Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 7

In the loss case, the premium earned on Policy(1,2) is 190 by the standard premium accrual procedure.
Using the No Arbitrage Principle, since the total premium earned on the two policies during year 1
must be 100, the premium earned on Policy(2,2) must be -90.
Similarly, in the no-loss case, the premium earned on Policy(2,2) should be all 10, because no coverage
remains. No Arbitrage forces the premium earned on Policy(1,2) to be 90, because the sum must be
100.
If you are uncomfortable with earning all of the premium for Policy(2,2) in the no-loss case in year 1,
consider what happens to the pair of policies in year 2 given that there was no loss in year 1.  The
coverage is identical to the coverage afforded by a one year deferred  Policy(1,1), so the earned
premium in year 2 must be the same: 100.  In fact, the coverage during year 2 for Policy(1,2) alone is
the same as for a Policy(1,1) because we are given that there was no loss in year 1.  So the premium
earned on  Policy(1,2) during year 2 must be 100.   Since the total premium earned is also 100, no
premium can have been earned on Policy(2,2).  Over the life of the Policy(2,2) $10 must be earned; if
none is earned in year 2, all of it must have been earned in year 1.
2.2  Reconciling Total Earnings
The total amount of pure premium earned during the life of the policy is always equal to the initial pure
premium.  If some “negative premium” is earned during one period, it is recovered in later periods (or
is balanced by some “over-earning” in prior periods).   The total change in the RPR from contract
inception to contract termination is the a priori pure premium.  This is an important point. The negative
premium earned is not “new” premium, the written premium stays the same,  it is just earned in adifferent pattern.
It is should be noted that this process is nothing more than a “mark to market” of the outstanding
UEPR.
The UEPR for a given policy is amortized over the policy’s term.  This amortization occurs according
to some schedule.  Commonly, for most lines of business this amortization is done linearly over the
term; this produces the familiar pro-rata earning pattern.  This pattern is theoretically correct for a
policy with no aggregate deductible, no aggregate limit, and an underlying loss process that is
compound with Poisson frequency.  For a further discussion of compound distributions see for example
Ross’s text [6].  For certain lines of business (e.g. extended warranty, ocean marine cargo cover, credit
insurance on a declining balance) other amortization patterns and, hence, earning patterns are used.
The “adequate unearned premium reserve” process described above can be thought of as adjusting this
amortization schedule to include the latest data.
Traditionally, one thinks of unearned premium reserves flowing into loss reserves and surplus as the
policy term progresses.  Sometimes the losses occur slower than expected, and an unexpectedly large
portion of this flow goes to surplus; other times losses occur faster than expected, and (unfortunately)
in these cases surplus may flow into loss reserves.  In the example we worked through above, it is the
unearned premium reserve, not the loss  reserve, that has become inadequate and requires
supplementation from surplus.  This is discussed further in Section 7.4: Is It Loss or Is It Premium?