# Portfolio Symmetry and Momentum

## Monica Billio

Ca Foscari University of Venice - Department of Economics

## Ludovic Cales

Paris School of Economics - Université Paris-1 Panthéon-La Sorbonne; University of Lausanne - Institute of Banking and Finance (IBF); University of Venice - Department of Economics

## Dominique Guegan

Ecole Normale Superieure de Cachan

October 30, 2009

University Ca' Foscari of Venice, Dept. of Economics Research Paper No. 05/WP/2009

Abstract:
This paper presents a theoretical 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. 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.

Number of Pages in PDF File: 23

Keywords: Graph Theory, Momentum, Dynamic Portfolio, Quantum Probability, Spectral Analysis

JEL Classification: C14, C44

working papers series

#### Suggested Citation

Billio, Monica, Cales, Ludovic and Guegan, Dominique, Portfolio Symmetry and Momentum (October 30, 2009). University Ca' Foscari of Venice, Dept. of Economics Research Paper No. 05/WP/2009. Available at SSRN: http://ssrn.com/abstract=1349701 or http://dx.doi.org/10.2139/ssrn.1349701

#### Contact Information

##### Monica Billio (Contact Author)
Ca Foscari University of Venice - Department of Economics ( email )
Cannaregio 873
Venice, 30121
Italy
##### Ludovic Cales
Paris School of Economics - Université Paris-1 Panthéon-La Sorbonne ( email )
48 Boulevard Jourdan
Paris, 75014 75014
France
University of Lausanne - Institute of Banking and Finance (IBF) ( email )
CH-1015 Lausanne
Switzerland
University of Venice - Department of Economics ( email )
Cannaregio 873
Venice, 30121
Italy
##### Dominique Guegan
Ecole Normale Superieure de Cachan ( email )
61 avenue du President Wilson
94235 Cachan
France
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