Catastrophe Risk

Study guide: Exam 9

Catastrophe Risk is the risk of a very large loss from a single event — hurricane, earthquake, flood, wildfire, severe convective storm, or a man-made event such as terrorism — that damages many insured exposures at once. Its defining feature is correlation: one event strikes many policies together, so pooling does not diversify it the way it diversifies independent claims.

$\text{EP}(x) = P(\text{annual loss} > x)$

$\text{Return period}(x) = \frac{1}{\text{EP}(x)}$

The exceedance probability curve $\text{EP}(x)$ is the standard description of catastrophe risk. The occurrence curve (OEP) looks at the largest single event in a year, the aggregate curve (AEP) at the year's total. The area under the AEP curve is the average annual loss (Catastrophe Expected Loss Cost), and the loss at a chosen return period is the PML.
How it is modelled. Grossi and Kunreuther describe a catastrophe model in four parts: hazard (where events occur, how often, how intense), inventory (the exposed properties), vulnerability (damage given intensity) and loss (the financial result after policy terms). See Catastrophe Modelling.
Two kinds of uncertainty. Aleatory uncertainty is the inherent randomness of events and cannot be reduced; epistemic uncertainty comes from incomplete knowledge of the hazard, vulnerability and exposure, and can be. Different vendor models give materially different answers for the same book (Model Risk).
Why it is capital-hungry. Because the Law of Large Numbers does not apply across a correlated book, catastrophe risk needs capital out of proportion to its expected loss: it supports little leverage and needs high margin ratios. It is the main reason insurers buy reinsurance (Reinsurance Pricing) and manage accumulations (Catastrophe Exposure Management).
Capital markets. Securitization moves catastrophe risk to investors through CAT Bonds and other Insurance-Linked Securities. A cat bond's principal is held in a collateral trust, largely removing counterparty credit risk. Its trigger — indemnity (the sponsor's own losses), industry index, parametric (physical event measures) or modelled loss — trades basis risk (the payout not matching the sponsor's loss) against moral hazard, transparency and speed of settlement.
Public mechanisms share part of the risk, for example the Florida Hurricane Catastrophe Fund; a changing climate shifts the hazard itself (Climate Risk).

Why Pooling Fails: Independent Versus Correlated Homes An insurer covers $1{,}000$ homes, each with a $1\%$ annual chance of a $200,000 total loss. Compare the mean and standard deviation of annual losses if (a) the losses are independent (fires) and (b) a single event destroys every home at once with probability $1\%$ (a storm).
Answer The mean is the same in both: $1{,}000 \times 0.01 \times \$200{,}000 = \$2$ M.
 $\begin{align*} \sigma_{(a)} &= \$200{,}000\sqrt{1{,}000(0.01)(0.99)} \\ &= \$0.63\text{M} \\ \sigma_{(b)} &= \$200\text{M}\sqrt{0.01(0.99)} \\ &= \$19.9\text{M} \end{align*}$ 
The coefficient of variation is $0.31$ for independent losses and $9.95$ for correlated ones — about $32$ times as large. Adding homes shrinks the relative volatility in (a) but not at all in (b). The expected loss is identical; the capital needed is not.

Occurrence Versus Aggregate Exceedance A region faces two independent perils, each striking at most once a year: a hurricane costing $60$ with probability $0.05$ , and a hailstorm costing $30$ with probability $0.10$ . Find the AAL, the occurrence and aggregate exceedance probabilities at $50$ and $70$ , and the return period of a year's losses exceeding $70$ .
Answer
 $\begin{align*} \text{AAL} &= 0.05(60) + 0.10(30) \\ &= 6 \\ \text{OEP}(50) &= P(\text{hurricane}) \\ &= 0.05 \\ \text{AEP}(70) &= P(\text{both occur}) \\ &= 0.05(0.10) \\ &= 0.005 \end{align*}$ 
$\text{AEP}(50) = 0.05$ as well, since the year's total exceeds $50$ exactly when the hurricane occurs; but $\text{OEP}(70) = 0$ , because no single event exceeds $70$ . The year's total exceeds $70$ once in $1/0.005 = 200$ years. A per-occurrence cover attaching at $70$ would never pay; an aggregate cover attaching at $70$ would — which is why the two curves answer different questions.