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

Univ. of Southern Switzerland Working Paper

42 Pages Posted: 8 Nov 2001

See all articles by Markus Leippold

Markus Leippold

University of Zurich - Department of Banking and Finance; University of Zurich - Faculty of Economics, Business Administration and Information Technology

Paolo Vanini

University of Basel

Fabio Trojani

Swiss Finance Institute; University of Geneva

Date Written: April 2002

Abstract

We present a geometric approach to discrete time multiperiod mean variance portfolio optimization that largely simplies 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. Specically, 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 multi-periods optimal policies and mean variance frontiers by discussing specic issued related to AL portfolios: The impact of taking liabilities into account on the implied mean variance frontiers, the quantication of the impact of the investment horizon and the determination of the optimal initial funding ratio.

Keywords: Assets and Liabilities Portfolios, Minimum-Variance Frontiers, Dynamic Programming, Markowitz Model

JEL Classification: G11, G12, G28, D92, C60, C61

Suggested Citation

Leippold, Markus and Vanini, Paolo and Trojani, Fabio, A Geometric Approach to Multiperiod Mean Variance Optimization of Assets and Liabilities (April 2002). Univ. of Southern Switzerland Working Paper. Available at SSRN: https://ssrn.com/abstract=289601 or http://dx.doi.org/10.2139/ssrn.289601

Markus Leippold

University of Zurich - Department of Banking and Finance ( email )

Plattenstrasse 14
Zürich, 8032
Switzerland

University of Zurich - Faculty of Economics, Business Administration and Information Technology ( email )

Plattenstrasse 14
Zürich, 8032
Switzerland

Paolo Vanini

University of Basel ( email )

Petersplatz 1
Basel, CH-4003
Switzerland

Fabio Trojani (Contact Author)

Swiss Finance Institute ( email )

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

University of Geneva ( email )

Geneva
Switzerland

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