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 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 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.

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

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

Portfolio Symmetry and Momentum

URL des Documents de tavail : http://ces.univ-paris14.fr/cesdp/CESFramDP2009.htmRevised for European Journal of Operational Research.Documents de Travail du Centre d'Economie de la Sorbonne 2009.03 - ISSN : 1955-611XThis 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.Ce papier présente un cadre théorique pour la modélisation de l'évolution de portefeuilles dont les poids changent dans le temps. Ces portefeuilles sont appelés portefeuilles dynamiques. Dans un premier temps, à partir d'une politique d'investissement donnée, nous définissons l'ensemble des portefeuilles dans lesquels nous pouvons investir. Ensuite, après avoir introduit une distance entre ces portefeuilles, nous représentons cette politique d'investissement sous la forme d'un graphe. De cette façon, nous pouvons modéliser l'évolution d'un portefeuille dynamique comme un processus stochastique dans l'ensemble des portefeuilles investissables. Notre premier modèle est une marche aléatoire sur le graphe de la politique d'investissement choisie. En utilisant la théorie des graphes et la probabilité quantique, nous calculons la probabilité de ce portefeuille dynamique d'être situé dans les différentes régions du graphe. La distribution obtenue est appelée distribution spectrale. Elle dépend de la géométrie du graphe et donc des propriétés de la stratégie d'investissement. Nous appliquons ensuite ce cadre théorique à une stratégie similaire à celle utilisée par Jeegadeesh et Titman [JT1993]. Le portefeuille dynamique considéré correspond à la suite des portefeuilles qui donnent les meilleurs rendements pour une suite de périodes données, tout en respectant la stratégie d'investissement choisie. Ce portefeuille dynamique est appelé portefeuille dynamique optimal. Sous l'hypothèse que ce portefeuille dynamique suive une marche aléatoire, nous pouvons calculer sa distribution spectrale. Nous trouvons alors que la géométrie de cette stratégie est une source de momentum

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.

On the cross-sectional distribution of portfolios' returns

The aim of this paper is to study the distribution of portfolio returns across portfolios and for given asset returns. We focus on the most common type of investment considering portfolios whose weights are positive and sum up to 1. We provide algorithms and formulas from computational geometry and spline literature to compute the exact values of the probability density function, and of the cumulative distribution function at any point. We also provide closed-form solutions for the computation of its first four moments, and an algorithm to compute the higher moments. All algorithms and formulas allow for equal asset returns.JRC.B.1-Finance and Econom

Cross-Sectional Analysis through Rank-based Dynamic Portfolios

The aim of this paper is to study the cross-sectional eects present in the market using a new framework based on graph theory. Within this framework, we represent the evolution of a dynamic portfolio, i.e. a portfolio whose weights vary over time, as a rank-based factorial model where the predictive ability of each cross-sectional factor is described by a variable. Practically, this modeling permits us to measure the marginal and joint eects of different cross-section factors on a given dynamic portfolio. Associated to a regime switching model, we are able to identify phases during which the cross-sectional eects are present in the market