In this paper, we derive an analytical solution to the dynamic optimal portfolio choice problem in the case of an investor equipped with a power utility function of wealth. The results are established by solving the Bellman backward recursion under the assumption that the vector of asset returns follows a vector-autoregressive process with predictable variables. In an empirical study, the performance of the derived solution is compared with the one obtained by applying the numerical method. The comparison is performed in terms of the final wealth and its expected utility. It is documented that the application of the analytical solution to the multi-period portfolio choice problem leads to higher values of both the final wealth and the expected utility.@en

In this paper, using the shrinkage-based approach for portfolio weights and modern results from random matrix theory we construct an effective procedure for testing the efficiency of the expected utility (EU) portfolio and discuss the asymptotic behavior of the proposed test statistic under the high-dimensional asymptotic regime, namely when the number of assets p increases at the same rate as the sample size n such that their ratio p/n approaches a positive constant cin (0,1) as nto infty. We provide an extensive simulation study where the power function and receiver operating characteristic curves of the test are analyzed. In the empirical study, the methodology is applied to the returns of S&P 500 constituents.

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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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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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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

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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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 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 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 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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