Optimal Insurance Contract Research Articles

Optimal insurance with counterparty and additive background risk

AbstractIn this paper, we explore how to design the optimal insurance contracts when the insured faces insurable, counterparty, and additive background risk simultaneously. The target is to minimize the mean-variance of the insured’s loss. By utilizing the calculus of variations, an implicit characterization of the optimal ceded loss function is given. An explicit structure of the optimal ceded loss function is also provided by making full use of its implicit characterization. We further derive a much simpler solution when these three kinds of risk have some special dependence structures. Finally, we give a numerical example to illustrate our results.

Variance insurance contracts

We study the design of an optimal insurance contract in which the insured maximizes her expected utility and the insurer limits the variance of his risk exposure while maintaining the principle of indemnity and charging the premium according to the expected value principle. We derive the optimal policy semi-analytically, which is coinsurance above a deductible when the variance bound is binding. This policy automatically satisfies the incentive-compatible condition, which is crucial to rule out ex post moral hazard. We also find that the deductible is absent if and only if the contract pricing is actuarially fair. Focusing on the actuarially fair case, we carry out comparative statics on the effects of the insured's initial wealth and the variance bound on insurance demand. Our results indicate that the expected coverage is always larger for a wealthier insured, implying that the underlying insurance is a normal good, which supports certain recent empirical findings. Moreover, as the variance constraint tightens, the prudent insured cedes less losses, while the insurer is exposed to less tail risk.

Optimal insurance contract design with government disaster relief

AbstractI examine the design of optimal insurance contracts considering the possibility of government disaster relief payments. This work focuses on the impact of (risky and ambiguous) government disaster relief on the shape of optimal private insurance contracts. I demonstrate that the optimal insurance contract is a straight deductible contract in the case of a fixed probability of government relief. This result is robust to ambiguity in the probability of relief payments, even for ambiguity‐averse decision makers. If government disaster relief becomes more likely for larger losses, then the optimal insurance contract features coinsurance above a deductible. I also extend this analysis to more general stochastic dominance relationships between disaster relief and loss magnitude.

Optimal insurance contracts for a shot-noise Cox claim process and persistent insured's actions

We consider a continuous-time model in which an insurer proposes an insurance contract to a potential insured. Motivated by climate change and catastrophic events, we assume that the number of claims process is a shot-noise Cox process. The insurer selects the premium to be paid by the potential insured and the amount to be paid for each claim. In addition, the insurer can request some actions from the potential insured to reduce the number of claims. The insurer wants to maximize his expected total utility, while the potential insured signs the contract if his expected total utility for signing the contract is greater than or equal to his expected total utility when he does not sign the contract. We obtain an analytical solution for the optimal premium, the optimal amount to be paid for each claim, and the optimal actions of the insured. This leads to interesting managerial insights.

Flexible Weather Index Insurance Design with Penalized Splines

In this article, we propose a flexible framework for the design of weather index insurance (WII) based on penalized spline methods. The aim is to find the indemnity function that optimally characterizes the intricate relationship between agricultural production losses and weather variables and thus effectively improves policyholders’ utilities. We use B-spline functions to define the feasible set of the optimization problem and a penalty function to avoid the “overfitting” issue. The proposed design framework is applied to an empirical study in which we use precipitation and vapor pressure deficit (VPD) to construct an index insurance contract for corn producers in Illinois. Numerical evidence shows that the resulting optimal insurance contract effectively enhances policyholder’s utility, even in the absence of the government’s premium subsidy. In addition, the performance of our proposed index insurance is robust to a variety of key factors, and the general payment structure is highly interpretable for marketing purposes. All of these merits indicate its potential to increase efficiency of the agricultural insurance market and thus enhance social welfare.

A new robust insurance model considering the time of accident

The principle of maximum utility is generally adopted to design the optimal insurance contracts, which should consider the influence of different factors such as the probability of accident, premium, compensation, and so on. However, most literatures deal with these variables from a static perspective. This paper considers the accident probability and the value of insurance subject based on the time of accident, which is rarely involved in the previous studies and considers the utility function of the insurer and the policyholder from a dynamic perspective. Firstly, to make this model more universally applicable, we establish an insurance model that considers the time of the accident and different premium payment forms for policy-holders and insurers respectively. Next, we derive a robust premium insurance model based on min-max regret, in which the time of the accident can be assumed to be certain and uncertain respectively. Then, we conduct numerical experiments and analyze the utility of the policy-holders, demonstrating guarantee period and value of insurance subject are significant when insuring and derive the optimal coverage rate. These results also show that the insurance model that takes into account the time of accident performs better.

Optimal contracts under adverse selection for staple goods such as energy: Effectiveness of in-kind insurance

An income loss can have a negative impact on households, forcing them to reduce their consumption of some staple goods, such as energy. This can lead to health issues and, consequently, generate significant costs for society. We suggest that consumers can, to prevent these negative consequences, buy insurance to secure sufficient consumption of a staple good if they lose part of their income. This paper’s contribution is twofold. From the theoretical side, we develop a two-period/two-good principal–agent problem with adverse selection and endogenous reservation utility to model insurance with in-kind benefits. This model allows us to obtain semi-explicit solutions for the optimal insurance contract. Secondly, using recent aggregate statistics on households and their energy consumption, our model is applied to the context of fuel poverty. For this application, we can conclude that, even in the least efficient scenario from the households point of view, i.e., when the insurance is provided by a monopoly, this mechanism decreases significantly the risk of fuel poverty of households by ensuring them a sufficient consumption of energy. The effectiveness of in-kind insurance is highlighted through a comparison with income insurance, but our results nevertheless underline the need to regulate such insurance market.

Optimal insurance contract design with “No-claim Bonus and Coverage Upper Bound” under moral hazard

In this paper, an optimal insurance problem from the view of a risk-averse individual under moral hazard is considered. Based on the Principal-Agent theory, we introduce a combined incentive tool of “No-claim Bonus and Coverage Upper Bound” to encourage the insured to make a higher risk-reducing effort and obtain higher expected utility than in a basic contract. We confirm if marginal expected utility brought by the increase of risk-reducing effort decreases at the critical point of transition between two contracts, the combined incentive tool can restrain the moral hazard of the insured. Moreover, Pareto efficiency improvement of the tool is visually displayed by two case examples.

Optimal insurance contract specification in the upstream sector of the oil and gas industry

The upstream sector of the Oil and Gas (O&G) industry is recognized by its capital-intensive projects and complex and hazardous associated recovery and production processes, thus susceptible for large and financially damaging accidents. In this context, to avoid the risk and impact of high expenses, O&G companies usually acquire insurance contracts. In practice, although the contract format is typically pre-specified, its parameter magnitudes can be adjusted aiming at maximizing the company total wealth. Therefore, this work proposes a holistic methodology to assess the optimal parameter specification of an insurance contract in the upstream sector of the O&G industry. A non-convex stochastic optimization problem is constructed aiming at maximizing a risk-adjusted measure of the policyholder total wealth. The modeling takes into account the uncertainty on the financial loss of an accident by making use of the safety barriers and precursor information framework. The non-convex optimization problem is cast as an equivalent mixed-integer linear programming problem by combining the scenario-based representation approach with a set of binary reformulation procedures. We illustrate the applicability of the proposed methodology with a set of numerical experiments. In a nutshell, we found that the proposed parameter specification methodology resulted in greater predictability when compared to two quantile-based specification policies and an uninsured company. In fact, the second best policy presented a standard deviation 103% higher than the proposed methodology. Furthermore, the model also provided greater protection against extreme events, since the second best policy presented a Conditional Value-at-Risk 41% higher than the proposed methodology.

배경위험 하에서의 최적보험계약과 해약환급금에 관한 연구

배경위험 하에서의 최적보험계약과 해약환급금에 관한 연구

Pareto-optimal insurance contracts with premium budget and minimum charge constraints

In view of the fact that minimum charge and premium budget constraints are natural economic considerations in any risk-transfer between the insurance buyer and seller, this paper revisits the optimal insurance contract design problem in terms of Pareto optimality with imposing these practical constraints. Pareto optimal insurance contracts, with indemnity schedule and premium payment, are solved in the cases when the risk preferences of the buyer and seller are given by Value-at-Risk or Tail Value-at-Risk. The effect of our constraints and the relative bargaining powers of the buyer and seller on the Pareto optimal insurance contracts are highlighted. Numerical experiments are employed to further examine these effects for some given risk preferences.

Optimal insurance design under narrow framing

In this paper, we study insurance decisions when the policyholder evaluates insurance with narrow framing. We show that due to aversion to risk on the net insurance payoff, i.e., insurance indemnity minus insurance premium, narrow framing reduces insurance demand. This helps explaining the observed low insurance demand in many insurance markets. We also show that the optimal insurance contract involves a deductible and the coinsurance of losses above the deductible when transaction costs depend on the actuarial value of the policy. Moreover, when the policyholder is loss averse over the net insurance payoff, a fixed indemnity equal to insurance premium should be paid for a range of intermediate losses.

OPTIMAL INSURANCE CONTRACTS UNDER DISTORTION RISK MEASURES WITH AMBIGUITY AVERSION

AbstractThis paper presents analytical representations for an optimal insurance contract under distortion risk measure and in the presence of model uncertainty. We incorporate ambiguity aversion and distortion risk measure through the model of Robert and Therond [(2014) ASTIN Bulletin: The Journal of the IAA, 44(2), 277–302.] as per the framework of Klibanoff et al. [(2005) A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.]. Explicit optimal insurance indemnity functions are derived when the decision maker (DM) applies Value-at-Risk as risk measure and is ambiguous about the loss distribution. Our results show that: (1) under model uncertainty, ambiguity aversion results in a distorted probability distribution over the set of possible models with a bias in favor of the model which yields a larger risk; (2) a more ambiguity-averse DM would demand more insurance coverage; (3) for a given budget, uncertainties about the loss distribution result in higher risk level for the DM.

Variance Contracts

We study the design of an optimal insurance contract in which the insured maximizes her expected utility and the insurer limits the variance of his risk exposure while maintaining the principle of indemnity and charging the premium according to the expected value principle. We derive the optimal policy semi-analytically, which is coinsurance above a deductible when the variance bound is binding. This policy automatically satisfies the incentive-compatible condition, which is crucial to rule out ex post moral hazard. We also find that the deductible is absent if and only if the contract pricing is actuarially fair. Focusing on the actuarially fair case, we carry out comparative statics on the effects of the insured's initial wealth and the variance bound on insurance demand. Our results indicate that the expected coverage is always larger for a wealthier insured, implying that the underlying insurance is a normal good, which supports certain recent empirical findings. Moreover, as the variance constraint tightens, the insured who is prudent cedes less losses, while the insurer is exposed to less tail risk.

Flexible Weather Index Insurance Design with Penalized Splines

In this paper we propose a flexible framework for the design of weather index insurance (WII) based on penalized spline methods. The aim is to find the indemnity function which optimally characterizes the intricate relationship between agricultural production losses and weather variables and thus effectively improves policyholders’ utilities. We use B-spline functions to define the feasible set of the optimization problem and a penalty function to avoid the “overfitting” issue. The proposed design framework is applied to an empirical study in which we use precipitation and vapor pressure deficit (VPD) to construct an index insurance contract for corn producers in Illinois. Numerical evidence shows that the resulting optimal insurance contract effectively enhances policyholder’s utility, even in the absence of government’s premium subsidy. In addition, performance of our proposed index insurance is robust to a variety of key factors, and the general payment structure is highly interpretable for marketing purposes. All these merits indicate its potential to increase efficiency of the agricultural insurance market and thus enhance social welfare.

Catastrophe Insurance and Solvency Regulation

This paper provides a welfare analysis of solvency regulation for catastrophe insurance. We consider an economy with risk-averse agents exposed to a common source of risk and an insurer owned by risk-averse shareholders. We show that the optimal insurance contract features full coverage and is Pareto optimal if and only if it includes participation in the insurer’s profit. When such participation is not available (as is often the case in the real world), we demonstrate that solvency regulation allowing insurer default or limited liability in the most catastrophic states is Pareto improving. We use a numerical application to analyze the welfare benefits of different default rates for a wide range of risk lines, risk correlations, and ratios of insureds to insurer’s shareholders. While allowing a small probability of default yields some welfare gains, we show that an excessive default probability may be highly detrimental to welfare. We find that a good rule of thumb is to maintain a default probability lower than the individual probability of loss.

Optimal Insurance for Prudent Risk Averters and Risk Lovers

This paper examines the structure of optimal insurance contracts for a broad class of insureds that includes both risk averters and risk lovers and by assuming that the insureds are prudent. We specify the difference in optimal contract form between risk averters and risk lovers. Treating these decision-makers as a unique group, we show that the optimal insurance form is dual limited stop-loss insurance with an upper limit, which implies that including risk lovers in the group of decision-makers changes the contract in the small loss part. We also consider the situation of contracts with a concave payoff, where the optimal contract form becomes limited change-loss insurance or limited dual change-loss insurance depending on the coefficient of variation of the retained loss. Finally, we derive the form of optimal contracts in the presence of a background risk.

Optimal Insurance Contracts with Limited Commitment and Unobservable Disability

We study a continuous-time model of an optimal long-term labor contract with a randomly arriving disability shock under two frictions: the agent cannot commit to a long-term contract and can falsely claim disability. We provide the explicit solution and analyze the optimal contract by using an optimal stopping approach. We show that the optimal contract can be implemented under a three-account trading system, which has different features from systems in the literature that are generated under only one of the two frictions. By using the three-account system, we investigate how the arrival intensity of the disability shock and the level of disability severity have an impact on the optimal contract. Finally, we also perform a quantitative analysis on the firm’s cost from providing inefficient contracts caused by the current long-term disability insurance practice.

Optimal XL-insurance under Wasserstein-type ambiguity

We study the problem of optimal insurance contract design for risk management under a budget constraint. The contract holder takes into consideration that the loss distribution is not entirely known and therefore faces an ambiguity problem. For a given set of models, we formulate a minimax optimization problem of finding an optimal insurance contract that minimizes the distortion risk functional of the retained loss with premium limitation. We demonstrate that under the average value-at-risk measure, the entrance-excess of loss contracts are optimal under ambiguity, and we solve the distributionally robust optimal contract-design problem. It is assumed that the insurance premium is calculated according to a given baseline loss distribution and that the ambiguity set of possible distributions forms a neighborhood of the baseline distribution. To this end, we introduce a contorted Wasserstein distance. This distance is finer in the tails of the distributions compared to the usual Wasserstein distance.

Pareto-Optimal Insurance Contracts With Premium Budget and Minimum Charge Constraints

In view of the fact that minimum charge and premium budget constraints are natural economic considerations in any risk-transfer between the insurance buyer and seller, this paper revisits the optimal insurance contract design problem in terms of Pareto optimality with imposing these practical constraints. Pareto optimal insurance contracts, with indemnity schedule and premium payment, are solved in the cases when the risk preferences of the buyer and seller are given by Value-at-Risk or Tail Value-at-Risk. The effect of our constraints and the relative bargaining powers of the buyer and seller on the Pareto optimal insurance contracts are highlighted. Numerical experiments are employed to further examine these effects for some given risk preferences.