Understanding Reinsurance Diagrams
I'm not sure if this is the right place to ask this question, but I'm having trouble understanding reinsurance diagrams. There are two ways to interpret the idea of inuring reinsurance that I've seen, and I'm not sure why they are being interpreted differently.
For a simple example, a 40% quota share inures to the benefit of a 50% XOL treaty in the layer 5M x 5M. There are two ways I can see this applying:
- Since there is no "overlap" in the 40% quota share and the 50% XOL, the XOL covers 50% of loss in the 5M x 5M layer, and the quota share covers 40% of the loss. The insured covers the other 10% in the layer.
- Since the quota share covers 40% of the losses, and the XOL treaty is meant to cover the layer of claims 5M x 5M, the XOL actually covers a different layer of ground up claims. In this case, the treaty still covers 2.5M in losses (50% of the 5M limit), but the start of the layer changes to 5M / (1-40%) = 8.33 M so the insured retains an initial 5M net of the quota share before the XOL treaty applies. The treaty limit ends at 12.5M (2.5M / ( 1-40%) + 8.33 M) so the XOL treaty, inured to the benefit of the quota share, covers 2.5M of loss.
Is one of these approaches correct? If we can decide between them, is there a situation in which one is correct / incorrect (for instance, if the treaties are placed with the same v. different reinsurers, if we are using ceding company v. reinsurer perspective)?
Comments
This is a great question and one for which in general we don't have a definite answer as reinsurance contracts are structured to suit each individual client's situation.
For your example above, the second version is correct in our opinion. This is because the problem says the quota share inures to the benefit of the XOL treaty. In other words, we need to apply the quota share first and then blindly apply the XOL to the primary insurer's loss net of the quota share. In this case the XOL is still covering the layer 5m xs 5m but it's applied to the primary insurer's loss net of the quota share. As you point out, this translates to a different (higher) layer of the ground up losses and so the XOL is paying out less than it would if applied by itself.
The first answer doesn't make a lot of sense because there would be no benefit to the XOL treaty, i.e. the XOL treaty is paying out the same loss as it would if it were the only treaty. The first answer would make sense if the question didn't say the quota share inures to the benefit of the XOL as in that case the treaties would apply concurrently and as you point out, there's no overlap in the percentages.
Our advice is to slow down in the exam and state your assumptions if the question is ambiguous. In general, if you have multiple XOL treaties covering different layers and you're not told which inures to which, then you're in situation one above whereas if you know which inures then the layers end up shifting and you're in situation two. Similar ideas hold if you have say an XOL treaty and a catastrophe treaty. If the cat treaty applies first then the XOL layer is shifted. Whereas if the XOL applies first then the cat treaty is applied to the remainder.
Hope this helps.
I think my confusion with the inuring reinsurance came about due to the situation you mention of multiple XOL treaties with unclear relationships to each other, each covering different percentages. This does help, and it helps to know that there is no definitive answer to how multiple reinsurance contracts "should" inure to each other but will be tailored to each client / exam question. Thank you!
Edit - I edited the picture. Originally I had made a small mistake on the top of the reinsurance diagram by stopping the QS at 20M. It's now fixed below, showing the QS doesn't have a limit.
Hold on...now I'm confused! In the above example ("For a simple example, a 40% quota share inures to the benefit of a 50% XOL treaty in the layer 5M x 5M"),
Shouldn't the reinsurance diagram look something like attached? My understanding would be the XoL layer covers losses between 8.33M and 16.67M (10/(1-40%)). This would mean, after including the QS, the losses to the insurer would be between (1-40%)*8.33M and (1-40%)*16.67M, or 5M-10M net. The XoL reinsurer would then pay 50% of the losses in this layer.
(As an example, suppose you have a 16.67M claim. The QS would pay 40%, or 6.67M. That leaves $10M left to pay, which is covered by the XoL treaty. The treaty would pay 50% of the layer of $5M-$10M which is $2.5M. In total, the 16.67M claim would be allocated: 6.67M QS, 2.5M 50% XoL, 7.5M retained. If the XoL treaty covers claims only up to $12M, I don't think the math would work out.)
Thanks for the clarification!
We're both perhaps guilty of complicating this problem. The solution depends on keeping track of the perspective (primary insurer, quota share reinsurer, XoL reinsurer etc.)
Let's assume we have a 20m loss
The 40% quota share inures to the benefit of the 5m xs 5m 50% XOL treaty. Applying the quota share first gives the 40% QS column in the left of your reinsurance diagram. The quota share insurer pays out 8m.
The remaining 12m is what would be reported to the XoL reinsurer who would consider the layer 5m xs of 5m and pay out 50% of the losses in that layer. We split the 12m in to 0m - 5m - 10m - 12m and so the XoL reinsurer pays out 2.5m which is 50% of the loss between 5m and 10m. So your 8.33m and 16.67m should be 5m and 10m respectively.
Due to the quota share, the XoL treaty is not triggered until we have a loss of at least 8.33m because ceding 40% of this leaves us with 5m which is the minimum to start the XoL paying out. Similarly, once losses exceed 16.67m, the XoL won't pay out any more because the loss net of the quota share exceeds 10m. So the XoL treaty is equivalent to a different XoL treaty between 8.33m and 16.67m which pays out 50%*(1-40%) of this layer.
Since we picked 20m for our loss size the XoL didn't benefit much (at all) from the quota share. Repeating this with a 10m loss, without the quota share present the XoL would cover 50%*(10m-5m) = 2.5m. But with the quota share inuring to the benefit of the XoL then the XoL covers 50% of the layer 5m to 10m on a loss of size 10m*(1-40%) = 6m. So the XoL only pays out 500,000 and hence benefits from the quota share.