Covariance Estimation For Markowitz Portfolio Optimization

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Covariance Estimation For Markowitz Portfolio Optimization Ka Ki Ng Nathan Mullen Priyanka Agarwal Dzung Du Rezwanuzzaman Chowdhury 1 4/7/2010

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Covariance Estimation For Markowitz Portfolio Optimization. Ka Ki Ng Nathan Mullen Priyanka Agarwal Dzung Du Rezwanuzzaman Chowdhury. Outline:. Michaud’s Resampling combined with Ledoit’s estimators Weight Descriptor Return Analysis Standard error Conclusion and Future Research. - PowerPoint PPT Presentation

Transcript of Covariance Estimation For Markowitz Portfolio Optimization

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Covariance Estimation For Markowitz Portfolio Optimization

Ka Ki NgNathan Mullen

Priyanka AgarwalDzung Du

Rezwanuzzaman Chowdhury

4/7/2010

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Outline:1. Michaud’s Resampling combined with Ledoit’s

estimators

2. Weight Descriptor

3. Return Analysis

4. Standard error

5. Conclusion and Future Research

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Resampling combined with Ledoit’s

1. Estimate (μ, Σ) from the observed data using estimators in Ledoit’s paper

2. Propose the distribution for the returns, e.g., Returns ~ N(μ, Σ) – meaning prices follow log-normal distributions

3. Resample n (large) of Monte Carlo scenarios

4. Solve the optimization problem for each Monte- Carlo scenario

5. Resampling allocation computed as the average of all obtained allocations.

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Performance

Std (unconstrained )

Resampling Combined (unconstrained)

Identity 18.12 18.18

Constant Correlation 13.10 19.9Market Model 11.08 21.64

PCA 9.13 21.34Shrinkage to identity 10.83 18.20Shrinkage to market 8.94 9.46

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Performance

Std (constrained )

StdResampling Combined

(constrained)Identity 18.11 18.18

Constant Correlation 15.19 19.9Market Model 12.59 20.11

PCA 10.36 20.43Shrinkage to market 9.53 9.02Shrinkage to identity 10.37 19.3

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Weight Descriptors – Lowest Weight

Ledoit’s values Our valuesIdentity 0.09 0.08

Constant Correlation -0.17 -0.14Industry Factor -1.08 -1.20Market Model -0.41 -0.36

PCA -0.84 -0.81Shrinkage to identity -1.23 -1.07Shrinkage to market -1.01 -0.89

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Weight Descriptors – Highest Weight

Ledoit’s values Our valuesIdentity 0.09 0.08

Constant Correlation 2.86 2.51Industry Factor 2.81 2.94Market Model 2.1 1.87

PCA 2.95 2.66Shrinkage to identity 1.19 1.03Shrinkage to market 3.81 3.0

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Rate of Returns on Investments• Average annual arithmetic return

– No reinvesting, i.e. start each year with the same amount of money

where

• Average annual geometric return– Time weighted– With reinvesting

n

iiyieldarithmetic r

nr

1,

1

nn

iiyieldgeometric rr/1

1,11

end

startendyield V

VVr

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Return Analysis

• New features for return analysis in software 1. Average annual return

(both geometric and arithmetic)

2. Plot of annual return3. Histogram of monthly return with Gaussian fit

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Compare to S&P 500• Compared the returns of all 16 estimators (for both

unconstrained and 20% constrained cases) and S&P 500– The comparison was incomplete when it was presented two

weeks ago

• Downloaded the S&P 500 data from CRSP

• Added an option for S&P 500 comparison in the software– Value-Weighted Return– Equal-Weighted Return

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Annual Return Plots

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Histogram of Returns• Typical returns have heavier tails than a Gaussian distribution• Gaussian fit

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Average Annual Geometric Return

ConstantCorr

Identity

Industry

Market PCA

Pseudo

Shrin

kage

Shrin

kageC

orr

Shrin

kageId

n

diag_si

gma

port_tw

oblock_sig

ma

portcorr_

sigma

portdiag

_sigm

a

portdiag

corr_sig

ma

portran

dom_sigm

a

twoblock_

sigma

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Unconstrained20% ConstrainedS&P 500 Equal-Weighted ReturnS&P 500 Value-Weighted Return

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Analysis• Return of identity matrix ≈ S&P 500 Equal-weighted return• Problems

1. 20% constrained results are much lower than 20%2. Most importantly, 20% constrained results are lower

than the unconstrained cases• Possible reasons for the error

1. q = 0.0153 is wrong2. Not on efficient frontier 3. Most likely: Models based on

previous data applying to future data. Bad predictor of future expected returns μ

121

2 1

BACBqA

BACqBCw

"For expected returns, we just take the average realized return over the last 10 years. This may or may not be a good predictor of future expected returns, but our goal is not to predict expected returns: it is only to show what kind of reduction in out-of-sample variance our method yields under a fairly reasonable linear constraint."

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Sample Output Text File

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Standard errorMethod 1: Assuming returns to be iid gaussian

Method 2: Get N bootstrap samples for monthly returns assuming uniform distribution

Method 3: Get N bootstrap samples for monthly log-returns assuming log-normal distribution

Bootstrap results are stable for N=1000

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Standard error- Conclusion1. Bootstrap technique confirmed our earlier belief that

the standard errors presented in the paper are monthly.

Though we agree it makes more sense to present annualized standard errors when standard deviations are annualized.

2. Our monthly SE results for uniform distribution bootstrap closely match with Ledoit’s (even for Identity and Psuedo-inverse estimators).

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Results:Unconstrained Standard Error (monthly)

Ledoit Bootstrap Uniform

Bootstrap Lognormal

IID Gaussian

Identity 0.44 0.42 0.23 0.22Constant Correlation 0.19 0.18 0.16 0.16Pseudo inverse 0.23 0.23 0.15 0.15Market Model 0.16 0.17 0.14 0.14Industry factors 0.17 0.17 0.11 0.12PCA 0.16 0.19 0.12 0.12Shrinkage to identity 0.17 0.19 0.12 0.12Shrinkage to market 0.15 0.18 0.11 0.11

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Results:Constrained Standard Error (monthly)

Ledoit Bootstrap Uniform

Bootstrap Lognormal

IID Gaussian

Identity 0.42 0.38 0.24 0.23Constant Correlation 0.29 0.29 0.19 0.19Pseudo inverse 0.32 0.31 0.17 0.16Market Model 0.27 0.27 0.15 0.16Industry factors 0.23 0.23 0.13 0.13PCA 0.22 0.25 0.13 0.13Shrinkage to identity 0.21 0.24 0.12 0.13Shrinkage to market 0.20 0.23 0.12 0.12

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Results:Unconstrained Standard Error (annualized)

Ledoit Bootstrap Uniform

Bootstrap Lognormal

IID Gaussian

Identity 0.44 1.45 0.80 0.76Constant Correlation 0.19 0.62 0.55 0.55Pseudo inverse 0.23 0.80 0.52 0.52Market Model 0.16 0.59 0.48 0.48Industry factors 0.17 0.59 0.38 0.42PCA 0.16 0.66 0.42 0.42Shrinkage to identity 0.17 0.69 0.42 0.42Shrinkage to market 0.15 0.62 0.38 0.38

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Results:Constrained Standard Error (annualized)

Ledoit Bootstrap Uniform

Bootstrap Lognormal

IID Gaussian

Identity 0.42 1.32 0.83 0.80Constant Correlation 0.29 1.00 0.66 0.66Pseudo inverse 0.32 1.07 0.59 0.55Market Model 0.27 0.94 0.52 0.55Industry factors 0.23 0.80 0.45 0.45PCA 0.22 0.87 0.45 0.45Shrinkage to identity 0.21 0.83 0.42 0.45Shrinkage to market 0.20 0.80 0.42 0.42

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Project goals and achievementsLedoit and Wolf [2002]

1. Implemented 8 estimators including ‘Shrinkage to market’

2. Results for standard deviation closely match those in the paper.

3. Fixed calculations for constrained portfolio

4. Solved standard error calculation puzzle

5. Added functionality for Return Analysis

6. Weight Analysis

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Project goals and achievements

Benninga & Disatnik [2002]

1. Implemented 8 estimators including ‘two block estimator’

2. Ran simulations for different time periods

3. Results for standard deviation closely match those in the paper.

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Future work:

1. Matlab code documentation and cleaning

2. Project report

3. Any further analysis Prof. Pollak suggests

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