Saturday, 2 July 2011

Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 6

Section 2: The “Adequate Pure Premium Reserve” Approach
In my opinion, the change in RPR is a correct measure for pure premium earned during the period, and
that the pure premium portion of the UEPR should be the RPR.  Applying this approach to the example
of the previous section: in the no-loss case we would earn premium of $90 during the first period, and
in the loss case we would earn premium of $190.
Under current accounting rules: in the loss case, since there is no more cover, all future premiums
would be accrued and earned in the current period
, so earned (pure) premium would be $190, just as
the “adequate pure premium reserve” approach indicates.   In the no-loss case, I believe that most
companies would simply earn half of the pure premium ($95) during the first year (and some might
recognize that they have a $5 premium deficiency, since the pure premium for year two is $100).
My view is that at policy inception we expected to earn $100, but that in fact we earned either $190 or
$90 depending on our experience. Before you agree with me too quickly, let’s look at another example:
Consider the expected change in the RPR for Policy(2,2) during year 1.  This policy, you will recall,
pays $1000 for the second loss in two years.  The pure premium for this policy is $10, so this is the
RPR at time 0.
After one year we are again in one of two states:
 Under US-GAAP, at least for reinsurers, this is the content of EITF93-6, Issue 3 “How should the ceding and assuming
companies account for changes in future coverage resulting from experience under the reinsurance contract?”State Probability RPR1
Loss Occurred 10% Remaining cover is Policy(1,1), so RPR1 = 100
No Loss Occurred 90% Since there can be only one loss per year, there
can now be no second loss: RPR1 = 0.
In the no-loss case, which occurs 90% of the time, the decrease in RPR is $10.  In the loss case, the
decrease in RPR is -$90.  The expected decrease in RPR is (0.9)(10) + (0.1)(-90) = 0.
The lemma tells us that this must be the expected value of the losses occurring during the first year.
Does this make sense?
Yes!  This policy pays only on the second loss, and since we assume there can be only one loss per
year, the second loss cannot occur during year 1.  That is why the expected losses during year 1 are
2.1 Standard Premium-Accrual Methodology Considerations
I am not certain how companies would account for the above cover today.  Some would argue that
since the second loss cannot occur in year 1, no premium should be earned in year 1 on this cover; they
would earn all 10 in year 2.  Others might earn 5 in the first year and 5 in the second year.
I would argue that in the no-loss case all 10 should be earned in the first year, but that in the loss case
negative 90 should be earned during the first year.  The “adequate pure premium reserve approach”implies that the amount of pure premium earned during a period must be that amount such that the
remaining RPR contains exactly the expected pure premium required for the remaining policy period.
At inception, the company’s expectation was to earn nothing during year 1 on this policy because the
insured event could not occur during this period.  But in fact one of two things happened: they had
either an underwriting gain of 10 or an underwriting loss of 90.
The standard premium accrual procedure referred to before (i.e. accruing all future premium when no
more cover remains) together with the No Arbitrage Condition (described earlier) leads to the same
conclusion as the “adequate pure premium reserve approach”, as we will now illustrate.
Recall that the portfolio consisting of Policy(1,2) and Policy(2,2) together gave identical coverage to
the portfolio consisting of Policy(1,1) along with a one year deferred Policy(1,1).   So, by the No
Arbitrage Principle, the premiums and how they are earned should be the same.  During year 1, the
premium earned on  Policy(1,1) is equal to 100.  The premium earned during year 1 on each of
Policy(1,2) and Policy(2,2) depends on the results of year 1:
Loss case: probability = 10%
Policy(1,2) earned premium = 190 impliesÞ Policy(2,2) earned premium = -90
No-loss case: probability = 90%
Policy(2,2) earned premium = 10  impliesÞ Policy(1,2) earned premium = 90.

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