The Rethinking Of Adjustment Coefficient Maximization Premium Rule

Insurance plays more and more important role in the daily life as the development of national economic and the improvement of residents' income. Meanwhile, how to price the insurance policy becomes the core of insurance industry because of the fiercely competition. The insurance companies are eager to find out new technologies to measure the risk, under the force of both financial security and market competition. That brings a controversy in the actual field.A main topic in expositions of life insurance mathematics is the calculation of equivalence premium: given an insurance treaty such as whole life or life annuities, one is concerned with the identities coming from equating the discounted steam of payments from the insured to the company to the stream going the opposite way. Once the equivalence premium is calculated, one must of course add a loading premium. However, different results come from different premium rules. Especially, adjustment coefficient maximization premium rule which is typically used in non-life insurance is under a controversy in life insurance because there are many things such as interest, the large number of years and bonus payments lead to non-exponential asymptotic.In this paper, we do some research about the adjustment coefficient maximization premium:In section one, we introduce the definition of adjustment coefficient:where u is the free reserve at time 0, φ(u) = P(swp S_n > u) is ruin probability. We alsoproof that- when all exponential moments of the number of newly add policies exist- the adjustment coefficient exist in life insurance.Theory: Assume that the human's age is finite, that's A + K ≤M , M<∞, the premium payments and benefit payments is also finite, all exponential moments of N₀ exist.then, 3 7 s.t.- log (f>(u) 7= Urn ----- v .This is the theoretic foundation of this paper. We analyze the property of this theory and proof the existence and uniqueness of adjustment coefficient.Theory: Assume thatJ*\a) = iogE[exp{aYo(z)}] ,w(a) = Jjx\a)dF{x) ,ii^ t 00 as a t ?) (a < 00) then there exists an a.s. unique premium rule p*{.) —Po(-) + W\{-), Pi(x) =------;— sucn tnat(1) Epl(A) = 1 ,(2) the solution 7 of E[exp {7F(#)}] is independent of x ,(3) if 7 > 7* then ￡[exp {-yY^}] > 1 .In section two, we discuss the master degree graduates who bought the 5-year life insurance shortly after they get jobs. In the term of convenience, we assume that their ages are uniformly distributed from 25 to 29 and the benefit payment is 1 yuan when he/she dies. We get some key values according the life table of China, 1993 and compare the pricing risk among the expected value principle, the variance principle, and the adjustment coefficient maximization principal under the square loss:VarfXJ? -p) = Var[Y^} = EAa2(A) +T?VarA\pi(A)) .the values of this equation(loss) are 0.002881002 with expected value principle, 0.002870232 with the variance principle, and 0.002870001 with the adjustment coefficient maximization principal. We get an expected result that the pricing risk with adjustment coefficient maximization principal is smaller than the other two.In the third section, we analyze the fluctuation of interest rate. Because of the frequent payments in insurance policy, we use cash-stream method to calculate the premium reserveand the potential risk of the company:hhV = pil + nf-Y^lWl + rif-1}, h = 2,3,...,s, 1=1hV = {p(l + n)s - ^[6,(1 + r1y-l]}(l + r2)h^s -1=1 l=s+lh=l KWe get the formulaF(p,ri,r2) = -log￡[exp {7F1}] -paccording to the relation among p,ritr2, and get the results which describe the influence through partial derivation.