Re: Credibility Experts?

Kathy Gile ( (no email) )
Fri, 7 May 1999 20:37:52 -0500

The "familiar" formula Z = n/(n+K) is derived from a model which expresses Z
as a function of EXPOSURE (n), as
noted on page 377. The credibility formulas 4.17a and 4.17b involve an
entirely different model: a TIME SERIES of
observations in which n refers not to exposures but the nth period of
observation (see p. 42, eqn 4.15). In particular,
the C vlues, as estimates of the next period's values are a linear
combination of all of the prior observations, but
exposure is not directly involved. To see the real difference in the two
models, let n tend to infinity. In the exposure model,
Z tends to 1, a natural result. In the SECOND model, however, as n
increases, the prior C values become MORE credible
and Z gets smaller! See Exercise 4.4 on page 431. In partiular, if J=0, Z
tends to zero as n tends to infinity.

Brad Gile

-----Original Message-----
From: john.t.devereux@us.pwcglobal.com <john.t.devereux@us.pwcglobal.com>
To: casnet@lists.casact.org <casnet@lists.casact.org>
Date: Friday, May 07, 1999 1:56 PM
Subject: Credibility Experts?

>I was recently posed a credibility question that, given my current level of
>familiarity in the area, I've opted to post here for any potential comments
>and suggested responses. A life actuary acquaintance of mine who performs
>credit related work was reviewing the Credibility chapter (7) of the
>Foundations of Casualty Actuarial Science book. He was reviewing formula
>4.17A and formula 4.17B shown below:
>
>Z(subscript: 1) = 1 / [ 1 + K ] and
>Z(subscript: i+1) = 1 / [ 1 + 1 / (J + Z(subscript: i)) ]
>
>Basically, his understanding was, that as J --> 0, (J being defined as d/v
>or the change in variance over the variance) (i.e. minimizing the
>fluctuation), the above formulas should revert to the familiar Z(subscript:
>n) = n / [ n + K ]. The formulas actually simplify to Z(subscript: n) = 1
>/ [ n + K ] when J=0. What are we missing? Is there a gap in his logic?
>Is there no foundation for his assumption of n/[n+K]?
>
>If there is someone out there who can offer their wisdom, I would rather
>not (re)read the entire chapter and its references to brush up on my
>credibility theory. Thanks
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