For this question, the use of claim incidence = ix + (1 - kx) * qx as the same for all durations of policy seems to me at best questionable. Have I missed something? For example, we know that the deaths from age 50 to age 51 include those who incepted critical illness at age 48. However, those who incepted critical illness at age 48 could not have purchased the policy at age 50 - since we are assuming they were healthy at the time they started the policy. So for short policy durations we need to exclude those deaths from both qx and kx! Am I right?
Hmmmmm, so are you really just saying that someone who has just bought a policy is less likely to make a claim, because they must have been healthy when they bought the policy? (And this is complicated slightly by the fact that there are two causes for claim here...) Surely this is usually allowed for by using Select rates for the first few years of a policy (and then Ultimate rates subsequently)? So the claim incidence rate for an x=50 year old who bought the policy ages ago is: ix + (1 - kx) * qx whereas the claim incidence rate for an x=50 year old who has only just bought the policy is: i[x] + (1 - k[x]) * q[x] I don't think this invalidates the expression - it just makes use of the duration impact in the "usual" way, (rather than changing the expression).
Sharpening the question That sounds great, but in the example on page 30, we are having to assume no select effect, at least for ix and qx, due to lack of information. However, we are given in the example the proportions of kx that relate to different durations since inception of critical illness - so I think it would be appropriate to reduce kx accordingly for newer policies. The example does not adjust ix, qx or kx for duration.
I see what you mean - sorry, I wasn't looking at the specific example before! I think that first of all, we should appreciate the final two sentences on page 30 - this is an unlikely situation and what we're doing is probably over-simplified anyway. In terms of reducing the kx terms for newer policies, I think the problem here is that we don't have enough data to be able to do that. You could say that anyone who had a CI before the age of 50 should be excluded as they wouldn't have been able to buy a policy, but I don't think that's always true. Underwriting won't pick up all cases that have (at least the beginnings of) a CI at the application stage*, so it would be imprudent to reduce these kx terms down by too much. (*I suppose this scenario is particularly likely where initial underwriting is limited - eg for a group policy.) So given that we don't know how much to reduce them by, I guess we use the full kx values given. Unrealistic maybe, but we've been told that this scenario is somewhat unrealistic!
