The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, for example the mean vector and the covariance matrix, are unknown and have to be estimated by using historical data on asset returns. Our new approach employs the Bayesian posterior predictive distribution which is the distribution of future realizations of asset returns given the observable sample. The parameters of posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. By contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step, and lead to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application not only gives the solution of the considered optimization problem, but also provides the posterior predictive distribution of the optimal portfolio return which can be used to construct a prediction interval. A Bayesian efficient frontier, the set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well. Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios which is known to be overoptimistic.@en

We consider the estimation of the multi-period optimal portfolio obtained by maximizing an exponential utility. Employing the Jeffreys non-informative prior and the conjugate informative prior, we derive stochastic representations for the optimal portfolio weights at each time point of portfolio reallocation. This provides a direct access not only to the posterior distribution of the portfolio weights but also to their point estimates together with uncertainties and their asymptotic distributions. Furthermore, we present the posterior predictive distribution for the investor's wealth at each time point of the investment period in terms of a stochastic representation for the future wealth realization. This in turn makes it possible to use quantile-based risk measures or to calculate the probability of default, i.e the probability of the investor wealth to become negative. We apply the suggested Bayesian approach to assess the uncertainty in the multi-period optimal portfolio by considering assets from the FTSE 100 in the weeks after the British referendum to leave the European Union. The behaviour of the novel portfolio estimation method in a precarious market situation is illustrated by calculating the predictive wealth, the risk associated with the holding portfolio, and the probability of default in each period.

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We derive new results related to the portfolio choice problem for power and logarithmic utilities. Assuming that the portfolio returns follow an approximate log-normal distribution, the closed-form expressions of the optimal portfolio weights are obtained for both utility functions. Moreover, we prove that both optimal portfolios belong to the set of mean-variance feasible portfolios and establish necessary and sufficient conditions such that they are mean-variance efficient. Furthermore, we extend the derived theoretical finding to the general class of the log-skew-normal distributions. Finally, an application to the stock market is presented and the behaviour of the optimal portfolio is discussed for different values of the relative risk aversion coefficient. It turns out that the assumption of log-normality does not seem to be a strong restriction.

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In this paper, new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose, we study the weak convergence of linear spectral statistics of central and (conditionally) noncentral Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) noncentral Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one subvector is relatively small compared to the dimension of the other sub-vector. On the other hand, the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.@en

In this paper, we construct two tests for the weights of the global minimum variance portfolio (GMVP) in a high-dimensional setting, namely, when the number of assets p depends on the sample size n such that p/n → c ϵ (0, 1) as n tends to infinity. In the case of a singular covariance matrix with rank equal to q we assume that q/n → c ϵ (0, 1) as n → ∞. The considered tests are based on the sample estimator and on the shrinkage estimator of the GMVP weights. We derive the asymptotic distributions of the test statistics under the null and alternative hypotheses. Moreover, we provide a simulation study where the power functions and the receiver operating characteristic curves of the proposed tests are compared with other existing approaches. We observe that the test based on the shrinkage estimator performs well even for values of c close to one.

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In this paper we derive the optimal linear shrinkage estimator for the high-dimensional mean vector using random matrix theory. The results are obtained under the assumption that both the dimension p and the sample size n tend to infinity in such a way that p∕n→c∈(0,∞). Under weak conditions imposed on the underlying data generating mechanism, we find the asymptotic equivalents to the optimal shrinkage intensities and estimate them consistently. The proposed nonparametric estimator for the high-dimensional mean vector has a simple structure and is proven to minimize asymptotically, with probability 1, the quadratic loss when c∈(0,1). When c∈(1,∞) we modify the estimator by using a feasible estimator for the precision covariance matrix. To this end, an exhaustive simulation study and an application to real data are provided where the proposed estimator is compared with known benchmarks from the literature. It turns out that the existing estimators of the mean vector, including the new proposal, converge to the sample mean vector when the true mean vector has an unbounded Euclidean norm.

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In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix-variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large-dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.

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We study the distributional properties of the linear discriminant function under the assumption of normality by comparing two groups with the same covariance matrix but different mean vectors. A stochastic representation for the discriminant function coefficients is derived, which is then used to obtain their asymptotic distribution under the high-dimensional asymptotic regime. We investigate the performance of the classification analysis based on the discriminant function in both small and large dimensions. A stochastic representation is established, which allows one to compute the error rate in an efficient way. We further compare the calculated error rate with the optimal one obtained under the assumption that the covariance matrix and the two mean vectors are known. Finally, we present an analytical expression of the error rate calculated in the high-dimensional asymptotic regime. The finite-sample properties of the derived theoretical results are assessed via an extensive Monte Carlo study.

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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 optimal in the sense of minimizing the out-of-sample variance. Its asymptotic properties are investigated assuming that the number of assets p depends on the sample size n such that [Formula presented]→c∈(0,+∞) as n tends to infinity. The results are obtained under weak assumptions imposed on the distribution of the asset returns: only the existence of the fourth moments is required. Furthermore, we make no assumption on the upper bound of the spectrum of the covariance matrix. As a result, the theoretical findings are also valid if the dependencies between the asset returns are described by a factor model which appears to be very popular in the financial literature nowadays. This is also documented in a numerical study where the small- and large-sample behavior of the derived estimator is compared with existing estimators of the GMV portfolio. The resulting estimator shows significant improvements and it turns out to be robust if the assumption of normality is violated.

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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.

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For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix σ_n let S~n denote the usual sample covariance (centered by the mean) and S_n the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p>n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore-Penrose inverse of S_n and S~n. We consider the large dimensional asymptotics when the number of variables p→∞ and the sample size n→∞ such that p/n→c∈(1, +∞). We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of S~n differs in the mean from that of S_n.

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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.

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In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived underweak conditions on the asset returns.No assumption on the correlation structure between different time points is needed and no assumption on the distribution is imposed. All expressions are presented in terms of the conditional mean vectors and the conditional covariance matrices. If the multivariate process of the asset returns is independent, it is shown that in the case without a riskless asset the solution is presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. The process dynamics are included only in the shape parameter of the utility function. If a riskless asset is present, then the multi-period optimal portfolio weights are proportional to the single-period solutions multiplied by time-varying constants which are dependent on the process dynamics. Remarkably, in the case of a portfolio selection with the tangency portfolio the multi-period solution coincides with the sequence of the single-period solutions. Finally, we compare the suggested strategies with existing multi-period portfolio allocation methods on real data.

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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.

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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.

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