Characterizations of risk aversion in cumulative prospect theory

In this paper, we investigate the necessary and sufficient conditions for a decision maker to be monotone risk averse and left-monotone risk averse, respectively, in cumulative prospect theory (CPT). Our results show that the decision maker is more pessimistic than greedy if she is either monotone or left-monotone risk averse, which is similar to that of Chateauneuf et al. (Econ Theory 25(3):649–667, 2005) in the rank-dependent expected utility model. Detailed examples are presented to illustrate the main theorems. With this work, we make a progress in the characterizations of risk aversion in CPT, which is essential in understanding the features of CPT and its applications in finance and insurance.

     

A note on additive risk measures in rank-dependent utility

This note proves that risk measures obtained by applying the equivalent utility principle in rank-dependent utility are additive if and only if the utility function is linear or exponential and the probability weighting (distortion) function is the identity.     

A decision theory for partially consonant belief functions

Partially consonant belief functions (pcb), studied by Walley, are the only class of Dempster-Shafer belief functions that are consistent with the likelihood principle of statistics. Structurally, the set of foci of a pcb is partitioned into non-overlapping groups and within each group, foci are nested. The pcb class includes both probability function and Zadeh’s possibility function as special cases. This paper studies decision making under uncertainty described by pcb. We prove a representation theorem for preference relation over pcb lotteries to satisfy an axiomatic system that is similar in spirit to von Neumann and Morgenstern’s axioms of the linear utility theory. The closed-form expression of utility of a pcb lottery is a combination of linear utility for probabilistic lottery and two-component (binary) utility for possibilistic lottery. In our model, the uncertainty information, risk attitude and ambiguity attitude are separately represented. A tractable technique to extract ambiguity attitude from a decision maker behavior is also discussed.     

The beta stochastic utility (β-SU)

In this article, a new model for decision making under uncertainty is presented. Here, we model human attitude toward risks to show that an individual estimate of the expected utility of a lottery follows a generalized Beta distribution with a random error that follows a similar distribution. An individual is said to maximize his stochastic utility when requested to present his preference between risky lotteries. Hypothetically, risky lotteries are those exhibiting wider ranges of rewards where human estimate will not be below the utility of the lowest reward nor above the highest of the lottery. The Beta distribution is bounded and complies to such intuitive preconditions with a variance depending on such bounds. The proposed model will overestimate/underestimate the expected utility of a lottery according to the lottery probability mass and individuals' risk attitudes. By such estimation, our model conforms to the fourfold choice pattern. The model also explains the violations present as inconsistencies in the expected utility theory, such as Allais paradox, common consequence effect, common ratio effect, and the violation of betweenness that can be found in the fourfold choice pattern. For the validation purposes, 13 datasets from literature were collected and tested. The β-SU model fits the data at least as good as other approaches such as the CPT/StEUT and presents higher prediction log-likelihoods and less sum of squared errors in most of the cases, a matter that supports the proposition that human estimates of the expected utility may be drawn out of a generalized Beta distribution.     

Axiomatic rank-dependent means

This paper provides an axiomatization for the implicit rank-dependent mean value, which class includes a number of mean values that have appeared in statistics, in utility theory and in the theory of income inequality measurement.     

Separable and additive representations of binary gambles of gains

Two approaches are taken to a new utility representation of binary gambles that is called “ratio rank-dependent utility.” Both are based on known axiomatizations of a ranked-additive representation of consequence pairs (x, y) in binary gambles (x, C; y) of gains with C held fixed and of a separable one of the special gambles (x, C; e), where e denotes the status quo. The axiomatized version imposes the condition of status-quo event commutativity to get a functional equation that leads to the result. The other assumes, but does not axiomatize, a separable representation of the (C; y) portion of the gamble. These assumptions lead to two difficult functional equations that are solved in the mathematical literature, but the former only under the assumption that the function is twice differentiable. Three behavioral conditions are shown to force this new utility representation to reduce to the standard rank-dependent utility one for gains. They are co-monotonic trade-off consistency, ranked bisymmetry, and segregation, the latter requiring the addition of an operation of joint receipt.     

A functional equation arising from ranked additive and separable utility

All strictly monotonic solutions of a general functional equation are determined. In a particular case, which plays an essential role in the axiomatization of rank-dependent expected utility, all nonnegative solutions are obtained without any regularity conditions. An unexpected possibility of reduction to convexity makes the present proof possible.

     

含有背景风险的双目标投资组合模型研究

投资者进行投资实践时无不面临着背景风险。绝大多数以均值方差为框架的投资组合并没有考虑背景风险,其效用在实际应用中容易受到背景风险的影响。本文在含有交易费用的双目标函数模型中引入背景风险,从是否含有背景风险和背景风险偏好度大小两方面对投资组合问题展开研究,并使用智能算法得到模型的最优解,对模型进行实证分析。实证结果表明:1)当背景风险收益为0时,含有背景风险的投资组合比不含有背景风险的投资组合更能反映真实的投资环境。2) 当背景风险收益不为0时,含有背景风险的投资组合比不含有背景风险的投资组合得到更高的收益。因此,考虑背景风险后投资组合的构建优于不考虑背景风险投资组合的构建。     

Robust optimal risk sharing and risk premia in expanding pools

We consider the problem of optimal risk sharing in a pool of cooperative agents. We analyze the asymptotic behavior of the certainty equivalents and risk premia associated with the Pareto optimal risk sharing contract as the pool expands. We first study this problem under expected utility preferences with an objectively or subjectively given probabilistic model. Next, we develop a robust approach by explicitly taking uncertainty about the probabilistic model (ambiguity) into account. The resulting robust certainty equivalents and risk premia compound risk and ambiguity aversion. We provide explicit results on their limits and rates of convergence, induced by Pareto optimal risk sharing in expanding pools.     

等级依赖效用下最优投资问题的扩展研究

本文在文献[https://ssrn.com/abstract=3135695]的基础上,去掉了等级依赖效用投资者的概率加权函数的连续性和单调增加的严格性,以及其原始效用函数的连续可微性.通过引入一般单调函数的广义逆函数以及凹函数的超微分,克服了分析上所带来的新的困难,证明了新模型最优解的存在性并给出其显式表达.     

On expected utility for financial insurance portfolios with stochastic dependencies

The effect of background risks as human capital, market risks and catastrophic events has been considered in the literature in different contexts. In this note, we consider financial insurance portfolios with insurable risks and one background risk (uninsurable financial asset), such that the random losses and the background risk depend on environmental parameters. We study how dependencies between the risks influence the expected utility of the portfolio’s wealth distribution under risk aversion, when the environmental parameters are random. Stochastic bounds for the expected wealth are given from modeling the dependence between the parameters by different notions. Similar results are given for multivariate portfolios with n groups and multivariate risk aversion, besides an expected utility comparison result for the minimum and the total portfolio’s wealth.     

On cross-risk vulnerability

We introduce the notion of cross-risk vulnerability to generalize the concept of risk vulnerability introduced by Gollier and Pratt [Gollier, C., Pratt, J.W. 1996. Risk vulnerability and the tempering effect of background risk. Econometrica 64, 1109–1124]. While risk vulnerability captures the idea that the presence of an unfair financial background risk should make risk-averse individuals behave in a more risk-averse way with respect to an independent financial risk, cross-risk vulnerability extends this idea to the impact of a non-financial background risk on the financial risk. It provides an answer to the question of the impact of a background risk on the optimal coinsurance rate and on the optimal deductible level. We derive necessary and sufficient conditions for a bivariate utility function to exhibit cross-risk vulnerability both toward an actuarially neutral background risk and toward an unfair background risk. We also analyze the question of the sub-additivity of risk premia and show to what extent cross-risk vulnerability provides an answer.     

Background risk,bivariate risk attitudes,and optimal prevention

This paper extends Eeckhoudt et al.’s (2012) results for precautionary effort to bivariate utility function framework. We establish an equivalence between the agent’s precautionary effort motive and the signs of successive cross-derivatives of the bivariate utility function. We show that the introduction (or deterioration) of an independent background risk induces more prevention to protect against wealth loss provided the individual exhibits correlation aversion of some given order. The conditions on the individual’s risk preferences are given to generate some specific prevention behaviors in the univariate framework with multiplicative risks. Our conclusion also indicates that an increase in the correlation between wealth risk and background risk leads to a reduction in optimal prevention.     

The fundamental theorem of mutual insurance

The essence of mutual insurance is the notion that re-distributing risk in a pool of risks is more beneficial than taking the risk alone. Interpreting ‘more beneficial’ as an increase in utility and considering sequences of exchangeable risks, we are able to formalize this notion from the policyholder’s perspective and demonstrate its validity for various alternative preference functionals (e.g., expected utility, Choquet expected utility, and distortion risk measures). To obtain this result, we exploit that for a sequence of exchangeable risks the corresponding sequence of arithmetical averages is a reversed martingale.We conclude that pooling risks is fundamental for understanding the mechanisms of insurance because it favourably affects the utility of policyholders, and we refer to this phenomenon as the ‘utility-improving effect of risk pooling’. Moreover, we demonstrate that the utility of the policyholder is (strictly) increasing with the size of the risk pool.     

A new foundation for the mean–variance analysis

In this paper, I re-examine how the mean–variance analysis is consistent with its traditional theoretical foundations, namely, stochastic dominance and the expected utility theory. Then I propose a simplified version of the coarse utility theory as a new foundation. I prove that, by assuming risk aversion and the normality of asset variables, the simplified model is well behaved; indifference curves are convex and the opportunity set is concave. Therefore, there exist global optimal portfolios in the market. Finally, I prove that decision-making in accordance with the simplified model is consistent with the mean–variance analysis.     

Precautionary paying for stochastic improvements under background risks

In a two-dimensional framework, we propose a general two-period decision model which extends the temporal precautionary saving and effort model. We relate the role of cross-prudence to the impact of background risks on paying for stochastic improvements of the future risk. We find that the effect of background risks introduced in the first period is consistent to signing cross derivatives of bivariate utility functions, which is independent of the type of stochastic improvements brought by additional paying; however, when the background risk occurs in the second period, that is not the case.     

Transitivity in the small and in the large for states-additive SSB utilities

Starting with induced state-conditional preferences for an additive and potentially state-dependent SSB utility model, we investigate the effects of transitive preferences on various super-sets of a set of available lottery acts. Sufficient richness of the act set forces transitive preferences to be fully linear; failing such richness, transitive but nonlinear preferences may obtain.     

Axiomatic utility theories with the betweenness property

This paper focuses on the betweenness property of expected utility theory. We provide an axiomatization of the class of betweenness-conforming utility theories. Subclasses of betweenness-conforming preferences are axiomatized with substitution axioms of intermediate generality. The latter axioms incorporate specifically the effects of replacing a certain outcome with a lottery that is indifferent to it. Our representation is applied to the second-price auction mechanism where we show that its demand-revelation property under expected utility is not robust with respect to the class of betweenness-conforming preferences.I would like to thank Larry Epstein, Itzhak Gilboa, Joe Harington, Edi Karni, Mark Machina, Roger Myerson, Zvi Safra, Fernando Saldanha, Itzhak Zilcha and Steve Zucker for their useful comments and helpful discussions. This is a revised version of an earlier working paper (Chew [5]). A simple version of semi-weighted utility was presented in an exchange of views session at the Second International Conference on the Foundations of Utility and Risk Theories in Venice, June 1984. Versions of this paper have been presented at the University of California, San Diego (November 1985), the University of Pennsylvania (October 1985), Johns Hopkins University (October 1984), University of Arizona (January 1985) and Tel Aviv University (December 1984 and May 1985). I also wish to acknowledge the partial financial support from the National Science Foundation SES 8213602.     

情绪影响下选时博弈模型及其最优策略

依据Quiggin的秩依期望效用理论研究经典选时博弈问题。通过引入可以刻画局中人在博弈中情绪状态的非线性决策权重函数,将RDEU有限策略博弈扩展到连续博弈,构建了RDEU选时博弈模型。基于Riccati微分方程的解法,求出博弈模型中局中人的最优策略。最后,通过数值仿真,分析了不同情绪状态对局中人博弈决策行为的影响。研究发现,情绪对混合策略意义下的局中人最优策略有着显著的影响,在乐观情绪状态下,局中人对混合策略极易产生自信和较高的信任倾向,表现出"风险爱好者"行为;在悲观情绪状态下,局中人往往对混合策略缺乏自信和信任,表现出“风险厌恶者”行为。     

Anticipated utility: A measure representation approach

This paper presents axioms which imply that a preference relation over lotteries can be represented by a measure of the area above the distribution function of each lottery. A special case of this family is the anticipated utility functional. One additional axiom implies this theory. This functional is then extended for the case of vectorial prizes.The author is grateful to Chew Soo Hong, Larry Epstein, Joe Ostroy, Joel Sobel, Peter Wakker, and Menahem Yaari for their comments.