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I saw different definitions of these reinsurance treaties. What's the exact definition of an excess of loss reinsurance and stop-loss reinsurance? In Chi's paper, see https://www.researchgate.net/profile/Yichun_Chi/publication/228231915_Optimal_Reinsurance_under_VaR_and_CVaR_Risk_Measures_A_Simplified_Approach/links/00463527c227800853000000/Optimal-Reinsurance-under-VaR-and-CVaR-Risk-Measures-A-Simplified-Approach.pdf the standard stop-loss reinsurance is of the form $f(x) = (x-d)_{+}$, where $d \geq 0$ is the retention level and $(x)_{+} = \max\{x, 0\}$.

The limited stop-loss treaty is of the form $f(x) = \min\{(x-a)_{+}, b\}$. This is equivalent to $f(x) = (x-a)_{+} - (x-a-b)_{+}$.

The truncated stop-loss is $f(x)=(x-d)_{+}\mathbb{I}(x \leq m)$.

While in Peter Antal's work, see,

http://www.actuaries.ch/fr/downloads/aid!b4ae4834-66cd-464b-bd27-1497194efc96/id!96/Reinsurance_Script2009_v2.pdf

Stop-loss reinsurance is an excess of loss on the DI's aggregate annual loss, which is $$ S_{SL} = (\sum_{i = 1}^{N}X^{i}-D)^{+} - (\sum_{i = 1}^{N}X^{i}-C-D)^{+}$$. And the excess of loss is of the form $$ S_{XL} = \sum_{i = 1}^{N}[(X^{i}-D)^{+} - (X^{i}-C-D)^{+}]. $$ I am confused. What's the exact definition of stop-loss reinsurance?

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A marine syndicate at Lloyds will insure 10 ships for 2018. The loss during 2018 on each ship will be $X_i$ for $i=1,\cdots,10$.

The syndicate has limited capital and could be wiped out if the losses are too large. For this reason they may want to take out re-insurance. There are 2 kinds:

(1) Stop-loss reinsurance puts a limit on the losses for the year, i.e. a limit on $\sum X_i$. If this exceed $D=100$ million dollars, only 100 million will be paid by the syndicate and the rest will be covered by the reinsurer. The reinsurance then is $(\sum X_i-D)^+$. If the reinsurer is only willing to provide reinsurance up to an upper limit C, then the other formula you gave applies.

(2) Another solution is to put a limit on the losses on each single ship of say 10 million dollars. This is called Excess of Loss insurance. In this case the formula is $\sum(X_i-D)^+$ with D=10. Again an upper limit to the reinsurance equal to C can be introduced, slightly modifying the formula.

In summary these are 2 types of reinsurance and each can have an upper limit $C$ or not. There is no inconsistency. (Also, if instead of $\sum X_i$ we write $x$ then we get different looking but equivalent formulas for stop-loss insurance).

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