Excess of Loss

Study guides: Exam FAM, Exam DISC-RM, Exam 9

Excess of Loss (XS or XOL) is non-proportional reinsurance in which the reinsurer pays the part of each loss — per risk or per occurrence — above the cedant's retention $R$ (the attachment point), up to a limit $L$ . The cover is written " $L$ xs $R$ " and priced as a loss cost, or rate, on subject premium.

$Y = \min\left(\max(X - R,\, 0),\; L\right)$

$E[Y] = E[X \wedge (R + L)] - E[X \wedge R]$

$X$ is the loss to one risk (per risk) or the total from one occurrence (per occurrence), and $E[X \wedge u]$ is the limited expected value. Expected layer loss is expected claim count times $E[Y]$ — the layer of the severity distribution between $R$ and $R+L$ .
Forms. Property per risk: one location. Casualty per occurrence: a working layer, penetrated often; an exposed excess layer, attaching below some policy limits; or a clash layer above them. Property catastrophe: per occurrence across many risks, with Reinstatements. The aggregate version is Aggregate Excess of Loss.
Experience rating (burn cost). Trend each historical loss individually before applying the layer. Excess losses grow faster than trend because a loss just below $R$ trends into the layer. Sum the layer losses by year, apply excess development factors (the layer reports and develops later than gross), and divide by on-level subject premium. Decide how ALAE enters: pro rata with loss, or added to loss so the limit applies to the sum.
Exposure rating prices the current profile rather than past experience. For property, use an exposure curve: $G\big((R+L)/IV\big) - G\big(R/IV\big)$ by insured-value band. For casualty, use a severity curve or ILFs: $\big[E[X \wedge \min(PL, R+L)] - E[X \wedge \min(PL, R)]\big] / E[X \wedge PL]$ by policy limit $PL$ . In both cases the factor multiplies subject premium times the expected loss ratio. Experience and exposure indications are blended by credibility.
Price language: rate $=$ premium $\div$ subject premium; rate on line $=$ premium $\div L$ ; payback $= L \div$ premium, in years.
XS takes away the large-loss volatility a Quota Share leaves in place, but ceded losses report late and develop steeply — see Reinsurance and Reinsurance Reserving.

Experience-Rating a Per-Risk Layer Price a property per-risk layer of $\$750$ K xs $\$250$ K. Historical large losses with trend factors to the treaty period and the layer's excess development factors:

2021: $\$400$ K, trend $1.26$ , LDF $1.00$

2022: $\$230$ K, trend $1.20$ , LDF $1.02$

2023: $\$900$ K, trend $1.14$ , LDF $1.10$

2024: $\$300$ K, trend $1.09$ , LDF $1.30$

On-level subject premium is $\$4.0$ M in each year. Find the loss cost, and compare trended layer losses with untrended ones.
Answer Trend first, then layer (in $ thousands):

2021: $504$ , layer $254$

2022: $276$ , layer $26$ (untrended, $230$ misses the layer)

2023: $1{,}026$ , layer $750$ (capped)

2024: $327$ , layer $77$
 $\begin{align*} \text{Developed} &= 254(1.00) + 26(1.02) \\ &\quad + 750(1.10) + 77(1.30) \\ &= 254.0 + 26.5 + 825.0 + 100.1 \\ &= 1{,}205.6 \end{align*}$ 
 $\begin{align*} \text{Loss cost} &= \frac{1{,}205.6}{16{,}000} \\ &= 7.54\% \end{align*}$ 
Leverage: untrended layer losses are $150 + 0 + 650 + 50 = 850$ , so trend raised them by $1{,}107/850 - 1 = 30.2\%$ . Ground-up losses rose only $2{,}133/1{,}830 - 1 = 16.6\%$ . Applying the layer before trending would understate the loss cost badly.

Exposure-Rating a Casualty Layer with ILFs A cedant's general liability book: $\$10$ M of subject premium on $\$1$ M-limit policies and $\$5$ M on $\$2$ M-limit policies. The expected loss and ALAE ratio is $65\%$ , with ALAE pro rata. Loss ILFs: $\text{ILF}(300\text{K}) = 1.35$ , $\text{ILF}(1\text{M}) = 1.75$ , $\text{ILF}(2\text{M}) = 2.00$ . Find the loss cost of a $\$700$ K xs $\$300$ K per-occurrence treaty.
Answer The layer tops out at $\$1$ M, so for both limits the numerator is $\text{ILF}(1\text{M}) - \text{ILF}(300\text{K}) = 0.40$ .
 $\begin{align*} \text{EF}_{1\text{M}} &= \frac{0.40}{1.75} \\ &= 0.2286 \end{align*}$ 
 $\begin{align*} \text{EF}_{2\text{M}} &= \frac{0.40}{2.00} \\ &= 0.2000 \end{align*}$ 
 $\begin{align*} E[\text{Layer loss}] &= 0.65\left[10(0.2286) + 5(0.2000)\right] \\ &= 0.65(3.286) \\ &= \$2.136\text{M} \end{align*}$ 
Loss cost $= 2.136/15 = 14.2\%$ of subject premium. The $\$2$ M policies put a smaller share of their expected loss in the layer, because part of it sits between $\$1$ M and $\$2$ M, above this treaty.