Optimal insurance premium and critical points

(

| ) (

[I X I I X I  

E (42a) Compared Equation (42) with Equation (22), we find the above optimal insurance problem resembles the problem in Zhou and Wu (2008). The solution process for equation (42) is similar but simple to Equation (39). The detail is remained to further work.

C. Optimal insurance premium and critical points

Following Raviv (1979) and Gollier (1987), the insurance problem is solved via two steps. First, given a fixed premium P, we solve the optimal indemnity

)

;

* (

P x I

I  as a function of P. The second step determines the optimal insurance premium P . The insurance contractual forms have been developed in the above ^* subsections A and B. Thus, this study proceeds to find the optimal premium. Since the two deductibles d_L and d_H depend on P, this study firstly finds the two deductibles as the premium function, d_L d_L(P) and d_H d_H(P). Since the calculation process of the optimal P for Table 3 is similar to Table 2, this study only presents the calculation for Table 2. Additionally, the insurance premium can be solved via the relation Ph(xR). This also presents that the critical point R VaR can be directly found via P.

First, we can directly find the insurance premium P for Panel C and Panel F, as follows.

R x dx x f R

x R R R

Carg



VaR (43)

R x dx x f R

x dx

x f x

R R

R R

F 





 ( ) [ ( VaR)] ( )

arg VaR

VaR

0 (44)

Based on R and _C R , we obtain _F P_C h(xR_C) and P_F h(xR_F). Note that

D F

E P P

P

P_C    and P_B P_F. Next, using the premium formula Ph(xR) can find the critical points d_L d_L(P) in Panels B and d_H d_H(P) in Panel D. However, besides RVaR, Panel A has two critical points d_L and d_H, indicating that the calculation process for Panel A is more complicated than the others. Accordingly, the first step finds d_H d_H(d_L,P) as a function of d_L and P, using the premium principle Ph(xR). Subsequently, maximizing the insurer’s expected utility finds

) (P d

d_L _L and hence d_H d_H(d_L)d_H(P). Once all the critical points in Table 2 are represented as a function of P, maximizing the insurer’s expected utility with respect to P by common calculus can obtain the optimal premium P and the ^* corresponding six contractual forms. Nevertheless, P should meet the VaR and ^* CVaR constraints. Since Panels B, C, E and F never violate the CVaR constrict, the check of CVaR constraint focuses only Panel A and D. Finally, comparing the expected utilities among the six insurance contracts can elect the “exactly” and

“uniquely” optimal insurance contract.

Table 1: The Hamiltonians and their corresponding R^*(x)

Table 1: The Hamiltonians and their corresponding R^*(x) (continued)

Table 2: Optimal contractual forms of R^*(x) for insured’s CVaR constraint

Table 3: Optimal contractual forms of I^*(x) for insured’s CVaR constraint

5. Conclusions

We have developed the optimal insurance forms under the insured’s and the insurer’s CVaR constraints, respectively, assuming that the insured is risk averse and the insurance premium is a function of expected indemnity. Nevertheless, the CVaR value is based on the calculated VaR value. So, the CVaR constraint frequently accompanies the VaR constraint. Consequently, the developed insurance contacts simultaneously meet the VaR and CVaR constraints. The main findings are as follows.

First, the insurance contractual forms with the insured’s VaR and CVaR constraints are analogous to the insurer’s ones. Second, the possible contracts have six different forms, including three single deductible insurances and three double deductible insurances. Meanwhile, the three double deductibles contain one standard

form like the result in Huang (2006), and the remained two insurances are viewed as the degenerative forms. Third, one of the six insurances is a standard deductible insurance where the VaR and CVaR constraints are not binding. Besides the standard deductible insurance, the representative indemnity schedule has two deductibles, where the lower and the higher are respectively below and above a particular threshold (expected indemnity plus VaR). Moreover, the remained four contractual forms are the degenerative forms of the representative double deductible insurance.

The future works can consider the following directions. First, the utility function and loss probability distribution are further specified. This leads to an explicit insurance form. Next, besides the optimal contractual form, one can further determine the optimal coverage levels on the specific insurances including the proportional coinsurance, deductible insurance, upper-limit insurance. Finally, comparing our

results with the previous studies provides an intuitive explanation.

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