Graves/Castillo - LDF adjustments

[Examinator]

Three adjustments to LDFs are discussed by both Graves/Castillo and the All 10 manual. The first is a credibility weighting of statewide link ratios to countrywide up to 75 months. The second is bringing statewide link ratios to the countrywide variation level. The credibility formula is then given.

Are the adjusted statewide link ratios calculated in step two then used in step one's credibility weighting? Or are there two different weightings going on here? The third step, adjusting for changing ALAE proportions, seems to make sense applying at the end. It's just that the first two steps, while distinct, seem to be two pieces of the same single step. Thanks.

[jk]

Are the adjusted statewide link ratios calculated in step two then used in step one's credibility weighting? Or are there two different weightings going on here?The latter. The adjustment is made only for purposes of calculating K, and you revert to the original, unadjusted link ratios when you do the credibility weighting and calculate the LDF's.

At least, that's the way I took it. Most of the practice problems seemed to give K, so I'm hoping they won't make us grind through an actual calculation of it. We did enough of that in Course 4!

[Examinator]

Your explanation is how I first interpretted both the text and the All 10, but I thought I was misreading it. All the problems I've found have either provided you with K or have given you both the between variance and within variance. I don't think we'll have to calculate it (as in Part 4, like you mentioned), but I'm just trying to get it to be sound in my head. Thanks.

[Galois]

Do you know off the top of your head, how they came up with
\frac{L_{i}}{L_{i}+\frac{Process Variance}{Structure Variance}}

How exactly does this make sense? Why compare it to Losses?

[Vorian Atreides]

Loss Development Factors vary solely with claim severity (frequency is irrelevant), so using amount of losses would make sense to determine level of credibility.

Formula is the Buhlmann credibility formula where K = EPV/VHM = Process Var/Structure Var (ala Finger).

[Galois]

Ok, here goes nothing.
Buhlmann Credibility


Var[E(\sum_{i=1}^{n}X_{i}|A)]=N^{2}*Var[E(X|A)]
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E[Var(\sum_{i=1}^{n}X_{i}|A)]=N*E[Var(X|A)]
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the link ratio is a measure of the expected development of each dollar of loss.
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the EVP and VHM of the link ratios can be seen as the EVP and VHM of the individual loss dollars
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Buhlmann Credibility dictates : Z = \frac{V[E(X|A)]}{V(X)}
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For total losses, the Buhlman Credibility could be expressed as Z = \frac{V[E(Losses|State)]}{V(CWLosses)}
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Z=\frac{V[E(Losses|State)]}{V[E(Losses|State)]+E[V(Losses|State)]}
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Z=\frac{V[E(\sum_{i=1}^{N}LossDollar_{i}|State)]}{V[E(\sum_{i=1}^{N}LossDollar_{i}|State)]+E[V(\sum_{i=1}^{N}LossDollar_{i}|State)]
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Z=\frac{N^{2}*V[E(LossDollar|State)]}{N^{2}*V[E(LossDollar|State)]+N*E[V(LossDollar|State)]}
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Z=\frac{N}{N+\frac{E[V(LossDollar|State)]}{V[E(LossDollar|State)]}}
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Z=\frac{N}{N+K}


Now I just have to find some time late May or June to review why Buhlmann credibility is a linear approximation of Bayesian credibility.

[Vorian Atreides]

Now I just have to find some time late May or June to review why Buhlmann credibility is a linear approximation of Bayesian credibility.
FWIW, this is discussed in the Exam 4/C syllabus material. Mahler's notes for this were very helpful to show when they are exactly identical and when they are different. If you're adept at translating textbook-ese into English, Loss Models covers this as well.

[Galois]

Excellent. Thank you.