CAS 5 2002 - Question 27, part a

[Halfmoon]

Can someone explain to me why the PDF gotten in CAS solutions is 1.0476 instead of 1.05?

If there was no rate change at 7-1-2000, I would agree with the answer. But wouldn't you look at the $90M and say half has gone through audit and half has not...45M + 45M*(1.1) = 94.5M. 94.5M/90M = 1.05.

CSM gives a different answer (>1.05) while CAS exam solutions & ALL 10 show 1.0476. As far as I can tell no one questioned CAS's solution, either. Am I completely missing something?

[egg]

Writings are uniform throughout the year. With your formula half the year has already gone through the audit yet it equals the second half of the year.

The set up x = the total premium written for the year:
(1/2)(1.1)x + (1/2)x=90M
The first half of the left size shows has the 1st half going through the audit the second half has not. Solve for x:
x=85.714M. This was the unaudited premium. The audited premium is 10% higher or 94.29M. 94.29M/90M = 1.0476 or the Premium development factor.

[Halfmoon]

I agree with you completely, except in the problem there is also a +10% rate change which inflates the 2nd half premium by the same amount as the audited...thus my 45M/45M split.

I get the 1.05 PDF you get egg, but two other places say it should be 1.0476. Formula used is 1.1 / [ .5*1.1 + .5*1.0] = 1.0476. I think the fact there is a rate change is being ignored...I just can't find a place/person to verify.

Thanx for looking at this with me!

[Halfmoon]

Oh nooo.... I've noticed a flaw in my thinking. Rereading the question..'Written premiums are uniform by month.' This would support using the 1.0476 factor. I was thinking in terms of exposures being uniform by month.

Wait a second...if exposures aren't uniform by month, doesn't the parallelogram method fail?

[egg]

The rate change is accounted for separately and not in the PDF.

As for the exposures not being uniform on the top of Page 98 of McClenahan says

the parallelogram method is based upon the assumption that exposures are written uniformly over the calendar period. In cases where material changes in exposure level have occured over the period, or where there is a non-uniform pattern to the written exposures, the parallelogram method may not produce a reasonable approximation to the on-level earned premium.