Abstracts from Mario Ghossoub’s Papers on Risk
Research
- Equilibria and efficient allocations in risk-sharing markets.
- The effect of non-standard preferences of the agents (e.g., ambiguity, probability weighting, etc.), non-linear pricing (e.g., Choquet pricing), and market frictions (e.g., transaction costs) on the shape of Pareto optima and the structure of equilibria.
- Applications to specific risk-sharing markets, such as (centralized) insurance markets and decentralized risk-sharing markets (e.g., peer-to-peer insurance).
Citations from Google Scholar on 2023-12-23.
Ghossoub et al. (2023)
Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions
We consider the problem of determining an upper bound for the value of a spectral risk mea- sure of a loss that is a general nonlinear function of two factors whose marginal distributions are known, but whose joint distribution is unknown. The factors may take values in complete separa- ble metric spaces. We introduce the notion of Maximum Spectral Measure (MSP), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSP admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem, but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distribu- tions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined.
Birghila et al. (2023)
Optimal insurance under maxmin expected utility
We examine a problem of demand for insurance indemnification, when the insured is sensi- tive to ambiguity and behaves according to the Maxmin-Expected Utility model of Gilboa and Schmeidler [43], whereas the insurer is a (risk-averse or risk-neutral) Expected-Utility maximizer. We characterize optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, marginal utility of wealth, belief heterogeneity, as well as ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how the singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a R´enyi ambiguity set.
Boonen and Ghossoub (2020a)
Boonen and Ghossoub (2020b)
Bilateral Risk Sharing with Heterogeneous Beliefs and Exposure Constraints
This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank- dependent utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level ofbeliefheterogeneity.We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue’s Decomposition Theorem.
Boonen and Ghossoub (2019), 26 citations
On the existence of a representative reinsurer under heterogeneous beliefs.
This paper studies a one-period optimal reinsurance design model with n reinsurers and an insurer. The reinsurers are endowed with expected-value premium principles and with heterogeneous beliefs regarding the underlying distribution of the insurer’s risk. Under general preferences for the insurer, a representative reinsurer is characterized. This means that all reinsurers can be treated collectively by means of a hypothetical premium principle in order to determine the optimal total risk that is ceded to all reinsurers. The optimal total ceded risk is then allocated to the reinsurers by means of an explicit solution. This is shown both in the general case and under the no-sabotage condition that avoids possible ex post moral hazard on the side of the insurer, thereby extending the results of Boonen et al. (2016). We subsequently derive closed-form optimal reinsurance contracts in case the insurer maximizes expected net wealth. Moreover, under the no-sabotage condition, we derive optimal reinsurance contracts in case the insurer maximizes dual utility, or in case the insurer maximizes a generic objective that preserves second-order stochastic dominance under the assumption of a monotone hazard ratio.
Ghossoub (2019a), 11 citations
Budget-constrained optimal insurance without the nonnegativity constraint on indemnities.
In a problem of Pareto-efficient insurance contracting (bilateral risk sharing) with expected- utility preferences, Gollier [28] relaxes the nonnegativity constraint on indemnities and argues that the existence of a deductible is only due to the variability in the cost of insurance, not the nonnegativity constraint itself. In this paper, we find support for a similar statement in problems of budget-constrained optimal insurance (i.e., demand for insurance). Specifically, we consider a setting of ambiguity (unilateral and bilateral) and a setting of belief heterogeneity. We drop the nonnegativity constraint and assume no cost (or a fixed cost) to the insurer, and we derive closed-form solutions to the problems that we formulate. In particular, we show that optimal indemnities no longer include a deductible provision; and
Ghossoub (2019b), 24 citations
Budget-constrained optimal insurance with belief heterogeneity
We re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss. For ease of comparison with the classical literature, we adopt the original setting of Arrow [4], but allow for divergence in beliefs between the insurer and the insured; and in particular for singularity between these beliefs, that is, disagreement about zero-probability events. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Under a mild consistency requirement between these beliefs that is weaker than the Monotone Likelihood Ratio (MLR) condition, we characterize the optimal indemnity and show that it has a simple two-part structure: full insurance on an event to which the insurer assigns zero probability, and a variable deductible on the complement of this event. As an example, we examine the important special case of an Esscher premium principle.
Ghossoub (2019c), 34 citations
Optimal insurance under rank-dependent expected utility
We re-visit the problem of optimal insurance design under Rank-Dependent Expected Utility (RDEU) examined by Bernard et al. [12] and Xu et al. [72]. Unlike the latter, we do not impose the no sabotage condition on admissible indemnities, that is, the comonotonicity of indemnity functions and retention functions with the loss. Rather, in a departure from the aforementioned work, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, hence automatically ruling out any ex post moral hazard that could otherwise arise from possible misreporting of the loss by the insured. Hence, monotonicity properties of indemnification schedules become of second-order concern. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. [12] and Xu et al. [72]. We then extend the setting by allowing for a distortion premium principle, with a distortion function that differs from that of the insured, and we provide a characterization of the optimal retention in that case.
Ghossoub (2017), 43 citations
Arrow’s Theorem of the Deductible with Heterogeneous Beliefs
In Arrow’s classical problem ofdemand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer—or decision maker (DM)—is a deductible contract when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU maximizer. In Arrow’s framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This article reexamines Arrow’s problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer’s and the DM’s subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow’s classical is then obtained as a special case.
Amarante et al. (2017)
Contracting on Ambiguous Prospects
We study contracting problems where one party perceives ambiguity about the relevant contingencies. We show that the party who perceives ambiguity has to observe only the revenue/ loss generated by the prospect object of negotiation, but not the underlying state. We, then, introduce a novel condition (vigilance), which extends the popular monotone likelihood ratio property to settings featuring ambiguity. Under vigilance, optimal contracts are monotonic and, thus, produce the right incentives in the presence of both concealed information and hidden actions. Our result holds irrespectively of the party’s attitude towards ambiguity. Sharper results obtain in the case of global ambiguity-loving behaviour.
Ghossoub (2016a)
Cost-efficient contingent claims with market frictions,
In complete frictionless securities markets under uncertainty, it is well-known that in the absence of arbitrage opportunities, there exists a unique linear positive pricing rule, which induces a state-price density (e.g., Harrison and Kreps in J Econ Theory 20(3):381– 408, 1979). Dybvig (J Bus 61(3):369–393, 1988; Rev Financ Stud 1(1):67–88, 1988)showed that the cheapest way to acquire a certain distribution of a consumption bundle (or security) is when this bundle is anti-comonotonic with the state-price density, i.e., arranged in reverse order of the state-price density. In this paper, we look at extending Dybvig’s ideas to complete markets with imperfections represented by a nonlinear pricing rule (e.g., due to bid-ask spreads). We consider an investor in a securities market where the pricing rule is “law- invariant” with respect to a capacity (e.g., Choquet pricing as in Araujo et al. in Econ Theory 49(1):1–35, 2011; Chateauneuf et al. in Math Financ 6(3):323–330, 1996; Chateauneuf and Cornet in Submodular financial markets with frictions, 2015; Cerreia-Vioglio et al. in J Econ Theory 157:730–762, 2015). The investor holds a security with a random payoff X and his problem is that ofbuying the cheapest contingent claimY on X, subject to some constraints on the performance of the contingent claim and on its level of risk exposure. The cheapest such claim is called cost-efficient. If the capacity satisfies standard continuity and a property called strong diffuseness introduced in Ghossoub (Math Op Res 40(2):429–445, 2015), we show the existence andmonotonicity of cost-efficient claims, in the sense that a cost-efficient claim is anti-comonotonic with the underlying security’s payoff X. Strong diffuseness is satisfied by a large collection of capacities, including all distortions of diffuse probability measures. As an illustration, we consider the case of a Choquet pricing functional with respect to a capacity and the case of a Choquet pricing functional with respect to a distorted probability measure. Finally, we consider a simple example in which we derive an explicit analytical form for a cost-efficient claim.
Ghossoub (2016b), 14 citations
Optimal Insurance with Heterogeneous Beliefs and Disagreement about Zero-Probability Events
In problems of optimal insurance design, Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub only gave a characterization of these optimal indemnity schedules in the special case of an MLR. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub as corollaries. Finally, we formalize Marshall’s argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any” shape.
Amarante and Ghossoub (2016)
Optimal insurance for a minimal expected retention: The case of an ambiguity-seeking insurer
In the classical expected utility framework, a problem of optimal insurance design with a premium constraint is equivalent to a problem of optimal insurance design with a minimum expected retention constraint. When the insurer has ambiguous beliefs represented by a non-additive probability measure, as in Schmeidler, this equivalence no longer holds. Recently, Amarante, Ghossoub and Phelps examined the problem of optimal insurance design with a premium constraint when the insurer has ambiguous beliefs. In particular, they showed that when the insurer is ambiguity-seeking, with a concave distortion of the insured’s probability measure, then the optimal indemnity schedule is a state-contingent deductible schedule, in which the deductible depends on the state of the world only through the insurer’s distortion function. In this paper, we examine the problem of optimal insurance design with a minimum expected retention constraint, in the case where the insurer is ambiguity-seeking. We obtain the aforementioned result of Amarante, Ghossoub and Phelps and the classical result of Arrow as special cases. Keywords:
Amarante et al. (2015), 27 citations
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., [24]). 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., [26]). 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 [5] on the optimality of the deductible indemnity schedule.
Ghossoub (2015), 22 citations
Equimeasurable rearrangements with capacities
In the classical theory ofmonotone equimeasurable rearrangements of functions, “equimeasurability” (i.e. the fact the two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many prob- lems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knightian uncertainty, or ambiguity is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to situations of ambiguity. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of non-additive probabilities, or capaci- ties that satisfy a property that I call strong diffuseness. The latter is a strengthening of the usual notion of diffuseness, and these two properties coincide for additive measures and for submodular (i.e., concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty.
Bernard and Ghossoub (2009), 185 citations
Static Portfolio Choice under Cumulative Prospect Theory Static Portfolio Choice under Cumulative Prospect Theory
We derive the optimal portfolio choice for an investor who behaves according to Cumulative Prospect Theory. The study is done in a one- period economy with one risk-free asset and one risky asset, and the reference point corresponds to the terminal wealth arising when the entire initial wealth is invested into the risk-free asset. When it exists, the optimal holding is a function of a generalized Omega measure of the distribution of the excess return on the risky asset over the risk-free rate. It conceptually resembles Merton’s optimal holding for a CRRA expected-utility maximizer. We derive some properties of the optimal holding and illustrate our results using a simple example where the excess return has a skew-normal distribution. In particular, we show how a Cumulative Prospect Theory investor is highly sensitive to the skewness of the excess return on the risky asset. In the model we adopt, with a piecewise-power value function with different shape parameters, loss aversion might be violated for reasons that are now well-understood in the literature. Nevertheless, we argue that this violation is acceptable. Key-words: