The best gain-loss ratio is a poor performance measure

TL;DR

The paper critically examines the gain-loss ratio as a performance measure, highlighting its theoretical strengths but also its practical limitations and undesirable properties in continuous models.

Contribution

It demonstrates that the best gain-loss ratio is an acceptability index with a dual representation, but also shows its limitations in general and continuous models, questioning its effectiveness.

Findings

Best gain-loss ratio is an acceptability index with dual representation.

In continuous models, the best gain-loss is often infinite or unattainable.

Gain-loss ratio exhibits undesirable scale invariance properties.

Abstract

The gain-loss ratio is known to enjoy very good properties from a normative point of view. As a confirmation, we show that the best market gain-loss ratio in the presence of a random endowment is an acceptability index and we provide its dual representation for general probability spaces. However, the gain-loss ratio was designed for finite $\Omega$, and works best in that case. For general $\Omega$ and in most continuous time models, the best gain-loss is either infinite or fails to be attained. In addition, it displays an odd behaviour due to the scale invariance property, which does not seem desirable in this context. Such weaknesses definitely prove that the (best) gain-loss is a poor performance measure.