If there is a deductible and no limit, then the excess trend will be greater or equal to ground up trend. However, if there is a limit, it is not certain. If tau_s = 8% and ground up inflation = 10%, and the exam asked why is tau_s lower than ground up inflation, how would I answer this?

Thank you!

Var(S)=E(N)Var(X;L)+Var(N)[E(X;L)^2]

Isn't S = X*N? If so, isn't Var(S) = Var(XN) = E(X^2)*E(N^2) - [E(X)^2]*[E(N)^2] = [E(N)^2]*Var(X)+Var(N)*E(X^2) ?

I'm not able to get to Bahnemann's equation.

"Let N be the per policy claim count random variable, m be the number of exposures, and ϕ be the ground-up claim frequency per exposure. Let Y be the claim size including ALAE. Then E[N]=mϕ is the expected claim count and the expected loss and ALAE for a policy is given by E[N]⋅E[Y]. The per policy pure premium is p=ϕ⋅E[Y]."

Why isn't p here "per exposure" instead of "per policy" since ϕ is per exposure frequency?

Following that, in Example 6.3, the solution seems to define p as E(N)*E(Y), which also seems to conflict with the p=ϕ⋅E[Y] formula here.

Could you please prove the cdf for diminishing deductible and how do we interpret the cdf functions?

Thank you!

It seemed like guaranteed cost policy is not under Bahnemann Chapter 6. Should this particular question be moved to another article (just like question 2a)?

Thanks!

What is the formula for the limited second moment of a lognormal?

E[X^{2};l]

I see that they came up with 512,509,058 and 5,283,276,848 but I don't know how to calculate those numbers. I'm sure I wouldn't be expected to have this memorized on an exam, but I like to be able to carry out the example problems on my own.

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:

1. 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.
2. 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)?

This came up for me in reviewing the Chap 6 problem set 2, questions 3 & 4. The fixed ALAE in both of those problems is not taken into account when calculating the modified severity. In problem set 1 it was $0 so it didn't come up.

Thanks.