| 1. |
- Bauder, David, et al.
(författare)
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Bayesian inference of the multi-period optimal portfolio for an exponential utility
- 2020
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Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 175
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Tidskriftsartikel (refereegranskat)abstract
- 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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| 2. |
- Bauder, David, et al.
(författare)
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Bayesian mean-variance analysis : optimal portfolio selection under parameter uncertainty
- 2021
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Ingår i: Quantitative finance (Print). - : Informa UK Limited. - 1469-7688 .- 1469-7696. ; 21:2, s. 221-242
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Bodnar, Olha, senior lecturer, 1979-, et al.
(författare)
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Recent advances in shrinkage-based high-dimensional inference
- 2022
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Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 188
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Tidskriftsartikel (refereegranskat)abstract
- Recently, the shrinkage approach has increased its popularity in theoretical and applied statistics, especially, when point estimators for high-dimensional quantities have to be constructed. A shrinkage estimator is usually obtained by shrinking the sample estimator towards a deterministic target. This allows to reduce the high volatility that is commonly present in the sample estimator by introducing a bias such that the mean-square error of the shrinkage estimator becomes smaller than the one of the corresponding sample estimator. The procedure has shown great advantages especially in the high-dimensional problems where, in general case, the sample estimators are not consistent without imposing structural assumptions on model parameters.In this paper, we review the mostly used shrinkage estimators for the mean vector, covariance and precision matrices. The application in portfolio theory is provided where the weights of optimal portfolios are usually determined as functions of the mean vector and covariance matrix. Furthermore, a test theory on the mean-variance optimality of a given portfolio based on the shrinkage approach is presented as well.
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| 4. |
- Bodnar, Taras, et al.
(författare)
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A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function
- 2015
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Ingår i: Annals of Operations Research. - : Springer Science and Business Media LLC. - 0254-5330 .- 1572-9338. ; 229:1, s. 121-158
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Tidskriftsartikel (refereegranskat)abstract
- 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 under weak 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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| 5. |
- Bodnar, Taras, et al.
(författare)
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Central limit theorems for functionals of large dimensional sample covariance matrix and mean vector in matrix-variate skewed model
- 2016
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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 general skew normal distribution. 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 an inverse 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 sample size n approach to infinity such that p/n → c ∈ (0, 1).
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| 6. |
- Bodnar, Taras, et al.
(författare)
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Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions
- 2019
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Ingår i: Scandinavian Journal of Statistics. - : John Wiley & Sons. - 0303-6898 .- 1467-9469. ; 46:2, s. 636-660
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Tidskriftsartikel (refereegranskat)abstract
- 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 to 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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| 7. |
- Bodnar, Taras, et al.
(författare)
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Direct shrinkage estimation of large dimensional precision matrix
- 2016
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Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 146, s. 223-236
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Tidskriftsartikel (refereegranskat)abstract
- 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 -> infinity and the sample size n -> infinity so that p/n -> c is an element of (0, +infinity). 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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| 8. |
- Bodnar, Taras, et al.
(författare)
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Discriminant analysis in small and large dimensions
- 2020
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Ingår i: Theory of Probability and Mathematical Statistics. - Providence, Rhode Island : American Mathematical Society (AMS). - 1547-7363 .- 0094-9000. ; 100, s. 21-41
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Tidskriftsartikel (refereegranskat)abstract
- 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 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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| 9. |
- Bodnar, Taras, 1979-, et al.
(författare)
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Dynamic shrinkage estimation of the high-dimensional minimum-variance portfolio
- 2023
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Ingår i: IEEE Transactions on Signal Processing. - 1053-587X .- 1941-0476. ; 71, s. 1334-1349
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Bodnar, Taras, et al.
(författare)
-
Estimation of the global minimum variance portfolio in high dimensions
- 2018
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Ingår i: European Journal of Operational Research. - : Elsevier BV. - 0377-2217 .- 1872-6860. ; 266:1, s. 371-390
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Tidskriftsartikel (refereegranskat)abstract
- 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 p/n -> c is an element of (0, + infinity) 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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