We are now interested in the probability that the insurer's capital as defined by (15.1) remains non-negative for a finite period rather than permanently. We assume that the number of claims process
is a Poisson process with rate
, and consequently, the aggregate loss process is a compound Poisson process. Premiums are payable at rate
per unit time. We recall that the intensity of the process
is irrelevant in the infinite time case provided that it is compensated by the premium, see discussion at the end of Section 15.1.
In contrast to the infinite time case there is no general formula for the ruin probability like the Pollaczek-Khinchin one given by (15.8). In the literature one can only find a partial integro-differential equation which satisfies the probability of non-ruin, see Panjer and Willmot (1992). An explicit result is merely known for the exponential claims, and even in this case a numerical integration is needed (Asmussen; 2000).
First, in order to simplify the formulae, let us assume that claims have the exponential distribution with and the amount of premium is
. Then
Now, notice that the case
is easily reduced to
, using the formula:
Moreover, the assumption is not restrictive since we have
Table 15.17 shows the exact values of the ruin probability for exponential claims with
(see Chapter
13) with respect to the initial capital
and the time horizon
.
The relative safety loading
equals
. We see that the values converge to those calculated in infinite case as
is getting larger, cf. Table 15.2. The speed of convergence decreases as the initial capital
grows.
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0 | ![]() |
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0.757164 | 0.147954 | 0.025005 | 0.003605 | 0.000443 | 0.000047 |
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0.766264 | 0.168728 | 0.035478 | 0.007012 | 0.001288 | 0.000218 |
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0.769098 | 0.176127 | 0.040220 | 0.009138 | 0.002060 | 0.000459 |
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0.769229 | 0.176497 | 0.040495 | 0.009290 | 0.002131 | 0.000489 |
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0.769231 | 0.176503 | 0.040499 | 0.009293 | 0.002132 | 0.000489 |