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A wealth-requirement axiomatization of riskiness

Dean P. Foster, Sergiu Hart

2013
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Theoretical Economics
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We provide an axiomatic characterization of the measure of riskiness of gambles (risky assets) introduced by Foster and Hart (2009). The axioms are based on the concept of "wealth requirement." and Hart is constructive, in providing for each gamble the critical wealth level that separates "bad" investments (such as those leading to bankruptcy) from "good" ones (such as those leading to increasing wealth). 1 In the present paper, we provide an axiomatic approach to the Foster-Hart measure of
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... iness: we propose four basic axioms for a riskiness measure and show that the minimal function satisfying these axioms is precisely the Foster-Hart measure. Since "riskiness" does not appear to be a straightforward and obvious concept, one needs to have in mind a certain viewpoint and interpretation. The leading one that we propose is that of wealth requirement: the minimal wealth required to engage in a risky activity. Wealth requirements are common, for instance, in high-risk investments (such as hedge funds, in which one should not invest more than a certain proportion of one's wealth), in risky endeavors (such as getting a license for exploration of natural resources-e.g., oil and gas-or for building a large project-e.g., a new transportation system). After all, risks have to do with possible changes in wealth-whether gains or losses-and so it is natural to use the "wealth effects" to measure the riskiness. The first two axioms that we propose are standard (and are satisfied by the return, the spread, and many other objective measures): the Distribution axiom, which says that only the outcomes and their probabilities matter, and the Scaling axiom, which says, for instance, that doubling the gamble doubles its riskiness (which is measured in the same units as the outcomes). The fact that we are dealing with riskiness and wealth requirements is expressed in the other two axioms: the Monotonicity axiom, which says that decreasing some gains or increasing some losses must increase the wealth requirement, and the Compound Gamble axiom, which says that once the wealth effect is taken into account, the way the gamble is presented does not matter. Our result is that these axioms characterize the "critical wealth" for a certain class of von Neumann-Morgenstern utility functions, i.e., that wealth level where the decisionmaker is indifferent between accepting and rejecting the gamble. 2 Perhaps surprisingly, one of these functions turns out to be minimal for all gambles; that is, it bounds from below all wealth requirements for all gambles (one may thus refer to it as the critical critical wealth). This minimal wealth requirement is precisely the Foster-Hart measure of riskiness. In other words, any wealth requirement (that satisfies the axioms) must be at least as conservative as the one given by the Foster-Hart measure. 3 The paper is organized as follows. In Section 2, we present the formal model and the four axioms. The main result that the Foster-Hart measure is the minimal function satisfying the axioms is stated in Section 3. Our axioms characterize a family of riskiness measures, which turn out to be the critical wealth levels for a certain one-parameter family of utility functions (specifically: CRRA-γ with γ ≥ 1); see Section 4. In Section 5, 1 An alternative, "ordinal," approach-comparing gambles according to how often they are rejected by risk-averse decision-makers-is provided by Hart (2011): it yields precisely the two orders generated by the Aumann-Serrano and the Foster-Hart measures. 2 We emphasize that our axioms characterize not only these expected utility functions, but also the resulting "fixed point" where the certainty equivalent of the final wealth (which is the current wealth plus the gamble outcome) equals the current wealth. 3 For generalizations to sets of gambles and non-expected-utility models, see Michaeli (2012).

doi:10.3982/te1150
fatcat:gsfocl4zjbc5bdrciedyal2c2a