Optimal insurance contract under VaR and CVaR constraints

This study endogenously develops an optimal insurance contractual form for maximizing insured expected utility under VaR and CVaR constraints. We find that CVaR constraint does not affect the contractual form, but may increase minimum insurance premium requirement. Additionally, when the VaR constraint is binding, the optimal contract is a double deductible insurance. However, if the contract is restricted to a regular form (both indemnity schedule and retained loss schedule are continuously nondecreasing) for avoiding moral hazard problem, the optimal contract is a piecewise linear deductible insurance. Finally, we provide intuitive comparison between this study result and relevant studies.     

Ambiguity on the insurer’s side: The demand for insurance

Empirical evidence suggests that ambiguity is prevalent in insurance pricing and underwriting, and that often insurers tend to exhibit more ambiguity than the insured individuals (e.g., Hogarth and Kunreuther, 1989). Motivated by these findings, we consider a problem of demand for insurance indemnity schedules, where the insurer has ambiguous beliefs about the realizations of the insurable loss, whereas the insured is an expected-utility maximizer. We show that if the ambiguous beliefs of the insurer satisfy a property of compatibility with the non-ambiguous beliefs of the insured, then optimal indemnity schedules exist and are monotonic. By virtue of monotonicity, no ex-post moral hazard issues arise at our solutions (e.g., Huberman et al., 1983). In addition, in the case where the insurer is either ambiguity-seeking or ambiguity-averse, we show that the problem of determining the optimal indemnity schedule reduces to that of solving an auxiliary problem that is simpler than the original one in that it does not involve ambiguity. Finally, under additional assumptions, we give an explicit characterization of the optimal indemnity schedule for the insured, and we show how our results naturally extend the classical result of Arrow (1971) on the optimality of the deductible indemnity schedule.     

指数风险偏好下的最优保险策略

本文讨论了当投保个体和保险公司为指数风险偏好时，在保费约束下投保个体的最优保险策略问题。本文采用求解对偶优化问题的方法求解这个问题，并给出当损失服从指数分布时最优保险策略解的解析式。本文最后讨论了投保个体和保险公司风险厌恶程度以及保费预算变化对个体最优保险策略的影响。     

An optimal insurance design problem under Knightian uncertainty

This paper solves an optimal insurance design problem in which both the insurer and the insured are subject to Knightian uncertainty about the loss distribution. The Knightian uncertainty is modeled in a multi-prior g-expectation framework. We obtain an endogenous characterization of the optimal indemnity that extends classical theorems of Arrow (Essays in the Theory of Risk Bearing. Markham, Chicago 1971) and Raviv (Am Econ Rev 69(1):84–96, 1979) in the classical situation. In the presence of Knightian uncertainty, it is shown that the optimal insurance contract is not only contingent on the realized loss but also on another source of uncertainty coming from the ambiguity.     

On the demand generated by a smooth and concavifiable preference ordering

It is shown that if a consumer's preference ordering is strictly convex and is representable by means of a concave, twice continuously differentiable utility function, then the partial derivative of a demanded commodity with respect to its price is bounded from above in a neighborhood of a price vector at which the demand fails to be differentiable. In the case of two commodities, if the demand does not possess finite derivatives with respect to prices at a certain point, then the partial ‘derivative’ of a commodity with respect to its price is equal to minus infinity. The same result holds for n commodities under ‘almost every’ choice of coordinates in the commodity space. If preferences are weakly convex but the same representation assumption holds, demand may not be single-valued but own-price difference quotients are still bounded from above.     

Production planning and inventories optimization: A backward approach in the convex storage cost case

We study the deterministic optimization problem of a profit-maximizing firm which plans its sales/production schedule. The firm controls both its production and sales rates and knows the revenue associated to a given level of sales, as well as its production and storage costs. The revenue and the production cost are assumed to be respectively concave and convex. In Chazal et al. [Chazal, M., Jouini, E., Tahraoui, R., 2003. Production planning and inventories optimization with a general storage cost function. Nonlinear Analysis 54, 1365–1395], we provide an existence result and derive some necessary conditions of optimality. Here, we further assume that the storage cost is convex. This allows us to relate the optimal planning problem to the study of a backward integro-differential equation, from which we obtain an explicit construction of the optimal plan.     

Are generalized call-spreads efficient?

In order to explain coexistence of a deductible for low values of the loss and an upper limit for high values of the loss in insurance contracts, we consider the exchange of risk between two rank dependent expected utility maximizers. It is shown that if the insurer (insured) takes more into account the lowest outcomes – hence maximal losses – than the insured (insurer), then the optimal contract has an upper limit (includes a deductible for high values of the loss). If furthermore, the insured (insurer) neglects the highest outcomes while the insurer (insured) does not, the optimal contract includes a deductible (full insurance) for low values of the loss.     

Periodic learning about a hidden state variable

In active learning models the value function is necessarily convex in the priors. Hence, in combination with a concave objective, the decision problem need not become concave so that nonregularity problems are inherent. This paper considers an objective that unambiguously implies a quasi-convex decision problem and highlights the effect of the inherent nonregularities on active learning. A trigger policy for learning is shown to be optimal: the minimum amount of learning is optimal until uncertainty surpasses a critical value. At this trigger point the maximum amount of learning is chosen, uncertainty falls temporarily, and the cycle then repeats itself.     

A central limit theorem for the Hellinger loss of Grenander‐type estimators

We consider Grenander‐type estimators for a monotone function , obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.     

Isotonic estimators of monotone densities and distribution functions: basic facts

We present short proofs of some basic results from isotonic regression theory. A straightforward argument is given to show that the left continuous version of the concave majorant of the empirical distribution function maximizes the likelihood function f↦f (X,)… f (X _n ) within the class of non-increasing densities. Similarly, it is shown that the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of interval censored data has an interpretation in terms of the left derivative of a convex minor ant. Finally, a short proof is given to show that the number of vertices of the concave major ant of the uniform empirical distribution function is asymptotically normal with asymptotic mean and variance both equal to log n .     

Growth and the diffusion of ideas

In a recent model of growth developed by Lucas (Lucas, R., 2009. Ideas and growth. Economica 76, 1–19), a continuum of people interact in a random manner and copy each other’s productive ideas when it is economically beneficial to do so. This paper extends the Lucas model by assuming that each person’s productivity also experiences random shocks due to individual discovery. A nonlinear partial differential equation is derived for the distribution of income, which admits a traveling wave solution representing a growing economy. The growth rate is an increasing function of the rate of imitation. The growth rate is also an increasing but concave function of population size and reaches a plateau in the continuum limit. Hence the scale effect is bounded. The model is extended to account for a nonzero cost of imitation, with similar results. The mathematical tools presented in this paper should prove useful in developing idea-based models of growth.     

Gamma-Minimax Prediction for the Multinomial Distribution

Characterizations of gamma-minimax predictors for the linear combinations of the unknown parameter and the random variable having the multinomial distribution under arbitrary squared error loss are established in two situations – when the sample size is fixed and when the sample size is a realization of a random variable. It is always assumed that the available vague prior information about the unknown parameter can be described by a class of priors whose vector of first moments belongs to a suitable convex and compact set. Several known gamma-minimax and minimax results can be obtained from the characterizations derived in the present paper.     

Turnpike property and convergence rate for an investment model with general utility functions

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.     

Optimal dividend strategies with time-inconsistent preferences

This paper studies the optimal dividend strategies of an insurance company when the manager has time-inconsistent preferences. We consider the problem for a naive manager and a sophisticated manager, and analytically derive the optimal dividend strategies when claim sizes follow an exponential distribution. Our results show that the manager with time-inconsistent preferences tends to pay out dividends earlier than her time-consistent counterpart and that the sophisticated manager is more inclined to pay out dividends than the naive manager. Furthermore, we extend these results to the case with claim sizes following a mixed exponential distribution, and provide a numerical analysis to reveal the sensitivity of the optimal dividend strategies to changes in the premium, claims and surplus volatility.     

International tenders and futures hedging

This paper examines the optimal bidding and hedging decisions of a risk‐averse firm that takes part in an international tender. The firm faces multiple sources of uncertainty: exchange rate risk, risk of an unsuccessful tender, and business risk. The firm is allowed to trade unbiased currency futures contracts to imperfectly hedge its contingent foreign exchange risk exposure. We show that the firm shorts less (more) of the unbiased futures contracts when its marginal utility function is convex (concave) as compared with the case that the marginal utility function is linear. We further show that the curvature of the marginal utility function plays a decisive role in determining the impact of currency futures hedging on the firm's bidding behavior. Sufficient conditions that ensure the firm bids more or less aggressively than in the case without hedging opportunities are derived. Copyright © 2006 John Wiley & Sons, Ltd.     

SOCIAL SECURITY AND RETIREMENT DECISION: A POSITIVE AND NORMATIVE APPROACH*

Abstract Social insurance for the elderly is judged responsible for the widely observed trend towards early retirement. In a world of laissez‐faire or in a first‐best setting, there would be no such trend. However, when first‐best instruments are not available, because health and productivity are not observable, the optimal social insurance policy may imply a distortion on the retirement decision. The main point we make is that while there is no doubt that retirement systems induce an excessive bias towards early retirement in many countries, a complete elimination of this bias (i.e. a switch to an actuarially fair system) is not the right answer for two reasons. First, some distortions are second‐best optimal. This is the normative argument. Second, and on the positive side, the elimination of the bias might be problematic from a political perspective. Depending on the political process, either it may not be feasible or alternatively it may tend to undermine the political support for the pension system itself.     

Risk aversion for variational and multiple-prior preferences

The objective of this paper is to identify variational preferences and multiple-prior (maxmin) expected utility functions that exhibit aversion to risk under some probability measure from among the priors. Risk aversion has profound implications on agents’ choices and on market prices and allocations. Our approach to risk aversion relies on the theory of mean-independent risk of Werner (2009). We identify necessary and sufficient conditions for risk aversion of convex variational preferences and concave multiple-prior expected utilities. The conditions are stability of the cost function and of the set of probability priors, respectively, with respect to a probability measure. The two stability properties are new concepts. We show that cost functions defined by the relative entropy distance or other divergence distances have that property. Set of priors defined as cores of convex distortions of probability measures or neighborhoods in divergence distances have that property, too.     

Evaluating the Adequacy of Parametric Functional Forms in Estimating Monotone and Concave Production Functions

We consider situations where the a priori guidance provided by theoretical considerations indicates only that the function linking the endogenous and exogenous variables is monotone and concave (or convex). We present methods to evaluate the adequacy of a parametric functional form to represent the relationship given the minimal maintained assumption of monotonicity and concavity (or convexity). We evaluate the adequacy of an assumed parametric form by comparing the deviations of the fitted parametric form from the observed data with the corresponding deviations estimated under DEA. We illustrate the application of our proposed methods using data collected from school districts in Texas. Specifically, we examine whether the Cobb–Douglas and translog specifications commonly employed in studies of education production are appropriate characterizations. Our tests reject the hypotheses that either the Cobb–Douglas or the translog specification is an adequate approximation to the general monotone and concave production function for the Texas school districts.     

Optimal investment policy of the regulated firm

The main purpose of this paper is to investigate the optimal investment behaviour of a regulated firm and to show that the Averch-Johnson effect of overcapitalization does not necessarily occur in a dynamic model under the assumption of concave revenue function. There may be three cases of undercapitalization, overcapitalization, and neutralization. As the result, whether the Averch-Johnson effect appears or not is crucially dependent on the regulated rate of return and the firm's planning horizon.     

A Reexamination of Yardstick Competition

This paper shows that yardstick competition does not assist a regulator when lump-sum transfers are not costly and the regulator does not care about the distribution of income. Yardstick competition may discourage investment that would make efficient operation possible. The paper characterizes optimal regulatory schemes in a simple model and demonstrates that it may be optimal to limit the amount of information available to the regulator.