Catastrophe Loss

Study guide: Exam 5

Catastrophe Loss is loss arising from a single event, or a series of related events, that produces many claims at once — hurricane, earthquake, wildfire, hail, winter storm, terrorism. The defining feature for ratemaking is not the size of any one claim but the correlation across the book: the independence assumption that makes ordinary experience credible fails completely.

\text{Total Pure Premium} = \text{Non-Cat PP} + \text{Cat Load}

\text{Cat Load} = \frac{\text{Expected Annual Catastrophe Loss}}{\text{Exposures}}

• Catastrophe losses are removed from the experience period and replaced with a long-run expected load. A five-year experience period either contains the hurricane or does not, and neither answer is the expected cost.
• Two ways to build the load:
  • Long-run historical, for perils with enough history relative to their return period — hail, wind, winter storm. Werner's approach: express historical catastrophe losses as a ratio to a stable base (non-catastrophe losses, or amount of insurance years) over 10–30 years, and apply that ratio to the current book.
  • Modelled, for perils whose return periods exceed any usable history — hurricane, earthquake. A catastrophe model simulates an event set against the insurer's actual exposure to produce an average annual loss (AAL) and the full loss distribution.
• The exposure base must be current. Historical hurricane losses reflect the book as it was; the load must reflect where the insurer writes today, which is what a model does naturally and a historical ratio does only if the base is adjusted.
• Modelled output also drives capital and reinsurance decisions through the exceedance probability curve — the 1-in-100 and 1-in-250 PMLs — so the same model serves pricing, reinsurance purchasing and solvency work.
• Consistency of definition across years is essential. If a \$40M hail event is treated as a catastrophe in one year and left in the base experience in another, both the non-catastrophe trend and the load are corrupted.
• In reserving, catastrophe claims are triangulated separately: they report fast and settle in a distinctive pattern (rapid initial payment, then a long tail of disputed and litigated claims), quite unlike ordinary property claims.

Building a Homeowners Catastrophe Load Five years of experience: \$50{,}000{,}000 of losses on 10{,}000 house-years a year, of which \$8{,}000{,}000 came from a single hurricane. The insurer's catastrophe model gives an average annual hurricane loss of \$600{,}000 for the current book.

Compute the pure premium.

Answer

\begin{align*}
\text{Non-cat losses} &= \$50{,}000{,}000 - \$8{,}000{,}000 \\
&= \$42{,}000{,}000 \\[4pt]
\text{Non-cat PP} &= \frac{\$42{,}000{,}000}{5 \times 10{,}000} \\
&= \$840 \\[6pt]
\text{Cat load} &= \frac{\$600{,}000}{10{,}000} = \$60 \\[4pt]
\text{Total PP} &= \$840 + \$60 = \$900
\end{align*}

Compare with leaving the hurricane in: \$50{,}000{,}000 / 50{,}000 = \$1{,}000 per house-year — an 11\% overstatement, because the experience period happened to contain one hurricane where the modelled expectation is \$600{,}000 a year, or \$3{,}000{,}000 over five years.

Had the five years contained no hurricane, the same procedure would have produced \$900 again, while the unadjusted calculation would have said \$840. Stability across experience periods is the point of the exercise.

Why Historical Averages Fail for Hurricane A coastal insurer's ten-year catastrophe loss history: \$0, \$1M, \$0, \$4M, \$0, \$0, \$38M, \$2M, \$0, \$1M, on \$50{,}000{,}000 of annual premium.

Evaluate the ten-year average as a catastrophe load.

Answer \text{Ten-year average} = \frac{\$46{,}000{,}000}{10} = \$4{,}600{,}000 \text{ per year} \;(9.2\% \text{ of premium})

The number is arithmetically fine and actuarially useless:

• The median year is \$0.5M. Five of ten years had no catastrophe at all. The mean is driven entirely by one event.
• Drop the \$38M year and the average falls to \$0.9M — a 1.8\% load. Add a second such event and it doubles. An estimator that swings by a factor of five on one observation has no credibility.
• The return period exceeds the data. If a \$38M event is a 1-in-25-year loss for this book, ten years of history cannot estimate its frequency; observing one says almost nothing about whether the true rate is 1-in-10 or 1-in-50.

A modelled AAL solves all three: it simulates thousands of years of events against the current exposure, so the estimate does not depend on which storms happened to occur in the last decade, and it prices the current concentration rather than the historical one.