Siewert example

[Examinator]

I'm working a sample problem I got at the MAF seminar (solution on page 309) and I'm confused on something. Hopefully my notation makes sense. See below.

Severity relativity at age 12 limited at 100K, R(12,100K) = 0.970.
Severity relativity at age 24 limited at 100K, R(24,100K) = 0.955.

Unlimited LDF from 12 to 24, ULDF(12-24) = 1.143.
Limited LDF from 12 to 24, LLDF(12-24,100K) = 1.125.

Calculate the excess (of 100K) LDF from 12 to 24, XSLDF(12-24,100K).

The solution is given as follows.
XSLDF(12-24,100K) = ULDF(12-24) * [1 - R(24,100K)] / [1 - R(12,100K)]
XSLDF(12-24,100K) = 1.143 * [1 - .955] / [1 - .970] = 1.715
Looking at this solution, it makes sense and I agree.

However, I solved the problem this way.
ULDF(12-24) = R(12,100K) * LLDF(12-24,100K) + [1 - R(12,100K)] * XSLDF(12-24,100K)
Solving for XSLDF(12-24,100K) gives 1.725.

Writing my solution out, I'm wondering if there's some sort of discrepency with using LDF factors from 12 to 24 and severity relativities at 12 only. But, given this situation on an exam, I'd probably work the problem the same way. Am I applying my method incorrectly somehow? Thanks.

[Quack]

Your method is sound except that based on Siewert's example on page 232 right before formula (4.10), I think you should be using the severity relativity at 24 months. That still gives a different answer than using just the unlimited development and severity relativities and I can't explain why that is. I would recommend learning the first method in any case, because there could be an exam question where you are not given the limited development factor.

[KidCA]

Examinator, you're spot on. Quack, crack is whack.

The two formulas are equivalent. Multiply them out.

With that, rearrange this formula:

LLDF = ULDF * DR (where D is my delta for change)
= ULDF * R24/R12
do a gymnastics routine, and,
R24 = R12*LLDF/ULDF

In the problem, solve for R24 with the GIVENS. You get .95472, not exactly .955. Now, redo the problem with this exact number. You'll get your 1.725.

In your method, Examinator, there was no R24 to give a screwy rounding error and you got the exact answer.

[The Sad Man]

Writing my solution out, I'm wondering if there's some sort of discrepency with using LDF factors from 12 to 24 and severity relativities at 12 only.

So I take it the Severity factors can be used to calculate ANY LDF(ie. 12 to 24, 12 to ult, 36-48) as long as the severity is taken at the first point of each development (12,12,36 in the 3 examples I gave)?

So if we have
U_LDF_x to y and L_LDF_x to y and Severity_x

then x and y can be anything and the direct developement equations will work?

[Examinator]

Examinator, you're spot on. Quack, crack is whack.

The two formulas are equivalent. Multiply them out.

With that, rearrange this formula:

LLDF = ULDF * DR (where D is my delta for change)
= ULDF * R24/R12
do a gymnastics routine, and,
R24 = R12*LLDF/ULDF

In the problem, solve for R24 with the GIVENS. You get .95472, not exactly .955. Now, redo the problem with this exact number. You'll get your 1.725.

In your method, Examinator, there was no R24 to give a screwy rounding error and you got the exact answer.Huh. I didn't even consider that the two methods were equal. I guess they are, since one is derived from the other (I think) in the actual text. If this were an essay on the exam, this should get full credit. Booyah.

[Quack]

I went back and did the algebra and you're right KidCA, it's definitely the beginning severity relativity. I have some seminar notes that confirmed it. I was confused because Siewert's numerical example uses the wrong severity relativity, but algebra doesn't lie. Bad author!