Title: Best Gain-Loss Ratio is a Poor Performance Measurement
Authors: Sara Biagini (Universita di Pisa) and Mustafa C. Pinar


Gain-loss ratio is known to enjoy very good properties from a normative point of view. As a confirmation, we prove a dual representation of the best market gain-loss in the presence of a random endowment for general probability spaces. The dual representation is then the key to show best gain-loss is an acceptability index. However, gain-loss ratio was designed for and works best only for finite Omega. In most general Omega, and continuous time models best gain-loss is either infinite or fails to be attained. In addition, it displays an odd beheviour in a risk neutral situation due to scale invariance, which does not seem desirable in a portfolio optimization context. Such weaknesses definitely prove that best gain-loss is a poor performance measurement. Keywords: Gain-loss ratio, acceptability indexes, incomplete markets, martingales, quasi-concave optimization, duality methods, market modified risk measures.