The multivariate symmetric stable distribution is a heavy-tailed elliptically contoured law that has many important applications in signal processing and financial mathematics. The family includes the sub-Gaussian stable distribution as a special case. This work addresses the problem of feasible estimation of the parameters of the multivariate symmetric stable distribution. We present a method based on the application of the method of moments to the empirical characteristic function. Further, we show almost sure convergence of our estimators, discover their limiting distribution, and demonstrate their finite-sample performance.

In this paper, we investigate the distributional properties of the estimated tangency portfolio (TP) weights assuming that the asset returns follow a matrix variate closed skew-normal distribution. We establish a stochastic representation of the linear combination of the estimated TP weights that fully characterizes its distribution. Using the stochastic representation we derive the mean and variance of the estimated weights of TP which are of key importance in portfolio analysis. Furthermore, we provide the asymptotic distribution of the linear combination of the estimated TP weights under the high-dimensional asymptotic regime, i.e., the dimension of the portfolio p and the sample size n tend to infinity such that p/n & RARR;c & ISIN;(0,1). A good performance of the theoretical findings is documented in the simulation study. In an empirical study, we apply the theoretical results to real data of the stocks included in the S & P 500 index.

In this paper, we construct a shrinkage estimator of the global minimum variance (GMV) portfolio by combining two techniques: Tikhonov regularization and direct shrinkage of portfolio weights. More specifically, we employ a double shrinkage approach, where the covariance matrix and portfolio weights are shrunk simultaneously. The ridge parameter controls the stability of the covariance matrix, while the portfolio shrinkage intensity shrinks the regularized portfolio weights to a predefined target. Both parameters simultaneously minimize, with probability one, the out-of-sample variance as the number of assets p and the sample size n tend to infinity, while their ratio p/n tends to a constant c > 0. This method can also be seen as the optimal combination of the well-established linear shrinkage approach of Ledoit and Wolf (2004) and the shrinkage of the portfolio weights by Bodnar et al. (2018). No specific distribution is assumed for the asset returns, except for the assumption of finite moments of order 4 + ε for ε > 0. The performance of the double shrinkage estimator is investigated via extensive simulation and empirical studies. The suggested method significantly outperforms its predecessor (without regularization) and the nonlinear shrinkage approach in terms of the out-of-sample variance, Sharpe ratio, and other empirical measures in the majority of scenarios. Moreover, it maintains the most stable portfolio weights with uniformly smallest turnover.

We suggest a new method for integrating volatility information for estimating the value-at-risk and conditional value-at-risk of a portfolio. This new method is developed from the perspective of Bayesian statistics and is based on the idea of volatility clustering. By specifying the hyperparameters in a conjugate prior based on two different rolling window sizes, it is possible to quickly adapt to changes in volatility and automatically specify the degree of certainty in the prior. This gives our method an advantage over existing Bayesian methods, which are less sensitive to such changes in volatilities and usually lack standardized ways of expressing the degree of belief. We illustrate our new approach using both simulated and empirical data. The new method provides a good alternative to other well-known homoscedastic and heteroscedastic models for risk estimation, especially during turbulent periods, when it can quickly adapt to changing market conditions.

In this paper, new results in random matrix theory are derived, which allow us to construct a shrinkage estimator of the global minimum variance (GMV) portfolio when the shrinkage target is a random object. More specifically, the shrinkage target is determined as the holding portfolio estimated from previous data. The theoretical findings are applied to develop theory for dynamic estimation of the GMV portfolio, where the new estimator of its weights is shrunk to the holding portfolio at each time of reconstruction. Both cases with and without overlapping samples are considered in the paper. The non-overlapping samples corresponds to the case when different data of the asset returns are used to construct the traditional estimator of the GMV portfolio weights and to determine the target portfolio, while the overlapping case allows intersections between the samples. The theoretical results are derived under weak assumptions imposed on the data-generating process. No specific distribution is assumed for the asset returns except from the assumption of finite 4+ε, ε>0, moments. Also, the population covariance matrix with unbounded largest eigenvalue can be considered. The performance of new trading strategies is investigated via an extensive simulation. Finally, the theoretical findings are implemented in an empirical illustration based on the returns on stocks included in the S&P 500 index.

The main contribution of this paper is the derivation of the asymptotic behavior of the out-of-sample variance, the out-of-sample relative loss, and of their empirical counterparts in the high-dimensional setting, i.e., when both ratios p/n and p/n tend to some positive constants as 𝑚 → ∞ and 𝑚 → ∞, where p is the portfolio dimension, while n and m are the sample sizes from the in-sample and out-of-sample periods, respectively. The results are obtained for the traditional estimator of the global minimum variance (GMV) portfolio and for the two shrinkage estimators introduced by Frahm and Memmel (2010) and Bodnar et al. (2018). We show that the behavior of the empirical out-of-sample variance may be misleading in many practical situations, leading, for example, to a comparison of zeros. On the other hand, this will never happen with the empirical out-of-sample relative loss, which seems to provide a natural normalization of the out-of-sample variance in the high-dimensional setup. As a result, an important question arises if the out-of-sample variance can safely be used in practice for portfolios constructed from a large asset universe.

We study the optimal portfolio allocation problem from a Bayesian perspective using value at risk (VaR) and conditional value at risk (CVaR) as risk measures. By applying the posterior predictive distribution for the future portfolio return, we derive relevant quantities needed in the computations of VaR and CVaR, and express the optimal portfolio weights in terms of observed data only. This is in contrast to the conventional method where the optimal solution is based on unobserved quantities which are estimated. We also obtain the expressions for the weights of the global minimum VaR (GMVaR) and global minimum CVaR (GMCVaR) portfolios, and specify conditions for their existence. It is shown that these portfolios may not exist if the level used for the VaR or CVaR computation are too low. By using simulation and real market data, we compare the new Bayesian approach to the conventional plug-in method by studying the accuracy of the GMVaR portfolio and by analysing the estimated efficient frontiers. It is concluded that the Bayesian approach outperforms the conventional one, in particular at predicting the out-of-sample VaR.

Financial portfolios and diversification go hand in hand. Diversification is one of, if not, the best risk mitigation strategy there is. If an investment performs poorly, then it will not impact the performance of the portfolio much due to diversification. Modern Portfolio Theory (MPT) is a framework for constructing diversified portfolios. However, MPT relies on unknown parameters that need to be estimated. By using estimates, estimation uncertainty is introduced to the allocation problem. This thesis contains five papers which provide results on how to deal with estimation uncertainty in very large sample portfolios from the MPT framework. These results provide tools to better understand the investment process and the empirical results that can be observed.

Paper I explores all of the portfolios that can be placed in the framework of MPT. The paper provides the sampling distribution for all optimal portfolios and their characteristics. This is done by assuming that the returns follow a multivariate normal distribution. Furthermore, the high-dimensional asymptotic joint distribution for the quantities of interest is derived. A simulation study shows that the high-dimensional distribution can provide a good approximation to the finite sample one.

Paper II continues on the idea of paper I. It considers the quadratic utility allocation problem from paper I with an additional risk-free asset in the portfolio. The portfolio is usually known as the Tangency Portfolio (TP). The distribution of the sample TP weights is derived under a skew-normal distribution. Results show that skewness implies a bias in the finite sample TP weights. The bias dissapears in the high-dimensional distribution.

Paper III takes on a practical aspect of investing, namely how to transition from one portfolio to another. A reallocation scheme is developed, which minimizes the out-of-sample variance of the Global Minimum Variance (GMV) portfolio, given a holding portfolio. The holding portfolio is the portfolio which an investor currently owns. An extensive simulation study show that the reallocation scheme can provide accurate estimates of the portfolio variance. Furthermore, an empirical application shows that the scheme provides the smallest out-of-sample variance in comparison to a number of benchmarks. The theoretical results from this paper are implemented in the DOSPortfolio R-package.

Paper IV derives properties of two different performance measures for three different high-dimensional GMV portfolio estimators. The measures are the out-of-sample variance and loss. The former is always used as an evaluation metric in empirical applications. The results show that the latter metric, the out-of-sample loss, does not need the same stringent assumptions as the out-of-sample variance in the high-dimensional setting. Using the out-of-sample loss, the performance of the three different portfolios can be ordered. This order is verified in a simulation study and an empirical application.

Paper V extends the results of papers III and IV. It introduces Thikonov regularization to the GMV portfolio weights as well as linear shrinkage. A simulation study shows that the method is preferable to a number of benchmarks. Furthermore, an empirical application shows that it can provide the smallest out-of-sample variance and provide good characteristics for the portfolio weights.

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population covariance matrix of the asset return distribution. In this paper, we characterize the exact sampling distribution of the estimated optimal portfolio weights and their characteristics. This is done by deriving their sampling distribution by its stochastic representation. This approach possesses several advantages, e.g. (i) it determines the sampling distribution of the estimated optimal portfolio weights by expressions, which could be used to draw samples from this distribution efficiently; (ii) the application of the derived stochastic representation provides an easy way to obtain the asymptotic approximation of the sampling distribution. The later property is used to show that the high-dimensional asymptotic distribution of optimal portfolio weights is a multivariate normal and to determine its parameters. Moreover, a consistent estimator of optimal portfolio weights and their characteristics is derived under the high-dimensional settings. Via an extensive simulation study, we investigate the finite-sample performance of the derived asymptotic approximation and study its robustness to the violation of the model assumptions used in the derivation of the theoretical results.

In this paper the concept of quantile-based optimal portfolio selection is introduced and a specific portfolio connected to it, the conditional value-of-return (CVoR) portfolio, is proposed. The CVoR is defined as the mean excess return or the conditional value-at-risk (CVaR) of the return distribution. The portfolio selection consists solely of quantile-based risk and return measures. Financial institutions that work in the context of Basel 4 use CVaR as a risk measure. In this regulatory framework sufficient and necessary conditions for optimality of the CVoR portfolio are provided under a general distributional assumption. Moreover, it is shown that the CVoR portfolio is mean-variance efficient when the returns are assumed to follow an elliptically contoured distribution. Under this assumption the closed-form expression for the weights and characteristics of the CVoR portfolio are obtained. Finally, the introduced methods are illustrated in an empirical study based on monthly data of returns on stocks included in the S&P index. It is shown that the new portfolio selection strategy outperforms several alternatives in terms of the final investor wealth.