An Economic Index of Relative Riskiness

Abstract

In their seminal works, Arrow (1965) and Pratt (1964) defined two aspects of risk aversion: absolute risk aversion and relative risk aversion. Based on their definitions, we define two aspects of risk: absolute risk and relative risk. We consider situations in which, by making an investment, an agent exchanges a certain amount of wealth w by a random distributed level of wealth w. In such situations, we define absolute risk as the riskiness of a gamble that is distributed as w −w, and relative risk as the riskiness of a security that is distributed as w/w. We measure absolute risk by the Aumann and Serrano (2008) index of riskiness and relative risk by an equivalent index that we develop in this paper. The two concepts of risk do not necessarily agree on which one of two investments is riskier, and hence they capture two different aspects of risk.

Comparing Local Risks by Acceptance and Rejection

“…Moreover, we show that several measures of riskiness defined on ordinary assets that are compatible with the acceptance dominance order coincide with our indices in the continuous‐time setup. For instance, the Aumann and Serrano () “Economic Index of Riskiness,” the Foster and Hart () “Operational Measure of Riskiness,” and the Schreiber () “Economic Index of Relative Riskiness” all coincide with our indices of riskiness in the continuous‐time setup.…”

“…Aumann and Serrano () characterize their index of riskiness of additive gambles by two axioms, duality and homogeneity, and Schreiber () characterizes an index of riskiness of multiplicative gambles based on similar axioms. We take a similar approach and characterize our measures of local riskiness by similar axioms.…”

“…Another interesting measure of riskiness of multiplicative gambles was proposed by Schreiber (). We refer here to the adjusted index of Schreiber (), , which is defined as follows. Given a multiplicative gamble r , …”

“…We focus on investments with infinitely short investment time horizons. Following Schreiber (), we distinguish between absolute and relative risks that arise from investing in such securities. The terms absolute and relative relate to the wealth of investors.…”

A wealth-requirement axiomatization of riskiness

“…24 In fact, Multiplicative Compounding can be interpreted as compounding the relative returns. 25 However, they are not equivalent in the Aumann-Serrano framework; see Schreiber (2012). γ = 2) and so K(g) = R 2 (g) for all g; i.e., K is the critical wealth for the CRRA-2 utility u 2 (x) = −1/x.…”