Aggregate Risk and Capital

Individual Risk Model

Aggregate heterogeneous policy risks and expose the effect of common dependence.

Individual Risk Model

The individual model keeps each insured risk visible. For $m$ policies,

S=\sum_{i=1}^{m}Y_i, \qquad Y_i=I_iB_i,

where $I_i$ indicates whether a claim occurs and $B_i$ is its amount. The simplest form allows at most one claim per risk during the period.

1. One risk

Let $I_i\sim\operatorname{Bernoulli}(p_i)$ , independent of $B_i$ , with

E[B_i]=\mu_i,\qquad \operatorname{Var}(B_i)=\sigma_i^2.

Then

E[Y_i]=p_i\mu_i,

\operatorname{Var}(Y_i) =p_i\sigma_i^2+p_i(1-p_i)\mu_i^2.

The first term is claim-size variation when a claim occurs; the second is occurrence variation.

2. Independent portfolio

If $Y_i$ are mutually independent,

E[S]=\sum_{i=1}^{m}p_i\mu_i,

\operatorname{Var}(S)= \sum_{i=1}^{m}\left[p_i\sigma_i^2+p_i(1-p_i)\mu_i^2\right].

Three-risk example

Risk	$p_i$	Mean claim	Claim SD	Expected loss	Loss variance (£²)
A	2%	£10,000	£5,000	£200	$2.460\times10^6$
B	5%	£5,000	£3,000	£250	$1.638\times10^6$
C	1%	£50,000	£30,000	£500	$33.750\times10^6$
Total				£950	$37.848\times10^6$

Portfolio SD is about £6,152. Risk C supplies more than half the expected loss and almost 89% of variance. Risk contribution, not policy count, identifies concentration.

3. Dependence adds covariance

Without independence,

\operatorname{Var}(S) =\sum_i\operatorname{Var}(Y_i) +2\sum_{i<j}\operatorname{Cov}(Y_i,Y_j).

Small positive covariance across many pairs can dominate individual variances. Sources include weather, geography, supply-chain inflation, court decisions, and shared policyholders.

Let $Z$ indicate a catastrophe state. Conditional on $Z$ , risks may be independent with probabilities $p_i(Z)$ ; marginally they are dependent because the same $Z$ shifts many probabilities. This gives a mechanism and a scenario to validate, rather than inserting one arbitrary correlation coefficient.

4. More than one claim per policy

If a policy can have multiple claims, replace Bernoulli $I_i$ with count $N_i$ :

Y_i=\sum_{j=1}^{N_i}X_{ij}.

The portfolio becomes a sum of policy-level collective models. The individual/collective distinction is therefore about retained detail, not mutually exclusive theories.

5. Approximation by a collective model

When $m$ is large and each $p_i$ is small, total claim count $\sum I_i$ may be approximated by Poisson with

\lambda=\sum_i p_i.

Severity becomes a probability-weighted mixture of risk severities. This simplifies calculation but can lose concentration and dependence information. Validate the aggregate tail, not only the mean.

6. Portfolio decisions

The individual model is useful for:

pricing/risk classification with policy covariates;
concentration by geography or sum insured;
policy-specific deductibles and limits;
accumulation and scenario analysis;
explaining which segments drive capital.

It is less useful when risk-level identifiers are unstable or the available data only support aggregate counts and severities.

Practice

One risk has $p=0.1$ , claim mean £2,000, and claim SD £1,000. Find its expected loss and variance.
Two risks each have variance 4 and covariance 1. Find variance of their sum.
Why might a Poisson approximation preserve expected count but understate catastrophe tail?

Answers

Mean £200; variance $=0.1(1{,}000^2)+0.1(0.9)(2{,}000^2)=460{,}000\,£^2$ .
$4+4+2(1)=10$ .
It can discard shared-event dependence, treating many claims from one catastrophe as independent arrivals.

Verify both models in the Computation Lab.

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Collective Risk Model

Derive aggregate-loss moments and transforms, then see exactly where independence enters.

Aggregate Risk Computation Lab

Reconcile analytical moments with transparent R and Python simulations.