Taras Bodnar

Elliptically contoured models in statistics and portfolio theory

Preliminaries.- Basic Properties.- Probability Density Function and Expected Values.- Mixtures of Normal Distributions.- Quadratic Forms and other Functions of Elliptically Contoured Matrices.- Characterization Results.- Estimation.- Hypothesis Testing.- Linear Models.- Skew Elliptically Contoured Distributions.- Application in Portfolio Theory.- Author Index.- Subject Index.

Econometrical analysis of the sample efficient frontier

The efficient frontier is a parabola in the mean-variance space which is uniquely determined by three characteristics. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we derive tests and confidence sets for all possible arrangements of these characteristics. Note that all of our results are based on the exact distributions for a finite sample size. Moreover, we determine a confidence region of the whole efficient frontier in the mean-variance space. It is shown that this set is bordered by five parabolas.

Bayesian Estimation of the Global Minimum Variance Portfolio

In this paper we consider the estimation of the weights of optimal portfolios from the Bayesian point of view under the assumption that the conditional distributions of the logarithmic returns are normal. Using the standard priors for the mean vector and the covariance matrix, we derive the posterior distributions for the weights of the global minimum variance portfolio. Moreover, we reparameterize the model to allow informative and non-informative priors directly for the weights of the global minimum variance portfolio. The posterior distributions of the portfolio weights are derived in explicit form for almost all models. The models are compared by using the coverage probabilities of credible intervals. In an empirical study we analyze the posterior densities of the weights of an international portfolio.

Direct shrinkage estimation of large dimensional precision matrix

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables p ? ∞ and the sample size n ? ∞ so that p / n ? c ? ( 0 , + ∞ ) . The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. Using this result, we construct a bona fide optimal linear shrinkage estimator for the precision matrix in case c < 1 . At the end, a simulation is provided where the suggested estimator is compared with the estimators proposed in the literature. The optimal shrinkage estimator shows significant improvement even for non-normally distributed data.

The distribution of the sample variance of the global minimum variance portfolio in elliptical models

In this article, we consider the variance of the global minimum variance portfolio. Assuming a matrix variate elliptically contoured distribution for the portfolio asset returns, we give the exact distribution and the density of an estimator of the portfolio variance. These results have many useful applications. They permit statements about the performance of the estimator. Moreover, it is possible to derive confidence intervals and to construct a test for the hypothesis that the global minimum variance is less than or equal to a certain value. We illustrate our results in an empirical study dealing with the daily returns of three developed countries.

On the exact solution of the multi-period portfolio choice problem for an exponential utility under return predictability

In this paper we derive the exact solution of the multi-period portfolio choice problem for an exponential utility function under return predictability. It is assumed that the asset returns depend on predictable variables and that the joint random process of the asset returns and the predictable variables follow a vector autoregressive process. We prove that the optimal portfolio weights depend on the covariance matrices of the next two periods and the conditional mean vector of the next period. The case without predictable variables and the case of independent asset returns are partial cases of our solution. Furthermore, we provide an exhaustive empirical study where the cumulative empirical distribution function of the investor’s wealth is calculated using the exact solution. It is compared with the investment strategy obtained under the additional assumption that the asset returns are independently distributed.

On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix

In this work we construct an optimal linear shrinkage estimator for the covariance matrix in high dimensions. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The developed distribution-free estimators obey almost surely the smallest Frobenius loss over all linear shrinkage estimators for the covariance matrix. The case we consider includes the number of variables p ? ∞ and the sample size n ? ∞ so that p / n ? c ? ( 0 , + ∞ ) . Additionally, we prove that the Frobenius norm of the sample covariance matrix tends almost surely to a deterministic quantity which can be consistently estimated.

On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory

In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e, the Markowitz mean-variance problem as well as the problems based on the mean-variance utility function and the quadratic utility.Conditions are derived under which the solutions of these three optimization procedures coincide and are lying on the efficient frontier, the set of mean-variance optimal portfolios. It is shown that the solutions of the Markowitz optimization problem and the quadratic utility problem are not always mean-variance efficient. The conditions for the mean-variance efficiency of the solutions depend on the unknown parameters of the asset returns. We deal with the problem of parameter uncertainty in detail and derive the probabilities that the estimated solutions of the Markowitz problem and the quadratic utility problem are mean-variance efficient. Because these probabilities deviate from one the above mentioned quadratic optimization problems are not stochastically equivalent. The obtained results are illustrated by an empirical study.

Sample efficient frontier in multivariate conditionally heteroscedastic elliptical models

In this paper, the asymptotic distributions of sample parameters of the efficient frontier and sample characteristics of optimal portfolios are derived. This is done assuming the asset returns to follow a k-dimensional VARMA–GARCH process. Moreover, estimators of the mean vector and the covariance matrix of the asset returns are suggested, which are asymptotic independent normally distributed within the considered class of stochastic processes.

Estimation of the Global Minimum Variance Portfolio in High Dimensions

We estimate the global minimum variance (GMV) portfolio in the high-dimensional case using results from random matrix theory. This approach leads to a shrinkage-type estimator which is distribution-free and it is optimal in the sense of minimizing the out-of-sample variance. Its asymptotic properties are investigated assuming that the number of assets