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).

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