“loss participation” = – 20% * min(0; 100% – 80% – 17%) = – 20% * min(0; 3%) = 0 If on the other hand, LR = 125%, Ceded LR = 120%, RI commission = 10%, there will be some non-zero loss participation: “Loss participation” = – 20% * min(0; 100% – 120% – 10%) = – 20% * min(0; –30%) = 6% Please note that with this definition, the reinsurer’s loss participation is always positive or zero.

When modeling the risk exposure for this contract, we differentiate between basic losses below certain threshold and individual large losses above this threshold. Whereas basic losses are modeled on the annual aggregate basis, large losses are modeled individually through a combination of loss frequency and loss severity probabilistic assumptions. For the sake of simplicity, we assume that there are no loss development patterns, i.e. losses are fully settled within one year. Hence, loss discounting effects do not play any role.

Let’s assume that from the analysis of loss and premium statistics from the previous years, we can derive the following assumptions:

# <u>Basic losses</u>

- Losses below the threshold of € 3m

- Distribution assumption: ultimate LR follows a lognormal distribution

- Expected ultimate basis loss ratio = 80%

- Standard deviation of the ultimate basic loss ratio = 12%

# <u>Individual large losses</u>

- Losses above the threshold of € 3m

- Distribution assumptions: number of losses p.a. (loss frequency) follows a Poisson distribution, loss severity follows a truncated Pareto distribution

- Expected number of individual large losses above € 3m (Poisson Lambda) = 0.5

- Shape parameter of Pareto Distribution (Pareto Alpha) = 2.4

- Lower threshold of Pareto distribution = € 3m, upper threshold = € 100m

To calculate ERD we carry out a Monte Carlo simulation which generates 100,000 scenarios for the total ultimate loss ratio. The realization probability for each scenario is equal 1 / 100,000 = 0.001%. For each scenario we then apply the contract conditions to calculate the reinsurer’s ultimate result and then the severity of reinsurer’s net economic loss (if any).

First we calculate reinsurer’s ultimate result which is the reinsurer’s result after the insured loss has been settled, i.e. has achieved its ultimate value.

Reinsurer’s ultimate result = 100% - “ceded LR” – “RI comm” + “loss participation” Now, we recalculate the ultimate result into its present value.

Present value of reinsurer’s result = “Reinsurer’s ultimate result” + “discounting adjustment” Please notice, that according to our previous assumption, the loss will be fully settled within one year, i.e. there will be no loss development over the time and the resulting discounting adjustment can be assumed to be zero. However, for the sake of completeness we provide the formula for a general case.

Finally, we recalculate the present value of reinsurer’s result into the severity of reinsurer’s net economic loss Severity of reinsurer’s net economic loss (as percentage of the ceded premium) = – min (0; “Present value of reinsurance results”) = – min (0; 100% – “ceded LR” – “RI comm” + “loss participation”) The following example for both loss scenarios Ultimate LR = 80% and Ultimate LR = 125% will illustrate the above formula.

As explained above, for the Ultimate LR = 80% Ceded LR = 80%, RI Commission = 17% and loss participation = 0; 19