Mitigating Estimation Risk in Asset Allocation: Diagonal Models Versus 1/N Diversification

Date01 August 2016
Published date01 August 2016
The Financial Review 51 (2016) 403–433
Mitigating Estimation Risk in Asset
Allocation: Diagonal Models Versus 1/N
Chris Stivers
University of Louisville
Licheng Sun
Old Dominion University
Recent literature suggests that optimal asset-allocation models struggle to consistently
outperform the 1/Nna¨
ıve diversification strategy, which highlights estimation-risk concerns.
We propose a dichotomous classification of asset-allocation models based on which elements
of the inverse covariance matrix that a model uses: diagonal only versusfull matrix. We argue
that parsimonious diagonal-only strategies, which use limited information such as volatility
or idiosyncratic volatility, are likely to offer a good tradeoff between incorporating limited
information while mitigating estimation risk. Evaluating five sets of portfolios over 1926–
2012, we find that 1/Nis generally not optimal when compared with these diagonal strategies.
Keywords: portfolio optimization, na¨
ıve diversification, idiosyncratic volatility
JEL Classifications: C13, C51, G11, G12
Corresponding author: Strome College of Business, Old Dominion University, Norfolk, VA 23456;
Phone: (757) 683-6552; Fax: (757) 683-3258; E-mail:
We would like to thank the Editor (Richard Warr), an anonymous referee, Loan Dang, Revansiddha
Khanapure, and seminar participants from Old Dominion University and the 2015 Financial Management
Association annual meetings for helpful comments. We also thank Kathy Mikell for editorial assistance.
C2016 The Eastern Finance Association 403
404 C. Stivers and L. Sun/The Financial Review 51 (2016) 403–433
1. Introduction
Since the publication of the seminal work by Markowitz (1952), mean-variance
analysis has been acclaimed as the cornerstone of modern finance portfolio theory.
In recent years, however, the theoretical elegance of the Markowitz paradigm has
been overshadowed by some serious doubts about its usefulness in practice due to
estimation risk. Estimation risk arises when researchers use plug-in estimates of the
mean and variance, based on sample information, and treat them as if they are the
true population moments. This plug-in method does not take into account that sample
moments are noisy estimators of the true moments. Since the portfolio weights
are complex nonlinear functions of the sample moments, this approach tends to
generate large long and short positions in a portfolio. This is commonly known as
the “wacky weights” problem, which seems to be contradictory to the intuition of
diversification that underlies the Markowitz paradigm.1
Many studies have attempted to address this issue. For example, Pastor (2000)
and Pastor and Stambaugh (2000) use Bayesian methods to account for parameter
uncertainty. Kan and Zhou (2007) propose the use of three funds (the risk-free asset,
the sample tangency portfolio, and the sample global-minimum-variance portfolio)
to mitigate the impact of estimation error. Tuand Zhou (2010) incorporate economic
objectives into Bayesian priors and find that the resulting portfolio rules substantially
outperform other asset-allocation strategies.
In spite of these collective efforts by researchers to overcomethe estimation-risk
problem, DeMiguel, Garlappi and Uppal (2009) present striking evidence that favors
ıve portfolio strategy. Specifically, they compare the out-of-sample
performance of 14 asset-allocation models, many of them specifically designed to
overcome estimation risk, against the benchmark 1/Nportfolio rule using seven
empirical data sets. They conclude that “of the various optimizing models in the
literature, there is no single model that consistently delivers a Sharpe ratio or a CEQ
(certainty equivalent) return that is higher than that of the 1/Nportfolio” (DeMiguel,
Garlappi and Uppal, 2009, p. 1947).
The apparent success of the 1/Nna¨
ıve diversification portfolio rule seems at-
tributable to two nontrivial factors. First,by construction, the 1/Nrule is not subject to
the estimation risk that has plagued many of its competitors since it does not require
the use of any sample information to estimate model parameters. Second, the 1/N
rule has low portfolio turnover and incurs low transaction costs.
However, it seems intuitive that parsimonious use of the data is likely to be
optimal over the 1/Nstrategy that uses no information. The practical issue is how to
balance the tradeoff between incorporating limited data information while mitigating
estimation risk.
At least two recent articles have made some progress in this direction. First,
Tu and Zhou (2011) propose a strategy that optimally combines the 1/Nportfolio
1Please see Brandt (2010) and the references therein for a more comprehensive discussion of this issue.

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