These examples apply the formulas from the concept pages to concrete numbers. All figures are illustrative and original, chosen to make the mechanics clear.
01
Quota share result
Problem. A cedant cedes 40% of a portfolio under a quota share. Gross written premium is 5000, the ceding commission is 25%, and the year’s loss ratio is 60%. Find the reinsurer’s technical result and combined ratio.
Solution
Ceded premium \(=0.40\times 5000=2000\).
Ceding commission \(=0.25\times 2000=500\).
Ceded losses \(=0.60\times 2000=1200\).
Technical result \(=2000-500-1200=300\); combined ratio \((1200+500)/2000=85\%\).
02
Surplus treaty shares
Problem. A surplus treaty has a retained line of 200 and 9 lines of capacity (surplus 1800, total 2000). Three policies have sums insured 150, 600 and 2500. Find the ceded shares and the recovery on a total loss of the second policy.
Solution
Policy 150 \((\le 200)\): fully retained, ceded 0.
Policy 600: retained 200, ceded 400, ceded share \(400/600=66.7\%\); a total loss of 600 recovers \(0.667\times 600=400\).
Policy 2500 \((>2000)\): retained 200, ceded 1800, and 500 is left unplaced (facultative); treaty share \(1800/2500=72\%\).
03
Per-risk excess of loss
Problem. A per-risk layer 400 xs 100. Three risk losses occur: 80, 250 and 900. Find each recovery and the layer premium at a 12% rate on line.
Solution
Loss 80 \((<100)\): recovery 0.
Loss 250: \(\min(\max(250-100,0),400)=150\).
Loss 900: \(\min(\max(900-100,0),400)=400\) (full limit).
Layer premium \(=0.12\times 400=48\).
04
Catastrophe XoL with reinstatement
Problem. A catastrophe layer 10 xs 10 with one reinstatement at 100%. Base premium is 1000. A catastrophe causes a 14 loss. Find the recovery and the reinstatement premium.
Solution
Recovery \(=\min(\max(14-10,0),10)=4\).
Fraction of limit used \(=4/10=40\%\).
Reinstatement premium \(=100\%\times 1000\times 0.40=400\).
The limit is then reinstated for the rest of the year.
05
Sliding-scale commission
Problem. A quota share pays a sliding-scale commission of 35% at a 45% loss ratio, down linearly to 20% at a 70% loss ratio. Find the commission at an actual loss ratio of 55%.
Solution
Slope \(=(35-20)/(70-45)=0.6\%\) of commission per 1 point of loss ratio.
At 55% (10 points above 45%): commission \(=35\%-10\times 0.6\%=29\%\).
06
Burning-cost pricing
Problem. Trended and developed losses to a layer over five years total 720; on-levelled subject premium over the same years totals 12000. Find the burning-cost rate, a technical rate with 25% total loading, and the premium on next year’s subject premium of 3000.
Solution
Burning Cost \(=720/12000=6.0\%\).
Technical rate \(=6.0\%/(1-0.25)=8.0\%\).
Layer premium \(=8.0\%\times 3000=240\).
07
Aggregate stop-loss
Problem. A stop-loss covers the annual loss ratio between 80% and 110% on a portfolio with premium 2000. The realized loss ratio is 95%. Find the recovery.
Solution
Recovery in points \(=\min(\max(95-80,0),30)=15\) points.
Recovery \(=15\%\times 2000=300\).
