Sliding Scale Commissions

Study guide: Exam 9

A Sliding Scale Commission is a ceding commission on a proportional treaty that varies inversely with the treaty's actual loss ratio, between a stated minimum and maximum. A provisional commission is paid as premium is ceded and then adjusted as losses mature. It is the most common adjustable feature of a Quota Share.

$c(\text{LR}) = \min\left(c_{\max},\; \max\left(c_{\min},\; c_0 + s\,(\text{LR}_0 - \text{LR})\right)\right)$

$E[c] = \sum_i p_i \, c(\text{LR}_i)$

$c_0$ is the commission at loss ratio $\text{LR}_0$ , and $s$ the slide: $1{:}1$ means one point of commission per point of loss ratio. The slide may differ in different ranges, e.g. $1{:}1$ near the minimum and $0.5{:}1$ toward the maximum. The minimum $c_{\min}$ applies at and above a high loss ratio, and the maximum $c_{\max}$ at and below a low one. $p_i$ is the probability of loss-ratio outcome $\text{LR}_i$ from an aggregate distribution.
What it does. Within the slide, the reinsurer's loss-plus-commission ratio moves by only $1 - s$ per point of loss ratio, and the cedant is rewarded for good experience. Outside the slide it does nothing: above the minimum-commission point the reinsurer bears every extra point of loss, and below the maximum it keeps the extra profit.
Never price it at the expected loss ratio alone. Only for a plan balanced around the expected loss ratio is that a fair shortcut. Clark's illustration: with the minimum commission of $25\%$ set at a $65\%$ loss ratio equal to the expected, the naive commission is $25\%$ and the technical ratio $90\%$ . Weighting over the aggregate distribution gives an expected commission of about $31\%$ and a technical ratio of about $96\%$ . Bad years cannot push the commission below the minimum, while good years raise it.
Historical adjusted loss ratios make a reasonability check. They are distorted by past catastrophes and thin years, and they miss outcomes the history does not contain.
Carryforward. Past deficits above the minimum-commission loss ratio can be carried into the current year. Priced for a single year, this simply shifts the slide by the carryforward. Priced over a multi-year block, it cuts the variance roughly in proportion to the number of years. Neither view is standard. See Reinsurance Contract Provisions, and compare Profit Commission and Loss Corridors.

Reading a Two-Slope Scale Terms: minimum commission $22\%$ at a $68\%$ loss ratio or higher; sliding $1{:}1$ to $30\%$ at $60\%$ ; then sliding $0.5{:}1$ to a maximum of $36\%$ at $48\%$ or lower. Find the commission at loss ratios of $75\%$ , $64\%$ , $55\%$ and $40\%$ .

Answer Check the scale: from $68\%$ to $60\%$ is $8$ points at $1{:}1$ , so $22 + 8 = 30\%$ . From $60\%$ to $48\%$ is $12$ points at $0.5{:}1$ , so $30 + 6 = 36\%$ .

$75\%$ : above $68\%$ , so the minimum, $22\%$ .

$64\%$ : $22 + (68 - 64) = 26\%$ .

$55\%$ : $30 + 0.5(60 - 55) = 32.5\%$ .

$40\%$ : below $48\%$ , so the maximum, $36\%$ .

The reinsurer's loss-plus-commission ratios are $97\%$ , $90\%$ , $87.5\%$ and $76\%$ . The $1{:}1$ band holds that ratio flat at $90\%$ , and the ratio moves again outside it.

Expected Commission Versus the Plug-In Same scale. The treaty loss ratio is $45\%$ ( $p = 0.15$ ), $55\%$ ( $0.30$ ), $64\%$ ( $0.30$ ) or $85\%$ ( $0.25$ ). Find the expected commission and the reinsurer's expected technical ratio, and compare with the plug-in.
Answer
 $\begin{align*} E[\text{LR}] &= 0.15(45) + 0.30(55) + 0.30(64) + 0.25(85) \\ &= 63.7\% \end{align*}$ 
Commissions: $36\%$ (at $45\%$ , the maximum), $32.5\%$ , $26\%$ and $22\%$ (at $85\%$ , the minimum).
 $\begin{align*} E[c] &= 0.15(36) + 0.30(32.5) + 0.30(26) + 0.25(22) \\ &= 5.40 + 9.75 + 7.80 + 5.50 \\ &= 28.45\% \end{align*}$ 
The plug-in gives $22 + (68 - 63.7) = 26.3\%$ , for a technical ratio of $63.7 + 26.3 = 90.0\%$ .

The true expected technical ratio is $63.7 + 28.45 = 92.15\%$ , so the plug-in understates the reinsurer's cost by more than two points. The plug-in behaves as if the $1{:}1$ slide ran on forever, $c = 90 - \text{LR}$ . Against that line:

The floor in the $85\%$ year adds $0.25(22 - 5) = 4.25$ points.

The flatter slide and the maximum in the good years take back only $0.15(45 - 36) + 0.30(35 - 32.5) = 2.10$ points.

The net is $+2.15$ points, and the asymmetry comes from the right-skewed loss ratio: the bad tail runs further past the minimum than the good tail runs past the maximum. To hit a $90\%$ technical ratio, the reinsurer must lower the scale or the minimum.