1) an insurer writes a single policy running from 7/1 to 6/30,
2) the policy includes an aggregate deductible of 100 zlotys (or whatever your
favorite currency may be),
3) the policy includes no per claim deductibles,
4) losses *gross* of the deductible are 120 zlotys,
5) losses are incurred uniformly over the policy period,
6) losses are paid subsequent to the policy period, and
7) the premium charged for the policy is 20 zlotys.
Along comes the 12/31 date in the middle of the policy period. Earned premium
through that date is presumably 10 zlotys (feel free to question this
presumption, though). What is the appropriate loss reserve *net* of the
deductible as of 12/31?
Argument 1: If the policy were terminated at 12/31 and the policy included no
language altering the deductible in the event of such termination, then the 60
zlotys of gross incurred losses would fail to exceed the deductible and the net
reserve to the insurer should therefore be 0 zlotys. (But since the unearned
premium reserve of 10 zlotys will be insufficient to cover the 20 zlotys of net
losses that occur in the second half of the policy period, there is a
deficiency in the UPR of 10 zlotys.)
Argument 2: If the policy were terminated at 12/31 and the policy included no
language altering the deductible in the event of such termination, then the
presumption should be that the deductible would in effect be pro-rated, in
which case the 60 zlotys of incurred losses would exceed the (pro-rated) 50
zloty deductible and the net reserve should therefore be 10 zlotys.
Argument 3: Losses net of the deductible (i.e., 20 zlotys) should, in
accordance with a notion of matching losses and premium (and assuming that
premium should be earned uniformly over the policy period), should be reserved
as if they occur uniformly, in which case the net reserve should again be 10
Argument 4: Losses net of the deductible (still 20 zlotys) should, in
accordance with a notion of matching gross losses and net losses, should be
reserved as if they occur uniformly, in which case the net reserve should yet
again be 10 zlotys.
I expect that most actuaries and accountants would feel most comforable with a
reserve of 10 zlotys based on either Argument 3 or Argument 4, but I find
Argument 1 strangely compelling. Should I try to get more sleep, or are there
other actuaries out there who share my fondness for Argument 1?
Senior Consulting Actuary
Ernst & Young LLP
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