Báo cáo môn Đầu tư tài chính Optimal Versus Naive Diversification: How Inefficient is the 1N Portfolio Strategy?

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:

The 07 empirical datasets are listed:
Description of the Asset-Allocation
Models Considered

Definition:
+ R
t
: the N vector of excess returns (over the risk-free asset) on
the N risky assets available for investment at date t.
+ μ
t
(N dimensional vector): the expected returns on the risky
asset in excess of the risk free rate.
+ :the corresponding N×N variance-covariance matrix of
returns
+ The sample counterparts of μ
t
, given by and
+ 1
N
: N dimensional vector of ones.
+ I
N
to indicate the N × N identity matrix
+ x
t
: the vector of portfolio weights invested in the N risky assets
+ M: the length over which these moments are estimated.
+ T: the total length of the data series.
Almost all the models that we consider deliver portfolio

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

student-t with mean and variance
3.2 Bayes-Stein shrinkage portfolio (bs):
This model is designed to handle the error in
estimating expected returns by using estimators of the
form:
3.3 Bayesian portfolio based on belief in an asset-
pricing model (dm)
These portfolios are a further refinement of shrinkage
portfolios because they address the arbitrariness of the
choice of a shrinkagetarget, ¯μ, and of the shrinkage
factor, φ, by using the investor’s belief about the
validity of an asset-pricing model.
We implement the Data-and-Model approach using
three different asset-pricing models: the Capital Asset
Pricing Model (CAPM), the Fama and French (1993)
three-factor model, and the Carhart (1997) four-factor
model.
4. Portfolios with moment restrictions:
4.1 Minimum-variance portfolio (min)
4.2 Value-weighted portfolio implied by the market
model (vw)
4.3 Portfolio implied by asset pricing models with
unobservable factors (mp)
5. Shortsale-constrained portfolios
We have the sample-based mean-variance-constrained
(mv-c), Bayes-Stein-constrained (bs-c), and minimum-
variance-constrained (min-c).
To interpret the effect of shortsale constraints, observe
that imposing the constraint xi ≥ 0, i = 1, , N in the
basic mean-variance optimization, following

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

seem to be very effective at dealing with estimation error.

About the portfolios that are based on restrictions on the moments of
returns (“min”, “vw” & “mp”)

Constraints alone do not improve performance sufficiently (“mv-c”,
“bs-c”)

Strategies combine constraints with shrinkage of expected returns are
more effective in reducing effect of estimation of error (“min-c”, “g-
min-c”)
Summary of findings from the empirical
dataset

Sharpe ratio: There is no single strategy that always dominates the 1/N
strategy

CEQ: No strategy from the optimal model is consistently better than
the benchmark 1/N strategy.

Turnover: Only the “vw” strategy is better than the 1/N strategy (hold
the market portfolios and does not trade at all)
Why the strategies from the various optimizing
models do not perform better relative to the 1/N
strategy ?
Following approach proposed by Kan & Zhou (2007)
The smallest number of estimation periods necessary for
the mv portfolio to outperform the 1/N

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.