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

Báo cáo môn Đầu tư tài chính Optimal Versus Naive Diversification:How Inefficient is the 1N Portfolio Strategy? Target of the research: The objective in this paper is to understand the conditions under which meanvariance optimal portfolio models can be expected to perform well even in the presence of estimation risk. Evaluating the outofsample performance of the samplebased meanvariance portfolio rule and its various extensions designed to reduce the effect of estimation error relative to the performance of the naive portfolio diversification rule (1N).

Optimal Versus Naive

Diversification:

How Inefficient is the 1/N

Portfolio Strategy?

LỚP TCDN NGÀY – K22 NHÓM 20:

1.Trương Thúy Diệu 2.Trần Thị Bích Kiều 3.Nguyễn Anh Văn 4.Huỳnh Quang Sơn

 Target of the research: The objective in this paper

is to understand the conditions under which variance optimal portfolio models can be expected

mean-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:

+ Rt: the N vector of excess returns (over the risk-free asset) on

the N risky assets available for investment at date t.

+ μt (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 μt, given by and

+ 1N: N dimensional vector of ones.

+ IN to indicate the N × N identity matrix

+ xt : 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 μt and

1 Naive portfolio:

The naive (“ew ” or “1/ N ”) strategy that we consider

involves holding a portfolio weight wewt = 1/N in each of the

N risky assets μt ∝ 1N 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, μ, and of the shrinkage factor,

φ, by using the investor’s belief about the validity of an

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

 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

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.

Using simulated data to analyze how the performance

of strategies in empirical results depend on N & M

 The out-of-sample Sharpe ratio of the sample-based mean-variance strategy is much lower than that of the 1/N strategy  Errors in estimating erode all the gains from optimal

 Models that have been proposed in the literature to deal with the problem of estimation error typically do not outperform the 1/N benchmark

that consistently delivers a Sharpe ratio or a CEQ return that is higher than that of the 1/N portfolio

 Second, the 1/N naive-diversification rule should serve at least

as a first obvious benchmark