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A geometric approach to multiperiod mean variance optimization of assets and liabilities

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  • Leippold, Markus
  • Trojani, Fabio
  • Vanini, Paolo

Abstract

We present a geometric approach to discrete time multiperiod mean variance portfolio optimization that largely simplifies the mathematical analysis and the economic interpretation of such model settings. We show that multiperiod mean variance optimal policies can be decomposed in an orthogonal set of basis strategies, each having a clear economic interpretation. This implies that the corresponding multi period mean variance frontiers are spanned by an orthogonal basis of dynamic returns. Specifically, in a k-period model the optimal strategy is a linear combination of a single k-period global minimum second moment strategy and a sequence of k local excess return strategies which expose the dynamic portfolio optimally to each single-period asset excess return. This decomposition is a multi period version of Hansen and Richard (1987) orthogonal representation of single-period mean variance frontiers and naturally extends the basic economic intuition of the static Markowitz model to the multiperiod context. Using the geometric approach to dynamic mean variance optimization we obtain closed form solutions in the i.i.d. setting for portfolios consisting of both assets and liabilities (AL), each modelled by a distinct state variable. As a special case, the solution of the mean variance problem for the asset only case in Li and Ng (2000) follows directly and can be represented in terms of simple products of some single period orthogonal returns. We illustrate the usefulness of our geometric representation of multiperiods optimal policies and mean variance frontiers by discussing specific issued related to AL portfolios: The impact of taking liabilities into account on the implied mean variance frontiers, the quantification of the impact of the investment horizon and the determination of the optimal initial funding ratio.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Economic Dynamics and Control.

Volume (Year): 28 (2004)
Issue (Month): 6 (March)
Pages: 1079-1113

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Handle: RePEc:eee:dyncon:v:28:y:2004:i:6:p:1079-1113

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References

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  1. Chen, Andrew H Y & Jen, Frank C & Zionts, Stanley, 1971. "The Optimal Portfolio Revision Policy," The Journal of Business, University of Chicago Press, vol. 44(1), pages 51-61, January.
  2. Keith V. Smith, 1967. "A Transition Model For Portfolio Revision," Journal of Finance, American Finance Association, vol. 22(3), pages 425-439, 09.
  3. Grossman, Sanford J. & Vila, Jean-Luc, 1992. "Optimal Dynamic Trading with Leverage Constraints," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 27(02), pages 151-168, June.
  4. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-57, August.
  5. Duan Li & Wan-Lung Ng, 2000. "Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation," Mathematical Finance, Wiley Blackwell, vol. 10(3), pages 387-406.
  6. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
  7. Samuelson, Paul A, 1969. "Lifetime Portfolio Selection by Dynamic Stochastic Programming," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 239-46, August.
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Citations

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Cited by:
  1. Aivaliotis, Georgios & Palczewski, Jan, 2014. "Investment strategies and compensation of a mean–variance optimizing fund manager," European Journal of Operational Research, Elsevier, vol. 234(2), pages 561-570.
  2. Xiangyu Cui & Xun Li & Duan Li, 2013. "Unified Framework of Mean-Field Formulations for Optimal Multi-period Mean-Variance Portfolio Selection," Papers 1303.1064, arXiv.org.
  3. Chiu, Mei Choi & Li, Duan, 2006. "Asset and liability management under a continuous-time mean-variance optimization framework," Insurance: Mathematics and Economics, Elsevier, vol. 39(3), pages 330-355, December.
  4. Wei, J. & Wong, K.C. & Yam, S.C.P. & Yung, S.P., 2013. "Markowitz’s mean–variance asset–liability management with regime switching: A time-consistent approach," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 281-291.
  5. Qian Zhao & Jiaqin Wei & Rongming Wang, 2013. "Mean-Variance Asset-Liability Management with State-Dependent Risk Aversion," Papers 1304.7882, arXiv.org.
  6. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2013. "On the equivalence of quadratic optimization problems commonly used in portfolio theory," European Journal of Operational Research, Elsevier, vol. 229(3), pages 637-644.
  7. Wu, Huiling & Li, Zhongfei, 2012. "Multi-period mean–variance portfolio selection with regime switching and a stochastic cash flow," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 371-384.
  8. Baumann, Roger T. & Müller, Heinz H., 2008. "Pension funds as institutions for intertemporal risk transfer," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1000-1012, June.
  9. Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2012. "On the Exact Solution of the Multi-Period Portfolio Choice Problem for an Exponential Utility under Return Predictability," Papers 1207.1037, arXiv.org.
  10. Leippold, Markus & Vanini, Paolo & Ebnoether, Silvan, 2006. "Optimal credit limit management under different information regimes," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 463-487, February.
  11. Wang, J. & Forsyth, P.A., 2010. "Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation," Journal of Economic Dynamics and Control, Elsevier, vol. 34(2), pages 207-230, February.
  12. Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2012. "A Closed-Form Solution of the Multi-Period Portfolio Choice Problem for a Quadratic Utility Function," Papers 1207.1003, arXiv.org.
  13. Castellano, Rosella & Cerqueti, Roy, 2014. "Mean–Variance portfolio selection in presence of infrequently traded stocks," European Journal of Operational Research, Elsevier, vol. 234(2), pages 442-449.
  14. Yao, Haixiang & Lai, Yongzeng & Li, Yong, 2013. "Continuous-time mean–variance asset–liability management with endogenous liabilities," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 6-17.
  15. Briec, Walter & Kerstens, Kristiaan, 2009. "Multi-horizon Markowitz portfolio performance appraisals: A general approach," Omega, Elsevier, vol. 37(1), pages 50-62, February.
  16. Basak, Suleyman & Chabakauri, Georgy, 2009. "Dynamic Mean-Variance Asset Allocation," CEPR Discussion Papers 7256, C.E.P.R. Discussion Papers.
  17. Wang, J. & Forsyth, P.A., 2011. "Continuous time mean variance asset allocation: A time-consistent strategy," European Journal of Operational Research, Elsevier, vol. 209(2), pages 184-201, March.

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