Estimation of portfolio efficiency in nonconvex settings: A free disposal hull estimator with non-increasing returns to scale

Abstract

Traditional data envelopment analysis (DEA) for estimating the portfolio efficiency requires that the portfolio frontier (theoretical frontier) is concave to ensure that the DEA efficiency in probability converges to the portfolio efficiency as the portfolio sample increases. However, in practice, the DEA efficiency is likely to overestimate the portfolio efficiency because some nonconvex settings may cause the portfolio frontier to be nonconcave. In this paper, we employ a free disposal hull (FDH) estimator by combining the free disposability and non-increasing returns to scale (NIRS) assumptions, namely FDH-NIRS estimator, to explore its theoretical nature and applications in the estimation of portfolio efficiency. First, we apply the directional distance function (DDF) to develop the portfolio frontier-, DEA-, FDH-, and FDH-NIRS-based models in the mean and value-at-risk (VaR) framework, and also show the differences between these models. Second, we transform the FDH-NIRS model into a linear equivalence model and further demonstrate that the FDH-NIRS efficiency in probability converges to the portfolio efficiency. Third, we extend the FDH-NIRS estimator to the framework of multiple return and risk measures with the purpose of showing its generalizability. Finally, we verify the validity of the proposed model and estimator by simulations.