Aggregate Loss Models: Compound Distributions for Exam STAM

The Collective Risk Model

The aggregate loss S = X_1 + X_2 + ... + X_N, where N is the random claim count (frequency) and the X_i are independent, identically distributed claim amounts (severity), independent of N. This is the collective risk model, also called the compound distribution model. It separates the randomness of how many claims occur from the randomness of how large each claim is.

The moments of S follow from the law of total expectation and total variance. E[S] = E[N]*E[X]. Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. The first term captures the variability in claim sizes (even if N were fixed), and the second captures the variability in claim count (even if severity were fixed).

For Poisson frequency (where E[N] = Var(N) = lambda), the variance simplifies to Var(S) = lambda*E[X^2]. This is because lambda*Var(X) + lambda*(E[X])^2 = lambda*(Var(X) + (E[X])^2) = lambda*E[X^2]. This simplification is heavily tested on Exam STAM and saves considerable computation.

The Individual Risk Model

The individual risk model treats each policy separately: S = sum from i=1 to n of I_i*X_i, where I_i ~ Bernoulli(q_i) indicates whether policy i has a claim and X_i is the claim amount if a claim occurs. The policies are assumed independent. E[S] = sum of q_i*E[X_i] and Var(S) = sum of q_i*(E[X_i^2] - q_i*(E[X_i])^2).

The individual risk model is more detailed but computationally demanding for large portfolios. In practice, the collective risk model is preferred because it requires only aggregate frequency and severity information rather than policy-level data. The two models give different results when individual policies have different characteristics (different q_i or different X_i distributions), but they converge for homogeneous portfolios.

Panjer Recursion

When N belongs to the (a, b, 0) class and the severity distribution is discretized onto non-negative integers, the aggregate distribution can be computed recursively. Let f(y) = P(X = y) be the discretized severity PMF and g(x) = P(S = x) be the aggregate loss PMF. The recursion is:

g(x) = (1/(1 - a*f(0))) * sum from y=1 to x of (a + b*y/x) * f(y) * g(x-y), for x = 1, 2, 3, ...

The starting value is g(0) = P_N(f(0)), the probability generating function of N evaluated at f(0). For Poisson frequency, g(0) = e^(-lambda*(1-f(0))). For negative binomial, g(0) = (1/(1+beta*(1-f(0))))^r.

The Panjer recursion efficiently computes the entire distribution of S without simulation or repeated convolutions. It is a key computational tool for Exam STAM, especially for computing stop-loss premiums, retention levels, and probability of ruin in small-scale problems.

Discretizing the Severity Distribution

To apply the Panjer recursion, the continuous severity distribution must be discretized. The rounding method assigns f(0) = F(h/2) and f(y) = F(yh + h/2) - F(yh - h/2) for y = 1, 2, ..., where h is the span (unit of discretization). The unmodified method uses f(y) = F((y+1)h) - F(yh). Smaller spans produce more accurate results but require more computational steps.

The choice of discretization method and span affects the accuracy of the resulting aggregate distribution. On Exam STAM, the span is usually specified, and you are expected to compute f(y) for the first several values and then apply the Panjer recursion to find g(x) for the required values.

Normal and Translated Gamma Approximations

For large portfolios, the Central Limit Theorem justifies approximating S as Normal(E[S], Var(S)). This approximation is computationally simple and requires only the mean and variance of S. It is used for computing P(S > d) and stop-loss premiums E[(S-d)+] = (E[S] - d)*(1 - Phi(z)) + sqrt(Var(S))*phi(z), where z = (d - E[S])/sqrt(Var(S)).

The normal approximation can be inaccurate for skewed aggregate distributions, which are common in insurance (few large claims create right skewness). The translated gamma approximation matches three moments (mean, variance, and skewness) of S using a shifted gamma distribution: S is approximated by Y + k, where Y ~ Gamma(alpha, beta). The parameters are: alpha = 4/gamma^2, beta = sqrt(Var(S))*gamma/2, and k = E[S] - 2*sqrt(Var(S))/gamma, where gamma is the coefficient of skewness of S.

The translated gamma approximation is more accurate than the normal approximation for positively skewed distributions and is preferred on Exam STAM when the skewness is known or can be computed.

Stop-Loss Premiums and Applications

The stop-loss premium at retention d is E[(S-d)+] = E[S] - E[S ^ d], where S ^ d = min(S, d). For an insurer, this represents the expected payment under an aggregate excess-of-loss reinsurance contract with retention d. Computing stop-loss premiums requires knowledge of the full aggregate loss distribution (from Panjer recursion) or an approximation (normal or translated gamma).

Using the exact distribution from Panjer: E[(S-d)+] = sum from x=d+1 to infinity of (x-d)*g(x) = E[S] - sum from x=0 to d of x*g(x) - d*(1 - G(d)), where G(d) = sum from x=0 to d of g(x). In practice, the sum is truncated at a large enough value that the tail probabilities are negligible.

Stop-loss premiums connect frequency models, severity models, and aggregate loss models into a complete framework for pricing reinsurance and setting reserves. They are among the most important calculations on Exam STAM.

Study Tips and Common Pitfalls

The most common mistake with the Panjer recursion is using incorrect (a, b) values or an incorrect starting value g(0). Always identify the frequency distribution first, determine a and b, compute g(0), and verify your setup before proceeding with the recursion. Each step builds on previous values, so an early error propagates through the entire calculation.

Another frequent error is confusing E[S] = E[N]*E[X] with E[S] = n*E[X] (using n instead of E[N]). The formula uses the expected frequency, not a fixed number. For the variance formula, remember that the Poisson simplification Var(S) = lambda*E[X^2] applies only when the frequency is Poisson. Practice at least 5 Panjer recursion problems and 3 stop-loss premium calculations to build computational fluency.