Distributional properties of portfolio weights

Okhrin, Yarema; Schmid, Wolfgang

doi:10.1016/j.jeconom.2005.06.022

Cited by 196 publications

(158 citation statements)

References 27 publications

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“…This, of course, does not constitute a proof, but rigorous proofs exist in similar, and even much more complicated, problems in the theory of disordered systems [60], [61], [62], which lends support to our conjecture. Furthermore, the results to be derived below agree with the rigorous results by [30], [39], [43] in the thermodynamic limit, and also with the numerical experiments by [36].…”

Section: The Replica Methodssupporting

confidence: 82%

“…One of the main results of the present calculation will be to derive the expectation value of q 0 for an arbitrary covariance matrix and verify its universality (its independence of the covariance matrix and the return), a possibility first raised on the basis of numerical experiments by [27]. As for the second moment, its behaviour can be inferred from the results in [30], [39], or [43]: in the thermodynamic limit the variance of q 0 vanishes, in the parlance of the theory of disordered systems, q 0 self-averages. In addition to the estimation error, we may even worry about whether the optimization can be carried out at all, that is whether the estimated covariance matrix σ ij preserves the positive definiteness of the true σ ij , so that to remain invertible.…”

Section: Sensitivity To Noise and The Replica Methodsmentioning

confidence: 76%

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Replica approach to mean-variance portfolio optimization

2016

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We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for r = N/T < 1, where N is the dimension of the portfolio and T the length of the time series used to estimate the covariance matrix. At the critical point r = 1 a phase transition is taking place. The out of sample estimation error blows up at this point as 1/(1 − r), independently of the covariance matrix or the expected return, displaying the universality not only of the critical exponent, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample variance is found to vanish at the critical point inversely proportional to the divergent estimation error.

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Section: The Replica Methodssupporting

confidence: 82%

Section: Sensitivity To Noise and The Replica Methodsmentioning

confidence: 76%

Replica approach to mean-variance portfolio optimization

2016

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“…Unlike the papers by Mori (2004) or Okhrin and Schmid (2006), we do not provide the full covariance matrix of the portfolio weight estimator (although it can be computed using similar methods). Instead, we study a simpler quantity (the MSE introduced below) which is a single number allowing for straightforward comparison between estimation methods and, as we show in Section 3, offers a good proxy for the variability of portfolio weights.…”

Section: Properties Of Portfolio Weights Estimatormentioning

confidence: 99%

“…The first term is the sum of variances of the portfolio weights estimatorŵ, i.e., the trace of the covariance matrix ofŵ (Mori (2004) and Okhrin and Schmid (2006) derive formulas for this covariance matrix in the case without a riskless asset).…”

Section: Mean Square Errormentioning

confidence: 99%

“…The first contribution is the derivation of closed-form formulas that relate the stability of portfolio estimator to the mean and the covariance matrix of asset returns (analogous formulas in the model without a riskless asset can be derived from results in Mori (2004) and Okhrin and Schmid (2006)). Our results rely on finite sample distributions of the mean and covariance estimators.…”

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confidence: 99%

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Theoretical and empirical estimates of mean–variance portfolio sensitivity

Palczewski

2014

European Journal of Operational Research

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This paper studies properties of an estimator of mean-variance portfolio weights in a market model with multiple risky assets and a riskless asset. Theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent. The sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified. It turns out that the relative contribution of the covariance matrix error depends mainly on the Sharpe ratio of the market portfolio and the sampling frequency of historical data. Theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets. An appropriately crafted bootstrapping method is employed to compute the empirical mean square error. Empirical and theoretical estimates are in good agreement, with the empirical values being, in general, higher. (JEL: C13, C52, G11) Keywords: Investment analysis, asset allocation, mean-variance portfolio, estimation error, bootstrapThe mean-variance framework, introduced by Markowitz (1952), opened a new era in asset management. The selection of optimal portfolio was formulated as seeking a balance between the risk, measured by the variance of the portfolio return, and the gain, measured by the expectation of the return. A mean-variance * Corresponding author, tel. +44 113 34 35180, fax. +44 113 34 35090

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Statistical Properties and Tests of Efficient Frontier Portfolios

Adcock

2015

Quantitative Financial Risk Management

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Distributional properties of portfolio weights

Cited by 196 publications

References 27 publications

Replica approach to mean-variance portfolio optimization

Replica approach to mean-variance portfolio optimization

Theoretical and empirical estimates of mean–variance portfolio sensitivity

Statistical Properties and Tests of Efficient Frontier Portfolios

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