UNCERTAINTY AVERSION AND PORTFOLIO INERTIA

In stock markets, we often observe portfolio inertia, i.e., a situation in which some stocks are not traded or not priced for a few minutes or longer. This is neither an exceptional situation in which some stock price soars too high to be priced, nor the one where some stock price plummets too much to be traded. By introducing the concept of ‘Knightian uncertainty’, Dow and Werlang (1992) account for the existence of portfolio inertia, which has not been accounted for under the concept of ‘risk’. This paper provides a characterization of the spread between buying and selling prices based on a parameter proposed by Ozaki and Streufert (1999, 2001) that enables us to estimate the attitude towards Knightian uncertainty, and shows that an increase (a decrease) in Knightian uncertainty expands (shrinks) the interval in which an investor never changes her initial position. Furthermore, we analyse the effect of an increase in Knightian uncertainty on portfolio inertia based on Epsilon‐contaminations.     

Model uncertainty and liquidity

Extreme market outcomes are often followed by a lack of liquidity and a lack of trade. This market collapse seems particularly acute for markets where traders rely heavily on a specific empirical model such as in derivative markets like the market for mortgage backed securities or credit derivatives. Moreover, the observed behavior of traders and institutions that places a large emphasis on “worst-case scenarios” through the use of “stress testing” and “Value-at-Risk” seems different than Savage expected utility would suggest. In this paper, we capture model-uncertainty using an Epstein and Wang [Epstein, L.G., Wang, T., 1994. Intertemporal asset pricing under Knightian uncertainty. Econometrica 62, 283–322] uncertainty-averse utility function with an ambiguous underlying asset-returns distribution. To explore the connection of uncertainty with liquidity, we specify a simple market where a monopolist financial intermediary makes a market for a propriety derivative security. The market-maker chooses bid and ask prices for the derivative, then, conditional on trade in this market, chooses an optimal portfolio and consumption. We explore how uncertainty can increase the bid–ask spread and, hence, reduces liquidity. Our infinite-horizon example produces short, dramatic decreases in liquidity even though the underlying environment is stationary. We show how these liquidity crises are closely linked to the uncertainty aversion effect on the optimal portfolio. Effectively, the uncertainty aversion can, at times, limit the ability of the market-maker to hedge a position and thus reduces the desirability of trade, and hence, liquidity.     

General equilibrium, wariness and efficient bubbles

Wary consumers overlook gains but not losses in remote sets of dates or states. As preferences are upper but not lower Mackey semi-continuous, Bewley?s (1972) [4] result on existence of equilibrium whose prices are not necessarily countably additive holds. Wariness is related to lack of myopia and to ambiguity aversion (and, therefore, to Bewley?s (1986) [6] work on Knightian uncertainty). Wary infinite lived agents have weaker transversality conditions allowing them to be creditors at infinity and for bubbles to occur in positive net supply assets completing the markets. There are efficient allocations that can only be implemented with asset bubbles.     

Innovation and stock market performance: A model with ambiguity-averse agents

Empirical evidence on stock prices shows that firms investing successfully in radical innovation experience higher stock returns. This paper provides a model that sheds light on the relationship between the degree of firm innovativeness and stock returns, the movements of which capture expectations on firm’s profitability and growth. The model is grounding on Neo-Schumpeterian growth models and relies on the crucial assumption of radical innovation, characterized by “ambiguity” or Knightian uncertainty: due to its uniqueness and originality, no distribution of probability can be reasonably associated with radical innovation success or failure. Different preferences (α-maxmin, Choquet) are here compared. Results show that the assumption of ambiguity-aversion is crucial in determining higher returns in the presence of radical innovation and that the specific definition of expected utility shapes the extent of the returns. This result holds also in the case of endogenous innovation; risk attitude plays no role.     

Search, common knowledge, and competition

This paper analyzes the effects of buyer search costs and seller private and common knowledge on seller competition. It shows that lack of common knowledge results in the equilibrium price continuously decreasing to the perfectly competitive one as buyer search costs for price decrease from positive for all buyers to zero for all buyers, even if each market agent's uncertainty (in the private knowledge) is small. At the same time, if the uncertainty of each seller about buyer valuations is small, the effects of a small change in the search costs or of information structure on pricing may be large (but continuous).     

Model Uncertainty,Ambiguity and the Precautionary Principle: Implications for Biodiversity Management

We analyze ecosystem management under ‘unmeasurable’ Knightian uncertainty or ambiguity which, given the uncertainties characterizing ecosystems, might be a more appropriate framework relative to the classic risk case (measurable uncertainty). This approach is used as a formal way of modelling the precautionary principle in the context of least favorable priors and maxmin criteria. We provide biodiversity management rules which incorporate the precautionary principle. These rules take the form of either safety margins and minimum safety standards or optimal harvesting under precautionary approaches.     

Irreversible investment and Knightian uncertainty

When firms make a decision about irreversible investment, they may not have complete confidence about their perceived probability measure describing future uncertainty. They may think other probability measures perturbed from the original one are also possible. Such uncertainty, characterized by not a single probability measure but a set of probability measures, is called “Knightian uncertainty.” The effect of Knightian uncertainty on the value of irreversible investment opportunity is shown to be drastically different from that of traditional uncertainty in the form of risk. Specifically, an increase in Knightian uncertainty decreases the value of investment opportunity while an increase in risk increases it.     

Implementing Arrow-Debreu equilibria by trading infinitely-lived securities

Summary. We show that Arrow-Debreu equilibria with countably additive prices in infinite-time economy under uncertainty can be implemented by trading infinitely-lived securities in complete sequential markets under two different portfolio feasibility constraints: wealth constraint, and essentially bounded portfolios. Sequential equilibria with no price bubbles implement Arrow-Debreu equilibria, while those with price bubbles implement Arrow-Debreu equilibria with transfers. Transfers are equal to price bubbles on initial portfolio holdings. Price bubbles arise in sequential equilibrium under the wealth constraint if some securities are in zero supply or negative prices are permitted, but cannot arise with essentially bounded portfolios.Received: 19 November 2003, Revised: 24 February 2004, JEL Classification Numbers: D50, G12, E44.Correspondence to: Jan WernerWe acknowledge helpful discussions with Roko Aliprantis, Subir Chattopaydhyay, Steve LeRoy, Manuel Santos, and seminar participants at Brown University, University of Pennsylvania, NBER Workshop in General Equilibrium Theory, SITE 2000, the 2000 World Congress of the Econometric Society, and Federal Reserve Bank of Kansas City. The views expressed herein are those of the authors and do not necessarily reflect the views of Federal Reserve Bank of Kansas City or the Federal Reserve System.     

Intertemporal equilibria with Knightian uncertainty

We study a dynamic and infinite-dimensional model with incomplete multiple prior preferences. In interior efficient allocations, agents share a common risk-adjusted prior and subjective interest rate. Interior efficient allocations and equilibria coincide with those of economies with subjective expected utility and priors from the agents? multiple prior sets. A specific model with neither risk nor uncertainty at the aggregate level is considered. Risk is always fully insured. For small levels of ambiguity, there exists an equilibrium with inertia where agents also insure fully against Knightian uncertainty. When the level of ambiguity exceeds a critical threshold, full insurance no longer prevails and there exist equilibria with inertia where agents do not insure against uncertainty at all. We also show that equilibria with inertia are indeterminate.     

The possibility of informationally efficient markets

A rational expectations equilibrium with positive demand for financial information does exist under fully revealing asset price—contrary to a wide-held conjecture. Whereas a continuum of investors is inconsistent with fully revealing equilibrium, finitely many investors with average portfolios demand information in equilibrium if they can adjust portfolio size in an additive signal-return model. More information diminishes the expected excess return of a risky asset so that investors who only have a choice of portfolio composition or whose asset endowments strongly differ from the average portfolio are worse off. Under fully revealing price, information market equilibria both with and without information acquisition are Pareto efficient.     

Market equilibrium with heterogeneous recursive-utility-maximizing agents

Summary This paper considers a heterogeneous agent Lucas style exchange economy. For a class of recursive utility functions containing the standard additive expected utility functions, I demonstrate that there exist market equilibria characterized by stationary (ergodic) Markov processes for consumption, portfolio holdings, asset prices and the unobserved utilities. No assumptions about market completeness are made, and there are no restrictions on the underlying information filtration.Other contributions of this paper include: (i) an existence and uniqueness theorem of intertemporal utility for the general class of recursive generators; (ii) the optimum principle as well as its corresponding Euler equation derived for the agent's consumption and portfolio choice problem under recursive utility, and (iii) a single-agent equilibrium asset pricing formula which generalizes that of Epstein and Zin (1989).This paper is a part of my PhD dissertation at the University of Toronto. I would like to thank Larry Epstein for his enthusiastic supervision, helpful discussion and valuable comments. Thanks also to Tan Wang and especially Darrell Duffie for valuable comments.     

Uncertainty and central bank transparency: A non-Bayesian approach

We use a non-Bayesian approach to uncertainty, where “ambiguity” is taken into account, in order to analyze the issue of central bank transparency, and we underline that the use of such an approach may greatly change the results. We reconsider a specific argument against transparency found in the literature. We show that, in the presence of ambiguity, the argument can become a case in favor of transparency, which seems more in accordance with some stylized facts. Reduced Knightian uncertainty associated with increased transparency can contribute to making transparency beneficial.     

A SMOOTH AMBIGUITY MODEL OF THE COMPETITIVE FIRM

This paper examines the optimal production decision of the competitive firm under price uncertainty when the firm's preferences exhibit smooth ambiguity aversion. Ambiguity is modeled by a second‐order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the price risk. Ambiguity preferences are modeled by the (second‐order) expectation of a concave transformation of the (first‐order) expected utility of profit conditional on each plausible subjective distribution of the price risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity‐averse firm optimally produces less in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. In the case that the price risk is binary, we show that ambiguity and greater ambiguity aversion always adversely affect the firm's production decision.     

On relative and partial risk attitudes: theory and implications

This paper develops context-free interpretations for the relative and partial Nth degree risk attitude measures and show that various conditions on theses measures are utility characterizations of the effects of scaling general stochastic changes in different settings. It is then shown that these characterizations can be applied to generalize comparative statics results in a number of important problems, including precautionary savings, optimal portfolio choice, and competitive firms under price uncertainty.     

Refundable Annuities (Annuity Options)

Individuals can insure themselves perfectly against uncertainty about the length of life by purchasing deferred annuities early in life. In the absence of other uninsurable uncertainties (e.g., income), there will be no residual purchases or sales of annuities later in life, thereby avoiding any  adverse-selection . In contrast, the presence of such uncertainties creates an active residual annuity market based on the arrival of new information. We characterize the equilibrium in the residual annuity market and propose a new financial instrument,  refundable annuities  with a guaranteed refund price, which enables individuals who hold a portfolio of such annuities to better adjust their optimum consumption plan to different realizations. Refundable annuities are shown to be equivalent to  annuity options , that is, options that, if exercised, enable the purchase of annuities later in life at a predetermined price. Holding a variety of refundable annuities is ( ex ante ) welfare enhancing.     

Hedging a Portfolio of Derivative Securities: A Simulation Approach

We illustrate a numerical simulation method to decompose a portfolio of derivative securities in a linear combination of dynamical risk factors. The price of the portfolio and its sensitivities are linear functions of these factors.
The method generalizes the static hedging theory proposed by Madan and Milne (1994) and applies to a dynamically complete, arbitrage free market with purely Brownian fluctuating assets. The extension to a class of market models whose volatility dynamics shows long memory and scaling behaviour is discussed and shown to be possible.
(J.E.L.: G12).     

Stochastic growth: a duality approach

We re-examine the representative agent's optimal consumption and savings under uncertainty in the presence of investment constraints using martingale representation and convex analysis techniques. This framework allows us to explicitly quantify precautionary savings which induces a higher average growth rate than in a certainty setup. We provide a closed form solution for a Cobb-Douglas economy. The effect of uncertainty on portfolio selection is analyzed. Consumption growth rate and risk free interest rate exhibit a U-shaped relationship. Uncertainty negatively affects expected consumption growth rate; such a result seems to be supported by empirical evidence.     

Optimal portfolio positioning under ambiguity

This paper analyzes the optimality of financial portfolios when the investor has a utility with ambiguity aversion. It provides a general result about the optimal portfolio profile under ambiguity, in the Anscombe–Aumann framework, using the Maccheroni et al. (2006) approach which includes Gilboa and Schmeidler's (1989) multiple prior preferences and Hansen and Sargent's (2011) multiplier preferences. The paper then details the CRRA case with an ambiguity index based on relative entropy. Such findings have practical applications in structured portfolio management. Indeed, it is important to take account of uncertainty about the true values of financial parameters when determining the best portfolio profile.     

PRICE CONSTRAINTS,FORWARD MARKETS AND THE COMPETITIVE FIRM UNDER PRICE UNCERTAINTY*

This article proposes a model of the competitive firm simultaneously facing price constraints and forward markets under price uncertainty. The incorporation of a forward market is shown to be very important because a risk-averse firm will set its production decision to the forward price regardless of its attitude toward risk. In addition, we show that risk aversion is a sufficient condition for a decrease in risk to reduce the amount hedged when risk is reduced through a mean-preserving price squeeze.     

Search and Knightian uncertainty

Suppose that “uncertainty” about labor market conditions has increased. Does this change induce an unemployed worker to search longer, or shorter? This paper shows that the answer is drastically different depending on whether an increase in “uncertainty” is an increase in risk or that in true uncertainty in the sense of Frank Knight. We show in a general framework that, while an increase in risk (the mean-preserving spread of the wage distribution that the worker thinks she faces) increases the reservation wage, an increase in Knightian uncertainty (a decrease in her confidence about the wage distribution) reduces it.