Werner & Modlin - Chapter 14 (Approx. Change in Avg. Rate Diff.)

[Vypa]

On page 278 of W&M, the text steps through the calculations of the approximated change in average rate differential method. On first glance everything seems to make sense, but when you look at the second footnote:

(Tot9) = (9) Weighted by (6)

This does not appear to be correct either in application or in practice. When I calculate it out, the value of (Tot9) equals the total of column 8 divided by the total of column 7, or the weighted average of the Proposed Differential over the weighted average of the Current Differential. This is exactly how it is defined in the text, which I would normally simply chalk-up the misleading footnote as a mistake on their part, however, when reviewing Ken Fikes' solution for 2010.28 from Old Exam 5, he does the same thing in the solution as the footnote [i.e. he takes the weighted average of the (Prop Differential / Current Differential) as opposed to taking the weighted average of the Prop Differential, taking the weighted average of the Current Differential and then taking the ratio of the two.]

So, my question is, has anybody else noticed this inconsistency and does anyone know the proper calculation. Is it:

(1) A weighted average of Prop Differential / Current Differential (Fikes and footnote).

or

(2) The ratio of the weighted average of both the Prop Differential and the Current Differential (Explanation in text).

Thanks in advance for any discussion.

[actuarialista]

On page 278 of W&M, the text steps through the calculations of the approximated change in average rate differential method. On first glance everything seems to make sense, but when you look at the second footnote:

(Tot9) = (9) Weighted by (6)

This does not appear to be correct either in application or in practice. When I calculate it out, the value of (Tot9) equals the total of column 8 divided by the total of column 7, or the weighted average of the Proposed Differential over the weighted average of the Current Differential. This is exactly how it is defined in the text, which I would normally simply chalk-up the misleading footnote as a mistake on their part, however, when reviewing Ken Fikes' solution for 2010.28 from Old Exam 5, he does the same thing in the solution as the footnote [i.e. he takes the weighted average of the (Prop Differential / Current Differential) as opposed to taking the weighted average of the Prop Differential, taking the weighted average of the Current Differential and then taking the ratio of the two.]

So, my question is, has anybody else noticed this inconsistency and does anyone know the proper calculation. Is it:

(1) A weighted average of Prop Differential / Current Differential (Fikes and footnote).

or

(2) The ratio of the weighted average of both the Prop Differential and the Current Differential (Explanation in text).

Thanks in advance for any discussion.

Vypa, I think the difference is that we're supposed to use (1) when we are weighting by current variable premium and (2) when we are weighting by exposures. (this distinction is implied by pages 276-7 of the text and also was mentioned at Ken's live seminar last year). So the footnote is wrong b/c that example is weighted by exposures. However, Ken's solution to 2010 #28 is correct b/c that one really is weighted by current variable premium.

Would be glad to hear what others have to say on this.

[actuarialista]

Vypa, I think the difference is that we're supposed to use (1) when we are weighting by current variable premium and (2) when we are weighting by exposures. (this distinction is implied by pages 276-7 of the text and also was mentioned at Ken's live seminar last year). So the footnote is wrong b/c that example is weighted by exposures. However, Ken's solution to 2010 #28 is correct b/c that one really is weighted by current variable premium.

Would be glad to hear what others have to say on this.

I still think my general comment above about when to use (1) and when to use (2) is correct. However, I looked more closely at the TIA solution for Old Exam 5, 2010 #28. Notice that in their solution (and in the CAS solution), there is a column headed "Expense"--that is a mistake and it should be "Exposures". Actually, the solutions do calculate the average change in differential using exposures as weights (not current variable premium). And they correctly use Vypa's (2) rather than (1). Vypa, I think you may be wrong when you say "he does the same thing in the solution as the footnote [i.e. he takes the weighted average of the (Prop Differential / Current Differential) as opposed to taking the weighted average of the Prop Differential, taking the weighted average of the Current Differential and then taking the ratio of the two.]". It looks to me like he is in fact taking the wgtd avg of the current diff and the wgtd avg of the prop diff. The answers are .965 and .946 respectively. He then divides the two averages.

[Vorian Atreides]

I'll address the text and what it's saying. Keep in mind that the Exhibits in the Appendix can add to the material presented in the main part of the text. That is, the text may talk about different methods and present one of those methods in the main part of the discussion and present the other one in the appendix. This is case here where the text focused on weighting using exposures, but provide one example of weighting by premium.

First, depending on what you're working with will (help) determine whether to weight with exposures or with (current variable) premium.

Notice that in Appendix E, the Pure Premium Exhibit (noted in the upper right-hand corner of the page) is weighting out the relativities and is using exposures as the weight.
Blank for spacing.
The Loss Ratio Exhibit is weighting out the changes in the relativities and is using (current variable) premium as the weight. Note that this should make sense to do since the classification with the most premium is going to drive the overall change (and not the exposures). To see why this must be the case, consider the (overly simplistic) example where there are equal exposures in all of the classification but the amount of premium in each isn't equal.
Second, you should understand the following thoroughly:

How approximating the changes using the average factors/average change will not be as accurate as using the extension of exposures method.
Understand the flaws inherent in using Exposures as weights (note that in some cases, this is the best option for estimating the overall change)
How using premiums can "correct" for this flaw.
And what adjustments are needed to use premium as the weight for relativities correctly. (NOTE: The text makes mention of this fact, but doesn't provide a direct example of it. Given the CAS's direction in using more of the higher levels of Bloom's Taxonomy, this could be part of a question asked. The correction is really easy and simple to do and can make a definite difference on this sort of problem.)
It seems that the above list is where the problem starts for the OP.

Finally, for the Exam, note that you can use either exposures or current premiums (with the appropriate modifications, see page 274-275) to weight out relativities; but you must use premiums (without a need to make the mentioned adjustment--you should see why this is the case) to weight out the changes to the relativities (as shown in Appendix E, Loss Ratio Exhibit).

[Vypa]

I still think my general comment above about when to use (1) and when to use (2) is correct. However, I looked more closely at the TIA solution for Old Exam 5, 2010 #28. Notice that in their solution (and in the CAS solution), there is a column headed "Expense"--that is a mistake and it should be "Exposures". Actually, the solutions do calculate the average change in differential using exposures as weights (not current variable premium). And they correctly use Vypa's (2) rather than (1). Vypa, I think you may be wrong when you say "he does the same thing in the solution as the footnote [i.e. he takes the weighted average of the (Prop Differential / Current Differential) as opposed to taking the weighted average of the Prop Differential, taking the weighted average of the Current Differential and then taking the ratio of the two.]". It looks to me like he is in fact taking the wgtd avg of the current diff and the wgtd avg of the prop diff. The answers are .965 and .946 respectively. He then divides the two averages.

Thanks for your thoughtful input. Your contribution has helped to reconcile the differences between (1) and (2) above. However, I have to disagree as to how the solution for 2010.28 in the TIA manual is constructed. At least in my version which is the current version as far as I know.

Ken is taking the ratio of the Proposed Discount Relativity to the Current Discount Relativity which he lables as Discount Change. He then weights this ratio by the exposures (400 and 600 for Discount = Yes and Discount = No respectively). This is his off-balance factor, which I think is accurate if he had weighted the ratio by current variable premium, not exposures.