“Probability of risk” aversion is principally concerned with reactions to scaling up of probabilities of non-zero values of a non-positive random variable by a common factor. Decreasing probability-of-risk aversion is defined and shown to be equivalent to ordinary risk aversion. Implications of this for insurance are pointed out. The sort of scaling involved is the same as that involved in “self-protection,” and it is shown that, for any expenditure on self-protection, say x, a concave utility function will prefer a coinsurance policy, costing x, which leaves probabilities unchanged, but scales down loss amounts by the same proportion as probabilities are scaled under self-protection. Properties of several comparative concepts of decreasing risk aversion are established. Derivatives of the certainty equivalent (CE) are used to elucidate well-known comparative static results in models of expected utility maximization. Finally, the study proves that concavity of the CE implies convexity of the coefficient of absolute risk aversion and examines the role of curvature of the CE in exploring relationships between properties of risk vulnerability, properness, and standardness.
The notion of risk aversion was originally developed with reference to the Expected Utility model. de Finetti (1952), Pratt (1964) and Arrow (1965) associated the concavity of the von Neumann-Morgenstern utility function with some relevant aspects of the decision-maker’s preferences. In particular, risk aversion can be defined in terms of risk premium (i.e., the difference between the expected value and the certainty equivalent of a lottery). With reference to the EU model the risk premium is nonnegative for all lotteries if and only if the von Neumann-Morgenstern utility function is concave. However, with reference to the EU model, other relevant aspects of the preferences also depend on the concavity of the utility function: for instance, if we compare two lotteries of which one has been obtained from the other through mean preserving spreads, the less risky lottery is (weakly) preferred for all pairs of lotteries of this kind if and only if the von Neumann-Morgenstern utility function is concave. Moreover, the EU model does not imply that a randomization of lotteries matters (for instance, according to the EU model, a lottery whose consequences are a randomization of the outcomes of two equally preferred lotteries is indifferent to them), while the possibility that a decision-maker prefers not to be involved in an additional lottery could be considered as a kind of risk aversion. Taking into consideration more general models than the EU model, it is no longer true that ris
Summary: We study the risk-aversion behavior of an agent in the dynamic framework of consumption/investment decision making that allows the possibility of bankruptcy. Agent's consumption utility is assumed to be represented by a strictly increasing, strictly concave, continuously differentiable function in the general case and by a HARA-type function in the special case treated in the paper. Coefficients of absolute and relative risk aversion are defined to be the well-known curvature measures associated with the derived utility of wealth obtained as the value function of the agent's optimization problem. Through an analysis of these coefficients, we show how the change in agent's risk aversion as his wealth changes depends on his consumption utility and the other problem parameters, including the payment at bankruptcy. Moreover, in the HARA case, we can conclude that the agent's relative risk aversion is nondecreasing with wealth, while his absolute risk aversion is decreasing with wealth only if he is sufficiently wealthy. At lower wealth levels, however, the agent's absolute risk aversion may increase with wealth in some cases.
Since the classic paper by Friedman and Savage (1948), it has generally been accepted that the observed fact that individuals or firms' participate in unfair lotteries and other forms of unfair risk taking' may be explained by a section in the individual's utility function in which the individual shows risk preference rather than risk aversion. In their model, Friedman and Savage specify a utility function which is, in turn, concave, convex, and concave, thus allowing for simultaneous purchase of insurance (risk aversion) and participation in lotteries (risk preference). However, while their specified utility function is indeed capable of explaining observed behavior, it does seem to be rather unsatisfactory in that it is an ad hoc specification. It is the purpose of this paper to suggest a set of circumstances which give rise to a FriedmanSavage-type utility function. In particular, it is shown that when certain capital market imperfections exist the utility function defined over intermediate wealth should be distinguished from and may have different properties than the one defined on final wealth. Then, even if we accept the common assumption made in the literature that individuals are risk averse, that is, that their utility function is concave over final wealth, it is still possible that they participate in unfair gambling (and, of course, may also purchase insurance).
In this chapter we study the risk-aversion behavior of an agent in the dynamic framework of consumption/investment decision making that allows the presence of a subsistence consumption level and the possibility of bankruptcy. Agent’s consumption utility is assumed to be represented by a strictly increasing, strictly concave, continuously differentiable function in the general case and by a HARA type function in the special case treated in the chapter. Coefficients of absolute and relative risk aversion are defined to be the well-known curvature measures associated with the derived utility of wealth obtained as the value function of the agent’s optimization problem. In the HARA case the agent’s absolute risk aversion decreases with wealth if his wealth is greater than some boundary level, while at lower wealth levels it increases with wealth. We describe the dependence of this boundary on the value assigned to bankruptcy. Furthermore, the agent’s relative risk aversion in the HARA case inherits the monotonicity behavior from his consumption utility provided his wealth is greater than another boundary level. At smaller wealth levels, however, the relative risk aversion is increasing not only in the HARA case, but also in the general case. Finally, in the HARA case we describe the agent’s optimal investment policy in terms of his wealth for different values of problem parameters.
If an individual's utility of wealth function is strictly concave [u'(w) > 0 and u"(w) < 0], he will be averse to risk. Arrow has suggested two ways of representing the intensity of risk aversion at each value of wealth by measures of the local concavity of the utility function. The first measure, which may be called "absolute risk aversion," is defined as rl(w) = -u"(w)/u"(w); the second measure, "proportional risk aversion," is defined as r2(W) = -Wu"(W)/ u'(w) .1 Arrow [1] and Pratt [4] have justified the use of these functions as measure of risk
its measurement.!* Risk Aversion and Its Measurement The concept of risk aversion provides one of the … see that risk aversion is equivalent to the concavity of u(-) and that strict risk aversion is equivalent … Arrow-Pratt coefficient of absolute risk aversion at x is defined as r(x) = —u"(x)/u'(x). The Arrow—Pratt
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