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

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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.

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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.

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