Risk Vulnerability and the Tempering Effect of Background Risk
提出风险脆弱性概念，即加入不公平背景风险会使风险规避者对其他独立风险更规避，并证明其等价于不良风险不会因独立不公平风险而变得可取，且降低风险资产需求。
We examine in this paper a new natural restriction on utility functions, namely that adding an unfair background risk to wealth makes risk-averse individuals behave in a more risk-averse way with respect to any other independent risk. This concept is called risk vulnerability. It is equivalent to the condition that an undesirable risk can never be made desirable by the presence of an independent, unfair risk. Moreover, under risk vulnerability, adding an unfair background risk reduces the demand for risky assets. Risk vulnerability generalizes the concept of properness (individually undesirable, independent risks are always jointly undesirable) introduced by Pratt and Zeckhauser (1987). It implies that the two first derivatives of the utility function are concave transformations of the original utility function. Under decreasing absolute risk aversion, a sufficient condition for risk vulnerability is local properness, i.e. r'' ≥ r'r, where r is the Arrow-Pratt coefficient of absolute risk aversion.