Minimum Variance Portfolios

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Minimum Variance Portfolios in the U.S. Equity
Market Chicago Quantitative Alliance Steven
Thorley PhD, CFA April 5, 2006
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• Minimum Variance Portfolios in the U.S. Equity
  Market
• Motivation
• Data mining critiques plague most empirical
  research on market anomalies.
• What happens if we just follow the normative
  prescription of Markowitz, popularized in the
  1960s? Specifically, construct mean-variance
  optimized portfolios.
• Avoid macro data mining (what we collectively
  think we know about stock returns now as opposed
  to then) by focusing on minimum variance
  portfolios that do not include an expected return
  input.
• Further, restrict the risk model (i.e.,
  covariance matrix estimation) to historical
  security returns. Pre-specified factor models
  are not allowed.

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Principal Components Covariance matrix estimation
technique 1
The asymptotic principle components procedure of
Conner and Korajczyk (1988) employs an eigenvalue
decomposition of the T by T security return
cross-product matrix in contrast to the security
return covariance matrix. Specifically, we supply
the transposed historical security return matrix
to the SAS procedure PRINCOMP with the number of
statistical factors set to K 5. The key output
of the principal components algorithm are the
eigenvectors associated with the K highest
eigenvalues of the return cross-product matrix,
which combine to form the T by K factor return
matrix F. Regression analysis then yields a K by
N matrix B of factor exposures and a residual
return matrix E. The standard factor model
equation is then used to calculate the PC version
of the security covariance matrix where
diag is the matrix diagonalization function.
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Bayesian Shrinkage Covariance matrix estimation
technique 2
While the principle components methodology
structures the off-diagonal elements of the
security covariance matrix to ensure
invertability, the procedure does not modify the
diagonal elements (i.e., security variance
estimates). Given 1,000 securities, there are
likely to be some fairly extreme values for
estimated security variances. This motivates our
use of the two-parameter Bayesian shrinkage
process of Lediot and Wolf (2003). Specifically,
from the historical sample covariance matrix, we
calculate the average value of the N diagonal
elements (security variance estimates) as well as
the average value of the N(N-1)/2 independent
off-diagonal elements (security covariance
estimates). These two scalar parameters are used
to populate an N by N Bayesian prior matrix
Oprior. The BS version of the security
covariance matrix is a weighted average of the
two-parameter prior matrix and the sample
covariance matrix O where ? is a scalar
shrinkage parameter based on a complex function
of the spread of variance and covariance values
around the averages.
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Numerical Portfolio Optimization

• Minimum variance optimization is conducted using
  the numerical routines in SAS Procedure NLP with
  cross-validation using Axioma. Portfolios are
  not true minimum variance because of three
  constraints
• 1) Budget constraint that the sum of the
  security weights is one (i.e., fully invested but
  non-margined equity portfolios).
• For comparability to the cap-weighted market
  portfolio, we focus short sell constrained
  long-only optimizations with a lower bound of
  zero on security weights, and an upper bound of
  3.0 percent/security.
• Where specified, we also apply market-neutrality
  constraints on three Fama/French factors size
  (market cap), value (book to market ratio), and
  momentum (prior year less prior month return).

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Data and Methodology

• We use CRSP and Compustat data, as well as
  factor returns from the Ken French Website. At
  the beginning of each month from January 1968 to
  December 2005 (456 months) we complete five steps
  and then roll the process forward
• Select the largest 1000 securities with
  sufficient historical data, including one year of
  prior monthly returns (or one year of high
  frequency daily return data) and ex-ante factor
  exposures.
• Calculate the sample covariance matrix based on
  historical returns for each of the 1000
  securities.
• Structure the sample covariance using Bayesian
  Shrinkage or Principal Components.
• Feed the estimated covariance matrix into the
  optimizer and determine optimal portfolio weights
  for the minimum variance portfolio under various
  constraints.
• Use the realized security returns in the current
  month to track the performance of the optimized
  and market portfolios.

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Factor Neutrality
The Fama/French (1993) factors, excluding CAPM
beta, plus Jegadeesh and Titman (1993) momentum,
have been canonized as the key market-wide
determinants of realized portfolio returns in the
U. S equity market. We apply ex-ante neutrality
constraints to the optimized portfolio on these
three factors size (market cap), value (book to
market ratio), and momentum (prior year less
prior month return). Ex-ante neutrality is
measured by either Direct measures of factor
exposure for the 1,000 securities normalized each
month to a cross-sectional mean of zero and
standard deviation of one. The minimum variance
portfolio is constrained to have the same
security-weighted factor exposure as the
cap-weighted market portfolio. Following Daniel
and Titman (1998) we also estimate security
factor sensitivities using multivariate
regression analysis of past security returns on
the returns of the Fama/French factors Mkt-RF
(market excess return), SMB (small-minus-big
market-capitalization), HML (high-minus-low
book-to-market), and UMD (up minus down).
Optimized portfolios are constrained to have
ex-ante estimated factor sensitivities of zero on
SBM, HML, and UMD.
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• Minimum Variance Portfolios in the U.S. Equity
  Market
• Some important new perspectives
• Optimized portfolios have more value-factor
  exposure over time than is generally appreciated
  using common ex-ante measurements of exposure
  (e.g., holdings data).
• Also, even after accounting for the positive
  ex-post value-factor exposure, optimized
  portfolios have positive alphas if you use high
  frequency data and allow for some shorting

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• Minimum Variance Portfolios in the U.S. Equity
  Market
• Conclusions
• The long-only minimum variance portfolio has
  about 25 percent less realized risk (standard
  deviation) than the cap-weighted market
  portfolio.
• Despite the lower risk, the average return on the
  minimum variance portfolio over time actually
  exceeds the average return on the market. The
  risk-return advantages are statistically and
  economically significant.
• Constraints imposing ex-ante neutrality with
  respect to the well-accepted cross-sectional
  drivers of stock returns leads to only a modest
  reduction in the Sharpe ratios of minimum
  variance portfolios.
• Despite the ex-ante neutrality constraints, an
  ex-post exposure to the value factor in most time
  periods explains much of the superior performance
  of minimum variance portfolios in the U.S. equity
  market.