Actuarial Notes

Extraordinary Losses

Study guide: Exam 5

Extraordinary Losses are losses too rare and too large to be priced from a few years of experience — individual shock losses and catastrophes. In ratemaking they are removed from the historical data and replaced with a provision based on a long-term view, so that rates reflect their expected cost without jumping after a bad year and falling after a quiet one.

\text{ELF} = 1 + \frac{\sum \text{Excess losses}}{\sum \text{Non-excess losses}}

\text{PP} = \text{PP}_{\text{capped, non-cat}} \times \text{ELF} + \text{PP}_{\text{cat}}

  • Shock losses are single high-severity claims — Werner's examples are a large multi-claimant liability claim, a total loss on an exceptionally valuable home, and the total permanent disability of a young worker. Each claim is capped at a threshold (the non-excess loss); the portion above it is excess. The excess loss factor (ELF) is the long-run ratio of excess to non-excess losses, applied to each year's capped losses. See Large Loss.
  • Choosing the threshold balances keeping as much actual experience as possible against removing volatility: the basic limit (the indication then assumes everyone buys the basic limit, and for a loss ratio indication premium is restated to basic limits too — higher limits are priced with ILFs), a percentile of the size-of-loss distribution such as the 99th, or for property a percentage of the amount of insurance.
  • Choosing the period: long enough to be stable, not so long the data are irrelevant — Werner contrasts about ten years for a medium-sized homeowners insurer's large fire losses with twenty for a small personal umbrella writer. Ideally the procedure censors trended losses, or indexes the threshold, since layers above a fixed threshold inflate faster.
  • Catastrophes are events that produce a large number of claims — hurricanes, tornadoes, hail, earthquakes, wildfires, winter storms, explosions, oil spills, some terrorist attacks. Werner cites the ISO Property Claim Services definition: at least \$25 million of direct insured property loss affecting a significant number of policyholders and insurers. See Catastrophe Loss.
    • Non-modeled perils that recur over decades (hail for auto comprehensive): a 10–30 year ratio of catastrophe to non-catastrophe losses, or catastrophe losses to exposures or amount-of-insurance years, applied to projected values.
    • Modeled perils too sporadic for any history (hurricane, earthquake): the catastrophe model's expected annual loss for the current exposure.
  • Order of adjustments: a modeled provision is already at future cost and ultimate value, so add it to non-catastrophe losses after those are developed and trended — adding it before over-adjusts it. Catastrophes are also excluded from trend data, and extraordinary losses from the data used to select development factors. Removing extraordinary losses without adding a provision back understates the rate.

An Excess Loss Factor from Twelve Years A liability line's reported losses over twelve years total \$600{,}000{,}000, including 30 claims above the \$500{,}000 threshold with ground-up value \$27{,}000{,}000. The latest accident year's reported losses are \$52{,}000{,}000, including two shock losses of \$2{,}400{,}000 and \$1{,}100{,}000.

Compute the excess loss factor and the latest year's adjusted losses.

Answer

\begin{align*}
\text{Excess} &= \$27{,}000{,}000 - 30 \times \$500{,}000 \\
&= \$12{,}000{,}000 \\
\text{Non-excess} &= \$600{,}000{,}000 - \$12{,}000{,}000 \\
&= \$588{,}000{,}000 \\
\text{ELF} &= 1 + 12 / 588 = 1.0204
\end{align*}
\begin{align*}
\text{Latest excess} &= \$1{,}900{,}000 + \$600{,}000 \\
&= \$2{,}500{,}000 \\
\text{Non-excess} &= \$52{,}000{,}000 - \$2{,}500{,}000 \\
&= \$49{,}500{,}000 \\
\text{Adjusted} &= \$49{,}500{,}000 \times 1.0204 \\
&= \$50{,}509{,}800
\end{align*}

The raw \$52{,}000{,}000 would overstate the year by 2.9\% because it happened to contain two shock losses; a year with none would be loaded up to its long-run level by the same factor.

Building a Homeowners Pure Premium with Two Catastrophe Provisions Developed and trended non-catastrophe pure premium is \$620. Over 25 years, hail losses (trended) averaged \$0.21 per \$1{,}000 of amount-of-insurance years; the projected average amount of insurance is \$400{,}000. The hurricane model gives an expected annual loss of \$1{,}800{,}000 on 20{,}000 house-years at the forecast cost level.

Compute the pure premium. What goes wrong if the hurricane provision is added before a 1.13 trend-and-development factor?

Answer

\begin{align*}
\text{Hail PP} &= \$0.21 \times 400 = \$84 \\
\text{Hurricane PP} &= \$1{,}800{,}000 / 20{,}000 = \$90 \\
\text{Total PP} &= \$620 + \$84 + \$90 \\
&= \$794
\end{align*}

The hail provision rests on a long history because hail recurs often enough to be measured; the hurricane provision comes from a model because thirty years of history may not contain the storm that matters.

The model output is already at the future cost level and ultimate. Adding it to the historical losses before the 1.13 factor would turn \$90 into \$101.70 — an over-adjustment of \$11.70 per house-year, or 1.5\% of the pure premium.