Linear statistical inference for global and local minimum variance portfolios

Traditional portfolio optimization has been often criticized since it does not account for estimation risk. Theoretical considerations indicate that estimation risk is mainly driven by the parameter uncertainty regarding the expected asset returns rather than their variances and covariances. This is also demonstrated by several numerical studies. The global minimum variance portfolio has been advocated by many authors as an appropriate alternative to the traditional Markowitz approach since there are no expected asset returns which have to be estimated and thus the impact of estimation errors can be substantially reduced. But in many practical situations an investor is not willing to choose the global minimum variance portfolio, especially in the context of top down portfolio optimization. In that case the investor has to minimize the variance of the portfolio return by satisfying some specific constraints for the portfolio weights. Such a portfolio will be called 'local minimum variance portfolio'. Some finite sample hypothesis tests for global and local minimum variance portfolios are presented as well as the unconditional finite sample distribution of the estimated portfolio weights and the first two moments of the estimated expected portfolio returns. --Estimation risk,linear regression theory,Markowitz portfolio,portfolio optimization,top down investment,minimum variance portfolio

An analytical investigation of estimators for expected asset returns from the perspective of optimal asset allocation

In the present work I derive the risk functions of 5 standard estimators for expected asset returns which are frequently advocated in the literature, viz the sample mean vector, the James-Stein and Bayes-Stein estimator, the minimum-variance estimator, and the CAPM estimator. I resolve the question why it is meaningful to study the risk function in the context of optimal asset allocation. Further, I derive the quantities which determine the risks of the different expected return estimators and show which estimators are preferable with respect to optimal asset allocation. Finally, I discuss the question whether it pays to strive for the optimal portfolio by using time series information. It turns out that in many practical situations it is better to renounce parameter estimation altogether and pursue some trivial strategy such as the totally risk-free investment. --Asset allocation,Bayes-Stein estimator,CAPM estimator,James-Stein estimator,Minimum-variance estimator,Naive diversification,Out-ofsample performance,Risk function,Shrinkage estimation

Testing for the best alternative with an application to performance measurement

Suppose that we are searching for the maximum of many unknown and analytically untractable quantities or, say, the 'best alternative' among several candidates. If our decision is based on historical or simulated data there is some sort of selection bias and it is not evident if our choice is significantly better than any other. In the present work a large sample test for the best alternative is derived in a rather general setting. The test is demonstrated by an application to financial data and compared with the Jobson-Korkie test for the Sharpe ratios of two asset portfolios. We find that ignoring conditional heteroscedasticity and non-normality of asset returns can lead to misleading decisions. In contrast, the presented test for the best alternative accounts for these kinds of phenomena. --Ergodicity,Gordin's condition,heteroscedasticity,Jobson-Korkie test,Monte Carlo simulation,performance measurement,Sharpe ratio

Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix ∑ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for ∑ or another matrix Γ ∝ ∑. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the presented estimators are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the presented scale function leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine (2008) in the context of local asymptotic normality theory. --local asymptotic normality,M-estimator,R-estimator,robust covariance matrix estimator,scale-invariant function,S-estimator,shape matrix,Tyler's M-estimator

A generalization of Tyler's M-estimators to the case of incomplete data

Many different robust estimation approaches for the covariance or shape matrix of multivariate data have been established until today. Tyler's M-estimator has been recognized as the 'most robust' M-estimator for the shape matrix of elliptically symmetric distributed data. Tyler's Mestimators for location and shape are generalized by taking account of incomplete data. It is shown that the shape matrix estimator remains distribution-free under the class of generalized elliptical distributions. Its asymptotic distribution is also derived and a fast algorithm, which works well even for high-dimensional data, is presented. A simulation study with clean and contaminated data covers the complete-data as well as the incomplete-data case, where the missing data are assumed to be MCAR, MAR, and NMAR. --covariance matrix,distribution-free estimation,missing data,robust estimation,shape matrix,sign-based estimator,Tyler's M-estimator

Generalized Elliptical Distributions: Theory and Applications

The thesis recalls the traditional theory of elliptically symmetric distributions. Their basic properties are derived in detail and some important additional properties are mentioned. Further, the thesis concentrates on the dependence structures of elliptical or even meta-elliptical distributions using extreme value theory and copulas. Some recent results concerning regular variation and bivariate asymptotic dependence of elliptical distributions are presented. Further, the traditional class of elliptically symmetric distributions is extended to a new class of `generalized elliptical distributions' to allow for asymmetry. This is motivated by observations of financial data. All the ordinary components of elliptical distributions, i.e. the generating variate, the location vector and the dispersion matrix remain. Particularly, it is proved that skew-elliptical distributions belong to the class of generalized elliptical distributions. The basic properties of generalized elliptical distributions are derived and compared with those of elliptically symmetric distributions. It is shown that the essential properties of elliptical distributions hold also within the broader class of generalized elliptical distributions and some models are presented. Motivated by heavy tails and asymmetries observed in financial data the thesis aims at the construction of a robust dispersion matrix estimator in the context of generalized elliptical distributions. A `spectral density approach' is used for eliminating the generating variate. It is shown that the `spectral estimator' is an ML-estimator provided the location vector is known. Nevertheless, it is robust within the class of generalized elliptical distributions. The spectral estimator corresponds to an M-estimator developed 1983 by Tyler. But in contrast to the more general M-approach used by Tyler the spectral estimator is derived on the basis of classical maximum-likelihood theory. Hence, desired properties like, e.g., consistency, asymptotic efficiency and normality follow in a straightforward manner. Not only caused by the empirical evidence of extremes but also due to the inferential problems occuring for high-dimensional data the performance of the spectral estimator is investigated in the context of modern portfolio theory and principal components analysis. Further, methods of random matrix theory are discussed. They are suitable for analyzing high-dimensional covariance matrix estimates, i.e. given a small sample size relative to the number of dimensions. It is shown that the Marchenko-Pastur law fails if the sample covariance matrix is used in the context of elliptically of even generalized elliptically distributed and heavy tailed data. But substituting the sample covariance matrix by the spectral estimator resolves the problem and the classical arguments of random matrix theory remain valid