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

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, 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:

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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

zero.

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

or

No-loss case: probability = 90%

Policy(2,2) earned premium = 10 impliesÞ Policy(1,2) earned premium = 90.

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