Journal of Applied Mathematics

Volume 2015, Article ID 186061, 18 pages

http://dx.doi.org/10.1155/2015/186061

## The Optimal Insurance Policy for the General Fixed Cost of Handling an Indemnity under Rank-Dependent Expected Utility

College of Economics and Management, Hunan Normal University, Changsha 410081, China

Received 20 July 2015; Revised 21 October 2015; Accepted 22 October 2015

Academic Editor: Walter Briec

Copyright © 2015 Liurui Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on Bernard et al.’s research, we focus on the Pareto optimal insurance design with the insured’s Rank-Dependent Expected Utility (RDEU). Compared with their previous work, our novelties are the more general fixed cost function of the insurer and the discussion of adverse selection and moral hazard. In particular, Bernard et al. only consider the case in which the fixed cost function of handling an indemnity is the linear function. However, the fixed cost function is not just linear functions in real insurance market. So, we explore the more general fixed cost function including both the linear and nonlinear functions. On the other hand, we consider adverse selection and moral hazard which are involved by Bernard et al. Leading adverse selection and moral hazard into our research makes our results more practical and meaningful. Moreover, we provide an insight into the sensitivity of an optimal solution for the insured’s initial wealth and the parameters related to the fixed cost function of handling an indemnity. We further compare the two different utility functions of the insured in terms of influence of optimal policy analysis.

#### 1. Introduction

In an economic system, both companies and individuals have to face uncertainty in the future. In such a situation, the redistribution of risk among the firms and individuals is facilitated. Usually, persons use insurance policies for reallocating risk. In the late 1960s, Mossin et al. initiated the insurance decision analysis [1–3], where they focused on relationally purchasing insurance from the viewpoint of an individual who has to face a specific risk, given his preference structure and wealth level. Their appealing research builds on the assumption that the insurance policy is exogenously specified. However, Borch [4] and Arrow [5, 6] object to Mossin’s opinion and firmly believe that the insurance contract is not exogenous. Sappington [7] provides a complete justification of principal-agent modeling for the insurance problem. He sets up the basic framework and the key model of game relationship between the principal and the agent. Furthermore, he builds a monitoring mechanism based on the prisoner’s dilemma and builds an incentive mechanism by competition.

In contrast with Sappington’s principal-agent model, the insured corresponds to the agent and the insurer corresponds to the principal. Actually, our problem is different from his. In his model, the agent first decides whether to accept or reject the contract. After the agent signs the contract, he can change his own expected utility by choosing the efficient level of effort. In our model, the agent first determines whether to accept or reject the contract too. But, once the agent agrees to and signs the insurance contract, he cannot change his own value function which is only determined by the random loss .

Although there are obvious differences between our models, our research is consistent with his basic framework. We can make a close link with his research. (1) As an agent, the insured should pay the upfront premium for sharing the loss with the insurer. This is as if the insured has to pay the “franchise fee” for the right to work for the principal; (2) in his research, the agent will accept the contract offered by the principal if and only if the subsequent self-interested behavior under the terms of the contract provides the agent with a level of expected utility that exceeds his reservation level, . The insured’s criterion of buying or not buying the insurance is where is equal to in his paper; (3) in his paper, this contract promises payments, , to the agent that are precisely the principal’s valuation of the agent’s performance minus some fixed constant . Formally, . This is consistent with the insurer’s safety loading. In particular, the insurer will price the indemnity in such a way that In a competitive insurance market, we can understand as the minimum price of the indemnity for a risk-neutral insurer to participate in the business.

Beyond insurance policy analysis, expected utility theory (EUT) has an underlying assumption that the decision maker is rational and uniformly risk averse, only considering the objective probability rather than the subjective probability [8]. In reality, however, various decision makers’ behaviors deviate from the implications of expected utility. Substantial experimental and empirical evidence identifies that expected utility theory is incompatible with human observed behavior. The abundant paradoxes lead to the development of a more realistic theory. It is dominant in prominent paradigms that Tversky and Kahneman propose prospect theory (PT) [9]. Later, they develop their prospect theory to cumulated prospect theory (CPT) since CPT is consistent with the first-order stochastic dominance [10]. In the context of CPT, they incorporate human emotion into their investigation.

The incentive for the optimal decision, especially to the optimal insurance policy, is extensively accepted. Quiggin [11] revolutionized the classical expected utility theory (EUT) by rank-dependent utility (RDU). His framework provides the theoretical background for the essentiality of design of insurance contracts. In 2000, Chateauneuf et al. [12] presented the Choquet expected utility framework and gave some results in the RDU framework as a special case. The Pareto efficient insurance contracts under RDU are illustrated in Carlier and Dana’s work [13, 14]. Dana and Scarsini [15] mention optimal risk sharing with background risk and briefly identify the case of RDU. He and Zhou [16, 17] emphasize the optimal insurance contract when the distortion is convex. Subsequently, Carlier and Dana [18] investigate two-person efficient risk-sharing problems about concave law-invariant utilities and give a characterized result which is valid for any RDU. Carlier and Dana [19] derive the optimal contingent claim for two significant decision frameworks, the RDU and the CP. However, these papers do not obtain the explicit solution while Bernard et al. [20] do so, when the utility function of the insured is concave.

Bernard et al.’s work inspires us to explore the optimal insurance design under Rank-Dependent Expected Utility. Compared with their research, our novelties are the generalization of the cost function and the discussion of adverse selection and moral hazard.

In detail, the key contribution of Bernard et al. is to get the explicit solution of the optimal insurance contract. But this result implicitly relies on the concavity of the extreme point function (equation (8) in the paper of Bernard et al. [20]). However, is concave, only when the fixed cost of handling the indemnity is a linear function of . So they only discuss the case in which the fixed cost function is linear; that is, . However, in a real insurance market, the different insurers have various cost functions including linear functions and other nonlinear functions. So, we pay attention to the more general cost functions to make the results more practical. Generalizing the fixed cost of handling the indemnity brings us a divers obstacle from Bernard and Zhou’s work: we are not sure about the convexity or concavity of the extreme point function ((32) in Section 3). In other words, this general cost function results in uncertain monotonicity of the extreme point function which is different from the proposition of the extreme point function in Bernard and Zhou’s article. Further, this uncertainty of monotonicity makes us have to discuss the different monotonic intervals of and five different relationships between , , and (see Figure 3) while Bernard and Zhou only need to consider one monotonic interval of and one relationship between , , and (see Figure 3). Although the various relationships lead to complicated discussion, these relationships make our novel results more general. In fact, Case in Figure 3 coincides with Bernard and Zhou’s Figure . Through discussing five different cases and the different monotonic intervals, we attain the explicit solution which is the general result applying to both the linear cost functions and nonlinear functions.

Another novel contribution is the discussion about two critical issues which are adverse selection and moral hazard while Bernad and Zhou do not involve them. In particular, we use the following bonus-malus systemto determine the premium , where is the premium in the previous period. We can estimate and by empirical data to decide the premium . If is the premium in the first period and we have no empirical data and , we have to decide relying on the indexes associated with , such as age, gender, and occupation. Based on this fixed , we research the optimal problem for indemnity under Rank-Dependent Expected Utility. Although we only offer a brief thought of how to decide and not carefully research it, this significant thought not only makes close link between these critical issues and our research but also offers the basic framework for further research.

Recently, Dhiab investigates the demand for insurance under the nonexpected utility theory [21]. He applies Rank-Dependent Expected Utility (RDEU) to the insurance contract. In his insurance context, agents behave not only according to their probability distribution but also according to their attitude towards risk.

Although Ben Dhiab’s research is similar to mine, there are obvious differences between our researches. The important difference is that we research the insurance problem from different angles. In particular, he researches the optimal insurance contract from the insured’s (or agent’s) point of view, so he only needs to maximize the RDEU of the insured without considering the utility function of an insurer and relative restrictive conditions. However, we study the optimal insurance policy from an insurer’s point of view. Thus, we set up the optimal insurance policy subjected to the restrictive condition associated with the utility function of the insurer. Meanwhile, the insurer’s utility functions and the optimal solutions change due to different insurers’ cost functions . So, our calculations and results are more complicated than his.

Another difference is that he does not present the quantified relationship between the indemnity function and the random loss when the optimal insurance policies are partial insurance and overinsurance. However, reveal the accurately quantified relationship between the indemnity function and the random loss in Proposition 9.

Furthermore, there are two main differences on the technological detail. (1) He supposes that the probability weighting function (probability distortions) is always concave or convex. But, the -shape probability weighting function is more reasonable than the concave or convex probability weighting function, because Kahneman and Tversky used sufficient experiments and evidences to extensively demonstrate that not only do people often overweight low-probability and certain outcomes but also the individual’s attitude to the risk always changes. So, we employ the -shape probability weighting function; (2) Dhiab only discusses two states of the nature which are the loss with probability and no-loss with probability . We explore the more general and complicated case in which the loss is a random variable on . The general means that we cannot attain the optimal solution by directly calculating and simply discussing as Dhiab does. So we have to use quantile function for solving the optimal problem.

This paper is organized as follows: In Section 2, we set up critical models. Section 3 explores the optimal solutions. In Section 4, numerical analysis is performed. Section 5 summarizes the conclusions. In Section 6, we introduce the further research. The paper ends with an appendix containing the proofs.

#### 2. The Model

In this section, we focus on the Pareto optimal insurance contract where the insured has Rank-Dependent Expected Utility [11] preference.

##### 2.1. The Basic Setting

###### 2.1.1. The Original Insurance Problem

Let be a probability space. An economic agent, called a policyholder or the insured, is endowed with the initial wealth and has to face the a nonnegative random loss with support in . The initial wealth of the insurer is . The loss is a random variable with the probability density function . The insured should pay the upfront premium for sharing the loss with the insurer. If the insured stands to the loss , the insurer will pay out . We treat as the indemnity function of the loss or coverage function. The indemnity principle compensates for the insured’s loss when an accident happens. According to this rule, the policyholder cannot collect more money than his actual loss. Hence, we assume . The constraint condition implies that if there is no loss there will be no reimbursement. As for the cost of the insurer, we state that it includes two part, the administrative expenses or other expenses and deadweight loss related to the insured and the insurer. So, we suppose that the cost of the insurer consists of fixed and variable components, which depends on the size of the insurance payment. We denote the cost by . And assume that , , and

We further suppose that the utility function of the insurer is . It is easy to see that the insurer’s final wealth is Here, we suppose and .

Consider there exit finite states of the world: the wealth level . And further assume that you can assign probabilities to each of these outcomes. You are fairly optimistic about your future, so you assign a probability to the wealth level (). . We call the series of wealth outcomes a prospect and represent this situation using the following convenient format: The value function represents the preferences in the EU model. For a prospect , preferences can be represented by a functional such that where is a utility function, strictly increasing and unique up to an affine positive transformation function. The value function is linear in probabilities.

It is necessary to mention that when , namely, the insurer is risk-neutral, the utility function has an important proposition that the expected utility function equals the utility of the expected value; that is, , where is a prospect (see [22]). We suppose, for simplicity, that there are only two states of the world: low wealth and high wealth . And further assume that you can assign probabilities to each of these outcomes. You are fairly optimistic about your future, so you assign a probability to low wealth and a probability to high wealth . This situation can be represented by a prospect as the following convenient format: Then, it is easy to write

Due to a risk-neutral utility function, from Figure 1, we can attain .