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A Geometric Approach to Multiperiod Mean Variance Optimization of Assets and Liabilities

  • Markus LEIPPOLD

    (Swiss Banking Institute, University of Zurich)

  • Fabio TROJANI

    (Institute of Finance, University of Southern Switzerland)

  • Paolo VANINI

    (Institute of Finance, University of Southern Switzerland)

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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Paper provided by International Center for Financial Asset Management and Engineering in its series FAME Research Paper Series with number rp48.

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Date of creation: Apr 2002
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Handle: RePEc:fam:rpseri:rp48
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  1. 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.
  2. 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.
  3. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
  4. Keith V. Smith, 1967. "A Transition Model For Portfolio Revision," Journal of Finance, American Finance Association, vol. 22(3), pages 425-439, 09.
  5. He, Hua & Pages, Henri F, 1993. "Labor Income, Borrowing Constraints, and Equilibrium Asset Prices," Economic Theory, Springer, vol. 3(4), pages 663-96, October.
  6. Duan Li & Wan-Lung Ng, 2000. "Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation," Mathematical Finance, Wiley Blackwell, vol. 10(3), pages 387-406.
  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.
  8. 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.
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