Risk seeking with diminishing marginal utility in a non-expected utility model (original) (raw)
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Economic Theory, 2005
This paper studies monotone risk aversion, the aversion to monotone, meanpreserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f . Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel & Lehmann in Non-parametric Statistics. We present a characterization of the pairs (u; f ) of monotone risk averse decision makers, based on an index of greediness G u of the utility function u and an index of pessimism P f of the probability perception function f : the decision maker is monotone risk averse if and only if P f¸Gu . The index of greediness (non-concavity) of u is the supremum of u 0 (x)=u 0 (y) taken over y · x. The index of pessimism of f is the in¯mum of 1¡f(v) 1¡v = f(v) v taken over 0 < v < 1. Thus, G u¸1 , with G u = 1 i® u is concave. If P f¸Gu then P f¸1 , i.e., f is majorized by the identity function. Since P f = 1 for Expected Utility maximizers, P f¸Gu forces u to be concave in this case; thus, the characterization of risk aversion as P f¸Gu is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P f = 1.
Risk-aversion concepts in expected- and non-expected-utility models
Insurance: Mathematics and Economics, 1996
The non-expected-utility tbeories of decision under risk have fovond the appearance of new notions of incieasing risk like ntonotone increasing risk (based on the notion of comonotonic random vahablcs) or new notions of risk aversion like averaion to monotone increasing risk, in better agreement witb tbese new theories. After a survey of atl tbc possible notions of increasing risk and of risk aversion and tbeir intrinsic de^nitions, we show tbat cODtivy to expected-utility tbeory where all the notions of risk avenion bave the same characterization (u concave), in the ftunework of rank-dependent expected utility (one of the most well known of the non-expectedutility models), tbe characterizations of all these notions of risk aversion are different. Moreover, we sbow tbat, even in tbe expected-utility framcworic, the new notion of monotone increasing risk can give better answers to some pn^lems of comparative sutics sucb as in ponfoUo cboice or in partial insurance. This new notion also can suggest more intuitive approaches to inequalities measurement.
Risk Aversion in the Small and in the Large under Rank-Dependent Utility
arXiv: Mathematical Finance, 2015
Under expected utility the local index of absolute risk aversion has played a central role in many applications. Besides, its link with the "global" concepts of the risk and probability premia has reinforced its attractiveness. This paper shows that, with an appropriate approach, similar developments can be achieved in the framework of Yaari's dual theory and, more generally, under rank-dependent utility.
A μ-σ-Risk Aversion Paradox and Wealth Dependent Utility
Journal of Risk and Uncertainty, 2001
We report a surprising property of μ-σ-preferences: the assumption of nonincreasing relative risk aversion implies the optimal portfolio being riskless. We discuss a solution of that paradox using wealth dependent utility functions in detail. Using the revealed preference theory we show that (general, i.e. not necessary μ-σ) wealth dependent utility functions can be characterized by Wald's axiom.
Characterization of left-monotone risk aversion in the RDEU model
Insurance: Mathematics and Economics, 2012
We extend the characterization of the left-monotone risk aversion developed by Ryan (2006) to the case of unbounded random variables. The notion of weak convergence is insufficient for such an extension. It requires the solution of a host of delicate convergence problems. To this end, some further intrinsic properties of the location independent risk order are investigated. The characterization of the rightmonotone risk aversion for unbounded random variables is also mentioned. Moreover, we remove the gap in the proof of the main result in Ryan (2006).
Journal of Mathematical Economics, 2004
This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant to the Quiggin-Yaari Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitz's Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose.
All investors are risk averse expected utility maximizers
2013
Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer with an explicitly derived concave non-decreasing utility function. This result enables us to infer the utility and risk aversion of agents from their investment choice in a non-parametric way. We relate the property of decreasing absolute risk aversion (DARA) to distributional properties of the terminal wealth and of the financial market. Specifically, we show that DARA is equivalent to a demand for a terminal wealth that has more spread than the opposite of the log pricing kernel at the investment horizon.
Expected utility theory and inner and outer measures of loss aversion
Journal of Mathematical Economics, 2016
We introduce a weak rank dependent utility (RDU) model, with one extra parameter compared to the canonical expected utility (EUT) model, which makes many of the same predictions as cumulative prospect theory (CPT). The model extends a set of nonconvex preferences to its maximal inner convex subset, satisfies stochastic dominance principles, resolves the Allais paradox, predicts CPT 4-fold pattern of risk attitudes, and characterizes reference dependent preferences. Unlike extant RDU models that transform probability weighting functions, our model transforms ranked choice sets while leaving objective probabilities intact. Cumulative prospect theory's (CPT) loss aversion index is a special case of the interior solution for the extra parameter for unconstrained utility maximization, and it is driven by tail probabilities in our model. We provide several examples to show how popular formulae for the loss aversion index can be classified into inner and outer measures of loss aversion via an approximate Radon-Nikodym formulation of the model. This resolves sources of disparity in estimating the loss aversion index with experimental data. We show that under extant approaches, the loss aversion index is best estimated by a mixture of inner and outer measures of itself. Furthermore, we identify a CPT paradox: The utility loss aversion index is unmeasurable under CPT nonexpected utility framework for mixed lotteries; but measurable for same under the expected utility paradigm adapted to our model.
Risk aversion under preference uncertainty
Finance Research Letters, 2012
We show that if an agent is uncertain about the precise form of his utility function, his actual relative risk aversion may depend on wealth even if he knows his utility function lies in the class of constant relative risk aversion (CRRA) utility functions. We illustrate the consequences of this result for optimal asset allocation: poor agents that are uncertain about their risk aversion parameter invest less in risky assets than wealthy investors with identical risk aversion uncertainty.
Risk aversion in the theory of expected utility with rank dependent probabilities
Journal of Economic Theory, 1987
Expected utility with rank dependent probability theory is a model of decisionmaking under risk where the preference relations on the set of probability distributions is represented by the mathematical expectation of a utility function with respect to a transformation of the probability distributions on the set of outcomes. This paper defines, based on Gateaux differentiability, measures of risk aversion for such preferences which characterize the relation "more risk averse" and applies these measures to the analysis of unconditional and conditional portfolio choice problems. Journal of Economic, Liter&we Classification Numbers: 026,521.