Covariance Estimation For Markowitz Portfolio

Optimization

Team = {Ka Ki Ng, Nathan Mullen, Priyanka Agarwal,Dzung Du, Rez Chowdhury}

Presentation by Rez

Outline

• 1. Covariance estimator code implementation this week

• 2. Overview of each estimator implemented– Results (std. dev. values)

• 3. More Results (some extra plots)• 4. Conclusion & Future Work…

This week’s achievements

• Total of 14 estimators from 2 papers

• Implemented 3 more estimators from Ledoit & Wolf paper– Only 2 estimators left from this paper…

• Principals Components and Industry Factors– PCA code is almost working… some bugs :(

This week’s achievements

• Disatnik and Benninga has 7 estimators– We already had one working. – This week we implemented the remaining 6…

Extra stuff that almost worked: - Almost got all the estimators to work with Ledoit and Wolf’s

constraint of 20% expected return on portfolio’s… some bugs that should be very easily fixable…

- Almost got all the estimators to work with short sales constraint… also should be relatively easy to debug…

Ledoit’s Standard Error Values

• Tried many things…• Seems that Ledoit has developed his own

method for estimating standard error for stock returns that are not necessarily assumed to be gaussian…

• The method and the code is buried in his big list of papers

• We have some leads and may get this to work…

Some methodology work

• Developed an algorithm to NOT look into the future for stock picking like Ledoit– It should also hold cash (“risk-free” rate)

positions for stocks that drop out in investment horizon just like Ledoit

– Yet to be implemented, but hopefully in the future…

Identity Estimators

• Covariance matrix is scalar multiple of identity matrix– Ledoit uses the mean of the diagonal values from the sample matrix

for this…

– Ledoit GMVP std dev = 17.75

– Our GMVP std dev = 18.43

Constant Correlation

- Every pair of stocks has the same correlation coefficient.- N + 1 parameters (N variance, 1 covariance)- Ledoit GMVP std dev = 14.27- Our GMVP std dev = 13.22

Shrinkage to Identity• The scalar multiple of identity matrix is the shrinkage target

– Ledoit GMVP std dev = 10.21– Our GMVP std dev = 9.87

Shrinkage factor is stable around 0.1. One can argue this implies robustness of the model in some sense…

Note on these results…

• Our implementation gives slightly better (lower) risk values than Ledoit– Why?

• We look into the future (Benniga method)– If stocks don’t drop out, variance (volatility) is

reduced– Also, Ledoit’s using cash positions should be a factor

» The cash positions can be relatively pretty big…

Benninga estimators

• Diagonal estimator– Simply the diagonal of the sample matrix.

Everything else is zero.• Stalking Horse. Terrible Estimator, but att least

it’s invertible…

Benninga GMVP std dev = 13.12

Our GMVP std dev = 15.01

Shrinkage to constant correlation matrix• The Constant Corr matrix is the shrinkage target

– Benninga GMVP std dev = 8.52

– Our GMVP std dev = 9.11

Shrinkage factor is sort of stable around 0.7… Maybe one can argue this implies robustness of the model in some sense…

Random average of sample and single index

• Uniformly random variable alpha goes from (0.5,1)– Why (0.5,1) instead of (0,1)?

– Benninga uses Ledoit and Wolf observation that there is more estimation error in the sample matrix than there is specification error in the single-index matrix.• So AT MOST sample gets half the weight• Whereas market index matrix gets AT LEAST half the weight

•Benninga GMVP std dev = 8.51•Our GMVP std dev = 9.08

Portfolio of sample, single index, and constant corr

• Equally weighted– Benninga GMVP std dev = 8.47– Our GMVP std dev = 9.00

Portfolio of sample, single index, and constant corr, diagonal

• Equally weighted– Benninga GMVP std dev = 8.46– Our GMVP std dev = 8.98

Portfolio of sample, single index, diagonal

• Equally weighted– Benninga GMVP std dev = 8.39– Our GMVP std dev = 8.97

Observation on these results

• The ranking of estimators based on risk is the same, with the very simple diagonal portfolio estimator being a very close second best!

• Our standard deviation values are slightly higher though…– Why?

– Our CSRP data window is 1970 to 1995 (Ledoit and Wolf)

– Main reason:• Benninga and Disatnik window is from 1964 to 2003.

– Also our periods go from August to July

» Benninga’s goes from January to December

Future Work

• We should definitely also run the models for the Benninga window of observation. All the values for the Benninga estimators are expected to match then…

» The End. - Thanks!