Gini Shortfall: A Gini mean difference-based risk measure

Abstract

A risk is the possibility of undesirable events happening in the future. A good risk measure is needed to quantify the risks one faces. Some well-known risk measures include the Value-at-Risk (VaR) and the Expected Shortfall (ES). VaR measures the lower bound for big losses in a loss distribution tail, while ES measures the average of losses surpassing the VaR. Unfortunately, there are drawbacks in using the stated risk measures, mainly that they do not provide information regarding the variability of losses in the distribution tail. This paper will introduce and explore Gini Shortfall (GS), a more comprehensive risk measure than VaR and ES. GS provides information on the variability of data in distribution tails measured with Tail-Gini functional, a tail variability measure based on the variability measure Gini Mean Difference or Gini functional. Another superiority of GS is that under certain conditions, it can satisfy the four criteria of coherency. A coherent risk measure is useful for companies or investors to determine the right business and investing strategies. This paper will also provide explicit formulas of GS for some loss-related distributions, namely the exponential, Pareto, and logistic distributions. These formulas are then applied to calculate risks from actual stock data.