Actuarial Models
Aggregate loss models
For S = Σ X_i with random claim count N, use E[S] = E[N]·E[X] and Var(S) = E[N]·Var(X) + Var(N)·(E[X])². Independence between N and the X_i is required. The compound Poisson mgf is exp(λ(M_X(t) − 1)); sums of independent Poissons are Poisson, and Poisson thinning splits a Poisson into independent Poissons.
Severity modifications
For an ordinary deductible d, the per-loss variable is Y = (X − d)₊, while the per-payment variable is Y divided by the indicator that X exceeds d. Therefore E[per payment] = E[per loss]/P(X > d). Limits, coinsurance, and inflation interact: inflation raises expected per-loss cost, but a fixed deductible changes the mix of paid claims.
Credibility and ruin
Limited-fluctuation credibility uses Z = √(n/n_F); Bühlmann credibility uses Z = n/(n + k) with k = E[σ²(θ)]/Var[μ(θ)]. In ruin theory the adjustment coefficient R solves the Lundberg equation c·R = λ(M_X(R) − 1), giving ψ(u) ≤ e^(−Ru). Panjer recursion applies to (a,b,0) frequency with arithmetic severity, so discretization is needed first.
Worked example
Severity is exponential with mean 500 and the deductible is 100. Then E[(X − 100)₊] = 500·e^(−100/500) = 409.37, and the per-payment mean is 409.37/0.8187 = 500, a direct consequence of memorylessness.
Traps
Mixing per-loss and per-payment expectations, applying Panjer to continuous severity, and using the normal approximation in a heavily skewed tail are the standard errors.