Tan Wang

Intertemporal Asset Pricing Under Knightian Uncertainty

In conformity with the Savage model of decision-making, modern asset pricing theory assumes that agents beliefs about the likelihoods of future states of the world may be represented by a probability measure. As a result, no meaningful distinction is allowed between risk, where probabilities are available to guide choice, and uncertainty, where information is too imprecise to be summarized adequately by probabilities. In contrast, Knight and Keynes emphasized the distinction between risk and uncertainty and argued that uncertainty is more common in economic decision-making. Moreover, the Savage model is contradicted by evidence, such as the Ellsberg Paradox, that people prefer to act on known rather than unknown or vague probabilities. This paper provides a formal model of asset price determination in which Knightian uncertainty plays a role. Specifically, we extend the Lucas (1978) general equilibrium pure exchange economy by suitably generalizing the representation of beliefs along the lines suggested by Gilboa and Schmeidler. Two principal results are the proof of existence of equilibrium and the characterization of equilibrium prices by an Euler inequality. A noteworthy feature of the model is that uncertainty may lead to equilibria that are indeterminate, that is, there may exist a continuum of equilibria for given fundamentals. That leaves the determination of a particular equilibrium price process to animal spirits and sizable volatility may result. Finally, it is argued that empirical investigation of our model is potentially fruitful.

Conditional preferences and updating

This paper axiomatizes updating rules for preferences that are not necessarily in the expected utility class. Two sets of results are presented. The first is the axiomatization of conditional preferences. The second consists of the axiomatization of three updating rules: the traditional Bayes rule, the Dempster–Shafer rule, and the generalized Bayes rule. The last rule can be regarded as the updating rule for the multi-prior expected utility (Gilboa and Schmeidler, J. Math. Econom. 18 (1989) 141). Operationally, it is equivalent to updating each prior by the traditional Bayes rule.

BELIEFS ABOUT BELIEFS WITHOUT PROBABILITIES

This paper constructs a space of states of the world representing the exhaustive uncertainty facing each player in a strategic situation. The innovation is that preferences are restricted primarily by regularity conditions and need not conform with subjective expected utility theory. The construction employs a hierarchy of preferences, rather than of beliefs as in the standard Bayesian model. Applications include the provision of foundations for a Harsanyi-style game of incomplete information and a rich framework for the axiomatization of solution concepts for complete information normal form games. Copyright 1996 by The Econometric Society.

Keynes Meets Markowitz: The Trade-Off Between Familiarity and Diversification

We develop a model of portfolio choice to nest the views of Keynes, who advocates concentration in a few familiar assets, and Markowitz, who advocates diversification. We use the concepts of ambiguity and ambiguity aversion to formalize the idea of an investors “familiarity” toward assets. The model shows that for any given level of expected returns, the optimal portfolio depends on two quantities: relative ambiguity across assets and the standard deviation of the expected return estimate for each asset. If both quantities are low, then the optimal portfolio consists of a mix of familiar and unfamiliar assets; moreover, an increase in correlation between assets causes an investor to increase concentration in familiar assets (flight to familiarity). Alternatively, if both quantities are high, then the optimal portfolio contains only the familiar asset(s), as Keynes would have advocated. In the extreme case in which both quantities are very high, no risky asset is held (nonparticipation). n nThis paper was accepted by Brad Barber, Teck Ho, and Terrance Odean, special issue editors.

Efficient Intertemporal Allocations with Recursive Utility

In this article, our objective is to determine efficient allocations in economies with multiple agents having recursive utility functions. Our main result is to show that in a multiagent economy, the problem of determining efficient allocations can be characterized in terms of a single value function (that of a social planner), rather than multiple functions (one for each investor), as has been proposed thus far (Duffie, Geoffard and Skiadas (1994)). We then show how the single value function can be identified using the familiar technique of stochastic dynamic programming. We achieve these goals by first extending to a stochastic environment Geoffards (1996) concept of variational utility and his result that variational utility is equivalent to recursive utility, and then using these results to characterize allocations in a multiagent setting.

Keynes Meets Markowitz: The Tradeoff Between Familiarity and Diversification

We develop a model of portfolio choice to nest the views of Keynes---who advocates concentration in a few familiar assets---and Markowitz---who advocates diversification across assets. We rely on the concepts of ambiguity and ambiguity aversion to formalize the idea of investors ``familiarity towards assets. The model shows that when investors are equally ambiguous about all assets, then the optimal portfolio corresponds to Markowitzs fully-diversified portfolio. In contrast, when investors exhibit different degrees of familiarity across assets, the optimal portfolio depends on (i)~the relative degree of ambiguity across assets and (ii)~the standard deviation of the estimate of expected return on each asset. If the standard deviation is low and the difference between the ambiguity about the familiar and unfamiliar assets is small, then the optimal portfolio is composed of a mix of both familiar and unfamiliar assets; moreover, an increase in correlation between assets causes investors to increase concentration in the assets with which they are familiar (flight to familiarity). Alternatively, if the standard deviation of the expected-return estimate is high and the difference between the ambiguity of familiar and unfamiliar assets is large, the optimal portfolio contains only the familiar asset(s) as Keynes would have advocated. In the extreme case in which the ambiguity about all assets and the standard deviation of the estimated mean are high, then no risky asset is held (non-participation). The model also has empirically testable implications for trading behavior: in response to a change in idiosyncratic risk the Keynesian portfolio always exhibits more trading than the Markowitz portfolio, while the opposite is true for a change in systematic volatility. In the equilibrium version of the model with heterogeneous agents who are familiar with different assets, we find that the risk premium of stocks depends on both systematic and idiosyncratic volatility, and that the equity risk premium is significantly higher than in the standard model without ambiguity.

Seasonally Varying Preferences: Theoretical Foundations for an Empirical Regularity

We investigate an asset pricing model with preferences cycling between high risk aversion and low EIS in fall/winter and the reverse in spring/summer. Calibrating to consumption data and allowing plausible preference parameter values, we produce returns that match observed equity and Treasury returns across the seasons: risky returns are higher and risk-free returns are lower or stable in fall/winter, and they reverse in spring/summer. Further, risky returns vary more than risk-free returns. A novel finding is that both EIS and risk aversion must vary seasonally to match observed returns. Further, the degree of necessary seasonal change in EIS is small.

The Becker-DeGroot-Marschak mechanism and generalized utility theories: Theoretical predictions and empirical observations

Karni and Safra [8] prove that the Becker-DeGroot-Marschak mechanism reveals a decision makers true certainty equivalent of a lottery if and only if he satisfies the independence axiom. Segal [17] claims that this mechanism may reveal a violation of the reduction of compound lotteries axiom. This paper empirically tests these two interpretations. Our results show that the second interpretation fits better with the collected data. Moreover, we show by means of some nonexpected utility examples that these results are consistent with a wide range of functionals.

Pricing of New Securities in an Incomplete Market: the Catch 22 of No-Arbitrage Pricing

There are two distinctly different approaches to the valuation of a new security in an incomplete market. The first approach takes the prices of the existing securities as fixed and uses no‐arbitrage arguments to derive the set of equivalent martingale measures that are consistent with the initial prices of the traded securities. The price of the new security is then obtained by appealing to certain criteria or on the basis of some preference assumption. The second method prices the new security within a general equilibrium framework. This paper clarifies the distinction between the two approaches and provides a simple proof that the introduction of the new security will typically change the prices of all the existing securities. We are left with the paradox that a genuinely new derivative security is not redundant, but the dominant pricing paradigm in derivative security pricing is the no‐arbitrage approach, which requires the redundancy of the security. Given the widespread practice of using the no‐arbitrage approach to price (or bound the price of) a new security, we also comment on some justifications for this approach.

Equilibrium with new investment opportunities

Abstract We develop a general equilibrium model where there are opportunities to abandon existing technologies and adopt new technologies. We study how technologies change endogenously in an equilibrium and the effect of technological changes on the equilibrium. We find that locally the equilibrium resembles that from Cox et al. (1985, Econometrica 53, 363–384); globally, the equilibrium can be viewed as the equilibria from CIR economies pasted together. We also find that, while the equilibrium consumption and security price processes are continuous, the equilibrium return processes experience jumps at the time of technological changes.