We're told we have a 25% quota share treaty which applies to 10m of subject premium. So the ceded premium is 2.5m. We're also told the ceding commission is 20%, so the reinsurer premium net of ceding commission is 2m = 2.5m - 2.5m*20%.
The question defines the ceded profit = ceded premium - ceded commission - ceded losses.
If this is more than 20% of the ceded premium then the excess is paid back via a profit commission (presumably at 100% but perhaps not).
Since we want a negative ceded profit, there is no excess to pay back. The ceded profit is negative when the ceded losses exceed 2m. Since we have a 25% quota share, this means the ground-up loss must have been 2m/25% = 8m.
From here, you can use the distribution details given in part d to find the probability of a ground-up loss of greater than 8m.
In your set up you are asking for the probability of the reinsurer returning excess profit to the primary insured. However, the question is asking for the probability that the reinsurer does not make a profit.
Notice this is not as straight forward as 1 - Pr(reinsurer returns excess profit to the primary insured) because if the ceded premium - ceding commission - ceded losses is say 10% of the ceded premium then the reinsurer has turned a profit but is not obligated to return any because it's not deemed excessive.
Comments
We're told we have a 25% quota share treaty which applies to 10m of subject premium. So the ceded premium is 2.5m. We're also told the ceding commission is 20%, so the reinsurer premium net of ceding commission is 2m = 2.5m - 2.5m*20%.
The question defines the ceded profit = ceded premium - ceded commission - ceded losses.
If this is more than 20% of the ceded premium then the excess is paid back via a profit commission (presumably at 100% but perhaps not).
Since we want a negative ceded profit, there is no excess to pay back. The ceded profit is negative when the ceded losses exceed 2m. Since we have a 25% quota share, this means the ground-up loss must have been 2m/25% = 8m.
From here, you can use the distribution details given in part d to find the probability of a ground-up loss of greater than 8m.
So I have
P(ced Prem - ced comm - ced L '> 20%CedPrem)
= P(2.5m-0.5m-0.25L>0.5m)
=P(6m>L)
What am I not understanding?
In your set up you are asking for the probability of the reinsurer returning excess profit to the primary insured. However, the question is asking for the probability that the reinsurer does not make a profit.
Notice this is not as straight forward as 1 - Pr(reinsurer returns excess profit to the primary insured) because if the ceded premium - ceding commission - ceded losses is say 10% of the ceded premium then the reinsurer has turned a profit but is not obligated to return any because it's not deemed excessive.
I got it now, thanks.