Estimation of portfolio efficiency via DEA Wenbin Liu a,b , Zhongbao Zhou a,n , Debin Liu a , Helu Xiao a a School of Business Administration, Hunan University, Changsha 410082, China b Kent Business School, University of Kent, Canterbury CT2 7PE, UK article info Article history: Received 9 December 2013 Accepted 10 November 2014 Available online 20 November 2014 Keywords: Portfolio evaluation Efficient portfolio frontier Diversification model DEA abstract Traditional DEA models and nonlinear (diversification) DEA models are often used in performance evaluation of portfolios. However, the diversification models are usually very complicated to compute except the very basic models. And the classic DEA models still need to be further justified and tested, since it is not clear whether they are over-linearised according to the diversification models. The existing studies on performance evaluation via the classic DEA models generally focus on the selection of inputs and outputs. In this work, we investigate theoretical foundation of DEA models for portfolios from a perspective of sampling portfolio. We show the classic DEA provides an effective and practical way to approximate the portfolio efficiency (PE). We further verify this approach through different portfolio models with various frictions and bounds on the market. Through the comprehensive simulations carried out in this study, we show that with adequate data sets, the classic DEA models can effectively sample portfolios to take into account sufficient diversification, and thus can be used as an effective tool in computing the PE for their performance assessments. This study can be viewed as a justification of the classic DEA performance assessments and the way to introduce other efficiency notions (like allocation efficiency, scale efficiency, etc) into assessment of portfolios. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction In financial studies, portfolio performance assessment is an important area [1–3]. Three of the most well-known performance measures, still in use today, are the Treynor index [4] of the excess return per unit of the systematic risk, the Sharpe index [5] of the excess return per unit of the total risk, and the Jensen index [6], which is defined as the difference between the actual portfolio return and the estimated benchmark return. Since these pioneer works, numerous studies have been carried out for measuring performance in two or more dimensions (i.e., risk and return). One important idea in portfolio evaluation is the portfolio frontier approach, which measures performance of a portfolio by some its distances to the efficient portfolio frontier. Markowitz's fundamental work [7] laid the base of the frontier approach under the mean-variance (MV) framework. This idea has been much extended afterwards, and the models further being developed along this idea are often referred as diversification models (or nonlinear DEA models). Morey and Morey proposed quadratic- constrained nonlinear DEA models [2]. They regarded the variance as an input and mean return as an output simultaneously over the same time horizons. By considering diversification, Briec et al. also developed a quadratic constraint nonlinear DEA model in the mean-variance framework for a single period (the so-called diversification model) and introduced the efficiency improvement possibility function [8]. Joro and Na proposed mean-variance- skewness nonlinear DEA models with the variance of returns as the only input and the mean and skewness as the outputs [9]. Briec and Kerstens extended the multi-horizon mean-variance portfolio analysis in [2] and accounted for diversification directly using nonlinear versions of DEA [10]. Lozano and Gutiérrez described several diversification linear models with one input and one output, which are quite different from the classic DEA models, but are consistent with the SSD in the sense that being efficient according to the proposed models is a necessary condi- tion for being SSD efficient [11]. Branda and Kopa investigated empirically influence of various risk measures on DEA efficiency and relations to SSD in [12]. Also, they developed a DEA-risk model with binary weights, which is equivalent to the pairwise SSD efficiency test although very complex to compute [13]. Lamb and Tee proposed diversification consistent DEA-risk models with an arbitrary number of risk and return measures and further exam- ined relationships between diversification, coherent measures of risk and stochastic dominance in [14]. However, they only used the positive parts of coherent measures as the inputs. Branda [15] proposed new efficiency tests in which general deviation measures Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/omega Omega http://dx.doi.org/10.1016/j.omega.2014.11.006 0305-0483/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail address: ZhongbaoZhou@gmail.com (Z. Zhou). Omega 52 (2015) 107–118