Billio et al. / International Review of Applied Financial Issues and Economics, Vol. 3, No. 4, 2011 700 Abstract- The aim of this paper is to propose an innovative score measuring the relative performance – in terms of return – of an asset allocation with respect to the alternative allocations offered to the manager. Intuitively, this score is defined as the percentage of alternative allocations outperformed by the manager’s allocation. In particular, considering the case of a manager investing according to the zero-dollar long/short equally weighted strategy, we study in details the computation and the properties of this score and we deal with the related combinatorial issues when the number of assets is large. Keywords: Performance measure; Order statistics; Generalized hyperbolic distribution JEL Codes: C13, C16, C46, C62 1. Introduction During the last two decades, the explosive growth of the asset management industry came with an increasing interest in the analysis of investment performance. The research in this area is axed on Sharpe-like ratios proposed in the 60’s [Sharpe (1966), Treynor (1965), Jensen (1968)] and it is expanding by developing the notion of performance as a reward counter- balanced by some risk. The main innovations focused on the definition and modeling of risk [Shadwick and Keating (2002), Darolles et al. (2009)]. Practically, the performance of a portfolio manager, over a given period, is usually computed as the ratio of his excess return over a risk measure [Grinblatt et al. (1994)]. The managers are then ranked according to these ratios, and the manager providing the highest and steadiest returns receives the best score. These measures are convenient because they require no assumption on the strategy of the portfolio managers. However, they suffer major drawbacks. First, these measures are relative to a peer’s performance and irrelevant if no peer is found. We generally © 2011 The S.E.I.F Press. All Rights Reserved. assume that the best score corresponds to a “good” portfolio allocation, with no guarantee on the goodness of this allocation. Secondly, as they are a ratio of two random variables, they suffer significant estimation errors [Lo (2002) and Christie (2007) among others] which prevent any performance comparison to be significant. In this paper, we differentiate three elements affecting the performance of a manager: the set of investable portfolios offered to this manager, the period of investment and the allocation method used. In general, when we measure the performance of a manager we aim to measure the performance of his allocation method, i.e. his ability to correctly choose his allocation among his set of investable portfolios and for any period. We can achieve this goal by fixing the two first elements. First, the set of investable portfolios is defined by the set of constraints on the portfolio positions. Those are usually stated by the investment policy of the manager. For instance, they can be affected by short-sale or diversification restrictions. Remark that, using Sharpe-like ratios, this issue is dealt with by comparing managers having the same constraints and investing with similar strategies, in other words these managers share the same set of investable portfolios. Secondly, the period of investment affects the return provided by an allocation method. Indeed, over two different periods, the use of the returns in order to compare the performance of two managers is known to be delicate. A manager using a certain allocation method would obtain different returns over the different periods while his allocation method may not be in question. This issue can be tackled by normalizing the returns with a proxy of the risk for the different periods. However, in the following, we propose an alternative way to deal with the normalization of the returns. Basically, we normalize an invested portfolio’s return by providing the percentage of investable portfolios that it outperforms. This normalized return, called score, has the advantage to require no peer to be interpreted and it is cross-sectional in the asset returns as it is cross-sectional in the set of the investable portfolio. This score is the subject of this paper. Finally, the performance of an allocation method over several periods can be obtained by computing its average score over these periods. This http://www.irafie.com A Cross-Sectional Score for the Relative Performance of an Allocation Monica Billio University Ca’Foscari of Venice, Italy Ludovic Calès University of Lausanne, Switzerland Dominique Guégan University Paris-1 Panthéon-Sorbonne, France