Insurance Portfolio

Study guides: Exam MAS-I, Exam 9

An insurance portfolio, or book of business, is the set of policies an insurer holds, whether in one line, one region or in total, whose combined losses the insurer bears. Risk is pooled, measured and managed at the portfolio level. What a policy costs the insurer depends on how its losses move together with everyone else's.

$S = \sum_{j=1}^{n} S_j$

$\text{Var}(S) = \sum_{j} \text{Var}(S_j) + \sum_{j \neq k} \text{Cov}(S_j, S_k)$

$S_j$ is policy $j$ 's loss for the period. When policies are independent, the covariance terms vanish and the coefficient of variation of $S$ falls like $1/\sqrt{n}$ . That is the pooling insurance depends on, and it breaks down when policies are correlated.
Catastrophe accumulation (Exam 9). A hurricane or earthquake causes many claims close together all at once, so policies in its footprint are correlated and the covariance terms do not vanish. Insuring 1,000 homes spread around a state may be acceptable where 1,000 homes on one stretch of coast are not. Cat exposure is therefore managed across the whole book. Cat models produce the expected loss cost (the average annual loss) and PMLs at chosen return periods, including how much one more location adds to the book's PML. The insurer controls concentrations through risk selection, premiums commensurate with the risk, and reinsurance (Catastrophe Exposure Management).
Heterogeneity (MAS-I). Policies differ in their underlying claim rates. A mixed Poisson model treats a policy's rate $\Lambda$ as a draw from a mixing distribution across the portfolio. A gamma mix gives negative binomial counts. A policy's own claims then tell you something about its $\Lambda$ , which is the basis of experience rating and credibility.
Marginal view. In risk-adjusted pricing, what a new policy adds to the book depends on how its losses interact with the existing policies. Marginal methods set its risk load from the change in portfolio variance or standard deviation. Adding account $n$ to portfolio $L$ changes the variance by $\text{Var}(n) + 2\,\text{Cov}(L, n)$ , so its covariance with the book counts, not just its standalone variance.
This is not the Portfolio page, which covers an Exam FM investment portfolio on the asset side of the balance sheet. An insurance portfolio is on the liability side.

Where Diversification Stops Each home in a coastal book has an expected annual loss of $\mu = \$1{,}000$ and a standard deviation of $\sigma = \$10{,}000$ . Any two homes' losses have correlation $\rho$ .

Find the portfolio's coefficient of variation for $n = 1{,}000$ and $n = 10{,}000$ homes, first with $\rho = 0$ and then with $\rho = 0.01$ .
Answer With $n$ equal variances and a common correlation:
 $\begin{align*} \text{Var}(S) &= n\sigma^2 + n(n-1)\rho\,\sigma^2 \\ \text{CV}(S) &= \frac{\sigma}{\mu}\sqrt{\frac{1 + (n-1)\rho}{n}} \end{align*}$ 
With $\rho = 0$ :

$n = 1{,}000$ : $\text{CV} = 10\sqrt{1/1{,}000} = 0.316$

$n = 10{,}000$ : $\text{CV} = 10\sqrt{1/10{,}000} = 0.100$

With $\rho = 0.01$ :

$n = 1{,}000$ : $\text{CV} = 10\sqrt{10.99/1{,}000} = 1.048$

$n = 10{,}000$ : $\text{CV} = 10\sqrt{100.99/10{,}000} = 1.005$

Independent risks diversify steadily. A correlation of just $0.01$ sets a floor of $\frac{\sigma}{\mu}\sqrt{\rho} = 1.0$ that growth can't get below. Adding 9,000 homes in the same footprint barely changes the relative risk. Reducing it further takes writing elsewhere or ceding risk to reinsurers.

What One Claim Says About a Policy A portfolio is $70\%$ low-risk policies with Poisson rate $0.05$ and $30\%$ high-risk policies with rate $0.20$ . The insurer can't tell them apart at underwriting. A policy has exactly one claim this year.

Find the probability it is high-risk and its expected claims next year.
Answer
 $\begin{align*} P(N{=}1 \mid \text{low}) &= 0.05\,e^{-0.05} \\ &= 0.04756 \\[4pt] P(N{=}1 \mid \text{high}) &= 0.20\,e^{-0.20} \\ &= 0.16375 \end{align*}$ 
 $\begin{align*} P(\text{high} \mid N{=}1) &= \frac{0.3(0.16375)}{0.7(0.04756) + 0.3(0.16375)} \\ &= \frac{0.04912}{0.08242} \\ &= 0.596 \end{align*}$ 
 $\begin{align*} E[\Lambda \mid N{=}1] &= 0.404(0.05) + 0.596(0.20) \\ &= 0.139 \end{align*}$ 
The portfolio average rate is $0.7(0.05) + 0.3(0.20) = 0.095$ . One claim raises this policy's expected frequency by almost half. Pricing it at the portfolio average would undercharge it, which is why experience rating uses this information.