Large Loss

Study guides: Exam 5, Exam GI-101

Large Losses (shock losses) are individual claims big enough that including them at face value would distort an experience analysis — a single claim moving a class's indicated rate by more than the underlying cost level justifies.

$\text{Capped Loss} = \min(X, M)$

$\text{Excess Load} = \frac{\text{Expected losses above } M}{\text{Expected losses below } M}$

The standard treatment has three steps: cap each claim at a threshold $M$ for the experience analysis, compute an excess load from a longer-term view of how often such losses occur, and add it back. The rate then covers the full expected cost without being hostage to whether a shock loss happened to occur in the experience period.
Ideally the threshold is the point at which losses become extraordinary and their inclusion makes rates volatile. It may be the basic limit — and then the premium in a loss ratio indication must be put on a basic-limits basis too. Otherwise Werner sets it to balance including as many losses as possible against minimizing volatility: for example at the $99$ th percentile of claim sizes, or at a percentage of losses; where the size-of-loss distribution varies a lot by policy (property), as a percentage of the amount of insurance rather than a fixed amount.
The excess load is computed over a long period because large losses are rare — Werner's illustrations are $10$ years for a medium-sized homeowners insurer's large fire losses and $20$ for a small personal umbrella insurer — but not so long that older data has become irrelevant. Ideally the procedure is run on losses trended to future levels, since higher layers are subject to greater inflation.
In reserving the parallel treatment applies: the actuary may exclude large claims from the initial projection, then add back a case-specific projection for the reported large claims and a smoothed provision for the IBNR portion of large claims (Friedland), or build the triangles on claims limited to, say, $\$500{,}000$ or $\$1$ M per occurrence. Leaving a $\$2$ M claim in an immature diagonal and multiplying it by a CDF of $4$ produces an $\$8$ M estimate from one claim.
Distinguish a large loss from a catastrophe: a large loss is one claim of unusual size, a catastrophe is many claims from one event. Both are excluded from base experience, but the loads are built differently.

Capping and Loading A general liability class has a $\$500{,}000$ per-occurrence cap. Over five years the class produced $\$14{,}000{,}000$ of losses on $20{,}000$ exposures, including one claim that settled at $\$1{,}200{,}000$ and another at $\$800{,}000$ . Ten years of wider data show that losses above $\$500{,}000$ have averaged $8\%$ of capped losses.

Compute the pure premium.
Answer Cap the two large claims, removing the excess:
 $\begin{align*} \text{Excess removed} &= (\$1{,}200{,}000 - \$500{,}000) \\ &\quad + (\$800{,}000 - \$500{,}000) \\ &= \$1{,}000{,}000 \\[4pt] \text{Capped losses} &= \$14{,}000{,}000 - \$1{,}000{,}000 \\ &= \$13{,}000{,}000 \end{align*}$ 
 $\begin{align*} \text{Capped PP} &= \frac{\$13{,}000{,}000}{20{,}000} = \$650 \\[4pt] \text{Loaded PP} &= \$650 \times 1.08 = \$702 \end{align*}$ 
The uncapped figure would have been $\$14{,}000{,}000/20{,}000 = \$700$ — close to the answer here, but only by coincidence. Had the five years contained no claim over $\$500{,}000$ , the uncapped pure premium would have been $\$600$ and the class would have been under-priced by $17\%$ ; had it contained a $\$5$ M claim, over-priced by a similar margin. Capping removes that dependence on luck.

A Shock Loss in an Immature Triangle Accident year $2024$ reported losses at $12$ months are $\$4{,}500{,}000$ , including a single $\$1{,}500{,}000$ claim reserved at policy limits. The selected reported CDF at $12$ months is $3.20$ , built on ordinary (non-large) claims. A smoothed provision for large claims not yet reported, from the long-term large-loss experience, is $\$700{,}000$ .

Estimate ultimate losses correctly.
Answer Wrong: develop everything.

$\$4{,}500{,}000 \times 3.20 = \$14{,}400{,}000$

This assumes the $\$1.5$ M claim will grow to $\$4.8$ M — impossible, since it is already at policy limits, and unjustified in any case because the CDF was built on ordinary claims.

Right: separate the two.
 $\begin{align*} \text{Ordinary} &= \$3{,}000{,}000 \times 3.20 = \$9{,}600{,}000 \\ \text{Large claim} &= \$1{,}500{,}000 \text{ (at limits, no development)} \\[4pt] \text{Expected additional large losses} &\approx \$700{,}000 \\[4pt] \text{Ultimate} &= \$9{,}600{,}000 + \$1{,}500{,}000 + \$700{,}000 \\ &= \$11{,}800{,}000 \end{align*}$ 
The provision for additional large claims not yet reported is essential — removing the known shock loss without adding an expected-large-loss load leaves the estimate short. The two errors are opposite in sign, and doing neither adjustment is not a safe middle ground.