State-dependent Utility Function Research Articles

Near-optimal asset allocation in financial markets with trading constraints

We develop a dual-control method for approximating investment strategies in multidimensional financial markets with convex trading constraints. The method relies on a projection of the optimal solution to an (unconstrained) auxiliary problem to obtain a feasible and near-optimal solution to the original problem. We obtain lower and upper bounds on the optimal value function using convex duality methods. The gap between the bounds indicates the precision of the near-optimal solution. We illustrate the effectiveness of our method in a market with different trading constraints such as borrowing, short-sale constraints and non-traded assets. We also show that our method works well for state-dependent utility functions.

Concavity, stochastic utility, and risk aversion

This paper studies the relation between concavity, stochastic or state-dependent utility functions, and risk aversion. Using the common definition of risk aversion, but modified for state-dependent preferences, we show that concavity does not imply risk aversion. Instead, it implies a weaker version of risk aversion, defined herein, and called risk aversion for independent gambles. Furthermore, to characterise the economic meaning of concavity, we define two new risk aversion notions, called uniform risk aversion and uniform risk aversion for independent gambles, respectively. We show that concavity is equivalent to uniform risk aversion for independent gambles, and that concavity plus some additional conditions are equivalent to uniform risk aversion.

Concavity, Stochastic Utility, and Risk Aversion

This paper studies the relation between concavity, stochastic or state dependent utility functions, and risk aversion. Using the common definition of risk aversion, but modified for state dependent preferences, we show that concavity does not imply risk aversion. Instead, it implies a weaker version of risk aversion, defined herein, and called risk aversion for independent gambles. Furthermore, to characterize the economic meaning of concavity, we define two new risk aversion notions, called uniform risk aversion and uniform risk aversion for independent gambles, respectively. We show that concavity is equivalent to uniform risk aversion for independent gambles, and that concavity plus some additional conditions are equivalent to uniform risk aversion.

A new approach to the rational expectations equilibrium: existence, optimality and incentive compatibility

Rational expectations equilibrium seeks a proper treatment of behavior under private information by assuming that the information revealed by prices is taken into account by consumers in their decisions. Typically agents are supposed to maximize a conditional expectation of state-dependent utility function and to consume the same bundles in indistiguishable states [see Allen (Econometrica 49(5):1173–1199, 1981), Radner (Econometrica 47(3):655–678, 1979)]. A problem with this model is that a rational expectations equilibrium may not exist even under very restrictive assumptions, may not be efficient, may not be incentive compatible, and may not be implementable as a perfect Bayesian equilibrium (Glycopantis et al. in Econ Theory 26(4):765–791, 2005). We introduce a notion of rational expectations equilibrium with two main features: agents may consume different bundles in indistinguishable states and ambiguity is allowed in individuals’ preferences. We show that such an equilibrium exists universally and not only generically without freezing a particular preferences representation. Moreover, if we particularize the preferences to a specific form of the maxmin expected utility model introduced in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), then we are able to prove efficiency and incentive compatibility. These properties do not hold for the traditional (Bayesian) Rational Expectation Equilibrium.

OPTIMAL PORTFOLIO UNDER STATE-DEPENDENT EXPECTED UTILITY

We derive the optimal portfolio for an expected utility maximizer whose utility does not only depend on terminal wealth but also on some random benchmark (state-dependent utility). We then apply this result to obtain the optimal portfolio of a loss-averse investor with a random reference point (extending a result of Berkelaar et al. (2004) Optimal portfolio choice under loss aversion, The Review of Economics and Statistics 86 (4), 973–987). Clearly, the optimal portfolio has some joint distribution with the benchmark and we show that it is the cheapest possible in having this distribution. This characterization result allows us to infer the state-dependent utility function that explains the demand for a given (joint) distribution.

Optimal Portfolio Under State-Dependent Expected Utility

We derive the optimal portfolio for an expected utility maximizer whose utility does not only depend on terminal wealth but also on some random benchmark (state-dependent utility). We then apply this result to obtain the optimal portfolio of a loss-averse investor with a random reference point (extending a result of Berkelaar et al. 2004). Clearly, the optimal portfolio has some joint distribution with the benchmark and we show that it is the cheapest possible in having this distribution. This characterization result allows us to infer the state-dependent utility function that explains the demand for a given (joint) distribution.

Optimal reinsurance and investment strategies for insurers with regime-switching and state-dependent utility function

This paper considers a proportional reinsurance-investment problem and an excess-of-loss reinsurance-investment problem for an insurer, where price processes of the risky assets and wealth process of the insurer are both described by Markovian regime switching. The target of the insurer is assumed to maximize the expected exponential utility from her terminal wealth with a state-dependent utility function. By employing the dynamic programming approach, the optimal value functions and the optimal reinsurance-investment strategies are derived. In addition, the impact of some parameters on the optimal strategies and the optimal value functions is analyzed, and lots of interesting results are discovered, such as the conclusion that excess-of-loss reinsurance is better than proportional reinsurance is not held in the regime-switching jump-diffusion model.

An expected utility theory for state-dependent preferences

This note is a generalization and improved interpretation of the main result of Karni and Schmeidler (An Expected utility theory for state-dependent preferences. Working paper no. 48-80 of the Foerder Institute for Economic Research, Faculty of Social Sciences, Tel Aviv University, 1980). A decision-maker is supposed to possess a preference relation on acts and another preference relation on state-prize lotteries, both of which are assumed to satisfy the von Neumann–Morgenstern axioms. In addition, the two preference relations restricted to a state of nature are assumed to agree. We show that these axioms are necessary and sufficient for the existence of subjective expected utility over acts with state-dependent utility functions and a subjective probability measure. This subjective probability measure is unique when conditioned on the set of states of nature in which not all the prizes are equally desirable.

The Impact of the Health Status on Consumption Behavior

How do individuals adjust their consumption behavior in response to a deterioration of their health status? In the light of demographic transition, when the number of individuals who reach a critical age where the probability of functional limitations increases dramatically, understanding how the individual health status influences consumption behavior deserves additional attention. While the theoretical framework is well established and builds on the concept of state dependent utility functions, previous empirical studies either focused on experimental designs and hypothetical questions or did not distinguish between the effects from aging and those effects from changes in the individual health status. This research paper uses empirical data, provided by the Health and Retirement Study and the related Consumption and Activities Mail Survey, to explore the consequences of functional problems in activities of daily living on consumption behavior of individuals above the age of 65; furthermore, the analysis is restricted to consumption of non durable and semi durable goods and dis-aggregates overall consumption into nine global consumption categories. First, the empirical analysis explores a system of seemingly unrelated regressions to estimate the impact of the health status on the relative composition of the consumption basket with respect to different categories of goods and services. Second, a fixed effects model is specified to focus on changes which occur within individuals and contribute to explanations how individual consumption behavior changes over time. Regression results show expected signs and significant coefficients for the impact of changes in the individual health on consumption behavior. Controlling for a powerful set of variables, findings from the empirical analysis separate out a change in consumption that can be solely associated to changes in the health status.

Optimal investment and consumption strategies with state-dependent utility functions and uncertain time-horizon

This paper considers the multi-period optimal strategies for an investment-only problem and an investment–consumption problem. The financial market is regime-switching and consists of one risk-free asset and multiple risky assets. The state process of the financial market is modeled by a finite-state Markov chain. Asset returns and utility functions are affected by the states of the financial market. The investment time-horizon is uncertain and exogenous. By adopting the dynamic programming approach, explicit expressions for optimal value functions and optimal investment and consumption strategies are derived. Moreover, some discussions and numerical examples are provided to illustrate our results, which extend some results in the existing literature to more general situations and show some interesting phenomena.

Pricing high-risk and low-risk insurance contracts with incomplete information and production costs

We consider the traditional model of an insurance market that consists of high-risk and low-risk individual customers who are identical except for their accident probabilities. Though insurers know the values of the high-risk and low-risk accident probabilities, each individual customer’s accident probability is unknown to the insurer. It is well known that if individual customers have state-independent utility functions, and insurers incur neither production costs nor interest costs, then in competitive markets with imperfect information on accident probabilities, if an equilibrium exists it entails separate contracts with the high-risk individuals obtaining complete insurance and low-risk individuals obtaining partial insurance. While in monopolistic markets with imperfect information on accident probabilities, the following four properties hold: (i) high-risk and low-risk individuals never purchase the same insurance policy (i.e., pooling is never optimal); (ii) the optimal contract for the high-risk individual is complete insurance; (iii) if the low-risk individual buys insurance, his/her utility is essentially the same as it would have been had he/she not purchased any insurance; and (iv) there exists a critical (finite) ratio of high- to low-risk individuals such that if the actual ratio exceeds the critical ratio, the low-risk individuals purchase no insurance. In this paper we will extend the traditional model by assuming that individual consumers have a common state-dependent utility function and assume insurers incur production costs that are proportional to the amount of insurance purchased and to the premium charged as well as interest costs. We derive results for both competitive markets and monopolistic markets with imperfect information on accident probabilities. We prove that even though pooling is never optimal in the traditional framework, it may be optimal in our model and high-risk individuals may optimally choose partial insurance. In addition, we develop extensions to the four properties listed above.

Insurance pricing with complete information, state-dependent utility, and production costs

We consider a group of identical risk-neutral insurers selling single-period indemnity insurance policies. The insurance market consists of individuals with common state-dependent utility function who are identical except for their known accident probability q. Insurers incur production costs (commonly called expenses or transaction costs by actuaries) that are proportional to the amount of insurance purchased and to the premium charged. By introducing the concept of insurance desirability, we prove that the existence of insurer expenses generates a pair of constants qmin and qmax that naturally partitions the applicant pool into three mutually exclusive and exhaustive groups of individuals: those individuals with accident probability q∈[0,qmin) are insurable but do not desire insurance, those individuals with accident probability q∈[qmin,qmax] are insurable and desire insurance, and those individuals with accident probability q∈(qmax,1] desire insurance but are uninsurable. We also prove that, depending on the level of q and the marginal rate of substitution between states, it may be optimal for individuals to buy complete (full) insurance, partial insurance, or no insurance at all. Finally, we prove that when q is known in monopolistic markets (i.e., markets with a single insurer), applicants may be induced to “over insure” whenever partial insurance is bought.

Optimal Portfolios in Lévy Markets under State-Dependent Bounded Utility Functions

Motivated by the so-called shortfall risk minimization problem, we consider Merton's portfolio optimization problem in a non-Markovian market driven by a Lévy process, with a bounded state-dependent utility function. Following the usual dual variational approach, we show that the domain of the dual problem enjoys an explicit “parametrization,” built on a multiplicative optional decomposition for nonnegative supermartingales due to Föllmer and Kramkov (1997). As a key step we prove a closure property for integrals with respect to a fixed Poisson random measure, extending a result by Mémin (1980). In the case where either the Lévy measure ν of Z has finite number of atoms or ΔSt/St−=ζtϑ(ΔZt) for a process ζ and a deterministic function ϑ, we characterize explicitly the admissible trading strategies and show that the dual solution is a risk-neutral local martingale.

Collective Risk-Taking Decisions with Heterogeneous Beliefs

Suppose that a group of agents having divergent expectations can share risks efficiently. We examine how this group should behave collectively to manage these risks. We show that the beliefs of the representative agent are in general a function of the group's wealth level, or equivalently, that the representative agent has a state-dependent utility function. We define the individual degree of pessimism as an index measuring probability differences across states. We show that the degree of pessimism of the representative agent is the mean of the individual ones weighted by their index of absolute risk tolerance. From this central result, we show how increasing disagreement on the state probability affects the state probability of the representative agent. We show that the divergence of opinions about the probability of occurence of a boom may help solving the equity premium puzzle.

Optimal full-hedging under state-dependent preferences

This paper examines the hedging behavior of a risk-averse individual who faces uncertainty in spot-market prices and has a state-dependent utility function. The model demonstrates how risk aversion, state-dependency of preferences, and dependence structure of spot-market prices and states of nature jointly determine the individual’s optimal hedge position. We stipulate two sets of necessary and sufficient conditions, one on the utility function and the other on the dependence structure of spot-market prices and states of nature, which yield the celebrated full-hedging theorem originally derived under state-independent preferences.

Corporate Spin-Offs and Closed-End Funds in a State-Preference Framework

Miller's clientele argument for market value subadditivity as an explanation for stockholder wealth gains in corporate spin-offs and discounts from the net asset value of closed-end fund shares is examined in the context of a simple state-preference model. It is shown that binding short sales constraints induce value subadditivity and thus render Miller's clientele argument valid. This is true regardless of whether or not divergence of opinion among investors or state-dependent utility functions exist. In the absence of binding short sales constraints, value additivity prevails and Miller's clientele argument is not viable. Although personal taxes are not considered in the model developed in this paper, it is shown that tax-timing options reinforce the existence of value subadditivity.

Paying for Safety: Voluntary Reduction of Residential Radon Risks

identify radon as a public health problem, to sponsor widespread testing, and to develop reliable, low-cost techniques for reducing indoor exposure levels. Data recently collected in Sweden make it possible to derive the first empirical estimates of homeowner willingness to pay to reduce radon health risks. This paper reports these estimates and discusses their implication for public policy. We conclude that the low value of estimated willingness to pay indicates that homeowners may have interpreted the technology-based recommended exposure as a safety threshold. After reviewing the nature and regulatory history of the radon hazard, we present a theoretical framework for modeling the value of reduced radon risk. The model employs a conventional state-dependent utility function associated with a discrete decision to mitigate or not to mitigate. In subsequent sections we describe the data collected from a sample of homeowners participating in a health department testing program and report parameter estimates from a binomial logit analysis. The estimated parameters form the basis for calculations of the probability of mitigating and willingness to pay for mitigation at various exposure levels and for various household characteristics.

A Transaction-Costs Explanation for Why the Poor Are More Likely to Commit Crime

ONE standard approach to criminal punishment says that the state should set the penalties for criminal offenses to reflect the full costs that these wrongful acts impose on society. Setting optimal penalties in this way, however, raises an intellectual puzzle. Why would we expect to observe any crime if a system of optimal penalties were in place? If the penalty is correctly set so that a shoplifter internalizes the full social cost of stealing a product (at least its foregone price), why doesn't the shoplifter simply buy it? It seems hard to believe that the transaction costs of waiting in line and purchasing the good legally would ever be greater than the costs of stealing the item. Even the most casual shoplifting involves some costs of planning, some effort at avoiding detection, some attempt to avoid capture if discovered, and, of course, some expected legal costs if apprehended, tried, and convicted. Burglary undoubtedly involves even greater costs. On the other hand, murder in the presence of optimal penalties might not be so difficult to explain. For example, difficulties may arise because of state-dependent utility functions. It is possible to offer several explanations for the persistence of crime even when penalties are set at the optimal level. If individuals are risk takers, then the true value to them of stealing an item may be greater than the value of purchasing it legally. Alternatively, Posner has argued that the concern with accidentally imprisoning an innocent individual'

A Revealed Preference Theory for Expected Utility

Abstract : Standard axiomatizations of expected-utility theory envision an agent with fixed probability assessments, who can be observed to choose actions from varying opportunity sets (for instance, pairs of lotteries). These axiomatizations also envision that the agent's preferences among these actions depend on the state of nature only through the state-dependent consequences of the actions, and that these consequences are clearly defined and observable. We suggest that this conception may be an unnecessarily restrictive basis for empirical testing, and instead study the pattern of choices from a fixed set of actions as probability assessments change. We show that maximization of the expectation of a general, state-dependent utility function places nontrivial restrictions on such a choice pattern. These restrictions are completely characterized by a discrete version of an integrability condition.

On the Theory of Rational Insurance Purchasing in a Continuous-Time Model

The determination of the optimal insurance coverage has raised a growing academic interest. The seminal works by Arrow [1963], Smith [1968] and Mossin [1968] contributed to establish the cornerstone results of insurance economics. Indeed, it is now a well-known result that a risk averse expected utility maximizer will never purchase full insurance coverage if the insurance premium is actuarially unfair A positive amount deductible is always optimal from the insured's point of view if the premium is based on expected insurance payments and includes a positive proportional loading. Arrow [1974] extended this result to the case of state-dependent utility functions. Some authors have however provided results which stand in sharp contrast with the above mentiond rule. Razin [1976], for instance, argued that the nature of results obtained very much depends on the decision framework. Using a binominal distribution of losses and Savage's minimax regret criterion as an alternative to the expected utility framework, he showed that it is always optimal to purchase insurance with a positive amount deductible even if the insurance premium is fair. In the same vein, Briys and Louberge [1985] obtained that, under the Hurwicz criterion, full insurance is very often optimal even under a non zero insurance loading. These results may seem conflicting. They nevertheless have a common basis: they focus on insurance in isolation. The optimal insurance coverage is determined in a single risk setting where consumption and portfolio decicions are not considered. In other words, previous studies assume either explicitly or implicitly that the insurance decision is perfectly separable from any other decisions. This strong assumption is obviously questionable. As recently pointed out by Doherty and Schlesinger [1983], the result by Smith and Mossin only holds if insurable and non insurable losses are non positively correlated. For a positive correlation between these losses more than full coverage is requested by risk averse individuals operating in incomplete markets.