Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 5

1.3 A Definition
The pure premium for a policy is equal to the expected losses at contract inception.  However, as time
passes the pure premium for the remaining expected losses will change. We will call the remaining
expected losses the required pure premium reserve (RPR).  This quantity will vary over time; when we
need to be more specific, we will call it the required pure premium reserve at time t (RPRt).  This value
RPRt, by the way, is exactly the amount that one of the hypothetical risk carriers from The Frictionless
World would require to assume the risk at time t.
So, at policy inception, the required premium reserve equals the pure premium for the policy.  At
policy termination, when no more losses can occur, the required premium reserve is zero. (Here and
throughout the paper we assume that losses are paid as they are incurred and that there is no reporting
                                                                                                                                                                                            
2
 In this first example, there can be only one loss per year so for the first year “at least one” implies “exactly one”.
3
 Such opportunities are also referred to as “free lunches”, but, alas, we all know that there is no such thing.lag).  The RPR is very similar to the unearned premium reserve (UEPR), but it has the following
difference: the UEPR contains premium elements other than pure premium (such as expense loads and
risk loads).  In The Frictionless World, an exactly adequate UEPR is equal to the RPR, so in the
following discussion the terms are used interchangeably.
The RPR may depend on loss experience, as the following continuation of the earlier example
illustrates:
The RPR for Policy(1,2) at time t = 0 is the pure premium, which we computed above as $190.
After one year, we are in one of two states:
State Probability RPR1
Loss Occurred 10% No more cover remains, so RPR1 = 0
No Loss Occurred 90% Remaining cover is Policy(1,1), so RPR1 = 100.
The decrease in the RPR during the first year is analogous to the (pure) premium earned during that
period.  The decrease in the RPR in the loss case is 190 and in the no-loss case is 90.  The probability
of the loss case is 10%, so the expected change in the RPR is  (0.1)(190) +  (0.9)(90) = 100.  This is
equal to the pure premium for a one-year cover (which is the coverage that you got during the first year
of Policy(1,2)).  Again, this is no coincidence.Lemma: The (a priori) expected value of the change in the RPR during a period is equal to the (a
priori) expected value of the losses occurring during that period.
Sketch of Proof: Consider a time period during the term; call this period D. Let B and A be the time
periods (during the contract term) before and after period D, respectively.
|-------B(efore)---------|--D(uring)--|------------------A(fter)--------------------|
<--------------------------------------Policy Term---------------------------------->
At contract inception, the RPR is equal to the expected losses occurring during the whole policy
period: B, D, and A combined.  And at contract inception, we expect the RPR at the start of period D to
be equal to the losses expected to occur during periods D and A.  Similarly, at contract inception we
expect the RPR at the start of period A to be equal to the losses expected to occur during period A.
It follows that the a priori expected change in RPR is equal to the a priori  expected value of losses
occurring during period D, which is what the lemma says.  QED.
In the above example, expected losses were $100 and the expected change in the RPR was also $100.
Notice that while the expected change was $100, an actual change of $100 is not possible in  this
example (it is either $90 or $190).