Approximate Solution To The Ruin Problem For Pareto Distributions

The classical risk surplus process of actuarial model iswhere u≥0 is the initial risk surplus at t = 0,{N(t),t≥0}the Poisson process with intensity A,and the X_i's(claim sizes)are independent and identically distributed(i.i.d.)non-negative random variables with cdf F_x(x), finite meanμ_x= E[X_i],and c = (1 +θ)λμ_x is the premium rate with loadingθ≥0.The ruin probability given surplus of u≥0 isψ(u) whereBecause of Beekman's convolution,it is well known thatψ(u) satisfies the following equation:where F_e^*k(u) is the kth convolution of F_e(u) and . SupposeF_x{x) is a Pareto cdf,and then it requires the computation of the convolution of Paretovariates.It is difficult to compute the ruin probability,and it is very important to find out some approximate solution.We have some datas of an insurance company,and want to analyze the ruin probability under Pareto cdf.Step 1.We should estimate the parameter of a. Method 1. method of moments.The estimator of parameterαis Method 2. maximum likelihood.The estimator of parameterαisMethod 3.Bayesian approach.The prior distribution is U(0,10). The estimator of parameter a isWith the results of the methods above,we could set parameterα= 5.Step 2.We should test the distribution of Pareto(5,l).The value of X²- is 2.7696,and it is less then 14.067,which is the critical of X²(7)-Definition 1. Maximal aggregate lossand it is the largest margin of loss and premium. ThenandWe could obtainand Theorem 1. Suppose L = L₁ + L₂ +…+ L_M,thenandApproximate solution.Method 1.We could use an exponential distribution,which shares the same mean with Pareto distribution.So approximate solution isMethod 2.Gamma approximate solution. The approximate solution isthenandLet (6),(7)equal(2),(3),we can obtain the values of m,n.Definition 2. Ruin intensityand we have Method 3.Constant ruin intensity. Supposeλ_x =λ,thenandso the approximate solution isMethod 4.Correct exponential ruin intensity. Letλ_x = kx^γ-1,x≥0, where k > 0,γ> 1 are parameters,and we obtainCompared with method 2,we could estimate k,γ.Supposeθ= 0.2We use method l,and obtain the resultand receive the approximate solution by (4). We use method 2,and obtain the resultand receive the approximate solution by (5). We use method 3,and obtain the resultand receive the approximate solution by (8). We use method 4,and obtain the resultand receive the approximate solution by (9).We find that method 4 is the best approximate solution.Definition 3. Flexibility coefficient.Flexibility coefficientand it denotes the reaction of ruin probability to the alteration ratio of capital. We compute the flexibility coefficientand obtain the following tabulation. We find that if we increase capital for 1% when the capital is arrived at 1.262,the ruin probability will decrease for more than 1%.