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Singular inverse Wishart distribution and its application to portfolio theory

Mazur, Stepan LU ; Bodnar, Taras and Podgorski, Krzysztof LU (2016) In Journal of Multivariate Analysis 143. p.314-326
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Mean–variance portfolio, Singular Wishart distribution, Sample estimate of precision matrix, Moore–Penrose inverse
in
Journal of Multivariate Analysis
volume
143
pages
314 - 326
publisher
Academic Press
external identifiers
  • wos:000366885300019
  • scopus:84945144193
ISSN
0047-259X
DOI
10.1016/j.jmva.2015.09.021
language
English
LU publication?
yes
id
f7f3b194-95b8-4f08-9765-db68184ed5b0 (old id 8228299)
date added to LUP
2015-11-20 15:20:50
date last changed
2017-08-06 04:06:01
@article{f7f3b194-95b8-4f08-9765-db68184ed5b0,
  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.},
  author       = {Mazur, Stepan and Bodnar, Taras and Podgorski, Krzysztof},
  issn         = {0047-259X},
  keyword      = {Mean–variance portfolio,Singular Wishart distribution,Sample estimate of precision matrix,Moore–Penrose inverse},
  language     = {eng},
  pages        = {314--326},
  publisher    = {Academic Press},
  series       = {Journal of Multivariate Analysis},
  title        = {Singular inverse Wishart distribution and its application to portfolio theory},
  url          = {http://dx.doi.org/10.1016/j.jmva.2015.09.021},
  volume       = {143},
  year         = {2016},
}