A Model of Optimum Tariff in Vehicle Fleet Insurance

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1 A Model of Optimum Tariff in Vehicle Fleet Insurance. Bouhetala and F.Belhia and R.Salmi Statistics and Probability Department Bp, 3, El-Alia, USTHB, Bab-Ezzouar, Alger Algeria. Summary: An approach about tariff in vehicle fleet insurance has been proposing, where we develop a non-linear mathematical model with constraints. The solution of the problem is an optimum cut rates for each class of ris. The model has been implementing on computer. ey words: Fleet insurance, Vehicle ris, Cut rate, Optimisation, Simulation. Introduction The individual sees often to protect of hazardous ris phenomena of his environment. A potential ris may be transferring to insurance company for cover, versus payment of premium. The calculus of this premium is an essential element at the insurer and requires actuarial techniques. In [RSST98], we find general principles to determinate a good premium. For the case of vehicle ris, an approach of the determination of an optimal premium, by using linear programming has given in [B0]. Recently, [NB04] show an asymptotic result concerning an estimator of extreme ris premium by using the adjusted premium principle given by [W96]. In this paper, we propose an optimal approach to determine cut rates of premium, by class of ris, in a vehicle fleet assurance. For a commercial reason, the reduction is an important parameter in the portfolio management strategy of company. We suggest a non-linear programming model with constraints to determine optimal cut rates. Factors and classes of ris The most signification factors of ris, considered by this problem are type of vehicle, traffic area, and usage and vehicle power. Each factor is a set of codes. A ris class is a combination of these factors. For instance, the class 0003 concerns vehicles for business use, of power from 7 to 0 horses and moving inside a determined area. The set of ris classes has been generating by applying trees method, as for instance the CART method.

2 3 3 Premium of ris Formally, a ris premium may be considering as a non-negative random variable X with distribution function F X. We associate to X a finite quantity (X ), called the premium of ris. The determination of (X ) needs a particular treatment and represents a reference point for commercial premiums calculus. Different principles of ris premium are given. A calculation of a good premium requires some proprieties of (X ). Let X, Y, Z be three arbitrary riss, the following proprieties are suggested, for a practice reason of the definition of ris premium principle (see [RSST98] ). P) α 0, ( α ) =α P) α 0, (α X) =α (X) P3) (X+Y) (X) + (Y) P4) α 0, (α +X) =α + (X) P5) X st Y then (X) (Y) ("st" order stochastic) P6) If for all a [0,], and for all (X) = (Y) ( a FX + (-a) F Z ) = ( a FY + (-a) F Z ). Each property admits a convenient interpretation. On the basic of these proprieties, principles of ris premium are defined by: A) Principle of the expected value: α 0, (X) = (+α ) E(X), (E(X)< ), A) Principle of variance: (X) = E(X) + a Var (X), A3) Principle of the deviation: (X) = E(X) + a A4) Principle of modified variance: Var ( X ) E ( X ) + a ( X ) = E( X ) 0 A5) Principle of the exponential:, Var(X), if E ( X ) > 0 if E ( X ) = 0 ax ) log E ( e (X) = a A6) Principle of adjusted premium: This principle concerns the case of extreme ris X (see [RSST98]). [W96] suggests the following principle. (X) = u p ( F ( x )) dx, X

3 33 Where p is a parameter of distortion and u is a value, appropriately determined in practice. For the case of this principle, an estimators family ( ˆ ), asymptotically normal has given in [NB04]. u n n N The following character of ris one or the other among of this principle may be chosen. As it is about the vehicle ris, we use expected value principle. 4 Modelling Given S the damage cost (random yearly cost) associate to th class of ris. The insurance supposed nowing the expectancy: E ( S ), =,,..., that can be calculated from damage historic. This, permits to determinate the net premium π, given by π = E, =,,...,. ( ) S = For a reason of commercial security, the insurer consider the premium π = ( +α ) π, =,,...,, where α is a loading parameter of net premium, which is, generally, fixed by the insurer according to his tariff strategy. Given N the total number of vehicles, separated into classes. The insurer gets N commercial premiums of a global value estimated to π. Let c be the cut rate relative to the th class of ris. The insurer yields part c c π of his premiums and conserve the other part (- ). Then, the total value of conserved premium becomes: π = S ( ) c π, with a total cost of damage: = To determine these cut rates, the company has to consider at least the quantity: ( ) Z= E c π S () =

4 34 4. The problem Hypothesis We consider the following assumptions: H) the portfolio is controlled for annual period, n = H) each class I, =,,,, including n vehicles (N = ) H3) in any class, riss insured are homogeneous and independents. H4) for each class of ris, the cost of damage S, is a non-negative random variable. c H5) the cut rates are unnown variables to determine. 4. Formulation of objective s function Let Y i be the cost of damage of i th vehicle associated to the th class of ris. n The global cost damage relative to class is, S = Y i i= Given: j any damage of the vehicle i. U i, the total yearly number of damages of vehicle i. Y ij is the random cost of j th damage relative to the vehicle i of the th class of ris (j =,,,U i ). Then, the total cost of damages is Y = Yi j, for =,,, and i =,,,n i U i j = Let N s be the random variable "number of damages of the th class Then n n N s =. U i i = U i S = Y = i = j = i j We admit also the following consideration: Y Y N s, (,..., are (i.i.d.), with : E l = E ( ) =,,, and l =,,, N s. Y Y N s, N s l = Y ) Y Y l,..., are independents from the variable N s =,,,.

5 35 An explicit calculation of () gives the following objective function, to minimize: Where Z = + c π E Ns E Y π + = = c π ( ) = ψ = + N s Y + var N s ( Y ) π E ( ) E ( ) ( ) E + ψ E var Y ( N s ) 4.3 Constraints 4.3. Constraint in relation to the technical equilibrium For the insurance company, the ratio S τ =, is an essential index to measure the π technical equilibrium. The quantities S and π represents, respectively, total cost of damages and total of the acquired premiums. In practice, a good tariff must verified, τ. Then, we have: Let = = S ( c ) π. τ ( S ( c ) π ) = Thus, the first constraint must be: c π = l τ 0. τ + Y τ π 0. = Ns l = = 4.3. Constraint in relation to the total cut rate: For more justice towards the customer, the insurer applies, class by class, a reduction rule. The average balanced rate must not exceed a given percentage ( τ ) (the maximum, fixed by the company, being 50%).

6 36 We have: = = c π π τ. Hence, the second constraint will be: = c π - τ 0 π = Constraints in relation to the cut rate of class In every class, the cut rate must be restrained between 0 and a proportion fixed by the expert of the company. According to this condition, we consider constraints: 0 c ξ 0 ξ <, =,,..., ξ 4.4 Mathematical model From 4. and 4.3, we have to resolve, the following program: Min c, c,..., c (Z = + c π E Ns E Y π + ) ψ = = Under constraints τ c π = = c π ( ) = + Y τ π 0 0 τ () < c π = N s l = = τ 0 0 τ < π = c ξ 0, 0 ξ < =,,...,

7 37 5 Comment The proposed model is a tool of a decision help, concerning the fixation of the cut rates in vehicle fleet insurance. The stochastic form of model needs the calculation of the tow first moments of the random variables N s and Y ; respectively, the particular case of Poisson and log-normal laws is taen into consideration. The model is implemented on computer, as interactive pacage, where the user has the choice to estimate the parameters of the model, by using the statistical database of the insurer, or by simulation as is showing by the fig., that represents the simulation module of calculus of cut rates. Ris Vehicle Individual Total Expected Expected Optimal cut rates class number premium premium number cost of damages Fig. Simulation module of calculus of optimal cut rates

8 38 References [RSST98] Rolsi, T., Schmidt, H., Schmidt, V., Teugels, J.: Stochastic Process for Insurance and Finance. John Wiley & Sons edition (998). [B0] Bouhetala,. : A linear Programming Model for an optimal basic premium in car insurance. Bulletin of the International Statistical Institute, Vol. III, pp (00). [NB04] Necir, A.,Bouhetala,.: Estimating the ris-adjusted premium for the largest claims reinsurance covers. COMPSTAT04, Proc. in Computational Statistics, Physica- Verlag, Heidelberg, New Yor, pp (004). [W96] Wang S.: Premium calculation by Transforming the Layer Premium Density, ASTIN Bulletin, 6, 7 9 (996).

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