Collective Risk Model

\[S_N=X_1+X_2+\cdots+X_N\]

Conditional Expectation and Variance

\[\begin{align*} E[X]&=E[E[X|Y]] \\ \mathrm{Var}(X)&=E[\mathrm{Var}(X|Y)]+\mathrm{Var}(E[X|Y]) \end{align*}\]

Compound Distribution

\[\begin{align*} E[S]&=E[N]E[X] \\ \mathrm{Var}(S)&=E[N]\mathrm{Var}(X)+\mathrm{Var}(N)E[X]^2 \\ M_S(t)&=M_N(\ln(M_X(t))) \end{align*}\]

Compound Poisson, \(\mathcal{CPN}(\lambda,F_X)\)

If \(N\sim\mathcal{PN}(\lambda)\), then \[\begin{align*} E[S]&=\lambda E[X] \\ \mathrm{Var}(S)&=\lambda E[X^2] \end{align*}\]

If \(S_i\sim\mathcal{CPN}(\lambda_i,F_i)\), then \(\sum S_i\sim\mathcal{CPN}\left(\sum\lambda_i,\dfrac{1}{\sum\lambda_i}\sum\lambda_iF_i\right)\)

Compound negative binomial, \(\mathcal{CNB}(k,p,F_X)\)

  • \(\mathrm{Var}(N)>E[N]\)
  • Appropriate for modeling heterogeneity of number of claims since if \(N\ |\ \lambda\sim\mathcal{PN}(\lambda)\) and \(\lambda\sim\mathcal{G}(\alpha,\beta)\), then \(N\) unconditionally is distributed as negative binomial or more technically, Poisson-Gamma mixture distribution, i.e. mixture of Poissons with gamma as the mixing distribution.

Compound binomial, \(\mathcal{CBN}(n,p,F_X)\)

Appropriate for a portfolio of \(n\) independent policies, each of which can have at most one claim.

Effects of Reinsurance

Proportional reinsurance

\[\begin{align*} S_I&=\sum_{i=1}^N\alpha X_i=\alpha S\\ S_R&=\sum_{i=1}^N(1-\alpha)X_i=(1-\alpha)S \end{align*}\]

Excess of loss reinsurance

\[\begin{align*} S_I&=\sum_{i=1}^NY_i=\sum_{i=1}^N\min(X_i,M)\\ S_R&=\sum_{i=1}^NZ_i=\sum_{i=1}^N\max(0,X_i-M)\\ &=\sum_{i=1}^{N_R}W_i \end{align*}\]

Approximations

Normal approximation

\[S\sim\mathcal{N}(E[S],\mathrm{Var}(S))\] not skewed and tail probabilities are underestimated

Translated gamma

\[S\sim Y+k\quad\text{where}\quad Y\sim\mathcal{G}(\alpha,\lambda)\] By matching moments, \(\alpha=\dfrac{4}{\mathrm{sk}(S)^2}\), \(\lambda=\sqrt{\dfrac{\alpha}{\mathrm{Var}(S)}}\) and \(k=E[S]-\dfrac{\alpha}{\lambda}\).

Premium Calculation

  • Expected value principle (EVP) \[P=(1+\underset{\substack{\uparrow \\ \text{relative security loading}}}{\theta})E[S]\]
  • Standard deviation principle (SVP) \[P=E[S]+\theta\mathrm{sd}(S)\]
  • Variance principle (VP) \[P=E[S]+\theta\mathrm{Var}(S)\]