Annals of Operations Research

, Volume 229, Issue 1, pp 121–158 | Cite as

A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function

  • Taras Bodnar
  • Nestor Parolya
  • Wolfgang Schmid


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.


Multi-period asset allocation Quadratic utility function Closed-form solution Tangency portfolio 



The authors are thankful to Professor Endre Boros, the Associate Editor, and an anonymous Reviewer for careful reading of the paper and for their suggestions which have improved an earlier version of this paper. The first and the third author appreciates the financial support of the German Science Foundation (DFG) via the projects BO 3521/3-1 and SCHM 859/13-1 ”Bayesian Estimation of the Multi-Period Optimal Portfolio Weights and Risk Measures”. We also thank David Bauder for his comments used in the preparation of the revised version of the paper.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Institute of Empirical EconomicsLeibniz University of HannoverHannoverGermany
  3. 3.Department of StatisticsEuropean University ViadrinaFrankfurt (Oder)Germany

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