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  #11  
Old 06-20-2018, 05:35 PM
Duke of Chalfont Duke of Chalfont is offline
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Sorry, was asking about a case where your full credibility standard is 1,082 claims, you expect 500, but observe 1,500. Is the credibility there 100% or sqrt(500/1082) = 68%?

At any rate, I do get the impression that there is a subtle difference between how LFC is generally applied in a mortality or a P&C context, and likely one that makes sense given the circumstances.

Looking at the SOA's credibility practice notes from 2009 (https://www.soa.org/Files/Research/P...eory-pract.pdf ), actual deaths/lapses are used in the simplified LFC formula on page I.6. That follows from an assumption that the deaths/lapses are Poisson distributed. However, there is an underlying assumption there that the true actual-to-expected ratio is that observed, instead of assuming it is just 100%.
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  #12  
Old 06-20-2018, 05:39 PM
Duke of Chalfont Duke of Chalfont is offline
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Quote:
Originally Posted by Colymbosathon ecplecticos View Post
Oh, and 1,082 isn't an arbitrary number.
Agreed. It is a number that follows from some rather arbitrary assumptions and inputs but that still works quite well in practice.
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  #13  
Old 06-20-2018, 06:17 PM
Colymbosathon ecplecticos's Avatar
Colymbosathon ecplecticos Colymbosathon ecplecticos is offline
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Quote:
Originally Posted by Duke of Chalfont View Post
Sorry, was asking about a case where your full credibility standard is 1,082 claims, you expect 500, but observe 1,500. Is the credibility there 100% or sqrt(500/1082) = 68%?
Yes, 68%.
Quote:
Looking at the SOA's credibility practice notes from 2009 (https://www.soa.org/Files/Research/P...eory-pract.pdf ), actual deaths/lapses are used in the simplified LFC formula on page I.6. That follows from an assumption that the deaths/lapses are Poisson distributed. However, there is an underlying assumption there that the true actual-to-expected ratio is that observed, instead of assuming it is just 100%.
Think about it this way:

You expected your full credibility standard is 1082, and you expected 0.001 deaths (your risk set was 1 life for one year with q=0.001). Oops! He dies.

So your unweighted estimate for q is q=1.0 (I agree with this.)

Your credibility should be sqrt(0.001/1082) which is about 0.001.

Using actual, we'd get sqrt(1/1082) which is about 0.03.

The credibility weighted estimate is 1(0.001) + 0.001(0.999) = 0.00196

Your estimate would be 1(0.03) + 0.001(0.97) = 0.03137

=============

Also, from a regulator's point of view, your method fails to give the insured credit for good experience and doubly punishes bad experience. That is unfairly discriminatory.
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  #14  
Old 06-21-2018, 12:39 AM
Duke of Chalfont Duke of Chalfont is offline
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Thanks, that helps. Though I do not think I agree with the conclusion from that example itself.

What I was clearly missing, and I am not sure how (guess it has been way too long since I saw anything P&C), is that the the A/E's you are talking about in P&C are loss ratios or whatever, not actual versus expected claims counts. Presumably, the expected # of claims are a better (or at least a fairer and more consistent) indicator of the "stability" of the observed loss ratio than the actual # of claims, and thus the preferred base to use for the credibility standard.

But when talking about mortality experience, the A/E's are the actual to expected deaths counts (potentially weighted, but can ignore that here). So, all else equal, more observed deaths really do mean a more stable observed A/E ratio in the LFC sense. If the expected deaths are 1000 but you observe 800, then the probability that the A/E is within a 5% relative tolerance of the estimate of 80% is about 84%, while with 1200 observed deaths that probability (around the 120% estimate) is 92% (if 1,082 deaths were required for full credibility, then the probability of the estimate being within a 5% tolerance would need to be above 90%). If we were focusing directly on the mortality rate, the conclusion would be the same.

With the example with 1 death, is the point supposed to be that the estimate of 0.03137 is too high? It probably is, but all sorts of assumptions (certainly Poisson, and even normality) breakdown with so few lives (1 in this case), so I am not sure I see any relevance.

Quote:
Originally Posted by Colymbosathon ecplecticos View Post
Also, from a regulator's point of view, your method fails to give the insured credit for good experience and doubly punishes bad experience. That is unfairly discriminatory.
I (now) see that from a P&C perspective, but similar concepts certainly do not necessarily carry over to a mortality application. For example, in the context of pricing an annuity purchase for a pension plan partially using its own experience, using expected deaths could be seen to be potentially overcharging groups with higher-than-average mortality (insufficient credibility placed on their experience given the relative "stability") and also potentially overcharging ones with lower-than-average mortality (too much emphasis placed on their inherently less "stable" experience). A single premium will be paid in this case, so no chance to adjust given future experience. But again, this all follows from more observed deaths leading much more directly into the "stability" of the A/E's or mortality rates.


This was really quite interesting. I never had thought about the practical considerations of credibility in P&C or how it differs from the mortality applications with which I have at least some familiarity. Though I do promise to stay out of P&C-related threads in the future!
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