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? 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. [...]... 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... 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. .. 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, ... 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 In summary, the various optimizing models, there is no single model that consistently delivers a Sharpe ratio or a CEQ return that is higher than that of the 1/N portfolio Further research First, improving the estimation of the. .. 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... 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 Using simulated data to analyze how the performance of strategies in empirical results depend on N & M Conclusions The out-of-sample Sharpe ratio of the sample-based meanvariance strategy is much lower than that of the 1/N strategy ...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 minimumvariance-constrained (min-c) To interpret the effect... that of the 1/N portfolio Further research First, improving the estimation of the moments of asset returns and to using not just statistical but also other available information about stock returns as the crosssectional characteristics of assets Second, the 1/N naive- diversification rule should serve at least as a first obvious benchmark ... 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 . 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. 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. 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