# Portfolio Symmetry and Momentum

## Author Info

• Monica Billio

()
(Università Ca' Foscari of Venice - Department of Economics)

• Ludovic Calès

()
(Università Ca' Foscari of Venice - Department of Economics, CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

• Dominique Guegan

()
(CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne, EEP-PSE - Ecole d'Économie de Paris - Paris School of Economics - Ecole d'Économie de Paris)

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## 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.

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## Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00645814.

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Publication status: Published, European Journal of Operational Research, 2011, 214, 3, 759-767
Handle: RePEc:hal:cesptp:halshs-00645814

Note: View the original document on HAL open archive server: http://halshs.archives-ouvertes.fr/halshs-00645814
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## Related research

Keywords: Finance - Graph theory - momentum - quantum probability - spectral analysis.;

Other versions of this item:

Find related papers by JEL classification:
• C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
• C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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## References

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1. Nicholas Barberis & Andrei Shleifer & Robert W. Vishny, 1997. "A Model of Investor Sentiment," NBER Working Papers 5926, National Bureau of Economic Research, Inc.
2. Jegadeesh, Narasimhan & Titman, Sheridan, 1993. " Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency," Journal of Finance, American Finance Association, vol. 48(1), pages 65-91, March.
3. K. Rouwenhorst, 1996. "International Momentum Strategies," Yale School of Management Working Papers ysm36, Yale School of Management, revised 01 Feb 2008.
4. Okunev, John & White, Derek, 2003. "Do Momentum-Based Strategies Still Work in Foreign Currency Markets?," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 38(02), pages 425-447, June.
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## Citations

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Cited by:
1. Monica Billio & Ludovic Calès & Dominique Guegan, 2012. "Cross-Sectional Analysis through Rank-based Dynamic Portfolios," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00707430, HAL.
2. repec:hal:journl:halshs-00707430 is not listed on IDEAS
3. Timo H. Leivo, 2012. "Combining value and momentum indicators in varying stock market conditions: The Finnish evidence," Review of Accounting and Finance, Emerald Group Publishing, vol. 11(4), pages 400-447.
4. Pätäri, Eero & Leivo, Timo & Honkapuro, Samuli, 2012. "Enhancement of equity portfolio performance using data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 220(3), pages 786-797.
5. Monica Billio & Ludovic Calès & Dominique Guegan, 2012. "Cross-Sectional Analysis through Rank-based Dynamic," Documents de travail du Centre d'Economie de la Sorbonne 12036, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.

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