Ruin Probabilities : Classical Versus Credibility Cary Chi-Liang Tsai Department of Finance, National Taiwan University Gary Parker Department of Statistics and Actuarial Science Simon Fraser University, Canada

Classical Discrete Time Risk Model u = U 0 : the initial surplus c : the amount of premiums received each period S n : the total claims of the first n periods assume W i is the aggregate claims in period i, and W 1, W 2, …, W n are non-negative i.i.d. random variables c= (1+η)×E[W], η is the relative security loading the time of ruin T = min{n: U n < 0} the probability of ruin Ψ(u) = Pr{T < ∞ | U 0 = u}

Experience Rating Constant premiums: Dynamic premiums: Renewal premiums charged in casualty insurance are usually based on past experience. c n+1 = (1+η)× E[W n+1 | W 1 =w 1, W 2 =w 2, …, W n =w n ]. The determination of c n+1 is based on credibility theory Question: Can the dynamic credibility premium scheme significantly affect the probability of ruin ?

Buhlmann’s Credibility Premium For j = 1, 2,…, n+1, the distribution of each W j depends on a parameter Θ Given Θ, the random variables W 1, W 2, …, W n+1 are conditionally independent and identically distributed. Denote µ(Θ) = E[W j |Θ] and ν(Θ) = Var[W j |Θ]. µ = E[µ(Θ)] = E[W j], a = Var[µ(Θ)] and ν = E[ν(Θ)] Buhlmann’s credibility factor Z n = n/(n+ ν /a), c n+1 = (1+η)×

One More Dynamic Premium Approach since Zn= n/(n+ ν /a) is increase to 1 as n goes to ∞, (the sample mean) as n approaches to ∞. If n is large, there is very little change from to when one more aggregate claim w n is observed, which implies the renewal premium c n+1 is very stable and the credibility impact disappear for large n. idea : consider the most recent k periods of claim experiences to renew the premium for the next period m = min (n, k), h = max (n-k, 0) + 1 = n – m + 1, Z n,m = m/(m+ ν/a),

Distribution Assumptions if X 1, X 2, …, X n are i.i.d. Exp( β ), ~ Gamma(n, β ) assume Λ follows Poisson(λ), and given Λ, W 1, W 2, …, W n+1 are independent Gamma(Λ, β ). µ(Λ) = E[W,j |Λ] = Λ β, ν(Λ) = Var[W j |Λ] = Λ β 2, µ = E[µ(Λ)] = E[W j] = λ β, ν = E[ν(Λ)] = λ β 2, and a = Var[µ(Λ)] = λ β 2. Zn = n/(n+1), Zn,m = m/(m+1)

Parameter Assumptions assume three sets of parameters for Gamma distribution (λ, β) = (100, 1) (high frequency and low severity), (λ, β) = (10, 10) (mid frequency and mid severity) and (λ, β) = (1, 100) (low frequency and high severity) Sample histograms of Λ and WiSample histograms of Λ and Wi (λ, β)E(Λ)E(Λ)Var(Λ)E(Wi)Var(Wi) (100,1) (10,10) (1,100)

Simulation Results perform Monte-Carlo simulation 1000 times the study period is 100 (that is, if Un > 0 for all n = 1, 2, …,100, then ruin is assumed not occur for this simulation). ruin probability = number of ruins/1000 the relative security loading η =0.1 Simulation results (tables and figures)tables and figures

Conclusions and Future Research “The credibility scheme can reduce ruin probability” is not always true, and only holds under some condition (large enough the initial surplus). Different lines of business of the insurer can yield significantly different ruin probabilities even though the claim amount of these business comes from the same distribution with different parameters but the same expectations. The low frequency and high severity case is far worse than the high frequency and low severity one Several lines of business portfolio, each line has its own claim distribution and parameters ==> apply the Buhlmann-Straub credibility theory to obtain the renewal premiums.