| The Insurance market is a typical market with asymmetric information. Neither buyer nor seller of insurance can acquire as much information as they wish because of the inherent property of insurance mechanisms. Participants of the Insurance Market can make use of asymmetric information easily to cause adverse selection and moral hazard, and induce that the insurance market circulate with low efficiency. From the insurant and the insurer with asymmetric information, we understand that, the main moral hazard the insurant faces is that the insurer refuses to compensate or defaults,while the main moral hazard the insurant faces is that the insurant influences the probility of incident and the extent of loss by his own behavior.According to the actual condition of the insurance industry and ponderance of problems ,we decide to analyse moral hazard arised by the behavior of the insurant.According to the view of premium in actuarial theory, we use experience method and tracing method to set up a stochastic insurance contract. For the sake of analysing moral hazard, when we design the contract, the premium is divided into the foundational premiumαand the adjustable premiumβπ. Thereinto, the adjustable premium is adjusted according to the actual loss of insuredπ.Furthermore the claimγπis the function of loss.The actual insured lossπis decided by the current insured amount w , the average loss rate(or the risk condition)in the past Eθ, the possible loss in the futureεand efforts on reducing and defending losse . According to the above viewpoint, when we implement the contract, we devide the premium into another two parts, i.e the fixed part and the stochastic part. The fixed partα+β( wEθ?e) is composed of foundational premium and one part of the adjustable premium, it is payed at the beginning of insurance.The stochastic part is the remaining part of adjustable premiumβε, it is payed or sent back according to the actual loss at the end of insurance. As long as the insurant claims on the insurance, the stochastic part can be realized. It will not affect the benefits of the insurer even if the insurant does not claim on the insurance. Finally, the proportion of the claimγwill be ascertained by coefficient of the adjustable premium. Based on the contract we talked above, by the hypothesis on risk-neutral principal and risk-averse agent with constant absolute risk aversion, we use parameterized distribution formulation of principal-agent theory to analyse moral hazard in markets of different competitiveness. It is concluded that: the first, in both monopolistic market and competitive market, moral hazard appears under incomplete information. Moral hazard leads to the decrease of the effort on reducing and defending loss. When the agent undertakes some risks, i.e. modify the coefficient of the adjustable premium and the proportion of the claim, moral hazard can be improved. The improved optimal effort distorts downward compared with complete information. The second, the foundational premium is different in the monopolistic market and the competitive market, because it affects the profit of the insurer, while the adjustable premium which affect moral hazard and the effort of the agent don't change. Concerning the first conclusion, we take an example of the monopoly market. Under complete information, the effort of the agent is decided by the cost of the effort. According to the economical condition and the demand for investment of the insurant, we have three types of insurance contracts: (1) whenγ<1,β=0.That is to say, when the contract is insufficient insurance or the claim is incomplete, the actual premium should be discounted on the base of foundational premium. (2) Whenγ=1,β=0.When it is full insurance or the claim is full, there is only foundational premium, it is unnecessary to adjust the premium. (3)Whenγ>1,β>0.When it is excessive insurance or the claim is excessive, the actual premium should be adjusted upward with the actual loss on the basis of foundational premium. Although excessive insurance is not allowed nowadays, we can regard it as a manner of investment for the insurant. The insurant injects capital into the premium and be compensated by the claim, moreover, the insurant can obtain Investment income from the insurer. Under incomplete information, in order to reduce moral hazard, the contracts mentioned above should be modified as follows: (1)γ>ρbσ2 (1 +ρbσ2),β>0.Compared with complete information,βshould increase whenγis the same. It is likely thatβis negative under complete information while positive under incomplete information with the sameγ. There is a special circumstance. Whenγ=1, the coefficient of adjustable premium is bigger than zero under incomplete information while it is just zero under complete information. When it is sufficient insurance, the actual premium should increase on the base of the foundational premium, so the agent undertake some risks. (2)γ<ρbσ2 (1 +ρbσ2),β<0. Whenγis the same, the value ofβshould be bigger compared with complete information. When adjusting the premium, the measure should be lower compared with complete information. (3)γ=ρbσ2 (1 +ρbσ2),β=0. There is only foundational premium when it is sufficient insurance under complete information. Whenγisρbσ2 (1 +ρbσ2), there is only foundational premium under incomplete information. Optimal effort depends on the cost of the effort, measure of absolute risk aversion and variance. Compared with complete information, optimal effort under incomplete information is lower. In the second conclusion, foundational premium depends on insurance amount, average loss rate in the past, the cost of the effect and reservation income. Under complete information, the foundational premium in the monopolistic marketα1 is higher than that in the competitive marketα2, because net income of the effort without insurance e 0 (b e02+ρσ2)2 is lower than that with insurance e 2 [b e22+ρ(be2)2σ2]2 (there be 2 =1 (γβ)2). Similarly, under incomplete information, the foundational premium in the monopolistic insurance marketα1SB is higher than that in the competitive insurance marketα2SB.This shows, the insurer extract profit from foundational premium in the monopolistic market. In competitive market, increasing the premium is no longer the main means to obtain profit.By comparing with two representative models, we find that the model in this paper is similar to most models but superior. In most models, moral hazard is settled by making the insurant to undertake some risks, under incomplete information, the insurer has to pay out some agency cost, so does our model. But, both the ex ante effort and the ex post effort are contained in our model, which is superior to common models. The contract we design is more concrete and adjacent to realism than common contracts. |
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