| 11. |
- Bodnar, Taras, et al.
(författare)
-
On the exact and approximate distributions of the product of a Wishart matrix with a normal vector
- 2013
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Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 122, s. 70-81
-
Tidskriftsartikel (refereegranskat)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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| 12. |
- Bodnar, Taras, et al.
(författare)
-
On the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions
- 2018
-
Ingår i: Theory of Probability and Mathematical Statistics. - : American Mathematical Society (AMS). - 0094-9000 .- 1547-7363. ; 99, s. 37-50
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation is derived for this product, in using which the characteristic function of the product and its asymptotic distribution under the double asymptotic regime are established. The application of obtained stochastic representation speeds up the simulation studies where the product of a singular Wishart random matrix and a singular normal random vector is present. We further document a good performance of the derived asymptotic distribution within a numerical illustration. Finally, several important properties of the singular Wishart distribution are provided.
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| 13. |
- Bodnar, Taras, et al.
(författare)
-
Singular inverse Wishart distribution and its application to portfolio theory
- 2016
-
Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 143, s. 314-326
-
Tidskriftsartikel (refereegranskat)abstract
- The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented.
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| 14. |
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| 15. |
- Bodnar, Taras, et al.
(författare)
-
Tangency portfolio weights for singular covariance matrix in small and large dimensions : estimation and test theory
- 2019
-
Ingår i: Journal of Statistical Planning and Inference. - : Elsevier. - 0378-3758 .- 1873-1171. ; 201, s. 40-57
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive the finite-sample distribution of the estimated weights of the tangency portfolio when both the population and the sample covariance matrices are singular. These results are used in the derivation of a statistical test on the weights of the tangency portfolio where the distribution of the test statistic is obtained under both the null and the alternative hypotheses. Moreover, we establish the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sample size increase to infinity. The theoretical findings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index.
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| 16. |
- Drin, Svitlana, 1977-, et al.
(författare)
-
A test on the location of tangency portfolio for small sample size and singular covariance matrix
- 2023
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper, we propose the test for the location of the tangency portfolio on the set of feasible portfolios when both the population and the sample covariance matrices of asset returns are singular. We derive the exact distribution of the test statistic under both the null and alternative hypotheses. Furthermore, we establish the high-dimensional asymptotic distribution of that test statistic when both the portfolio dimension and the sample size increase to infinity. We complement our theoretical findings by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented.
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| 17. |
- Drin, Svitlana, et al.
(författare)
-
A test on the location of tangency portfolio for small sample size and singular covariance matrix
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper, we propose the test for the location of the tangency portfolio onthe set of feasible portfolios when both the population and the sample covariancematrices of asset returns are singular. We derive the exact distribution of the teststatistic under both the null and alternative hypotheses. Furthermore, we establishthe high-dimensional asymptotic distribution of that test statistic when both theportfolio dimension and the sample size increase to infinity. We complement ourtheoretical findings by comparing the high-dimensional asymptotic test with anexact finite sample test in the numerical study. A good performance of the obtainedresults is documented.
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| 18. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
-
An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection
- 2020
-
Ingår i: Computational Economics. - : Springer. - 0927-7099 .- 1572-9974. ; 56, s. 773-794
-
Tidskriftsartikel (refereegranskat)abstract
- Covariance matrix of the asset returns plays an important role in the portfolioselection. A number of papers is focused on the case when the covariance matrixis positive definite. In this paper, we consider portfolio selection with a singu-lar covariance matrix. We describe an iterative method based on a second orderdamped dynamical systems that solves the linear rank-deficient problem approxi-mately. Since the solution is not unique, we suggest one numerical solution that canbe chosen from the iterates that balances the size of portfolio and the risk. The nu-merical study confirms that the method has good convergence properties and givesa solution as good as or better than the constrained least norm Moore-Penrose solu-tion. Finally, we complement our result with an empirical study where we analyzea portfolio with actual returns listed in S&P 500 index.
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| 19. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
-
Portfolio Selection with a Rank-deficient Covariance Matrix
- 2021
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz’ problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.
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| 20. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
-
Portfolio Selection with a Rank-Deficient Covariance Matrix
- 2024
-
Ingår i: Computational Economics. - : Springer. - 0927-7099 .- 1572-9974. ; 63, s. 2247-2269
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz' problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.
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