finish Industry Factors implementation• Finish the option to not look at the out-of-

sample data when selecting portfolio.• Calculate Standard errors by resampling• Add additional output data (write to file).