Covariance Estimation For
Markowitz Portfolio Optimization
Ka Ki Ng
Nathan Mullen
Priyanka Agarwal
Dzung Du
Rezwanuzzaman Chowdhury
2/24/2010
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Portfolio Selection Problem
• Consider N stocks whose returns are distributed with mean μ
and covariance matrix Σ
• Markowitz defines the portfolio selection problem as:
min w' w
w
w'1  1
subject to
w'   q
where q is the required expected return
• Solve it using Lagrange multipliers, the solution is:
C  qB 1
qA  B 1
w
 1
 
2
2
AC  B
AC  B
where A  1'  11, B  1'  1 , C   '  1
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Portfolio Selection Problem
• Input: expected stocks returns and covariance matrix of the
stock returns
• Output: the efficient frontier (i.e. the set of portfolios with
expected return greater than any other with the same or
lesser risk, and lesser risk than any other with the same or
greater return)
• Global minimum variance portfolio (GMVP) doesn’t require
the estimation of the expected stock returns
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Portfolio Selection Problem
• The problem is reduced to:
min w' w
w
w'1  1
subject to
w'   q
• The solution is:
C  qB 1
qA  B 1
w
 1
 
2
2
AC  B
AC  B
where A  1'  11, B  1'  1 , C   '  1
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 11
wGMVP 
1'  11
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Covariance Matrix Estimators
• Sample covariance matrix estimator:
1 t
S
( xi  xi )( xi  xi )'

t  1 i 1
• Covariance matrix estimator implied by Sharpe’s single-index
model:
2
F  s00
bb' D
where s002 is the sample variance of the market returns, b is
the vector of the slope estimates, and D is the diagonal matrix
containing residual variance estimates
• Ledoit and Wolf’s shrinkage (of S towards F) estimator:
  F  (1   ) S
(weighted average of the single-index model and the sample matrix)
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Data and Period of Study
• Ledoit and Wolf’s paper
– Use NYSE and AMEX stocks from August 1962 to July 1995
– For each year t from 1972 to 1994
• In-sample period: August of year t-10 to July of year t for
estimation
• Out-of-sample period: August of year t to July of year t+1
– Consider those with
• valid CRSP returns for the last 120 months and future 12
months (Disatnik and Benninga's paper)
• valid Standard Industrial Classification (SIC) codes
– The resulting number of stocks used for constructing the
GMVP varies across the years
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Number of Stocks Across 23 Years
2/24/2010
Ledoit’s paper
Our experiment
Min N
909
1146
Max N
1314
1309
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Experiments
• Used the estimated covariance matrices to compute the
portfolio weights
• Compute the returns of the portfolios for the out-of-sample
period
• Record the monthly returns for each portfolio
• Compute the variances of the monthly returns for each
portfolio over the 23-year period
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Risk of Minimum Variance Portfolio
Standard deviation
(unconstrained from
Ledoit’s paper)
Standard deviation
(unconstrained from
our experiment)
Pseudoinverse
12.37
12.10
Market Model
12.00
11.15
Shrinkage to market
9.55
8.95
Notes:
1.
2.
Unconstrained refers to global minimum variance portfolio
Standard deviation is annualized through multiplication by
and expressed in percents
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Optimal Shrinkage Intensity Estimate
• This intensity is α from Ledoit and Wolf’s shrinkage estimator:
  F  (1   ) S
(weighted average of the single-index model and the sample matrix)
• They all have values ≈ 0.8, that means there is four times as
much estimation error in the sample covariance matrix
as there is bias in the single index model
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