Ambiguity in Asset Pricing and Portfolio Choice: A Review of the Literature
Social Science Research Network
A growing body of empirical evidence suggests that investors' behavior is not well described by the traditional paradigm of (subjective) expected utility maximization under rational expectations. A literature has arisen that models agents whose choices are consistent with models that are less restrictive than the standard subjective expected utility framework. In this paper we conduct a survey of the existing literature that has explored the implications of decision-making under ambiguity for
... nancial market outcomes, such as portfolio choice and equilibrium asset prices. We conclude that the ambiguity literature has led to a number of significant advances in our ability to rationalize empirical features of asset returns and portfolio decisions, such as the empirical failure of the two-fund separation theorem in portfolio decisions, the modest exposure to risky securities observed for a majority of investors, the home equity preference in international portfolio diversification, the excess volatility of asset returns, the equity premium and the risk-free rate puzzles, and the occurrence of trading break-downs. JEL codes: G10, G18, D81. . This literature replaces rational expectations with beliefs updated through a rational learning rule, for instance Bayes' rule, in the light of the arrival of stochastic signals with non-zero correlation with relevant fundamentals. Under these approaches, investment decisions can differ from standard ones due to the difficulty of learning the true state given a complex generating process (for instance, subject to breaks or instability of other types, like regimes). A third approach focusses on alternative frameworks of rational decision-making. This literature entertains agents whose choices are consistent with models that are less restrictive than the standard (S)EU framework, in the sense that the underlying axioms are less demanding. In this area, particular attention has recently been dedicated to ambiguity and ambiguity aversion. 1 In this paper we conduct a systematic-albeit admittedly incomplete-review of the literature that has explored the implications of decision-making under ambiguity for financial market outcomes, such as portfolio choices and equilibrium asset prices. Under (S)EU, if preferences satisfy certain axioms, there are numerical utilities and probabilities that represent acts (decisions under uncertainty) by a standard weighted sum of the utilities/outcomes deriving from acts in the possible states of the world, where the weights are (subjective) probabilities for each of the states. As innocuous as this basic principle may seem, there is a long, rich tradition of questioning whether it describes behavior adequately. Keynes (1921) was the first to draw a distinction between the implications of evidence-the likelihood judgments that evidence implies-and the "weight" that should be attached to this evidence, or the confidence in the assessed likelihoods. Keynes wondered whether a single probability number could express both dimensions of evidence. Knight (1921) distinguished risk, or known probability, from uncertainty. He suggested that economic returns could be earned for bearing uncertainty but not for bearing risk. However, the modern attack to (S)EU as a descriptive theory was made most directly by Ellsberg's (1961) paradox that we describe in Section 2.1. Ellsberg's (1961) thought-provoking article stirred a debate and between the 1970s and 1980s led researchers to assemble massive experimental evidence that indicates that people generally prefer the least ambiguous acts. This implies that the experimental subjects take their own confidence in estimates of subjective probability into account when making decisions. Such a pattern is inconsistent with Savage's sure-thing principle of (S)EU, the axiom by which a state with a consequence common to a pair of acts is irrelevant in determining preference between the acts. Why is a survey of the literature on ambiguity in financial markets of any use? First, because the last decade has seen a tremendous growth in the number, breadth, and quality of papers that have exploited ambiguity in connection to research questions in financial economics. In fact, a quick scoring of our own references-as incomplete and deficient as they may be-reveals that out of a total of 195 references, 86 (i.e., a hefty 44%) have been either written (as indicated by the year of the working paper in our possession) or published on or after 2003. 2 Second, a thought-provoking paper by Al-Najjar and Weinstein (2009) has spurred a debate on whether ambiguity-based approaches to economic modelling would be as sensible as their growing popularity implies. Al-Najjar and Weinstein have been dismissive 1 Whether or not preferences reflecting ambiguity are "behavioral" is mostly a matter of tastes. However, many researchers that have used and critically commented ambiguity averse preferences (e.g., Backus et al., 2004) have noticed that most of these preferences do represent well-defined neo-classical preference orderings. Of course, this is not to be uncritically taken as a "good feature". On the one hand, the strong theoretical foundations for ambiguity averse preferences allow a researcher to use all the tools of neo-classical economics, particularly optimization and welfare analysis. On the other hand, when these preferences come to ignore aspects of human behavior stressed in other social sciences, particularly sociology and social psychology, they may imply a loss of realism. 2 Of course, this is only a very rough scoring system, but we doubt that the stunning percentage we have determined could be much affected by any other sensible assumption. Generalities and Definitions: What is Ambiguity? Let's introduce some bits of notation that will become handy later on. A decision problem is structured on a state space, an outcome space, and a preference relation. The state space Ω, whose elements are called states of nature, represents all the possible realizations of future uncertainty. Sets of states of nature, ⊂ Ω, are called events. The outcome space F contains the possible, random results of any conceivable decision. The outcome space can be rather abstract: although in many applications we can take F to be the set of real numbers (e.g., wealth), in principle it could be any relevant aspect of a decision problem. 5 A preference relation % is defined over the mappings from Ω to F, these mappings are called acts or decisions and they associate to each state of nature ∈ Ω a possible consequence () (or ). % means that the decision maker weakly prefers decision to decision ; ∼ means that the decision maker is indifferent between and Most of the time-we will clearly note when this is not the case-all preferences we consider are assumed to be complete (i.e., a decision maker is always able to rank decisions), reflexive ( % ) and transitive (i.e., if a decision maker prefers over and over , then she also prefers over ). We generally denote by a standard VNM utility index, and we label by BM standard Brownian motions. 2.