# Portfolio Symmetry and Momentum

23 Pages Posted: 1 Mar 2009 Last revised: 25 Apr 2012

See all articles by Monica Billio

## Monica Billio

Ca Foscari University of Venice - Dipartimento di Economia

## Ludovic Calès

Joint Research Center of the European Commission

## Dominique Guegan

Ecole Normale Superieure de Cachan

Date Written: October 30, 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.

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

JEL Classification: C14, C44

Suggested Citation

Billio, Monica and Calès, 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: https://ssrn.com/abstract=1349701 or http://dx.doi.org/10.2139/ssrn.1349701