ch6 doubts

Comments

• Graham

  January 2021 edited January 2021

  I linked to this discussion in the Final Comments box at the bottom of this wiki section: https://www.battleacts5.ca/wiki5/Friedland06.Diagnostics#Changes_in_Settlement_Rate

  You're correct that in general you can't tell whether average case O/S will go up or down, even if the settlement rate for small and large claims stays the same. I will make a note to clarify that. Thank you for pointing that out.

  The following analysis is beyond the scope of the syllabus but it highlights how difficult it is to draw general conclusions about the behavior of triangles.

  Whether the average case O/S goes up or down when settlement rate increases actually depends on the ratios of the 4 numbers in the formula. If we write the formula for the current average case O/S as:

  • (a-b)/(c-d)

  and write the average case O/S after speedup by a factor of S as:

  • (a-Sb)/(c-Sd)

  then taking the derivative of with respect to S, to find out when this function is increasing/decreasing, we can conclude

  • the average case O/S will go down when (ad-bc) < 0
  • the average case O/S will go up when (ad-bc) > 0

  Note that (ad-bc) = 0 when a/b = c/d. This means that paid-to-reported losses are in the same proportion as paid-to-reported counts. This is the case in my example. Here, the average case O/S will stay the same, except for the entries in the triangle where the average case O/S goes to 0 for claims that are closed earlier than they would have been before the speedup in settlement.
• keshav1996

  January 2021 edited January 2021

  In reference to your explanation above for change in average case o/s due to change in settlement rate, please clarify the following:

  I don't think that average case o/s should be affected by the speed of settlement when the settlement rate for amount of claims is equal to settlement rate for # of claims which is also evident from the fact that ad-bc above is independent of S (settlement rate).

  Ergo, the entries for average case o/s logically should not equal to 0 even after settlement rate increases. (So, why do they become zero then in the traingle??)

  
  

  Please check the following comment too (as an additional mathematical insight):

  However, suppose settlement rate for amounts is S and for # of claims is T (in denominator), say, then, we have two cases (if we take derivative in the same way as above):

  1. If S increases, then first derivative = -b/(c-Td) which is <0 implying avg case o/s decreases because greater amounts are paid early, so actuary will keep less case reserves for remaining claims partially settled.
  2. If T increases, then first derivative = d*(a-Sb)/(c-Td) which is >0 implying avg case o/s increases because greater # of claims close faster, so average case for remaining lesser no. of losses will increase.

  NOTE: Both cases above must satisfy the necessary condition that c-Td >0 i.e T<c/d or T < Reported counts/Closed counts (which is quite obvious).

  
  

  Thanks

  Keshav
• Graham

  January 2021

  I appreciate your interest in investigating this further but this line of discussion is outside the bounds of the syllabus. The forum needs to stay focused on topics directly relevant to the exam. My strong advice is return to this investigation at a later date, preferably after the exam.