1Statistical Analysis of Efficient Portfolios

Econ 424/Amath 540Eric Zivot

Eric Zivot 2006

Eric ZivotSummer 2011

Updated: August 16, 2011

Example Data: Monthly cc Returns on Boeing, Microsoft, Nordstrom and Starbucks

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Eric Zivot 2006

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q41999 2000 2001 2002 2003

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msftnordboeing

2Full Sample CER Model Estimates> muhat.vals

sbux msft nord boeing 0.01782 -0.00006364 0.003202 0.001688

> sigmahat.valssbux msft nord boeing

0.1353 0.1375 0.1325 0.1097> cor.mat

sbux msft nord boeing

Eric Zivot 2006

sbux msft nord boeing sbux 1.000000 0.295506 0.1525 0.008218msft 0.295506 1.000000 0.3833 0.007876nord 0.152500 0.383348 1.0000 0.258940

boeing 0.008218 0.007876 0.2589 1.000000

CER Model Estimates with Standard Errors

> rbind(muhat.vals,se.muhat)sbux msft nord boeingsbux msft nord boeing

muhat.vals 0.01782 -0.00006364 0.003202 0.001688se.muhat 0.01747 0.01775446 0.017110 0.014168

> rbind(sigmahat.vals,se.sigmahat)sbux msft nord boeing

sigmahat vals 0 13534 0 13753 0 1325 0 10975

Eric Zivot 2006

sigmahat.vals 0.13534 0.13753 0.1325 0.10975se.sigmahat 0.01236 0.01255 0.0121 0.01002

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95% Confidence Ellipses for and m

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sigma

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Global Minimum Variance Portfolio# global minimum variance portfolio> gmin.4 = globalMin.portfolio(er=muhat.vals,+ cov mat=cov mat)+ cov.mat=cov.mat)> summary(gmin.4)Call:globalMin.portfolio(er = muhat.vals, cov.mat = cov.mat)P tf li t d t 0 00543

Eric Zivot 2006

Portfolio expected return: 0.00543 Portfolio standard deviation: 0.07655 Portfolio weights:sbux msft nord boeing 0.241 0.1907 0.1252 0.443

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Portfolio Weights

Global Minimum Variance Portfolio

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sbux msft nord boeing

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Bootstrapping the Global Minimum Variance Portfolio

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5Bootstrapping the Global Minimum Variance Portfolio

Number of Replications: 100 Summary Statistics:

Observed Bias Mean SE er 0.00543 0.0003845 0.005815 0.009668sd 0.07655 -0.0045245 0.072027 0.009249

sbux 0.24100 0.0177585 0.258759 0.090101msft 0 19070 -0 0081261 0 182577 0 090327

Eric Zivot 2006

msft 0.19070 0.0081261 0.182577 0.090327nord 0.12525 0.0015642 0.126812 0.089178

boeing 0.44305 -0.0111965 0.431852 0.094154

Efficient Portfolio: Target = 0.015> eport.015 =+ efficient.portfolio(er=muhat.vals,+ cov.mat=cov.mat,+ target.return=0.015)> summary(eport.015)Call:efficient.portfolio(er = muhat.vals, cov.mat = cov.mat, target.return = 0.015)

Eric Zivot 2006

Portfolio expected return: 0.015 Portfolio standard deviation: 0.1104 Portfolio weights:

sbux msft nord boeing 0.7937 -0.1673 0.1435 0.2301

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Portfolio Weights

Efficient Portfolio: Target = 0.0150.

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sbux msft nord boeing

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Bootstrapping the Efficient Portfolio with Target=0.015

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7Bootstrapping the Efficient Portfolio with Target=0.015

Number of Replications: 100 Summary Statistics:

Observed Bias Mean SE er 0.0150 -1.735e-017 0.01500 0.0000sd 0.1104 -1.500e-002 0.09537 0.0293

sbux 0.7937 -4.139e-001 0.37986 0.2831msft -0 1673 2 781e-001 0 11084 0 2938

Eric Zivot 2006

msft 0.1673 2.781e 001 0.11084 0.2938nord 0.1435 -2.817e-002 0.11529 0.3653

boeing 0.2301 1.639e-001 0.39401 0.3013

Efficient Frontier

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Portfolio SD

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Bootstrapping the Efficient Frontier

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Portfolio SD

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Rolling weights in global min portfolio

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24 month rolling weights in global minimum variance portfolio: mt(n)

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Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q42000 2001 2002 2003

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9Rolling means and sds on global min portfolio

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24 month rolling means, t(n), and standard deviations, t(n).0.

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Rolling weights in efficient portfolio with target=0.015

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24 month rolling weights in efficient portfolio with target=0.015: xt(n)

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Rolling means and sds on efficient portfolio with target=0.015

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24 month rolling means, t(n), and standard deviations, t(n).

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Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q42000 2001 2002 2003

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