| 1. |
- Bauder, David, et al.
(author)
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Bayesian inference for the tangent portfolio
- 2018
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In: International Journal of Theoretical and Applied Finance. - : World Scientific Publishing Co. Pte. Ltd.. - 0219-0249. ; 21:8
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Journal article (peer-reviewed)abstract
- In this paper we consider the estimation of the weights of tangent portfolios from the Bayesian point of view assuming normal conditional distributions of the logarithmic returns. For diffuse and conjugate priors for the mean vector and the covariance matrix, we derive stochastic representations for the posterior distributions of the weights of tangent portfolio and their linear combinations. Separately we provide the mean and variance of the posterior distributions, which are of key importance for portfolio selection. The analytic results are evaluated within a simulation study, where the precision of coverage intervals is assessed.
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| 2. |
- Bodnar, Olha, senior lecturer, 1979-, et al.
(author)
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Robust Surveillance of Covariance Matrices Using a Single Observation
- 2013
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In: Sankhya - The Indian Journal of Statistics. - : Springer. - 0972-7671 .- 0976-3139. ; 76:2, s. 219-256
-
Journal article (peer-reviewed)abstract
- In this paper a new technique for monitoring shifts in covariance matrices of Gaussian processes is developed. The processes we monitor are obtained from the covariance matrices estimated using a single observation. These processes follow independent Gaussian distribution in the in-control state, thus allowing for application of standard control charts. Furthermore, in contrary to the existing literature, the suggested procedure is asymptotically robust to the shifts in the mean. The explicit out-of-control distribution for an arbitrary moment of the shift is derived. The performance of numerous multivariate control charts is evaluated in an extensive simulation study and applied to monitoring volatilities on financial markets.
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| 3. |
- Bodnar, Taras, et al.
(author)
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Bayesian Estimation of the Global Minimum Variance Portfolio
- 2015
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Other publication (other academic/artistic)abstract
- 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 distribution of the logarithmic returns is 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 andnon-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.
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| 4. |
- Bodnar, Taras, et al.
(author)
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Bayesian estimation of the global minimum variance portfolio
- 2017
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In: European Journal of Operational Research. - : Elsevier. - 0377-2217 .- 1872-6860. ; 256:1, s. 292-307
-
Journal article (peer-reviewed)abstract
- 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.
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| 5. |
- Bodnar, Taras, et al.
(author)
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Determination and estimation of risk aversion coefficients
- 2018
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In: Computational Management Science. - : Springer Science and Business Media LLC. - 1619-697X .- 1619-6988. ; 15:2, s. 297-317
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Journal article (peer-reviewed)abstract
- In the paper we consider two types of utility functions often used in portfolio allocation problems, i.e. the exponential utility and the quadratic utility. We link the resulting optimal portfolios obtained by maximizing these utility functions to the corresponding optimal portfolios based on the minimum value-at-risk (VaR) approach. This allows us to provide analytic expressions for the risk aversion coefficients as functions of the VaR level. The results are initially derived under the assumption that the vector of asset returns is multivariate normally distributed and they are generalized to the class of elliptically contoured distributions thereafter. We find that the choice of the coefficients of risk aversion depends on the stochastic model used for the data generating process. Finally, we take the parameter uncertainty into account and present confidence intervals for the risk aversion coefficients of the considered utility functions. The theoretical results are validated in an empirical study.
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| 6. |
- Bodnar, Taras, et al.
(author)
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Distribution of the product of a singular Wishart matrix and a normal vector
- 2014
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In: Theory of Probability and Mathematical Statistics. - : American Mathematical Society (AMS). - 0094-9000 .- 1547-7363. ; :91, s. 1-15
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Journal article (peer-reviewed)abstract
- In this paper we derive a very useful formula for the stochastic representation of the product of a singular Wishart matrix with a normal vector. Using this result, the expressions of the density function as well as of the characteristic function are established. Moreover, the derived stochastic representation is used to generate random samples from the product which leads to a considerable improvement in the computation efficiency. Finally, we present several important properties of the singular Wishart distribution, like its characteristic function and distributional properties of the partitioned singular Wishart matrix.
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| 7. |
- Bodnar, Taras, et al.
(author)
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Distribution of the product of singular Wishart matrix and normal vector
- 2014
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In: Theory of Probability and Mathematical Statistics. - 1547-7363. ; :91, s. 1-14
-
Journal article (pop. science, debate, etc.)abstract
- In this paper we derive a very useful formula for the stochastic representation of the product of a singular Wishart matrix with a normal vector. Using this result, the expressions of the density function as well as of the characteristic function are established. Moreover, the derived stochastic representation is used to generate random samples from product which leads to a considerable improvement in the computation efficiency. Finally, we present several important properties of the singular Wishart distribution, like its characteristic function and distributional properties of the partitioned singular Wishart matrix.
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| 8. |
- Bodnar, Taras, et al.
(author)
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On the exact and approximate distributions of the product of a Wishart matrix with a normal vector
- 2013
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In: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 122, s. 70-81
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Journal article (peer-reviewed)abstract
- In this paper we consider the distribution of the product of a Wishart random matrix and a Gaussian random vector. We derive a stochastic representation for the elements of the product. Using this result, the exact joint density for an arbitrary linear combination of the elements of the product is obtained. Furthermore, the derived stochastic representation allows us to simulate samples of arbitrary size by generating independently distributed chi-squared random variables and standard multivariate normal random vectors for each element of the sample. Additionally to the Monte Carlo approach, we suggest another approximation of the density function, which is based on the Gaussian integral and the third order Taylor expansion. We investigate, with a numerical study, the properties of the suggested approximations. A good performance is documented for both methods.
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| 9. |
- Bodnar, Taras, et al.
(author)
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Optimal Shrinkage-Based Portfolio Selection in High Dimensions
- 2023
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In: Journal of business & economic statistics. - : Informa UK Limited. - 0735-0015 .- 1537-2707. ; 41:1, s. 140-156
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Journal article (peer-reviewed)abstract
- In this article, we estimate the mean-variance portfolio in the high-dimensional case using the recent results from the theory of random matrices. We construct a linear shrinkage estimator which is distribution-free and is optimal in the sense of maximizing with probability 1 the asymptotic out-of-sample expected utility, that is, mean-variance objective function for different values of risk aversion coefficient which in particular leads to the maximization of the out-of-sample expected utility and to the minimization of the out-of-sample variance. One of the main features of our estimator is the inclusion of the estimation risk related to the sample mean vector into the high-dimensional portfolio optimization. The asymptotic properties of the new estimator are investigated when the number of assets p and the sample size n tend simultaneously to infinity such that p/n→c∈(0,+∞). The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely the existence of the 4+ε moments is only required. Thereafter we perform numerical and empirical studies where the small- and large-sample behavior of the derived estimator is investigated. The suggested estimator shows significant improvements over the existent approaches including the nonlinear shrinkage estimator and the three-fund portfolio rule, especially when the portfolio dimension is larger than the sample size. Moreover, it is robust to deviations from normality.
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| 10. |
- Bodnar, Taras, et al.
(author)
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Statistical Inference for the Expected Utility Portfolio in High Dimensions
- 2021
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In: IEEE Transactions on Signal Processing. - 1053-587X .- 1941-0476. ; 69, s. 1-14
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Journal article (peer-reviewed)abstract
- 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 c is an element of (0, 1) as n -> infinity. 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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