# A Model of Optimum Tariff in Vehicle Fleet Insurance

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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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