Equilibrium in Competitive Insurance Markets Under Adverse Selection and Yaari's Dual Theory of Risk

Under Yaari's dual theory of risk, we determine the equilibrium separating contracts for high and low risks in a competitive insurance market, in which risks are defined only by their expected losses, that is, a high risk is a risk that has a greater expected loss than a low risk. Also, we determine the pooling equilibrium contract when insurers are assumed non-myopic. Expected utility theory generally predicts that optimal insurance indemnity payments are nonlinear functions of the underlying loss due to the nonlinearity of agents' utility functions. Under Yaari's dual theory, we show that under mild technical conditions the indemnity payment is a piecewise linear function of the loss, a common property of insurance coverages.     

Public Versus Private Insurance with Dual Theory: A Political Economy Argument

This paper analyzes the political support for public insurance in the presence of a private insurance alternative. The public insurance is compulsory and offers a uniform insurance policy. The private insurance is voluntary and can offer different insurance policies. Adopting Yaari's [Econometrica, 55, 95–115, 1987] dual theory to expected utility (i.e., risk aversion without diminishing marginal utility of income), we show that adverse selection on the private insurance market may lead a majority of individuals to prefer public insurance over private insurance, even if the median risk is below the average risk (so that the median actually subsidizes high-risk individuals). We also show that risk aversion makes public insurance more attractive and that the dual theory is less favourable to a mixed insurance system than the expected utility framework. Lastly, we demonstrate how the use of genetic tests may threaten the political viability of public insurance.     

Saving under uncertainty: A bivariate non-expected utility approach

We adopt the multivariate non-expected utility approach proposed by Yaari [1986] to provide a characterization of the comparative statics effects of greater risk aversion and of mean-preserving increases in risk on saving and borrowing in the presence of income and interest rate risk.We show that in Yaari's model, it is possible to extend the applicability of the Diamond and Stiglitz [1974] and Kihlstrom and Mirman [1974] (DSKM) single-crossing property to establish a relationship between greater risk aversion and saving (or borrowing) on the basis of the individual's ordinal preferences as long as the two risks are independent. We also demonstrate that the comparative statics effects of a joint mean-preserving increase in random income and interest rate on saving and borrowing can be determined by an extension of the DSKM single-crossing property.     

Optimal portfolios under worst-case scenarios

In standard portfolio theories such as Mean–Variance optimization, expected utility theory, rank dependent utility heory, Yaari’s dual theory and cumulative prospect theory, the worst outcomes for optimal strategies occur when the market declines (e.g. during crises), which is at odds with the needs of many investors. Hence, we depart from the traditional settings and study optimal strategies for investors who impose additional constraints on their final wealth in the states corresponding to a stressed financial market. We provide a framework that maintains the stylized features of the SP/A theory while dealing with the goal of security in a more flexible way. Preferences become state-dependent, and we assess the impact of these preferences on trading decisions. We construct optimal strategies explicitly and show how they outperform traditional diversified strategies under worst-case scenarios.     

Life insurance decisions under recursive utility

In this paper, we generalize recursive utility to include lifetime uncertainty and utility from bequest. The generalization applies to discrete-time as well as continuous-time recursive utility, and it is an important step forward in the development of recursive utility. We formalize the problem of optimal consumption, investment, and life insurance choice under recursive utility, and we state a verification theorem with a generalized Hamilton-Jacobi-Bellman equation. Our generalization of recursive utility allows us to study optimal consumption, investment, and life insurance choice under separation of (market) risk aversion, elasticity of inter-temporal substitution, and elasticity of substitution between bequest and future utility. The separation gives rise to hump-shaped consumption patterns as observed in realized consumption.     

Increasing return response to changes in risk

Risk aversion theory is based on an individual's choice among risky assets with expected utility in its foundation. It is about investor behavior (i.e., investor choice), under normal circumstances, toward assets with various levels of risk. A positive and marginally diminishing relationship between risk and return exists. This study is about investor behavior related to their response (not choice) to risk. We present an argument and supporting evidence that investors’ return response to risk is increasing with the level of risk. Thus, investor behavior is subject to change and the level of risk is a determinant of such change. We also explain the negative time‐series correlation between risk and return.     

Communication in Private-Information Models: Theory and Computation

Communication and no-communication versions of a two-stage principal-agent model are compared. The models contain a risk-averse agent and two sources of private information, a shock to preferences followed by a productive action. Both models are formulated as linear programs, which are then used to compute solutions to examples. For the communication model, an alternative method of accounting for the utility from off-equilibrium strategies is derived. This method greatly reduces the size of the linear program. For the no-communication model a Revelation-Principle like proof is provided. In simple cases, a sufficient condition for communication to be valuable is derived. In these cases, communication improves risk-sharing in bad states of the world. In more complicated cases, computed examples demonstrate how communication may also alter labor supply. Further examples demonstrate how action and consumption lotteries may separate agents by their shock.     

Portfolio optimization under convex incentive schemes

We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function g of the terminal wealth. The manager’s own utility function U is assumed to be smooth and strictly concave; however, the resulting utility function U°g fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor.     

Multivariate utility maximization with proportional transaction costs

We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor’s preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for this include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer.     

Convex duality in optimal investment and contingent claim valuation in illiquid markets

This paper develops duality theory for optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into three terms, corresponding to the agent’s risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid when the market model satisfies an appropriate generalization of the no-arbitrage condition and the agent’s utility function satisfies an appropriate generalization of asymptotic elasticity conditions. When applied to classical liquid market models or models with bid–ask spreads, we recover well-known pricing formulas in terms of martingale measures and consistent price systems. Building on the general theory of convex stochastic optimization, we also obtain optimality conditions in terms of an extended notion of a “shadow price”. The results are illustrated by establishing the existence of solutions and optimality conditions for the nonlinear market models recently proposed in the literature. Our results allow significant extensions including nondifferentiable trading costs which arise, e.g., in modern limit order markets where the marginal price curve is necessarily discontinuous.     

Consumption-based capital asset pricing models: issues and controversies

This paper discusses the issues and controversies surrounding consumption-based capital asset pricing models (CCAPMs). While CCAPMs provide a chance to explain the phenomena observed in stock markets, their viability is jeopardized owing to the weak predictability of the equity premium and risk-free rate puzzles. Even given market frictions and market incompleteness, CCAPMs must test their validity constantly in the face of the formidable challenges of rival models. Measurement error with respect to time aggregation is also regarded as a major threat, causing the low volatility of consumption and eventually resulting in chaining itself to weak return predictability. In addition, the dual choice problem of portfolio and consumption rooted in CCAPMs guides us into how investors accumulate wealth through the financial market to reach the zenith of expected utility. This paper offers insights as well as understanding into the behavior of an agent and market phenomena in the context of a consumption-based economy.     

Using vNM expected utility theory to facilitate the decision-making in social ethics

The overall objective of this article is to demonstrate that in social ethics, certain problems related to decision-making are easier to resolve using conceptual tools borrowed from mathematics than using philosophical ethics theories, such as classical utilitarianism. With the help of a case study, the first part of the article will attempt to point out that if an agent bases his reasoning on the verbal and purely qualitative concepts of the classical utilitarian theory, he will find himself confronting ‘undecidable’ dilemmas, for which making a specific choice becomes almost arbitrary. The second part of the article proposes a more formal quantification of utility and attitude towards risk that can help the agent to overcome the uncertainties emanating from a strictly qualitative perception of the real world’s configuration. This method for decision-making is inspired by the works of Howard Raiffa, John von Neumann and Oskar Morgenstern.     

Optimal tax policy and expected longevity: a mean and variance utility approach

This paper studies the normative problem of redistribution between agents who can influence their survival probability through private health spending, but who differ in their attitude towards the risks involved in the lotteries of life to be chosen. For that purpose, a two-period model is developed, where agents’ preferences on lotteries of life can be represented by a mean and variance utility function allowing, unlike the expected utility form, some sensitivity to what Allais (Econometrica 21(4), 503–546, 1953) calls the ‘dispersion of psychological values’. It is shown that if agents ignore the impact of health spending on the return of their savings, the decentralization of the first-best utilitarian optimum requires intergroup lump sum transfers and group-specific positive taxes on health spending. Under asymmetric information, a differentiated taxation across agents is still required, but subsidizing health spending may be optimal as a way to solve the incentive problem.     

CAPM and Option Pricing With Elliptically Contoured Distributions

This article offers an alternative proof of the capital asset pricing model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modeling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modeling the thickness of tails associated with the possibility that asset returns take extreme values with nonnegligible probabilities. As summarized in this article, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Stein's lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is log‐elliptically distributed. The Black–Scholes call option price is a special case of this general result when the underlying is log‐normally distributed.     

Prudence, Demand Uncertainty, Background Risk, and the Law of Supply: A Nonexpected Utility Approach to the Firm

We identify two motives, prudence and risk aversion, which give rise to precautionary behavior for a quantity- or price-setting monopolist facing demand uncertainty who has dual theoretic preferences. We also analyze a piecewise linear profit function due to a tax on profits that varies with the profit level. We show that the comparative statics of greater risk (mean-preserving spread and mean-utility preserving spread) can be totally or partially determined by the Diamond-Stiglitz and Kihlstrom-Mirman single-crossing property. For example, for a prudent risk-averse quantity-setting dual theoretic monopolist, a mean-preserving spread will have the same impact on output under uncertainty as a fall in the state of demand under certainty. Finally, we find that, in contrast to expected utility, a stochastically larger state of demand (first-order stochastic dominance) will raise output even if background risk is present.     

The design of an optimal insurance contract for irreplaceable commodities

This paper discusses optimal insurance contract for irreplaceable commodities. To describe the dual impacts on individuals when a loss occurs to the insured irreplaceable commodities, we use a state-dependent and bivariate utility function, which includes both the monetary wealth and sentimental value as two arguments. We show that over (full, partial) insurance is optimal when a decrease in sentimental value will increase (not change, decrease, respectively) the marginal utility of monetary wealth. Moreover, a non-zero deductible exists even without administration costs. Furthermore, we demonstrate that a positive fixed reimbursement is optimal if (1) the premium is actuarially fair, (2) the monetary loss is a constant, and (3) the utility function is additively separable and the marginal utility of money is higher in the loss state than in the no-loss state. We also characterize comparative statics of fixed-reimbursement insurance under an additively separable preference assumption. JEL Classification G22 · D86 The author acknowledge funding from National Science Council in Taiwan (NSC93-2416-H-130-020).     

Heterogeneity and Lotteries in Monetary Search Models

We introduce ex ante heterogeneity into the Berentsen, Molico, and Wright monetary search model with lotteries. We show that their three main results regarding lotteries do not survive this modification of the environment.