Chapter 4 - Difference between random variables Z and W

[actuary-to-be]

I struggle to understand how is Z (the claim amount paid by Reinsurer) different to W (the conditional claim amount paid by Reinsurer).

We know that W=X-M|X>M (in other words, the amount paid by the reinsurer given that the reinsurer pays something). But how is this different to Z?

And subsequently, why is E(Z) ≠ E(W) and Var(Z) ≠ Var(W) ?

Any help would be appreciated.

Thank you

 

[Bharti Singla]

I struggle to understand how is Z (the claim amount paid by Reinsurer) different to W (the conditional claim amount paid by Reinsurer).

We know that W=X-M|X>M (in other words, the amount paid by the reinsurer given that the reinsurer pays something). But how is this different to Z?

And subsequently, why is E(Z) ≠ E(W) and Var(Z) ≠ Var(W) ?

Any help would be appreciated.

Thank you

Yes, both Z and W are the claim amounts paid by the insurer i.e. X-M (X is the total claim amount and M is the retention limit).
But, E(Z)≠E(W) because:

Lets take an example. Suppose, we have a sample of 6 total claims, among which 4 claims are above the retention limit. Following are the claim amounts paid by the reinsurer on these claims:-
X-M, X-M, 0, 0, X-M, X-M

Then, the unconditional mean claim amount paid by the reinsurer is:
E(Z) = 4(X-M)/6
[ Mean amount on all claims]

And the conditional mean claim amount paid by the reinsurer is:
E(W) = 4(X-M)/4
[Mean amount on claims in which it is involved]

Same approach for variance. This is what I know. Hope this helps.

 

[actuary-to-be]

This is very helpful!

Thanks Bharti!

 

[Chandrima]

I struggle to understand how is Z (the claim amount paid by Reinsurer) different to W (the conditional claim amount paid by Reinsurer).

We know that W=X-M|X>M (in other words, the amount paid by the reinsurer given that the reinsurer pays something). But how is this different to Z?

And subsequently, why is E(Z) ≠ E(W) and Var(Z) ≠ Var(W) ?

Any help would be appreciated.

Thank you

Z includes those counts too where reinsurer's contribution is not involved (i.e. it includes zero amount payments by reinsurer too). W considers only those counts where reinsurer's contribution is involved (i.e. it only counts non-zero amount payments by reinsurer). Say, an insurance company's retention limit is 1500 GBP. In one year, total 5 claims have come with amounts 400 GBP, 600 GBP, 1700 GBP, 500 GBP, 1800 GBP. Here, Z = {0, 0, 200, 0, 300} but W = {200, 300}. I hope now it is clear to you.