Note that the earning is initially slow (in fact it is instantaneously zero at contract inception) and then
rapidly increases towards the end of the first year. Also notice that the earnings in the last year are
expected to be small. The initial slow earning is due to the fact that it is very unlikely for two losses to
occur in a short period of time. The earnings expected later in the policy term are small because, a
priori, we expect to be off risk by then (by virtue of having already paid the loss!).
But again, this graph shows only the a priori expectation at contract inception. When the first loss
occurs, the RPR for the second-loss cover immediately jumps. In effect the policy then converts from a
second-loss cover to a first-loss cover; this is a manifestation of the “memory-less” feature of the
Poisson distribution. The first loss to occur is not a loss event for the cover, and no loss reserves arerequired. But suddenly the RPR is inadequate and an underwriting loss has been incurred (because
some “negative premium” has been earned – this premium will be earned back over the remainder of
Section 6: An Example with Expenses
In the real world, the UEPR contains many components in addition to the RPR’s pure premium. There
may be, for example, on-going contract maintenance expense
Effectively such expense forms an .
annuity that runs until contract termination. One quick example will give a flavor of the complications.
Recall the earlier example of an indefinite-term policy that pays $3000 when the loss occurs, has
annual loss probability of 10%, and no investment income. Assume that on-going contract
maintenance expense is $150 per year. Letting G stand for the expense-loaded premium, the premium
equation now reads:
G = 1/10 ($3000 + $150) + 9/10 (G + $150)
That is, one-tenth of the time we have expenses of $150 and a loss of $3000, and the other nine-tenths
of the time we have expenses of $150 and RPR1 = G (because of the indefinite term).
Solving for G, we find that G = $4500.
The company with this risk on its books suffers an underwriting loss (after expenses) of $150 each yearthat there is no loss, but has an underwriting gain of $1350 the year that the loss occurs!
The interested reader may find it amusing to work out the effect on this example of including 5%
investment income as before.
Section 7: Some Practical Ramifications of the Methodology
The preceding examples illustrate some theoretical issues, but the practicing actuary must consider the
broader practical effects of any change to common practice. Questions of materiality and practicality
also should be addressed.
7.1 Actuarial Reserve Opinions
The NAIC SAO Instructions for Property-Casualty  specify that the SCOPE paragraph include the
Reserve for Direct, Ceded, and Net Unearned Premiums. Also, these three items must be covered in
the OPINION and RELEVANT COMMENTS paragraphs. This applies to all insurers that write direct
and/or assumed contracts or policies (excluding financial guaranty, mortgage guaranty, and surety
contracts) with terms of thirteen months or more, which the insurer cannot cancel, and for which the
insurer cannot increase premiums during the term.
The insurer is required to establish an adequate unearned premium reserve. For each of the three most
recent policy years, the gross unearned premium reserve must be no less than the largest result of three
Had these expenses have been deferred policy acquisition expenses, there would be additional accounting complications.
What’s happening here is that we have an annuity with an expected life of ten years funding the expenses. When we have
a no-loss year, the expected life of the annuity stays at ten (instead of decreasing to nine) and we show an underwriting loss
of the difference. When we have a loss year, the expense annuity is no longer needed (its expected life drops from ten to
zero). The release of the reserve supporting this annuity yields the underwriting gain.tests. The three tests (in slightly simplified form) are:
1) The best estimate of the amounts refundable to the contract holders at the reporting date.
2) The gross premium multiplied by the ratio of (a) over (b) where:
(a) equals the projected future gross losses and expenses to be incurred during the unexpired term
of the contracts; and
(b) equals the projected total gross losses and expenses under the contracts.
3) The amount of the projected future gross losses and expense to be incurred during the unexpired
term of the contracts (as adjusted), reduced by the present value of the future guaranteed gross
The examples in this paper are intended to be non-cancelable and to have fixed premiums (generally
payable at contract inception, to avoid irrelevant complications). The example contract terms are more
than thirteen months, so except for the proscribed lines of business the rule applies. How do our
examples fare under these tests?
For simplicity, we shall assume that there are no refund provisions in the policy, so the Test 1 lower
bound on the unearned premium reserve is zero.
The second test requires that we estimate gross losses and expense. The examples in this paper for the
most part have been concerned with pure premiums (i.e., only the expected losses, with no provision
for expenses). Under the simplifying assumption that expenses are zero, Test 2 tells us to estimate the
projected future gross loss to be incurred, and to divide this by the projected total gross loss. This ratio
is then multiplied by the gross premium to obtain the second lower bound on the unearned premiumreserve.
The third test requires that the unearned premium reserve be at least as large as the expected future
losses and expenses to be incurred during the contract (as adjusted). The amount of the projected
future gross losses to be incurred is exactly the RPR at the statement date. The “adjustments” in
question are for future premiums (our examples have none) and for investment income up until the loss
is incurred but not beyond (our losses are immediately payable, so the example with investment income
complies). [The test also specifies a company-specific maximum interest rate. We will assume that 5%
meets this test.]
So in our examples, the RPR is the lower bound on the unearned premium reserve specified by Test 3.
7.2 Perspectives on Aggregate Deductible Business
In a multi-year contract with an aggregate deductible, the experience of the first few years can
influence the required premium reserve in two ways. First, the aggregate deductible may be depleted
faster or slower than planned; second, adverse or favorable experience during the initial period may
influence one’s view of the future ground-up experience. This paper addresses only the former.
There is an additional way to view such policies. The later years of a multi-year policy with an
aggregate deductible can be thought of as excess layers, each year/layer having a retention that depends
on the earlier years’ experience. If the total losses to date have been small, little of the aggregate
deductible has been eroded and the retention (the remaining aggregate deductible) for the later years is
higher. Since higher layers have lower premiums, the RPR is small. Similarly, if early experience hasbeen unfavorable, much of the aggregate deductible will have been eroded. The retention will be lower
and the RPR will be large. In essence, early experience determines which layers the later years’
coverage corresponds to.
7.3: What to Do about Negative Premium
In chapter 14 of the IASA text, David L. Holman and Chris C. Stroup discuss US-GAAP accounting
for P&C insurers. Under US-GAAP there is a notion of a premium deficiency reserve (PDR). Holman
and Stroup write: “Projections, therefore, are periodically updated, based on new information about
expected cash flows. GAAP requires that a premium deficiency be recognized if the sum of expected
loss and loss adjustment expenses, expected dividends to policyholders, maintenance costs, and
unamortized (or deferred) policy acquisition costs, exceed the related unearned premiums related
thereto.” If this is the case, the unamortized policy acquisition costs are reduced to make up the
shortfall. If that alone is not sufficient, a liability is reported for the remaining deficiency.
Interestingly, Canadian statutory accounting provides a line item (Line 15) for Premium Deficiency
(see chapter 18 of the IASA text). European actuaries speak of the “reserve for unexpired risks”, which
is similar in concept to a premium deficiency reserve.
So, under US-GAAP one might establish a PDR to handle “negative premium” earnings. Effectively,
negative premium is earned by the reduction of an asset (the unamortized policy acquisition cost)
and/or the establishment of an additional liability
Statutory accounting does not have the notion of a premium deficiency, although in principle one could
include one by using the write-in lines. However, due to US income tax regulation, there may be amaterial difference between treating the shortfall as premium or as some other type of liability. The
interested reader should see chapter 13 of the IASA text or the Almagro and Ghezzi paper from the
7.4: Is It Loss or Is It Premium?
The argument can be made that instead of altering the premium earning methodology, we should put up
loss reserves corresponding to the losses that are eroding the aggregate deductible. That is, there is an
increase in expected losses to the cover caused by events that have occurred prior to the statement date.
The amounts to be put up are not in dispute; they would be exactly the amount needed to make the
booked unearned premium reserve match the RPR. The difference is that these reserves would be
characterized as loss instead of premium.
But these reserves behave more like premium than loss in two important ways. First, they amortize
over the remaining policy period. To see the second reason, consider a two-trigger two-year policy. In
order for the policy to pay, two events, A and B, must occur during a two-year period. Say event A
occurs in year 1, and as a result some additional reserve (either a loss reserve or a premium deficiency
reserve) is needed. Suppose now you wanted to completely reinsure this risk. You could do this by
purchasing cover for event B. Observe that this reinsurance is completely prospective. Being
prospective, it should be funded from premium reserves, not loss reserves.
Claims-made policies and “sunset clauses” in reinsurance agreements can further blur the line between premium reserves
and loss reserves. Suppose that an event has occurred, but that it has not been reported yet. Assuming that a reserve is
appropriate should it be premium or loss? This reserve amortizes over the remaining reporting period (acts like premium).
On the other hand, the underlying loss event has already occurred. Is the reporting a second trigger?Section 8: Conclusions
We could use the “adequate pure premium reserve” approach to answer Ruy Cardoso’s question, which
was mentioned at the start of this article: Losses are certain at $10 per month. You cover $20 excess
$100 in aggregate. The contract begins 7/1/xx. What is the loss reserve at 12/31/xx?
Assuming no expenses or investment income, the UEPR would be $20 (because that is the RPR
remaining), and the loss reserve would be $0 (because no covered loss has occurred). No premium
(positive or negative) would have been earned to date. This question sparked a very lengthy and
enlightening discussion, and I urge the interested reader to look over the thread (link:
The “adequate pure premium reserve” approach outlined in this paper is internally consistent, even
though it leads to some controversial implications such as negative earned premium. But the idea of
negative earned (and written) premium already is used in some instances, such as the treatment of
ceded proportional reinsurance. US GAAP and Canadian accounting have a notion of a premium
deficiency reserve (PDR), and additionally in some European jurisdictions there is a notion of an
“unexpired risk reserve”. These entries could be used to record unexpected changes in the required
However, there are some operational problems with using what might be called “the negative premium
approach”: it might distort loss and expense ratios; it can make budgeting difficult; and for UStaxpayers, the treatment of the UEPR for US taxation is different than for other reserves, which could
lead to complications.
The good news is that “on average” the standard methodology should give the same results as this
method for a large book of uncorrelated risks, written evenly throughout the year. However, the type
of analysis outlined in this paper is probably justified for those risk carriers with a few large risks or for
single risks that are large enough to distort the book.
 Almagro, M.; and Ghezzi, T.L., “Federal Income Taxes --- Provisions Affecting Property/Casualty
Insurers,” PCAS LXXV, 1988
 American Academy of Actuaries, Property/Casualty Loss Reserve Law Manual – 1998, 1998
 Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; and Nesbit, C.J, Actuarial Mathematics
(Second Edition), 1997
 Casualty Actuarial Society, 1999 Yearbook, 1999
 Insurance Accounting and Systems Association, Property-Casualty Insurance Accounting (Sixth
 Ross, S.M., Introduction to Probability Models (Sixth Edition), 1997