## Saturday, 2 July 2011

### 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.
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
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
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])