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: