Saturday, 2 July 2011

Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 11


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
the contract).
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
7
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!
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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 [2] 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
                                                                                                                                                                                            
7
 Had these expenses have been deferred policy acquisition expenses, there would be additional accounting complications.
8
 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
premiums.
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
Proceedings [1].
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.
9
                                                      
9
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:
http://www.casact.org/RESEARCH/casnet.htm)
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
premium reserve.
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.
Bibliography
[1] Almagro, M.; and Ghezzi, T.L., “Federal Income Taxes --- Provisions Affecting Property/Casualty
Insurers,” PCAS LXXV, 1988
[2] American Academy of Actuaries, Property/Casualty Loss Reserve Law Manual – 1998, 1998
[3] Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; and Nesbit, C.J, Actuarial Mathematics
(Second Edition), 1997
[4] Casualty Actuarial Society, 1999 Yearbook, 1999
[5] Insurance Accounting and Systems Association, Property-Casualty Insurance Accounting (Sixth
Edition), 1994
[6] Ross, S.M., Introduction to Probability Models (Sixth Edition), 1997

Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 10


Section 5: The Continuous-Time Case
We now will quickly look at an example in continuous time.  For simplicity we shall go back to
discounting at 0% (no investment income).  For this example, we consider a constant loss amount of
$1000 per occurrence and we assume that the frequency of losses on an annual basis is Poisson with
                                                                                                                                                                                            
6
 The premium is made payable at the end of the year to remove timing effects.parameter 4/3.  Our cover, Policy(2,3), will have a term of three years and will pay for the second loss
during the period.  Let’s look at the probabilities of paying off in each of years 1, 2, and 3.
Since losses are Poisson, we have the following probabilities for year 1:
Losses during Year 1 Probability
0 e
-4/3
= 26.360%
1 4/3 e
-4/3
 = 35.146%
2 or more 1 – 7/3 e
-4/3
 = 38.494%
Notice that of these three possible outcomes for year 1, the most likely is that the second loss occurs
during the first year – even though we expect only one and one third losses per year.
For the first two years we have:
Losses during Years 1 and 2 Probability
0 e
-8/3
= 6.948%
1 8/3 e
-8/3
= 18.529%
2 or more 1 – (11/3) e
-8/3
 = 74.523%
During years 1 and 2 we will pay 74.523% of the time.  During year 1 we paid 38.494% of the time, so
it follows that during year 2 we will pay 74.523 – 38.494 = 36.029% of the time.To find the probability of paying in year 3, we observe:
Losses during Year 1, 2, and 3 Probability
0 e
-4
 = 1.832%
1 4 e
-4
 = 7.326%
2 or more 1 – 5 e
-4
 = 90.842%
So, the probability of paying in year 3 is 90.842% - 74.523% = 16.319%.
With  these probabilities we see that at contract inception, we expect to earn the $908.42 = $1000 *
90.842% of pure premium over three years in the following yearly pattern:  $384.94, $360.29, and
finally $163.19.
But as in the discrete time case, this expectation is only valid at contract inception. As soon as any time
has passed (or rather, once some period has passed and you know how many losses there were during
that period) the expected future pattern changes. Below is a graph showing the earning pattern expected
at contract inception:

Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 9


Section 4: An Indefinite-Term Example
In this example we will assume a 1/10 chance of loss each year and go back to the simplified model of
at most one loss per year.  Loss severity is assumed constant at $3000.  We will continue to ignore
investment income.  The policy that we consider in this example covers one loss, but has no time limit.
The policy will stay in effect until there is a loss, at which time it will pay $3000.
4.1  Pure Premium and Earning Patterns
What is the pure premium for this coverage?  Let P be this premium.  Then P must pay for two things.
One-tenth of the time there is a loss during year 1 of $3000 and RPR1 = 0.  The other nine-tenths of the
time, there is no loss in the first year and RPR1 is the pure premium for a policy that pays $3000whenever the loss occurs – but this is exactly what P is.  We have:
P = 1/10 ($3,000 + 0) + 9/10 (0 + P)
Solving for P, one finds P = $3000.
Upon reflection this is not very surprising, since $3000 will be paid out eventually (recall that we are
still ignoring investment income).  So the pure premium equals the expected loss, which is $3000.
How does one earn the premium for such a policy?  In the loss case, the premium earned in year 1 is
$3000; in the no-loss case the premium earned in year 1 is $0 (since RPR1 remains at $3000).  So, at
policy inception the expected earned premium for the first year is $300.
What about later years?  The answer depends on when you ask the question.
At the start of the first year, we expect to earn $270 during the second year and $243 during the third.
But these are the a priori expectations at the start of the first year; after one year has passed there has
been either one loss or no loss, and with this additional information the expected values for earned
premium change.
At the start of the second year there are two possibilities: either there was a loss in year 1 (in which
case no coverage remains) or there was no loss in year 1 (in which case there is coverage for year 2).
Also, since we are assuming no late reporting, you will know which case applies.  The conditional
expectation (given no loss in year 1) for the premium earned in year 2 is $300.  Similarly, theconditional expectation (given no loss is year 1) for the premium earned in year 3 is $270.  On the other
hand, the conditional expectation (given no loss in years 1 and 2) for the earned premium in year 3 is
$300.
The expected earning pattern at the start of any year, for that and subsequent years, is: ($300, $270,
$243, …) with each term being 9/10 of the previous term.  When a year passes without loss, each of
these terms shifts forwards. It should come as no surprise that this infinite geometric series sums to
$3000.
Why is no premium earned during no-loss years?  Because the RPR at the start of the no-loss year is
$3000, and it is also $3000 at the end of the year.  The change in the RPR, in this case 0, is the earned
premium.  During a loss year, the RPR is $3000 at the start of the year and it is $0 at the end of the year
(because no more coverage remains), so the amount earned during the year is $3000.
Note that the company shows no underwriting gain or loss, no matter what the outcome.  In the no-loss
case there is no movement in the reserves; in the loss case the RPR becomes the loss reserve.  This is a
consequence of the indefinite policy term.  Since the cover continues until there is a loss, having a noloss year only delays the inevitable payment; and without investment income the delay does not benefit
us.  We relax this restriction below.
4.2 Investment Income
Now let’s modify the last example to take into account investment income.  Assume that all losses are
paid at the end of the year, and that invested funds earn interest at a rate of 5%.Now the equation for the present value of the pure premium reads:
P = 1/10 * ($3000)/1.05 + 9/10 (P/1.05)
One tenth of the time we pay a loss of $3000 (discounted one year) and nine tenths of the time the
present value of RPR1 is P (discounted one year). Solving for P, we find that P = $2000.
How should this premium be earned?  Should the fact that we now consider investment income affect
how we earn the premium?
Suppose that we have a loss in year 1.  Then, as before, the RPR1 = 0, so we earn the full $2000 during
year 1.  We also have investment income of $100.  On the other hand, suppose that we have no loss in
the first year.  Then RPR1 = $2000, and again we have investment income of $100.  What should be
done with the investment income?
To investigate that question, we examine an alternative way to construct this same coverage.  Consider
an annual policy that pays $1000 at the end of the year if there is a loss, for a premium payable at the
end of the year
6
 of $100 (the pure premium for the policy).  In effect, this policy provides similar
coverage to the first year of the original policy, subject to a $2000 self-insured retention.  Imagine that
the insured sets aside this $2000 in a special account.  During the year, $100 in investment income is
earned on the $2000 (this is paid to the insurer as premium) and, if there is a loss, the $2000 set asideand the $1000 from the insurer combine to provide the $3000.
With a one-time premium of $2000 and a limit of $3000, the insurer has only $1000 at risk.  So in this
second setup, the insurer is entitled to only $100 (= $1000 * 10%) in annual pure premium.  This, as
we have seen, is the investment income generated by the one-time premium payment of $2000.
We see that the insured can obtain identical coverage in two ways: by setting aside the $2000 and
paying an annual premium of $100, or by paying a one-time premium of $2000.  The No Arbitrage
Principle says that since the two coverages are identical, their pure premiums must be equal.  In order
for this to work out, we need to view the investment income on (discounted) premium as premium – in
fact, this is implicit in the pricing equation.
Now we can determine the earning pattern for the original multiyear policy, and answer the question
about what to do with the investment income.  In a year with no loss, premium of $100 is earned.  In a
loss year, premium of $2100 (the original premium plus one year’s investment income) is earned.
This result is related to the “Paid-up Insurance Formula for Life Reserves”. (see for example, [3])

Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 8


Section 3: A Less Simplified Example
Now let’s start to relax the conditions that we imposed in Sections 1 and 2  for the first example.  We
now allow more than one loss in each year.  For simplicity, we will assume that in each year there are
0, 1, or 2 losses with probabilities 1/2, 1/3, and 1/6 respectively.  We will continue to ignore investment
income and will again assume a constant loss amount, but this time, to make the arithmetic simple, the
constant loss amount will be 216 instead of 1000.
The pure premiums for  Policy(k,n) may be computed as follows.  First compute the probability of
having exactly k losses by the end of year n; the result of this calculation
5
 is displayed in Table 1.
TABLE 1 (probability of exactly k losses in n years)
No. of losses One Year Two Years Three Years
0 50.00% 25.00% 12.50%
1 33.33% 33.33% 25.00%
2 16.67% 27.78% 29.17%
3 0.00% 11.11% 20.37%
4 0.00% 2.78% 9.72%
5 0.00% 0.00% 2.78%
6 0.00% 0.00% 0.46%
Then sum these to produce the probability of having at least k losses in n years; see Table 2 for these
values.
                                                      
5
 The probabilities are most easily computed recursively.  For example:  Pr(2,2) = 1/2*Pr(2,1) + 1/3*Pr(1,1) + 1/6*Pr(0,1).
That is, the only way to have exactly two losses at the end of year two is to have had no loss in year 2 AND exactly two
losses in year 1,  OR  exactly one loss in year 2 AND one loss in year 1,  OR  two losses in year 2 AND no loss in year 1.
(Here the events joined by AND are independent and the events joined by OR are mutually exclusive.)TABLE 2 (probability of at least k losses in n years)
No. of losses One Year Two Years Three Years
0 100.00% 100.00% 100.00%
1 50.00% 75.00% 87.50%
2 16.67% 41.67% 62.50%
3 0.00% 13.89% 33.33%
4 0.00% 2.78% 12.96%
5 0.00% 0.00% 3.24%
6 0.00% 0.00% 0.46%
Finally, multiply by the constant loss amount of 216 to compute the pure premiums shown in Table 3.
TABLE 3 (pure premiums for Policy(k,n))
Loss k n = 1 n = 2 n = 3
1 108 162 189
2 36 90 135
3 0 30 72
4 0 6 28
5 0 0 7
6 0 0 1
Consider Policy(2,3), which covers the second loss in three years.  The pure premium for this coverage
is $135.  How much of this premium do we expect to earn during the first year?
Half of the time there will be no loss the first year, and the RPR for the last two years of the policy
must be $90 – the pure premium for Policy(2,2).  So in this case $135 - $90 = $45 would be earned in
the first year.Similarly, one-third of the time there will be one loss during the first year; then the RPR for the last
two years must be $162 (the pure premium for  Policy(1,2), which is equivalent to the remaining
coverage) and $135 - $162 = -$27 would be earned during the first year.
Finally, one-sixth of the time there are two losses in year 1.  In this  case there is no more coverage
available.  The RPR for the last two years is zero, and the full $135 would be earned during year 1.
Combining the above calculations we find that  at policy inception  the expected earned premium for
year 1 is
1/2 ($45) + 1/3 (-$27) + 1/6 ($135) = $36.
Year 3’s expected earnings are similarly easy to calculate: during the first two years of the cover there
is a 1/2 * 1/2 = 1/4 chance that there have been no losses and a 1/2 * 1/3 + 1/3 * 1/2 = 1/3 chance of
exactly one loss.  From Table 2, we see that the pure premium for Policy (2,1) is 36 and for Policy
(1,1) is 108.  From this we see that at policy inception we expect to earn 1/4 ($36) + 1/3 ($108) = $45
during year 3.
During the life of the policy we will earn exactly $135.  If at policy inception we expect to earn $36 in
year 1 and $45 in year 3, it follows that we must expect at policy inception to earn $135 - $36 - $45 =
$54 during year 2.
Does this mean that we should earn the premium over the three years in this pattern: $36, $54, $45?No, because these are a priori expectations.  As we have seen in earlier sections, the premium earned
during year 1 need not equal the a priori  expected earned premium.  Also, at the end of year 1 our
expectations for the earnings in years 2 and 3 will probably be different than they were at inception.
The first two rows of Table 3 contain all of the information needed to compute the actual amount of
premium earned to date at the end of each year.  For example, suppose there is exactly one loss and it
occurs in year 2.  Then we should earn $45 in the first year, because when we start year two, the
remaining coverage is the second loss in two years – a Policy(2,2).  During year three we are in a firstloss position, so we need to earn $108 because at the start of year 3, the remaining coverage is the first
loss in one year – a Policy(1,1).  Since the total amount earned over the three years must be $135, we
find that the year two (actual) earnings must be negative 18.  So the actual earning pattern observed in
this case would be ($45, - $18, $108), which differs markedly from the a priori expectation.

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?

Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles part 6


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
4
, 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:
                                                      
4
 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 implies├× Policy(2,2) earned premium = -90
or
No-loss case: probability = 90%
Policy(2,2) earned premium = 10  implies├× Policy(1,2) earned premium = 90.

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).