Individual vs. Collective Risk Models
Compare individual and collective risk models for aggregate loss distributions on Exam STAM.
Individual Risk Model
The individual risk model considers a fixed portfolio of n independent policies. Each policy i has its own claim probability q_i and severity distribution. The random variable X_i represents the loss from policy i: it equals zero with probability 1 - q_i (no claim) and follows a severity distribution with probability q_i. Aggregate losses for the portfolio are S = X_1 + X_2 + ... + X_n, the sum across all n policies.
The mean of S is E[S] = sum of E[X_i] = sum of q_i * E[B_i], where B_i is the claim amount for policy i given that a claim occurs. The variance is Var(S) = sum of Var(X_i), since policies are independent. Each individual variance is Var(X_i) = q_i * E[B_i^2] - (q_i * E[B_i])^2 = q_i * (1-q_i) * (E[B_i])^2 + q_i * Var(B_i). This decomposition separates the variance from the uncertainty of whether a claim occurs from the variability of the claim amount given occurrence.
Strengths and Limitations of the Individual Model
The individual risk model is realistic because it preserves the heterogeneity of the portfolio. In a real insurance company, policies have different coverage levels, deductibles, and risk characteristics, and the individual model captures these differences. It is the natural framework for analyzing specific portfolios where each policy's contribution to total risk matters.
The main limitation is computational complexity. For n policies, computing the exact distribution of S requires convolving n different distributions, which is impractical for large portfolios. Even with identical policies, the computation grows quickly. For this reason, the individual model is mainly used for small portfolios or when the normal approximation is applied. The central limit theorem justifies approximating S as normal when n is large, with mean E[S] and variance Var(S) computed from individual policy parameters.
Collective Risk Model
The collective risk model simplifies by treating aggregate losses as S = X_1 + X_2 + ... + X_N, where N is a random variable representing the total number of claims across the portfolio and X_1, X_2, ..., X_N are independent and identically distributed (i.i.d.) claim amounts, independent of N. This formulation does not track which policy generates each claim. Instead, all claims are pooled and treated as draws from a common severity distribution.
The key assumptions are: (1) claim amounts X_i are i.i.d. with common distribution F_X, (2) the claim count N follows a specified frequency distribution (Poisson, negative binomial, or binomial), and (3) N is independent of the X_i values. These assumptions enable a powerful set of computational tools that make the collective model far more tractable than the individual model.
Moments of the Collective Model
The mean of S is E[S] = E[N] * E[X], derived from the law of iterated expectations: E[S] = E[E[S|N]] = E[N * E[X]] = E[N] * E[X]. The variance uses the law of total variance: Var(S) = E[Var(S|N)] + Var(E[S|N]) = E[N * Var(X)] + Var(N * E[X]) = E[N] * Var(X) + Var(N) * (E[X])^2.
For a Poisson frequency with E[N] = Var(N) = lambda, the variance simplifies to Var(S) = lambda * E[X^2], since lambda * Var(X) + lambda * (E[X])^2 = lambda * (Var(X) + (E[X])^2) = lambda * E[X^2]. This elegant result means the aggregate variance depends only on the second moment of the severity, not separately on its mean and variance.
For a negative binomial frequency with E[N] = r*beta and Var(N) = r*beta*(1+beta), the variance of S is r*beta*Var(X) + r*beta*(1+beta)*(E[X])^2 = r*beta*E[X^2] + r*beta^2*(E[X])^2. The extra term r*beta^2*(E[X])^2 reflects the additional variability from the overdispersed frequency distribution.
Computational Tools for the Collective Model
The moment generating function (MGF) of S in the collective model is M_S(t) = M_N(ln(M_X(t))), where M_N is the MGF of N and M_X is the MGF of X. This composition property enables derivation of the aggregate distribution in certain cases. For a compound Poisson model, M_S(t) = exp(lambda * (M_X(t) - 1)).
The Panjer recursion computes the exact PMF of S when the severity is discretized and the frequency belongs to the (a,b,0) class (Poisson, negative binomial, or binomial). Convolution computes the distribution of S by iteratively convolving the severity distribution with itself, but this is only practical for small values of N. Simulation provides a flexible alternative: generate N from the frequency distribution, then generate N claim amounts from the severity distribution and sum them. Repeating this thousands of times yields an empirical estimate of the distribution of S.
Connecting the Two Models
The individual and collective models can be reconciled. If all n policies in an individual risk model have the same claim probability q and the same severity distribution, the individual model becomes a special case of the collective model with N ~ Binomial(n, q). More generally, the collective model can approximate the individual model by choosing N to match the total expected claim count and X to match the portfolio's average severity characteristics.
The approximation is most accurate when individual policies are similar. When policies are heterogeneous, the collective model's assumption of i.i.d. claim amounts introduces some error, but this is often acceptable given the computational advantages. In practice, portfolios are often segmented into more homogeneous subgroups, each modeled collectively, and then aggregated.
Common Pitfalls
Students sometimes confuse the number of policies (n in the individual model) with the number of claims (N in the collective model). In the individual model, n is fixed and each policy generates at most one claim (in the basic formulation). In the collective model, N is random and represents the total claim count across the entire portfolio. Another common error is applying the collective model variance formula incorrectly by using Var(X) where E[X^2] is needed, or vice versa. Keep the formula Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2 clearly in mind and identify which moments you have been given.
Study Tips
Exam STAM heavily favors the collective risk model because of its computational tractability. Focus your study time on computing moments of aggregate losses under different frequency assumptions (Poisson, negative binomial, binomial) and practicing with the MGF composition formula. Be able to set up the collective model from a problem description, identify the appropriate frequency and severity distributions, and compute E[S] and Var(S). Also understand when and why the individual model is used, and how the normal approximation bridges both models for large portfolios.