Optimal Versus Naive Diversification:
How Inefficient is the 1/N Portfolio
Strategy?
L P TCDN NGÀY – K22Ớ
NHÓM 20:
1.Tr ng Thúy Di uươ ệ
2.Tr n Th Bích Ki uầ ị ề
3.Nguy n Anh Vănễ
4.Huỳnh Quang S nơ
Abstract:

Target of the research: The objective in this
paper is to understand the conditions under
which mean-variance optimal portfolio models
can be expected to perform well even in the
presence of estimation risk.

Evaluating the out-of-sample performance of the
sample-based mean-variance portfolio rule and
its various extensions designed to reduce the
effect of estimation error relative to the
performance of the naive portfolio diversification
rule (1/N).

Using three performance criteria to compare:
+ The out-of-sample Sharpe ratio;
+ The certainty-equivalent (CEQ) return for the
expected utility of a mean-variance investor;
+ The t urnover ( trading volume) for each portfolio
strategy.
The 14 models are listed:

weights have the main difference is how to estimate μ
t

and
1. Naive portfolio:
The naive (“ew ” or “1/ N ”) strategy that we consider
involves holding a portfolio weight w
ew
t
= 1/N in each of
the N risky assets. μ
t
∝ 1
N
for all t.
2. Sample-based mean-variance portfolio:
Markowitz model (“mv”), the investor optimizes the
tradeoff between the mean and variance of portfolio
returns.
3. Bayesian approach to estimation error:
The estimates of μ and are computed using the
predictive distribution of asset returns. This
distribution is obtained by integrating the conditional
likelihood, f(R/μ, ), over μ and
with respect to a certain subjective prior, p ( μ, ).
3.1 Bayesian diffuse-prior portfolio:
If the prior is chosen to be diffuse, that is,

, and the conditional likelihood
is normal, then the predictive distribution is a

Lagrangian:

6. Optimal combination of portfolios
-
6.1 The Kan and Zhou (2007) three-fund portfolio (vm-min)
Estimation risk cannot be diversified away by holding only a
combination of the tangency portfolio and the risk-free
asset, an investor will also benefit from holding some other
risky-asset portfolio; that is, a third fund.
6.2 Mixture of equally weighted and minimum-variance
portfolios (ew-min)
This strategy is a combination of the naive 1/N portfolio and
the minimum-variance portfolio
Methodology for Evaluating Performance

Out-of-sample Sharpe ratio of strategy k

Certainty-equivalent (CEQ) return

Turnover
Results from the Seven Empirical
Datasets Considered

Observations

The 1/N strategy outperforms the sample-based mean-variance
strategy if one were to make no adjustment at all for the presence of
estimation error.

Bayesian strategies, explicitly account for estimation error, do not

In which:
Is the expected loss from using a particular estimator of the optimal
weight

be the squared Sharpe ratio of the mean-variance portfolio
be the squared Sharpe ratio of the 1/N portfolio.
Observations

First, a large part of the effect of estimation error is
attributable to estimation of the mean

Second, and more importantly, the magnitude of the
critical number of estimation periods is striking.