Frequency and Severity Models

Mixtures and Heterogeneity

Represent distinct risk populations and latent states without mistaking flexibility for explanation.

Mixtures and Heterogeneity

A portfolio distribution can be wide because each risk is volatile, because risks belong to different groups, or both. Mixture models separate within-group and between-group variation.

1. Finite mixture

Let latent class $Z\in\{1,\ldots,K\}$ have $P(Z=k)=\pi_k$ , where $\sum_k\pi_k=1$ . Conditional on class $k$ , loss has density $f_k$ :

f_X(x)=\sum_{k=1}^{K}\pi_k f_k(x).

The moments expose the two sources of variation:

E[X]=\sum_k\pi_k\mu_k,

\operatorname{Var}(X) =\underbrace{\sum_k\pi_k\sigma_k^2}_{\text{within classes}} +\underbrace{\sum_k\pi_k(\mu_k-\mu)^2}_{\text{between classes}}.

Example: ordinary and complex claims

Suppose 90% of claims have mean £2,000 and SD £1,000; 10% have mean £20,000 and SD £10,000.

E[X]=0.9(2{,}000)+0.1(20{,}000)=£3{,}800.

The mixture SD is about £6,329. A single “average claim” hides both the rare high-cost class and the uncertainty over class membership.

2. Observed segmentation versus latent class

Approach	Example	Advantage	Risk
observed segmentation	vehicle type or injury indicator	interpretable and actionable	coding changes; small cells
latent finite mixture	unobserved simple/complex claim	flexible distribution shape	classes may not have unique meaning
hierarchical/random effect	region or policyholder risk varies continuously	partial pooling	distribution of random effects matters

Prefer observed, causally relevant information when available. Latent classes are mathematical devices unless external evidence gives them operational meaning.

3. Continuous mixture: Poisson–Gamma gives negative binomial

Suppose conditional claim count is Poisson:

N\mid\Lambda\sim\operatorname{Poisson}(\Lambda),

but latent rates differ across risks:

E[\Lambda]=\mu, \qquad \operatorname{Var}(\Lambda)=\frac{\mu^2}{k}.

Gamma mixing produces a negative-binomial marginal count with

E[N]=\mu, \qquad \operatorname{Var}(N)=\mu+\frac{\mu^2}{k}.

The first variance term is Poisson process variation; the second is heterogeneity.

Poisson–Gamma heterogeneity creates overdispersion

4. Mixture is not compound aggregation

Do not confuse:

\text{mixture: } X\sim F_Z

with

\text{compound sum: } S=X_1+\cdots+X_N.

A mixture selects a distribution or parameter state; a compound model sums a random number of losses. An insurance portfolio can require both.

5. Changing mix can imitate inflation

Suppose group-specific severities stay constant, but the high-cost group's weight rises from 10% to 20%:

0.9(2{,}000)+0.1(20{,}000)=3{,}800,

0.8(2{,}000)+0.2(20{,}000)=5{,}600.

Portfolio mean rises 47.4% with no within-group inflation. Severity trends should therefore be decomposed into price, coverage, and mix effects.

6. Identifiability and stability

Mixture likelihoods can have multiple local optima, near-empty components, and label switching. A component with tiny weight and huge mean can dominate the tail. Use:

several initial values and convergence checks;
minimum-volume or regularisation rules justified before seeing results;
holdout performance and tail sensitivity;
stability across valuation dates;
comparison with simpler observed segmentation.

A two-component model can fit many shapes. It does not prove that two real claim populations exist, nor that their fitted weights will remain stable.

Practice

A 70/30 mixture has component means 1 and 6. Find the mixture mean.
In a Poisson–Gamma mixture, what happens as $k\to\infty$ ?
Why can changing class weights bias a trend estimated from the unsegmented mean?

Answers

$0.7(1)+0.3(6)=2.5$ .
$\operatorname{Var}(\Lambda)=\mu^2/k\to0$ ; latent rates become homogeneous and the marginal count approaches Poisson variance $\mu$ .
The aggregate mean changes even when every class-specific mean is constant; composition is confounded with within-class trend.

Next, apply these distributions to contractually transformed losses.

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Fitting and Validating Loss Models

Fit the likelihood that generated the observations, then validate the decision-relevant centre and tail.

Reinsurance as a Loss Transformation

Translate treaty wording into gross, ceded, and retained random variables before pricing or capital analysis.