Degree of downside risk aversion and self-protection

This paper shows that, identifying individuals with their utility functions, − v ‴ ( x ) / v ″ ( x ) ≤ − u ‴ ( x ) / u ″ ( x ) for all x implies that individual v ’s optimal choice of self-protection expenditure is larger than individual u’s, provided that marginal increases in self-protection expenditure from u’s optimal choice are mean-preserving. The result clarifies the relationship between self-protection and downside risk aversion and underscores the interpretation of − u ‴ ( x ) / u ″ ( x ) as a measure of the strength of u’s downside risk aversion relative to his own risk aversion because a mean-preserving increase in self-protection expenditure is shown to effect a special combination of a downside risk increase and a mean-preserving contraction.