OPTIMAL INSURANCE DESIGN UNDER RANK‐DEPENDENT EXPECTED UTILITY

We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank‐dependent expected utility (RDEU) theory with a concave utility function and an inverse‐S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.     

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise‐linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box‐type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst‐case risk under distributional ambiguity. New closed‐form results for the worst‐case optimized certainty equivalent risk measures and optimal portfolios are provided for two‐ and three‐piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.     

PROBABILITY AND UNCERTAINTY: THE LEGACY OF GEORGESCU‐ROEGEN

In this paper we consider Georgescu‐Roegen's approach to uncertainty, showing that his characterization of expectations cannot be reduced to any probabilistic decision‐making model. Drawing upon Georgescu‐Roegen's lesson a lexicographical utility function is proposed and analysed in the mark of his own peculiar scientific methodology. It is demonstrated that such a formulation can be useful in solving the usual failure of the expected utility model, such as the Ellsberg paradoxes. The epistemic limits of our re‐construction are considered.     

Parametric Portfolio Policies in the Surplus Consumption Ratio

The surplus consumption ratio plays a central role as a state variable in successful attempts to explain the time series properties of stock and bond prices with consumption‐based asset pricing models. In this paper, optimal portfolio policies for a strategic investor who maximizes the conditionally expected utility of terminal wealth are parameterized as a polynomial in the surplus consumption ratio. Optimal portfolio policies are estimated using a method of moments estimator based on Euler equations. Unconditional portfolio policies are rejected in favor of conditional policies. Lower order polynomials are rejected in favor of higher order polynomials. Optimal stock and bond allocations are clearly countercyclical.     

Dynamically consistent alpha‐maxmin expected utility

The alpha‐maxmin model is a prominent example of preferences under Knightian uncertainty as it allows to distinguish ambiguity and ambiguity attitude. These preferences are dynamically inconsistent for nontrivial versions of alpha. In this paper, we derive a recursive, dynamically consistent version of the alpha‐maxmin model. In the continuous‐time limit, the resulting dynamic utility function can be represented as a convex mixture between worst and best case, but now at the local, infinitesimal level. We study the properties of the utility function and provide an Arrow–Pratt approximation of the static and dynamic certainty equivalent. We then derive a consumption‐based capital asset pricing formula and study the implications for derivative valuation under indifference pricing.     

A NOTE ON THE QUANTILE FORMULATION

Many investment models in discrete or continuous‐time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change‐of‐variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank‐dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well‐posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law‐invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.     

Risk‐Sensitive Control and an Optimal Investment Model

We consider an optimal investment model in which the goal is to maximize the long‐term growth rate of expected utility of wealth. In the model, the mean returns of the securities are explicitly affected by the underlying economic factors. The utility function is HARA. The problem is reformulated as an infinite time horizon risk‐sensitive control problem. We study the dynamic programming equation associated with this control problem and derive some consequences of the investment problem.     

Inference for large financial systems

We treat the parameter estimation problem for mean‐field models of large interacting financial systems such as the banking system and a pool of assets held by an institution or backing a security. We develop an asymptotic inference approach that addresses the scale and complexity of such systems. Harnessing the weak convergence results developed for mean‐field financial systems in the literature, we construct an approximate likelihood for large systems. The approximate likelihood has a conditionally Gaussian structure, enabling us to design an efficient numerical method for its evaluation. We provide a representation of the corresponding approximate estimator in terms of a weighted least‐squares estimator, and use it to analyze the large‐system and large‐sample behavior of the estimator. Numerical results for a mean‐field model of systemic financial risk highlight the efficiency and accuracy of our estimator.     

Convergence of optimal expected utility for a sequence of discrete‐time markets

We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The nth discrete‐time economy is generated by a scaled n‐step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that .     

Why do European consumers smoke? Responses from the rational addiction model

In this paper, we test whether European consumers are addictive smokers and, if this is the case, then whether such addictions can be explained by the rational addiction theory. To this end, we start from a non‐separable intertemporal utility function, which allows us to derive a demand function that is estimated using tobacco time‐series. The results are in accordance with the model of rational addiction for all European smokers. Thus, we observe the addictive character of tobacco consumption and, secondly, we note that the addiction is not the result of myopic consumer behaviour, but rather of the maximization of total utility, implying that consumers consider the future effects of their current decisions.     

UTILITY MAXIMIZATION IN A LARGE MARKET

We study the problem of expected utility maximization in a large market, i.e., a market with countably many traded assets. Assuming that agents have von Neumann–Morgenstern preferences with stochastic utility function and that consumption occurs according to a stochastic clock, we obtain the “usual” conclusions of the utility maximization theory. We also give a characterization of the value function in a large market in terms of a sequence of value functions in finite‐dimensional models.     

Two Sides of a Coin: Endogenous and Exogenous Effects of Corporate Diversification on Firm Value

We investigate the value effects of two types of corporate diversification – unexpected exogenous diversification and endogenous diversification. Combining Heckman's sample‐selection estimator with a two‐stage least squares estimator and a generalized method of moments instrumental variables estimator to control for both endogeneity and sample‐selection bias, we find that while an unexpected increase in diversification caused by exogenous shocks destroys firm value, an endogenous increase in diversification due to managerial decisions will enhance firm value, indicating a diversification premium from altering organizational structures.     

Identification and semiparametric estimation of a finite horizon dynamic discrete choice model with a terminating action

We study identification and estimation of finite-horizon dynamic discrete choice models with a terminal action. We first demonstrate a new set of conditions for the identification of agents’ time preferences. Then we prove conditions under which the per-period utilities are identified for all actions in the agent’s choice-set, without having to normalize the utility for one of the actions. Finally, we develop a computationally tractable semiparametric estimator. The estimator uses a two-step approach that does not use either backward induction or forward simulation. Our methodology can be implemented using standard statistical packages without the need to write specialized computational routines, as it involves linear (or nonlinear) projections only. Monte Carlo studies demonstrate the superior performance of our estimator compared with existing two-step estimation methods. Monte Carlo studies further demonstrate that the ability to identify the per-period utilities for all actions is crucial for counterfactual predictions. As an empirical illustration, we apply the estimator to the optimal default behavior of subprime mortgage borrowers, and the results show that the ability to identify the discount factor, rather than assuming an arbitrary number as typically done in the literature, is also crucial for obtaining correct counterfactual predictions. These findings highlight the empirical relevance of key identification results of the paper.     

A Note on Loss Aversion and Futures Hedging

This note examines the effect of loss aversion on the futures trading behavior of a short hedger. Using a modified constant‐absolute‐risk‐aversion utility function, I show that loss aversion has no effect in an unbiased futures market. It has different, predictable impacts when the futures market is in backwardation or contango. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 681–692, 2001     

Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear price impact. In our model, cross‐impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second‐order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.     

Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation

The mean‐variance formulation by Markowitz in the 1950s paved a foundation for modern portfolio selection analysis in a single period. This paper considers an analytical optimal solution to the mean‐variance formulation in multiperiod portfolio selection. Specifically, analytical optimal portfolio policy and analytical expression of the mean‐variance efficient frontier are derived in this paper for the multiperiod mean‐variance formulation. An efficient algorithm is also proposed for finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth.     

SENSITIVITY ANALYSIS OF NONLINEAR BEHAVIOR WITH DISTORTED PROBABILITY

In this paper, we propose a sensitivity‐based analysis to study the nonlinear behavior under nonexpected utility with probability distortions (or “distorted utility” for short). We first discover the “monolinearity” of distorted utility, which means that after properly changing the underlying probability measure, distorted utility becomes locally linear in probabilities, and the derivative of distorted utility is simply an expectation of the sample path derivative under the new measure. From the monolinearity property, simulation algorithms for estimating the derivative of distorted utility can be developed, leading to gradient‐based search algorithms for the optimum of distorted utility. We then apply the sensitivity‐based approach to the portfolio selection problem under distorted utility with complete and incomplete markets. For the complete markets case, the first‐order condition is derived and optimal wealth deduced. For the incomplete markets case, a dual characterization of optimal policies is provided; a solvable incomplete market example with unhedgeable interest rate risk is also presented. We expect this sensitivity‐based approach to be generally applicable to optimization problems involving probability distortions.     

ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES

We provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.     

The Impact of Bilateral Free Trade Agreements on Bilateral Foreign Direct Investment among Developed Countries

Previous studies find that a trade treaty positively impacts foreign direct investment (FDI). But does a trade treaty always have positive effects on FDI? What is the effect of bilateral free trade agreement (FTA) on bilateral FDI among developed countries? Based on the Knowledge‐Capital model, I hypothesize that bilateral FTA has negative effects on bilateral FDI in developed–developed country pairs, but positive effects in developed–developing country pairs. To test this hypothesis empirically, I conduct the within estimator, the Difference‐in‐Difference estimator and the Arellano–Bond estimator with panel data of bilateral FTA and outward FDI in 30 OECD countries and 32 non‐OECD countries between 1982 and 2005. The result supports the hypothesis. The existence of bilateral FTA decreases bilateral FDI in the OECD–OECD country pairs but increases bilateral outward FDI in the OECD–non‐OECD country pairs. The finding of negative effects of bilateral FTA on FDI is robust to different country classifications by gross national income (GNI) per capita and secondary school enrolment. Hence, the results are consistent with what Carr et al. (2001) predicts about the effects of trade cost on FDI in developed–developed country pairs and in developed–developing country pairs.     

On utility maximization under model uncertainty in discrete‐time markets

We study the problem of maximizing terminal utility for an agent facing model uncertainty, in a frictionless discrete‐time market with one safe asset and finitely many risky assets. We show that an optimal investment strategy exists if the utility function, defined either on the positive real line or on the whole real line, is bounded from above. We further find that the boundedness assumption can be dropped, provided that we impose suitable integrability conditions, related to some strengthened form of no‐arbitrage. These results are obtained in an alternative framework for model uncertainty, where all possible dynamics of the stock prices are represented by a collection of stochastic processes on the same filtered probability space, rather than by a family of probability measures.