Existence and computation of the Aumann–Serrano index of riskiness and its extension

Aumann and Serrano (2008) introduce the index of riskiness to quantify the risk of a gamble. We discuss for which gambles this index of riskiness exists by considering the acceptance behavior of CARA-agents. Since for several relevant distributions riskiness is not defined, we suggest an extension of riskiness to all gambles. We prove that this extension is unique and that it satisfies the central duality axiom. Finally, we derive closed-form solutions of extended riskiness and list some applications.