Shortfall deviation risk: an alternative for risk measurement

Marcelo Brutti Righi and Paulo Sergio Ceretta

Abstract

ABSTRACT

We present the shortfall deviation risk (SDR): a risk measure that represents the expected loss that occurs with a certain probability penalized by the dispersion of results that are worse than such an expectation. SDR combines expected shortfall (ES) and shortfall deviation (SD), which we also introduce, contemplating two fundamental pillars of the risk concept (the probability of adverse events and the variability of an expectation) and considering extreme results. We demonstrate that SD is a generalized deviation measure, whereas SDR is a coherent risk measure. We achieve the dual representation of SDR, and we discuss issues such as its representation by a weighted ES, acceptance sets, convexity, continuity and the relationship with stochastic dominance. Examples using real and simulated data allow us to conclude that SDR offers greater protection in risk measurement than value-at-risk and ES, especially in times of significant turbulence in riskier scenarios.