Yaari's Dual Theory Research Articles

Ordering Higher Risks in Yaari's Dual Theory

Ordering Higher Risks in Yaari's Dual Theory

The optimal payoff for a Yaari investor

Yaari's dual theory of choice under risk is the natural counterpart of expected utility theory. While optimal payoff choice for an expected utility maximizer is well studied in the literature, less is known about the optimal payoff for a Yaari investor. We perform a fairly general analysis and derive optimal payoffs in a variety of relevant cases. As a main contribution, we provide the optimal payoff for a Yaari investor under a variance constraint; thus, extending mean–variance optimization to distorted expectation–variance optimization. We also derive the optimal payoff for an investor who aims to outperform an external benchmark under the requirement that the payoff stays in the neighbourhood of this benchmark.

Preference Robust Optimization for Choice Functions on the Space of CDFs

In this paper, we consider decision-making problems where the decision maker's (DM's) utility/risk preferences are ambiguous but can be described by a general class of choice functions defined over the space of cumulative distribution functions (CDFs) of random prospects. These choice functions are assumed to satisfy two basic properties: (i) monotonicity w.r.t. the order on CDFs and (ii) quasiconcavity. We propose a maximin preference robust optimization (PRO) model where the optimal decision is based on the robust choice function from a set of choice functions elicited from available information on the DM's preferences. The current univariate utility PRO models are fundamentally based on Von Neumann--Morgenstein's (VNM's) expected utility theory. Our new robust choice function model effectively generalizes them to one which captures common features of VNM's theory and Yaari's dual theory of choice. To evaluate our robust choice functions, we characterize the quasiconcave envelope of $L-$Lipschitz functions of a set of points. Subsequently, we propose two numerical methods for the DM's PRO problem: a projected level function method and a level search method. We apply our PRO model and numerical methods to a portfolio optimization problem and report test results.

Concave/convex weighting and utility functions for risk: A new light on classical theorems

This paper analyzes concave and convex utility and probability distortion functions for decision under risk (law-invariant functionals). We characterize concave utility for virtually all existing models, and concave/convex probability distortion functions for rank-dependent utility and prospect theory in complete generality, through an appealing and well-known condition (convexity of preference, i.e., quasiconcavity of the functional). Unlike preceding results, we do not need to presuppose any continuity, let be differentiability.An example of a new light shed on classical results: whereas, in general, convexity/concavity with respect to probability mixing is mathematically distinct from convexity/concavity with respect to outcome mixing, in Yaari's dual theory (i.e., Wang's premium principle) these conditions are not only dual, as was well-known, but also logically equivalent, which had not been known before.

Generalized Shortfall Risk Measure Based on Insurance Premium

Utility-based shortfall risk (SR) measure proposed by (F\”ollmer and Schied, 2002) has been well studied in risk management and finance. In this paper, we revisit the concept from insurance premium perspective. We show under some moderate conditions that the indifference equation-based insurance premium calculation can be equivalently formulated as an optimization problem similar to the definition of SR and subsequently call the premium functional as a generalized shortfall risk measure (GSR). We then use the latter formulation to investigate the properties of the GSR with a focus on the case that the preference functional is a distorted expected value function based on prospect theory. Specifically, we exploit Weber's methods and techniques (Weber, 2006) for characterization of the shortfall risk measure to derive a relationship between properties of GSR based on the prospect theory and the underlying value function and weighting function in terms of convexity/concavity and positive homogeneity. We also investigate the GSR as a functional of cumulative distribution function of random loss/liability and derive local and global Lipschitz continuity of the function under Wasserstein metric, a property which is related to statistical robustness of the GSR. The results cover the premium risk measures based on the von Neumann-Morgenstern's expected utility and Yaari's dual theory of choice as special cases. Finally, we propose a computational scheme for calculating GSR.

The Optimal Payoff for a Yaari Investor

Yaari's dual theory of choice is the natural counterpart of expected utility theory. While the optimal payoff choice for an expected utility maximizer is well studied in the literature, substantially less is known about the optimal payoff for a Yaari investor. In the first part of the paper, we study well-posedness of the optimal payoff problem for a Yaari investor and derive explicit solutions in a variety of relevant cases. In the second part of the paper, we extend the results to derive optimal payoffs for Yaari investors who wish to beat an external benchmark.

Optimal Payoff under the Generalized Dual Theory of Choice

We consider portfolio optimization under a preference model in a single-period, complete market. This preference model includes Yaari's dual theory of choice and quantile maximization as special cases. We characterize when the optimal solution exists and derive the optimal solution in closed form when it exists. The payoff of the optimal portfolio is a digital option: it yields an in-the-money payoff when the market is good and zero payoff otherwise. When the initial wealth increases, the set of good market scenarios remains unchanged while the payoff in these scenarios increases. Finally, we extend our portfolio optimization problem by imposing a dependence structure with a given benchmark payoff and derive similar results.

Risk apportionment: The dual story

By specifying new model free preferences towards simple nested classes of lottery pairs, we develop the dual story to stand on equal footing with that of (primal) risk apportionment. The dual story provides an intuitive interpretation, and full characterization, of dual counterparts of such concepts as prudence and temperance. The direction of preference between these nested classes of lottery pairs is equivalent to signing the successive derivatives of the probability weighting function within Yaari's (1987) dual theory. We explore implications of our results for optimal portfolio choice and show that the sign of the third derivative of the probability weighting function may be naturally linked to a self-protection problem.

On the Construction of Optimal Payoffs

The quantile formulation for optimal portfolio selection problems under increasing law-invariant objectives allows to reduce any such problem to an optimization problem on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to solve in a unified way several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in Yaari's dual theory of choice. Extending previous work we also derive a reduction result in general form when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. Finally, we identify market conditions which ensure attainability of the optimal payoff by means of a static portfolio of puts and calls.

To Vaccinate or to Procrastinate? That is the Prevention Question.

Invoking Yaari's dual theory, we develop a model of individual vaccination decisions that incorporates quasi-hyperbolic discounting, risk aversion, and information. We test the resulting hypotheses for the flu season 2010/2011 using a representative German data set. We find a significant impact of time preferences on immunization decisions. The impact of the discount factor is significantly negative for exponential discounters. While present-biased individuals' demand for vaccination is not statistically different from the one of exponential discounters, future-biased individuals have a significantly higher probability to vaccinate. Stratification by gender reveals that these effects are entirely driven by men. That is, time preferences have no explanatory power for the vaccination decisions of women. This also applies to risk aversion, where more risk aversion implies a significantly higher probability to vaccinate for men but not women. All information measures turn out significant. Well-informed individuals have a much higher propensity to vaccinate than poorly informed individuals. If policy makers aim at improving immunization rates, then our results suggest that public policy should concentrate on providing easily accessible and concise information on the flu and the flu shot. Our results on time preferences and risk preferences imply a rather inactive role for public policy. Copyright © 2015 John Wiley & Sons, Ltd.

Risk Aversion in Travel Mode Choice with Rank-Dependent Utility

Using 2004 stated preference data on travel mode collected in the Zürich area, different parametric specifications of the rank-dependent utility function in a logit mixture model show that commuters are weakly averse to small-time losses. The results also justify Yaari's dual theory of choice under risk, that the utility function is linear on outcomes but that the perception of corresponding probabilities is biased. For leisure travel, the travelers are risk neutral to small losses of time.

Weak Insider Trading and Behavioral Finance

In this paper, we study the optimal portfolio selection problem for weakly informed traders in the sense of Baudoin [Stochastic Process. Appl., 100 (2002), pp. 109–145]. Apart from expected utility maximizers, we consider investors with other preference paradigms. In particular, we consider agents following cumulative prospect theory as developed by Tversky and Kahneman [J. Risk Uncertainty, 5 (1992), pp. 297–323] as well as Yaari's dual theory of choice [Econometrica, 55 (1987), pp. 95–115]. We solve the corresponding optimization problems, in both noninformed and informed case, i.e., when the agent has an additional weak information. Finally, comparison results among investors with different preferences and information sets are given, together with explicit examples. In particular, the insider's gain, i.e., the difference between the optimal values of an informed and a noninformed investor, is explicitly computed.

Dual theory of choice with multivariate risks

We propose a multivariate extension of Yaari's dual theory of choice under risk. We show that a decision maker with a preference relation on multidimensional prospects that preserves first order stochastic dominance and satisfies comonotonic independence behaves as if evaluating prospects using a weighted sum of quantiles. Both the notions of quantiles and of comonotonicity are extended to the multivariate framework using optimal transportation maps. Finally, risk averse decision makers are characterized within this framework and their local utility functions are derived. Applications to the measurement of multi-attribute inequality are also discussed.

Valuing future cash flows with non separable discount factors and non additive subjective measures: conditional Choquet capacities on time and on uncertainty

We consider future cash flows that are contingent both on dates in time and on uncertain states. The decision maker (DM) values the cash flows according to its 2 decision criterion: Here the payoffs' expectation with respect to a capacity measure. The subjective measure grasps the DM's behaviour in front of the future, in the spirit of de Finetti's (1930) and of Yaari's (1987) Dual Theory in the case of risk. Decomposition of the criterion into two criteria that represent the DM's preferences on uncertain payoffs and time contingent payoffs are derived from Ghirardato's (1997) results. Conditional Choquet integrals are defined by dynamic consistency requirements and conditional capacities are derived, under some conditions on information. In contrast with other models referring to dynamic consistency, ours doesn't collapse into a linear one because it violates a weak version of consequentialism.

Accident Liability with Rank Dependence

KritV Kritische Vierteljahresschrift für Gesetzgebung und Rechtswissenschaft , Seite 160 - 171

Tail Risk and pK-Tail Risk

This paper discusses the notion of tail risk, and the ability of a tail risk measure to reflect this kind of risk. In particular, Yamai and Yoshiba's (2001, 2002) notion of strict risk measure tail risk is discussed and linked with a different notion of tail risk, the pK-tail risk, which is the risk associated with the probability measure conditional on the event that the losses are at least as large as K. A subset of pK-tail risk measures that are free of strict risk measure tail risk is introduced. These notions are then extended to Yaari's (1987) dual theory and the distorted risk measures framework.

Yaari's dual theory without the completeness axiom

This note shows how Yaari (1987)'s dual theory of choice under risk naturally extends to the case of incomplete preferences. This also provides an axiomatic characterization of a large and widely studied class of stochastic orders used to rank the riskiness of random variables or the dispersion of income distributions (including, e.g., second order stochastic dominance, dispersion, location independent riskiness, see Chateauneuf, Cohen, and Meilijson, 1997).

A Unified Approach to Generate Risk Measures

The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation: E [φ (S, π)] = αE [v (S)].For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

A Unified Approach to Generate Risk Measures

The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation: E [φ (S, π)] = αE [v (S)].For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

Risk Measures and Theories of Choice

ABSTRACTWe discuss classes of risk measures in terms both of their axiomatic definitions and of the economic theories of choice that they can be derived from. More specifically, expected utility theory gives rise to the exponential premium principle, proposed by Gerber (1974), Dhaene et al. (2003), whereas Yaari's (1987) dual theory of choice under risk can be viewed as the source of the distortion premium principle (Denneberg, 1990; Wang, 1996). We argue that the properties of the exponential and distortion premium principles are complementary, without either of the two performing completely satisfactorily as a risk measure. Using generalised expected utility theory (Quiggin, 1993), we derive a new risk measure, which we call the distortion-exponential principle. This risk measure satisfies the axioms of convex measures of risk, proposed by Föllmer & Shied (2002a,b), and its properties lie between those of the exponential and distortion principles, which can be obtained as special cases.