Excess Insurance

Study guides: Exam DISC-RM, Exam DISC-IA, Exam 7, Exam 8

Excess Insurance is coverage that pays only the part of a loss above an attachment point — an underlying policy's limit, a self-insured retention or a deductible — up to its own limit. It responds to the large, infrequent losses and leaves the working layer to the underlying insurer or the insured.

$Y = \min\left(\max(X - a,\ 0),\ l\right)$

$E[Y] = E[X \wedge (a + l)] - E[X \wedge a]$

$X$ is the ground-up loss per occurrence, $a$ the attachment point and $l$ the excess limit (" $l$ xs $a$ "). $E[X \wedge u]$ is the limited expected value, so the cover is one layer of the loss.
Forms. Excess liability and umbrella policies over primary limits; excess over an SIR; specific and aggregate excess workers compensation for self-insured employers; per-risk property excess; and excess-of-loss reinsurance, the same structure bought by an insurer. An excess-over-SIR limit sits above the retention: a $\$1$ M limit over a $\$250$ K SIR covers $\$250$ K– $\$1.25$ M.
Pricing (Exam 8). Few claims reach the layer, so its own experience is rarely credible. The expected cost comes from a fitted severity curve, ILFs or exposure curves. Inflation, development and parameter risk all bear harder on it than on primary coverage — see Excess and Deductible Rating.
Reserving (Exam 7). Excess claims are reported later and develop further. A claim now carried below the attachment can later pierce it, so the layer's IBNR includes development on known claims. Layer development factors are leveraged and volatile. Sahasrabuddhe links development patterns, trend and claim-size models, so that the pattern for one layer can be converted into another's. See Reinsurance Reserving and Ceded Loss Reserve.
Excess versus umbrella. An excess liability policy adds limits above one or more underlying policies, either following their form or on its own terms, without broadening their coverage. An umbrella also covers some claims the underlying policies exclude, above a self-insured retention, and drops down when an underlying aggregate is exhausted. How each responds to a claim is worked on the umbrella page.
Contract terms change the liability. Whether ALAE is pro rata, inside the limit or excess of it; aggregate limits and reinstatements; an umbrella dropping down when underlying aggregates are exhausted; and latent claims exposure.

Allocating Losses to an Umbrella Layer An insured has a $\$1$ M primary liability policy and a $\$4$ M xs $\$1$ M umbrella. Three occurrences settle at $\$600{,}000$ , $\$1{,}800{,}000$ and $\$7{,}000{,}000$ . Allocate them.
Answer Apply $\min(\max(X - 1\text{M}, 0), 4\text{M})$ to each loss for the umbrella. The primary pays $\min(X, 1\text{M})$ , and anything above $\$5$ M stays with the insured.

$\$600$ K: primary $\$600$ K, umbrella $0$ .

$\$1.8$ M: primary $\$1$ M, umbrella $\$800$ K.

$\$7$ M: primary $\$1$ M, umbrella $\$4$ M, uninsured $\$2$ M.
 $\begin{align*} \text{Primary} &= 0.6 + 1.0 + 1.0 \\ &= \$2.6\text{M} \\[4pt] \text{Umbrella} &= 0 + 0.8 + 4.0 \\ &= \$4.8\text{M} \\[4pt] \text{Uninsured} &= \$2.0\text{M} \end{align*}$ 
The three pieces add back to the $\$9.4$ M ground-up total. One occurrence produced $83\%$ of the umbrella's losses — the reason excess results are so volatile.

Why the Excess Layer Develops More Three claims are valued at 24 and 60 months: A goes from $\$300$ K to $\$450$ K, B from $\$800$ K to $\$1.3$ M, and C from $\$1.5$ M to $\$2.1$ M. Compare the 24-to-60 development of the ground-up losses, of the primary layer up to $\$1$ M, and of the layer excess of $\$1$ M.
Answer
 $\begin{align*} \text{Ground-up factor} &= \frac{450 + 1{,}300 + 2{,}100}{300 + 800 + 1{,}500} \\ &= \frac{3{,}850}{2{,}600} \\ &= 1.481 \\[6pt] \text{Primary factor} &= \frac{450 + 1{,}000 + 1{,}000}{300 + 800 + 1{,}000} \\ &= \frac{2{,}450}{2{,}100} \\ &= 1.167 \\[6pt] \text{Excess factor} &= \frac{0 + 300 + 1{,}100}{0 + 0 + 500} \\ &= \frac{1{,}400}{500} \\ &= 2.800 \end{align*}$ 
The excess layer develops almost twice as far as the ground-up losses. Part of that comes from claim B, which showed nothing excess at 24 months and then pierced the attachment. Applying the ground-up factor instead would project the layer at $500 \times 1.481 = \$741$ K rather than $\$1.4$ M, understating its ultimate by almost half.