We examine in this paper a new natural restriction on utility function
s, namely that adding an unfair background risk to wealth makes risk-a
verse 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. Moreove
r, under risk vulnerability, adding an unfair background risk reduces
the demand for risky assets. Risk vulnerability generalizes the concep
t of properness (individually undesirable, independent risks are alway
s jointly undesirable) introduced by Pratt and Zeckhauser (1987). It i
mplies that the two first derivatives of the utility function are conc
ave 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 coeffi
cient of absolute risk aversion.