, Volume 76, Issue 8, pp 1105–1134 | Cite as

Asymptotic behavior of the estimated weights and of the estimated performance measures of the minimum VaR and the minimum CVaR optimal portfolios for dependent data

  • Taras Bodnar
  • Wolfgang SchmidEmail author
  • Taras Zabolotskyy


In this paper we derive the asymptotic distributions of the estimated weights and of estimated performance measures of the minimum value-at-risk portfolio and of the minimum conditional value-at-risk portfolio assuming that the asset returns follow a strictly stationary process. It is proved that the estimated weights as well as the estimated performance measures are asymptotically multivariate normally distributed. We also present an asymptotic test for the weights and a joint test for the characteristics of both portfolios. Moreover, the asymptotic densities of the estimated performance measures are compared with the corresponding exact densities. It is shown that the asymptotic approximation performs well even for the moderate sample size.


Efficient frontier Minimum VaR portfolio Minimum CVaR portfolio Parameter uncertainty Statistical inference  Asymptotic distribution  Matrix differentiation 



The authors are thankful to the Editor and the two anonymous Referees for careful reading of the paper and for their suggestions which have improved an earlier version of this paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Taras Bodnar
    • 1
  • Wolfgang Schmid
    • 2
    Email author
  • Taras Zabolotskyy
    • 3
  1. 1.Department of MathematicsHumboldt-University of BerlinBerlinGermany
  2. 2.Department of StatisticsEuropean University ViadrinaFrankfurt (Oder)Germany
  3. 3.Lviv Institute of BankingUniversity of Banking of the National Bank of UkraineLvivUkraine

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