Aggregate Loss Models: Compound Distributions for Exam STAM

The Collective Risk Model

The aggregate loss S = X1 + X2 + ... + XN, where N is the claim count (frequency) and Xi are iid claim amounts (severity), independent of N. This is the collective risk model. E[S] = E[N]*E[X] and Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. For Poisson frequency, Var(S) = lambda*E[X^2], a simplification that is heavily tested on Exam STAM.

The individual risk model treats each policy separately: S = sum of I_i*X_i, where I_i is a Bernoulli indicator for whether policy i has a claim and X_i is the claim amount. For large portfolios, the collective risk model is computationally simpler.

Panjer Recursion

When N belongs to the (a, b, 0) class and severity is discretized, the aggregate distribution can be computed recursively: g(x) = (1/(1-a*f(0))) * sum from y=1 to x of (a + b*y/x) * f(y) * g(x-y), starting from g(0) = P_N(f(0)). This recursion efficiently computes the entire distribution of S without simulation or convolutions.

The Panjer recursion is a key computational tool for Exam STAM, especially for computing stop-loss premiums and the probability of ruin in small-scale problems.

Normal Approximation

For large portfolios, the CLT justifies approximating S as Normal(E[S], Var(S)). The normal approximation is used for computing P(S > d) and stop-loss premiums E[(S-d)+]. The translated gamma approximation matches three moments (mean, variance, skewness) of S using a shifted gamma distribution and is more accurate than the normal approximation for skewed aggregate distributions.