# Portfolio Symmetry and Momentum

*Monica Billio* (),
*Ludovic Calès* () and
*Dominique Guegan*

Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) from HAL

**Abstract:**
This paper presents a theorical framework to model the evolution of a portfolio whose weights vary over time. Such a portfolio is called a dynamic portfolio. In a first step, considering a given investment policy, we define the set of the investable portfolios. Then, considering portfolio vicinity in terms of turnover, we represent the investment policy as a graph. It permits us to model the evolution of a dynamic portfolio as a stochastic process in the set of the investable portfolios. Our first model for the evolution of a dynamic portfolio is a random walk on the graph corresponding to the investment policy chosen. Next, using graph theory and quantum probability, we compute the probabilities for a dynamic portfolio to be in the different regions of the graph. The resulting distribution is called spectral distribution. It depends on the geometrical properties of the graph and thus in those of the investment policy. The framework is next applied to an investment policy similar to the Jeegadeesh and Titman's momentum strategy [JT1993]. We define the optimal dynamic portfolio as the sequence of portfolios, from the set of the investable portfolios, which gives the best returns over a respective sequence of time periods. Under the assumption that the optimal dynamic portfolio follows a random walk, we can compute its spectral distribution. We found then that the strategy symmetry is a source of momentum.

**Keywords:** spectral analysis; Finance; Graph theory; momentum; quantum probability; spectral analysis.; Théorie des graphes; probabilité quantique; analyse spectrale. (search for similar items in EconPapers)

**Date:** 2009-10

**Note:** View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00363383v2

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**Published** in Documents de Travail du Centre d'Economie de la Sorbonne 2009.03 - ISSN : 1955-611X. 2009

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**Related works:**

Journal Article: Portfolio symmetry and momentum (2011)

Working Paper: Portfolio Symmetry and Momentum (2011)

Working Paper: Portfolio Symmetry and Momentum (2009)

Working Paper: Portfolio Symmetry and Momentum (2009)

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**Persistent link:** https://EconPapers.repec.org/RePEc:hal:cesptp:halshs-00363383

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