Indexing gamble desirability by extending proportional stochastic dominance

Abstract

We axiomatically characterise two new orders of desirability of gambles (risky assets) that are natural extensions of the proportional stochastic dominance order to complete orders. These orders are represented by indices with parallels to the recently introduced Aumann-Serrano index of riskiness and the Foster-Hart measure of riskiness. The new indices are shown to be related to the concept of coherent measures of risk and to the Sharpe ratio.

Short-Term Investments and Indices of Risk

“…We study four specific decision problems in the paper: (1) acceptance/rejection, in which the agent faces a binary choice between accepting and rejecting the gamble (e.g., Hart, 2011); (2) capital allocation, in which the agent has a continuous choice of how much to invest in the gamble (e.g., Markowitz, 1952;Sharpe, 1964;Landsberger & Meilijson, 1993); (3) the optimal certainty equivalent, in which the agent evaluates how much an opportunity to invest in the gamble (according to the optimal investment level) is worth to the agent (e.g., Hellman & Schreiber, 2018); and (4) risk premium, in which the agent evaluates how much investing in the gamble is inferior to obtaining the gamble's expected payoff (Arrow, 1970). 1 A risk index is a function that assigns to each gamble a nonnegative number, which is interpreted as the gamble's riskiness.…”

“…In two influential papers, Aumann & Serrano (2008) and Foster & Hart (2009) presented two "objective" indices of riskiness of gambles, which are independent of the subjective utility of the agent. These indices are either based on reasonable axioms that an index of risk should satisfy (e.g., Artzner et al, 1999;Aumann & Serrano, 2008;Cherny & Madan, 2009;Foster & Hart, 2013;Schreiber, 2014;Hellman & Schreiber, 2018; see also the recent survey of Föllmer & Weber, 2015), or they are based on an "operative" criterion such as an agent never going bankrupt when relying on an index of risk in deciding whether to accept a gamble (Foster & Hart, 2009; and see also Meilijson, 2009 , for a discussion of operative implication of Aumann & Serrano's index of risk). 3 We argue that risk is a multidimensional attribute that crucially depends on the investment problem.…”

“…The optimal certainty equivalent function has been studied in Hellman & Schreiber, 2018, in which an axiomatic analysis is used to define riskiness indices that are relevant to this decision function.…”

“…That is, f CE ((u, w) , g) is interpreted as the certain gain for which the decision maker is indifferent between obtaining this gain for sure and having an option to invest in the gamble g, when the agent is allowed to optimally choose his investment level in g. Observe that one can express the RHS in (2) in terms of f CA and obtain the following equivalent definition of f CE as the unique solution to the equation u (w + f CE ) = E u w + f CA ((u, w) , g) • g . The function f CE has been studied axiomatically in Hellman & Schreiber (2018).…”