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Ambiguity, risk and asset returns in continuous time

01 Jul 2000-Research Papers in Economics (University of Rochester - Center for Economic Research (RCER))-
TL;DR: In this article, a continuous-time intertemporal version of multiple-priors utility, where aversion to ambiguity is admissible, is proposed for a representative agent asset market setting, which delivers restrictions on excess returns that admit interpretations reflecting a premium for risk and a separate premium for ambiguity.
Abstract: Existing models in stochastic continuous-time settings assume that beliefs are represented by a probability measure. As illustrated by the Ellsberg Paradox, this feature rules out a priori any concern with ambiguity. This paper formulates a continuous-time intertemporal version of multiple-priors utility, where aversion to ambiguity is admissible. When applied to a representative agent asset market setting, the model delivers restrictions on excess returns that admit interpretations reflecting a premium for risk and a seperate premium for ambiguity.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors propose a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility.
Abstract: We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

1,475 citations

Journal ArticleDOI
TL;DR: In this paper, a Benchmark Resource Allocation Problem with Model Misspecification and Robust Control Problems is discussed. But the problem is not addressed in this paper, and the following sections are included:
Abstract: The following sections are included:IntroductionA Benchmark Resource Allocation ProblemModel MisspecificationTwo Robust Control ProblemsRecursivity of the Multiplier FormulationTwo Preference OrderingsRecursivity of the Preference OrderingsConcluding Remarks

1,239 citations

Journal ArticleDOI
TL;DR: In this article, the authors characterize the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that, for all acts f and g,
Abstract: We characterize, in the Anscombe–Aumann framework, the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that, for all acts f and g, . The function u represents the decision maker's risk attitudes, while the index c captures his ambiguity attitudes. These preferences include the multiple priors preferences of Gilboa and Schmeidler and the multiplier preferences of Hansen and Sargent. This provides a rigorous decision-theoretic foundation for the latter model, which has been widely used in macroeconomics and finance.

1,014 citations

Journal ArticleDOI
TL;DR: This paper axiomatizes an intertemporal version of multiple-ptiors utility that is consistent with a rich set of possibilities for dynamic behavior under ambiguity and argues that dynamic consitency is intuitive in a wide range of situations.

900 citations


Cites background from "Ambiguity, risk and asset returns i..."

  • ...8) is provided in [7], where it is argued that it permits a three-way separation between the two noted aspects of preference and attitudes towards ambiguity....

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  • ...1) is essentially that adopted in [13], though without axiomatic foundations; a continuous-time counterpart is formulated in [7]....

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  • ...where P is the agent’s set of priors over the state space and the s-algebra Ft represents information available at time t:(1) An essential feature is that P is restricted by the noted axioms to satisfy not only the regularity conditions for sets of priors in the atemporal model, but also a property that (following [7]) we call rectangularity....

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  • ...The noted backward recursion underlies the dynamic consistency of preference, which in turn delivers tractability as demonstrated in [7,11,13]....

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Journal ArticleDOI
TL;DR: For example, the authors show that ambiguity-averse investors tend to take a worst-case assessment of quality when processing news of uncertain quality, and as a result, they react more strongly to bad news than to good news.
Abstract: When ambiguity-averse investors process news of uncertain quality, they act as if they take a worst-case assessment of quality. As a result, they react more strongly to bad news than to good news. They also dislike assets for which information quality is poor, especially when the underlying fundamentals are volatile. These effects induce ambiguity premia that depend on idiosyncratic risk in fundamentals as well as skewness in returns. Moreover, shocks to information quality can have persistent negative effects on prices even if fundamentals do not change. FINANCIAL MARKET PARTICIPANTS ABSORB a large amount of news, or signals, every day. Processing a signal involves quality judgments: News from a reliable source should lead to more portfolio rebalancing than news from an obscure source. Unfortunately, judging quality itself is sometimes difficult. For example, stock picks from an unknown newsletter without a track record might be very reliable or entirely useless—it is simply hard to tell. Of course, the situation is different when investors can draw on a lot of experience that helps them interpret signals. This is true especially for “tangible” information, such as earnings reports, that lends itself to quantitative analysis. By looking at past data, investors may become quite confident about how well earnings forecast returns. This paper focuses on information processing when there is incomplete knowledge about signal quality. The main idea is that, when quality is difficult to judge, investors treat signals as ambiguous. They do not update beliefs in standard Bayesian fashion, but behave as if they have multiple likelihoods in mind when processing signals. To be concrete, suppose that θ is a parameter that an investor wants to learn. We assume that a signal s is related to the parameter by a family of likelihoods: s = θ + � , � ∼ N � 0, σ 2 s � , σ 2 s ∈ � σ 2

749 citations

References
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01 Jan 1990
TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.
Abstract: 0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov's Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- 1. Gronwall's Lemma.- 2. Distributions.- 3. Convex Functions.- 4. Hausdorff Measures and Dimension.- 5. Ergodic Theory.- 6. Probabilities on Function Spaces.- 7. Bessel Functions.- 8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.

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Journal ArticleDOI
TL;DR: This paper showed that an equilibrium model which is not an Arrow-Debreu economy will be the one that simultaneously rationalizes both historically observed large average equity return and the small average risk-free return.

6,141 citations


"Ambiguity, risk and asset returns i..." refers background in this paper

  • ...An illustration of the potential usefulness of recognizing the presence of ambiguity is provided by the equity premium puzzle (Mehra and Prescott (1985)-the failure of the representative agent model to fit historical averages of the equity premium and the risk-free rate....

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Journal ArticleDOI
TL;DR: In this paper, a class of recursive, but not necessarily expected utility, preferences over intertemporal consumption lotteries is developed, which allows risk attitudes to be disentangled from the degree of inter-temporal substitutability, leading to a model of asset returns in which appropriate versions of both the atemporal CAPM and the inter-time consumption-CAPM are nested as special cases.
Abstract: This paper develops a class of recursive, but not necessarily expected utility, preferences over intertemporal consumption lotteries An important feature of these general preferences is that they permit risk attitudes to be disentangled from the degree of intertemporal substitutability Moreover, in an infinite horizon, representative agent context these preference specifications lead to a model of asset returns in which appropriate versions of both the atemporal CAPM and the intertemporal consumption-CAPM are nested as special cases In our general model, systematic risk of an asset is determined by covariance with both the return to the market portfolio and consumption growth, while in each of the existing models only one of these factors plays a role This result is achieved despite the homotheticity of preferences and the separability of consumption and portfolio decisions Two other auxiliary analytical contributions which are of independent interest are the proofs of (i) the existence of recursive intertemporal utility functions, and (ii) the existence of optima to corresponding optimization problems In proving (i), it is necessary to define a suitable domain for utility functions This is achieved by extending the formulation of the space of temporal lotteries in Kreps and Porteus (1978) to an infinite horizon framework A final contribution is the integration into a temporal setting of a broad class of atemporal non-expected utility theories For homogeneous members of the class due to Chew (1985) and Dekel (1986), the corresponding intertemporal asset pricing model is derived

4,218 citations


"Ambiguity, risk and asset returns i..." refers background in this paper

  • ...2 To explain this nomenclature, note that stochastic differential utility is the continuous-time counterpart of recursive utility (Epstein and Zin (1989))....

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  • ...Here W is an aggregator function (strictly increasing in its second argument) analogous to that appearing in Epstein and Zin (1989) and motivated there by the desire to disentangle risk aversion from other aspects of preference....

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Journal ArticleDOI
TL;DR: Dunford and Schwartz as discussed by the authors provided a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis, and emphasized the significance of the relationships between the abstract theory and its applications.
Abstract: This classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis. Dunford and Schwartz emphasize the significance of the relationships between the abstract theory and its applications. This text has been written for the student as well as for the mathematician—treatment is relatively self-contained. This is a paperback edition of the original work, unabridged, in three volumes.

2,890 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of finding an adapted pair of processes with values in Rd and Rd×k, respectively, which solves an equation of the form: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∪ t 1 [g(m, x, s, g(m)) + y(m)] dW s = X.

2,812 citations


"Ambiguity, risk and asset returns i..." refers background in this paper

  • ...By the existence and uniqueness result in Pardoux and Peng (1990), there exist unique solutions (V,, o-r) and (VQ, o-fQ) to (2.17) and (2.16) respectively....

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  • ...However, the arguments in Pardoux and Peng (1990) and Peng (1997) rely only on progressive measurability of the above map for each fixed (y, v)....

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