Expected utility with ambiguous probabilities and ?irrational? parameters

Neoclassical economists use expected utility theory to explain, predict, and prescribe choices under risk, that is, choices where the decision-maker knows – or at least deems it suitable to act as if she knows – the relevant probabilities. Expected utility theory has been subject to both empirical and conceptual criticism. This chapter reviews expected utility theory and the main criticism it has faced. It ends with a brief discussion of subjective expected utility theory, which is the theory neoclassical economists use to explain, predict, and prescribe choices under uncertainty, that is, choices where the decision-maker cannot act on the basis of objective probabilities but must instead consult her own subjective probabilities.

Muth’s Rational Expectations Hypothesis, That “…The Subjective Probability Distributions Are Distributed Around an Objective Probability Distribution, for a given Information Set…” is an Oxymoron. It Can Be Corrected by Amending It to Read “…The Subjective Probability Distributions Are Distributed Around a Subjective Consensus Probability Distribution, for a given Information Set…”

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.

Optimal Expected Utility Risk Measures

This paper introduces optimal expected utility (OEU) risk measures, investigates their main properties and puts them in perspective to alternative risk measures and notions of certainty equivalents. By taking the investor’s point of view, OEU maximizes the sum of capital available today and the certainty equivalent of capital in the future. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU is more sensitive than value at risk and average value at risk with respect to changes of the probability of a financial loss.

In this paper, we present a brief version of de Finetti-Ramsey's subjective probability theory and provide a rigorous yet intuitively plausible explanation of expected utility using elementary mathematics.In a final section, we take up the case of some "Paradoxes in Expected Utility Theory" and try to reconcile them with the help of subjective probabilities.

Introduction In deriving the Mean-Variance (M-V) rule and the Capital Asset Pricing Model (CAPM), one assumes explicitly or implicitly normal distributions of returns and risk aversion. With these assumptions, it emerges that the variance of the portfolio's return is the measure of the investment's risk. In Chapter 8, we saw that although normality of returns is statistically rejected, the economic loss from assuming normality when the distributions are actually not normal is relatively small. Moreover, in Chapter 4, we saw that even if the distributions of returns are not normal, the M-V rule can be employed as an approximation to expected utility, as long as risk aversion is assumed. Furthermore, it was shown empirically in Chapter 4 that the choices made by the M-V rule are almost fully correlated with the choices made by direct expected utility maximization for a wide set of risk-averse utility function. Considering the relatively small economic loss involved as a result of the normality assumption, in this chapter we assume that the M-V and the CAPM are derived based on this normality assumption, within the expected utility framework, rather than as an approximation to expected utility. However, even in this framework, as we saw in Chapters 9 and 10, the M-V and the CAPM are questionable for the following reasons: By the Prospect Theory (PT), people maximize a value function, which is defined on change of wealth (gains and losses) rather than on total wealth; by expected utility, the M-V rule and the CAPM are defined on total wealth. To derive the M-V rule and the CAPM, risk aversion is needed, whereas by PT, the value function has a risk-seeking segment. The risk- seeking segment of preference is steeper than the risk-aversion segment, implying loss aversion . Adopting the modified version of the PT, called Cumulative Prospect Theory (CPT), people employ cumulative decision weights . Thus, the cumulative distribution of returns is transformed, and in calculating the expected value (utility), the transformed distribution rather than the original distribution is employed. This seems to be a “deathblow” to the M-V rule and the CAPM because even if the original distribution of returns is normal, generally the transformed distribution is not. Moreover, although the normal distribution is symmetric, implying that skewness is equal to zero, the transformed distribution may be skewed, thus relying only on the mean, and variance may be misleading because there is ample evidence that investors like positive skewness and dislike negative skewness. For the preceding reasons, it would seem that the CPT paradigm and the M-V rule plus the CAPM paradigm cannot coexist. Moreover, the fact that Markowitz and Sharpe won the Noble Prize in Economics in 1990 for developing the M-V rule and the CAPM, and Kahneman won the Noble Prize in Economics in 2002 for introducing the PT, emphasizes the seeming contradiction between these two paradigms and makes it even more uncomfortable for researchers. If one adopts PT, it seems the M-V and the CAPM should be abandoned. If one employs the M-V and the CAPM, the important evidence regarding subjects’ choices, as clearly shown in PT experiments, are ignored.

9 - Prospect theory, asset pricing, and market dynamics

Expected utility theory, and the Bayesian probability theory on which it is based, form the normative basis of most work in medical decision analysis. Recent work in the psychology of judgments and decisions indicates that people do not conform to the axioms of this theory and that these deviations occur in clinical reasoning as well as in the psychology laboratory. At issue is what to do now. The authors argue that the important next steps lie at the interface between descriptive, prescriptive, and normative accounts, all of which affect each other. They point to examples in which the simplest application of supposedly normative theory seems inappropriate, and suggest ways in which the tension between normative and descriptive models may be resolved.

Recent work on risky choice modelling has sought to address the shortcomings of expected utility theory (EUT) by using non-expected utility theoretic (non-EUT) approaches. However, to date these approaches have been merely tested on stated choice data which is flexible and cheap. In this study, we empirically investigate the feasibility and validity of non-EUT approaches in a revealed preference (RP) context in which travel time distribution is extracted from observed historical travel time data, and subsequently present systematic comparisons between EUT, weighted utility theory, rank-dependent expected utility theory, and prospect theory (PT). The empirical evidence indicates that each non-EUT model has important behavioural insights to offer, moreover, EUT as well as non-EUT models can be applied to the RP context. However, the EUT and non-EUT model fits are generally similar with only PT providing a marginally improved model fit over EUT. The key findings presented in this study reinforce the importance of exploring non-EUT models within a revealed preference context before they can be applied reliably to modelling risky choices in the real world.

To quantify physicians’ risk and time preferences and explore the association between the preferences and their willingness to detect and disclose secondary findings (SFs) derived from genome-scale sequencing. We designed a web-based survey incorporating a multiple price list (MPL) as the instrument for risk and time preference measurement. The estimation was under the expected utility theory (EUT) and rank dependent utility (RDU) frameworks, respectively. The isoelastic and power function utility models were applied. We received responses from 87 physicians, among whom 46 completed the questionnaire (a completion rate of 52.9%). We observed positive risk-aversion coefficients under EUT (0.33, 95% CI 0.15–0.51) and RDU (0.51, 95% CI 0.32–0.71), suggesting that physicians were generally risk-averse. Respondents were likely to underestimate probabilities of low to moderate levels, and slightly overestimate high-level probabilities. Physicians supporting the detection and disclosure of SFs had a larger risk-aversion coefficient and a smaller discounting parameter than the non-supporters, suggesting that they were more risk-averse and discounted future utility less. Assuming heterogeneous risk perception, we found respondents underestimated low/moderate risk and slightly overestimated high risk. This study indicates that physicians who are risk-averse and discount future utility slightly are willing to detect and return SFs. The findings contribute to the debate surrounding SF disclosure and generate implications for shared decision-making in clinical genome-scale sequencing. • Physicians' risk and time preferences influenced their decisions on genomic SFs disclosure. • Physicians favouring SFs disclosure were more risk-averse than those opposing such detection and disclosure. • Physicians with lower time discounting rates showed greater willingness to return SFs to patients. • Physicians tended to underestimate low to moderate levels and slightly overestimate higher probabilities.

This paper discusses an alternative explanation for the empirical findings contradicting the positive relationship between risk (variance) and reward (expected return). We show that these contradicting results might be due to the false definition of risk-perception, which we correct by introducing Expected Downside Risk (EDR). The EDR parameter, similar to the Expected Shortfall or Conditional Value-at-Risk, measures the tail risk, however, fits and better explains the utility perception of investors. Our results indicate that when using the EDR as risk measure, both the positive and negative relationship between expected return and risk can be derived under standard conditions (e. g. expected utility theory and positive risk-aversion). Therefore, no alternative psychological explanation or additional boundary condition on utility theory is required to explain the phenomenon. Furthermore, we show empirically that it is a more precise linear predictor of expected return than volatility, both for individual assets and portfolios.

Bayesian statistical techniques are applicablewhen the information anduncertainty with respect to the parameters or hypotheses in question can be expressed by a probability distribution. This prior probability is also the focus of most of the criticism against the Bayesian school. My starting point is to join the critics in attacking a certain aspect of the prior probability: The probability attached to an uncertain event does not reflect the heuristic amount of information that led to the assignment of that probability. For example, when the information on the occurrence of two events is symmetric they are assigned equal prior probabilities. If the events are complementary the probabilities will be 1/2, independently of whether the symmetric information is meager or abundant.

Chapter 11 studies the second phase of the debate on expected utility theory (EUT), which commenced in May 1950, when Paul Samuelson, Leonard J. Savage, Jacob Marschak, Milton Friedman, and William Baumol initiated an intense exchange of letters. These economists argued about the exact assumptions underlying EUT, quarreled over whether these assumptions are compelling requisites for rational behavior under risk, and debated the nature of the cardinal utility function u featured in EUT. This correspondence modified the views of all five economists and transformed Samuelson into a supporter of EUT. In a prominent conference in Paris in May 1952, Friedman, Savage, Marschak, and Samuelson advocated EUT in the face of attacks from Maurice Allais and other opponents of the theory. The Paris conference and the publication of an Econometrica symposium on EUT in October 1952 marked the emergence of EUT as the mainstream economic model of decision-making under risk.

This paper proposes a risk measure for a portfolio of European-style derivative securities over a fixed time horizon under the Black–Scholes economy. The proposed risk measure is scenario-based along the same line as [3]. The risk measure is constructed by using the risk-neutral probability ([Formula: see text]-measure), the physical probability ([Formula: see text]-measure) and a family of subjective probability measures. The subjective probabilities are introduced by using Girsanov's theorem. In this way, we provide risk managers or regulators with the flexibility of adjusting the risk measure according to their risk preferences and subjective beliefs. The advantages of the proposed measure are that it is easy to implement and that it satisfies the four desirable properties introduced in [3], which make it a coherent risk measure. Finally, we incorporate the presence of transaction costs into our framework.