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# Portfolio Symmetry and Momentum

## Author

Listed:
• 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 - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

• Dominique Guegan

() (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

## 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.

## Suggested Citation

• Monica Billio & Ludovic Calès & Dominique Guegan, 2009. "Portfolio Symmetry and Momentum," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00363383, HAL.
• Handle: RePEc:hal:cesptp:halshs-00363383
Note: View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00363383v2
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File URL: https://halshs.archives-ouvertes.fr/halshs-00363383v2/document

## References listed on IDEAS

as
1. K. Geert Rouwenhorst, 1998. "International Momentum Strategies," Journal of Finance, American Finance Association, vol. 53(1), pages 267-284, February.
2. 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.
3. Barberis, Nicholas & Shleifer, Andrei & Vishny, Robert, 1998. "A model of investor sentiment," Journal of Financial Economics, Elsevier, vol. 49(3), pages 307-343, September.
4. repec:hrv:faseco:30747159 is not listed on IDEAS
5. 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.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. 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.
2. Monica Billio & Ludovic Calès & Dominique Guegan, 2012. "Cross-Sectional Analysis through Rank-based Dynamic Portfolios," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00707430, HAL.
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, October.
4. repec:eee:ejores:v:265:y:2018:i:2:p:655-672 is not listed on IDEAS
5. repec:hal:journl:halshs-00707430 is not listed on IDEAS
6. 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.

### Keywords

spectral analysis; Finance; Graph theory; momentum; quantum probability; spectral analysis.; Théorie des graphes; probabilité quantique; analyse spectrale.;

### 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

### NEP fields

This paper has been announced in the following NEP Reports:

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