A Cross-Sectional Performance Measure for Portfolio Management

Sharpe-like ratios have been traditionally used to measure the performances of portfolio managers. However, they are known to suffer major drawbacks. Among them, two are intricate : (1) they are relative to a peer's performance and (2) the best score is generally assumed to correspond to a "good" portfolio allocation, with no guarantee on the goodness of this allocation. Last but no least (3) these measures suffer significant estimation errors leading to the inability to distinguish two managers' performances. In this paper, we propose a cross-sectional measure of portfolio performance dealing with these three issues. First, we define the score of a portfolio over a single period as the percentage of investable portfolios outperformed by this portfolio. This score quantifies the goodness of the allocation remedying drawbacks (1) and (2). The new information brought by the cross-sectionality of this score is then discussed through applications. Secondly, we build a performance index, as the average cross-section score over successive periods, whose estimation partially answers drawback (3). In order to assess its informativeness and using empirical data, we compare its forecasts with those of the Sharpe and Sortino ratios. The results show that our measure is the most robust and informative. It validates the utility of such cross-sectional performance measure.Performance measure, portfolio management, relative-value strategy, large portfolios, absolute return strategy, multivariate statistics, Generalized hyperbolic Distribution.

Portfolio Symmetry and Momentum

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.Graph Theory, Momentum, Dynamic Portfolio, Quantum Probability, Spectral Analysis

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International audienc

A Performance Measure of Zero-Dollar Long/Short Equally Weighted Portfolios

Sharpe-like ratios have been traditionally used to measure the performances of portfolio managers. However, they suffer two intricate drawbacks (1) they are relative to a perr's performance and (2) the best score is generally assumed to correspond to a "good" portfolio allocation, with no guarantee on the goodness of this allocation. In this paper, we propose a new measure to quantify the goodness of an allocation and we show how to estimate this measure in the case of the strategy used to track the momentum effect, namely the Zero-Dollar Long/Short Equally Weighted (LSEW) investment strategy. Finally, we show how to use this measure to timely close the positions of an invested portfolio.Portfolio management, performance measure, generalized hyperbolic distribution.

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International audienc

Portfolio Symmetry and Momentum

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.Finance, Graph theory, momentum, quantum probability, spectral analysis.

A Cross-Sectional Performance Measure for Portfolio Management

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/CESFramDP2010.htmDocuments de travail du Centre d'Economie de la Sorbonne 2010.70 - ISSN : 1955-611XSharpe-like ratios have been traditionally used to measure the performances of portfolio managers. However, they are known to suffer major drawbacks. Among them, two are intricate : (1) they are relative to a peer's performance and (2) the best score is generally assumed to correspond to a "good" portfolio allocation, with no guarantee on the goodness of this allocation. Last but no least (3) these measures suffer significant estimation errors leading to the inability to distinguish two managers' performances. In this paper, we propose a cross-sectional measure of portfolio performance dealing with these three issues. First, we define the score of a portfolio over a single period as the percentage of investable portfolios outperformed by this portfolio. This score quantifies the goodness of the allocation remedying drawbacks (1) and (2). The new information brought by the cross-sectionality of this score is then discussed through applications. Secondly, we build a performance index, as the average cross-section score over successive periods, whose estimation partially answers drawback (3). In order to assess its informativeness and using empirical data, we compare its forecasts with those of the Sharpe and Sortino ratios. The results show that our measure is the most robust and informative. It validates the utility of such cross-sectional performance measure.Le ratio de Sharpe et ses dérivés ont été traditionnellement utilisés pour mesurer les performances des gestionnaires de portefeuille. Cependant, ils sont connus pour souffrir d'inconvénients majeurs. Parmi eux, deux sont importants : (1) la performance est mesurée par rapport à un autre portefeuille et (2) le meilleur score est généralement considéré comme correspondant à une "bonne" allocation, sans aucune garantie sur la notion de "bonne" allocation. Enfin (3) ces performances souffrent d'erreurs d'estimation significatives conduisant à l'incapacité à distinguer entre les performances de deux gérants. Dans cet article, nous proposons une mesure nouvelle de la performance d'un portefeuille permettant de répondre aux trois limitations précédentes. Tout d'abord, nous définissons le score d'un portefeuille sur une période unique comme le pourcentage de portefeuilles dont le rendement est dépassé par ce portefeuille. Ce score évite les problémes soulevés en (1) et (2). Ensuite, nous construisons un indice de performance dont l'estimation résoud en partie les questions soulevées en (3). Afin d'évaluer la qualité de cette nouvelle mesure, nous comparons ses prévisions avec celles des ratios de Sharpe et Sortino, sur un ensemble de données de marché