Optimal Versus Naive Diversification
and Valkanov, 2007) can represent a promising direction to pursue. Second,
given the inherent simplicity and the relatively low cost of implementing the
1/N naive-diversification rule, such a strategy should serve as a natural bench-
mark to assess the performance of more sophisticated asset-allocation rules.
This is an important hurdle both for academic research proposing new asset-
allocation models and for “active” portfolio-management strategies offered by
the investment industry.
The rest of the paper is organized as follows. In Section 1, we describe the
various models of optimal asset allocation and evaluate their performance. In
Section 2, we explain our methodology for comparing the performance of these
models to that of 1/N ; the results of this comparison for seven empirical datasets
are given in Section 3. Section 4 contains the analytical results on the critical
length of the estimation window needed for the sample-based mean-variance
policy to outperform the 1/N benchmark; and in Section 5 we present a similar
analysis for other models of portfolio choice using simulated data. The various
experiments that we undertake to verify the robustness of the findings are
described briefly in Section 6, with the details reported in a separate appendix
titled “Implementation Details and Robustness Checks,” which is available from
the authors. Our conclusions are presented in Section 7. The empirical datasets
we use are described in Appendix A and the proof for the main analytical result
is given in Appendix B.
1. Description of the Asset-Allocation Models Considered
In this section, we discuss the various models from the portfolio-choice liter-
ature that we consider. Because these models are familiar to most r eaders, we
provide only a brief description of each, and instead focus on explaining how
the different models are related to each other. The list of models we analyze is
summarized in Table 1, and the details on how to implement these models are
given in the separate appendix to this paper.
We use R
t
to denote the N -vector of excess returns (over the risk-free asset)
on the N risky assets available for investment at date t.TheN -dimensional
vector μ
t
is used to denote the expected returns on the risky asset in excess
of the risk-free rate, and
t
to denote the corresponding N × N variance-
covariance matrix of returns, with their sample counterparts given by
ˆ
μ
t
and
ˆ
t
,
respectively. Let M denote the length over which these moments are estimated,
and T the total length of the data series. We use 1
N
to define an N -dimensional
vector of ones, and I
N
to indicate the N × N identity matrix. Finally, x
t
is
the vector of portfolio weights invested in the N risky assets, with 1 − 1
N
x
t
invested in the risk-free asset. The vector of relative weights in the portfolio
with only-risky assets is
w
t
=
x
t
|1
N
x
t
|
, (1)
1921
at Shanghai Jiao Tong University on March 6, 2014http://rfs.oxfordjournals.org/Downloaded from
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