In this section, we present different approximations. We illustrate them on a common claim size
distribution example, namely the mixture of two exponentials claims with
,
and
(see Chapter 13). Their numerical comparison is given in Section 15.7.
The ruin probability in finite time can always be approximated by means of Monte Carlo simulations. Table 15.18 shows the output for mixture of two exponentials claims with ,
,
with respect to the initial capital
and the time horizon
. The relative safety loading
is set to
. For the Monte Carlo method purposes we generated
x
simulations.
We see that the values approach those calculated in infinite case as
increases, cf. Table 15.4. We note that the Monte Carlo method will be used as a reference method when comparing different finite time
approximations in Section 15.7.
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0.672550 | 0.428150 | 0.188930 | 0.063938 | 0.006164 | 0.000002 |
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0.718254 | 0.501066 | 0.256266 | 0.105022 | 0.015388 | 0.000030 |
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0.753696 | 0.560426 | 0.323848 | 0.159034 | 0.035828 | 0.000230 |
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0.765412 | 0.580786 | 0.350084 | 0.184438 | 0.049828 | 0.000726 |
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0.769364 | 0.587826 | 0.359778 | 0.194262 | 0.056466 | 0.001244 |
The following result due to Segerdahl (1955) is said to be a time-dependent version of the Cramér-Lundberg approximation given by (15.13). Under the assumption that , cf. relation (15.28), we have
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0.663843 | 0.444333 | 0.172753 | 0.070517 | 0.013833 | 0.000141 |
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0.663843 | 0.554585 | 0.229282 | 0.092009 | 0.017651 | 0.000175 |
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0.663843 | 0.587255 | 0.338098 | 0.152503 | 0.030919 | 0.000311 |
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0.663843 | 0.587260 | 0.359593 | 0.192144 | 0.049495 | 0.000634 |
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0.663843 | 0.587260 | 0.359660 | 0.194858 | 0.057143 | 0.001254 |
This method requires existence of the adjustment coefficient. This implies that only light-tailed distributions can be used. Numerical evidence
shows that the Segerdahl approximation gives the best results for huge values of the initial capital , see Asmussen (2000).
In Table 15.19, the results of the Segerdahl approximation for mixture of two exponentials claims with ,
,
with respect to the initial capital
and the time horizon
are presented. The relative safety loading
. We see that the approximation in the considered case yields quite accurate results for moderate
, cf. Table 15.18.
The idea of the diffusion approximation is first to approximate the claim surplus process by a Brownian motion with drift (arithmetic Brownian motion) by matching the first two moments, and next, to note that such an approximation implies that the first passage probabilities are
close. The first passage probability serves as the ruin probability.
The diffusion approximation is given by:
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1.000000 | 0.770917 | 0.223423 | 0.028147 | 0.000059 | 0.000000 |
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1.000000 | 0.801611 | 0.304099 | 0.072061 | 0.001610 | 0.000000 |
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1.000000 | 0.823343 | 0.370177 | 0.128106 | 0.011629 | 0.000000 |
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1.000000 | 0.829877 | 0.391556 | 0.150708 | 0.020604 | 0.000017 |
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1.000000 | 0.831744 | 0.397816 | 0.157924 | 0.024603 | 0.000073 |
We also note that in order to apply this approximation we need the existence of the second moment of the claim size distribution.
Table 15.20 shows the results of the diffusion approximation for mixture of two exponentials claims with ,
,
with respect to the initial capital
and the time horizon
. The relative safety loading
equals
. The results lead to the conclusion that the approximation does not produce accurate results for such a choice
of the claim size distribution. Only when
billion USD the results are acceptable, cf. the reference values in Table 15.18.
The idea presented above of the diffusion approximation ignores the presence of jumps in the risk process (the Brownian motion with drift is skip-free) and the value
in the moment of ruin. The corrected diffusion approximation takes this and other deficits into
consideration (Asmussen; 2000). Under the assumption that
, cf. relation (15.28), we have
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0.521465 | 0.426840 | 0.187718 | 0.065264 | 0.007525 | 0.000010 |
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0.587784 | 0.499238 | 0.254253 | 0.104967 | 0.016173 | 0.000039 |
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0.638306 | 0.557463 | 0.321230 | 0.157827 | 0.035499 | 0.000251 |
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0.655251 | 0.577547 | 0.347505 | 0.182727 | 0.049056 | 0.000724 |
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0.660958 | 0.584386 | 0.356922 | 0.192446 | 0.055610 | 0.001243 |
Similarly as in the Segerdahl approximation, the method requires existence of the moment generating function, so we can use it only for light-tailed distributions.
In Table 15.21 the results of the corrected diffusion approximation for mixture of two exponentials claims with ,
,
with respect to the initial
capital
and the time horizon
are given. The relative safety loading
is set to
. It turns out that corrected diffusion method gives surprisingly good results and is vastly superior
to the ordinary diffusion approximation, cf. the reference values in Table 15.18.
Let us recall the idea of the De Vylder approximation in infinite time: we replace the claim surplus process with the one with
,
and exponential claims with parameter
, fitting first three moments, see Section 15.3.6.
Here, the idea is the same. First, we compute
Next, we employ relations (15.27) and (15.28) and finally use the exact, exponential case formula presented in Section 15.5.1.
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0.528431 | 0.433119 | 0.189379 | 0.063412 | 0.006114 | 0.000003 |
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0.594915 | 0.505300 | 0.256745 | 0.104811 | 0.015180 | 0.000021 |
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0.645282 | 0.563302 | 0.323909 | 0.158525 | 0.035142 | 0.000215 |
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0.662159 | 0.583353 | 0.350278 | 0.183669 | 0.048960 | 0.000690 |
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0.667863 | 0.590214 | 0.359799 | 0.193528 | 0.055637 | 0.001218 |
Obviously, the method gives the exact result in the exponential case. For other claim distributions, the first three moments have to exist in order to apply the approximation.
Table 15.22 shows the results of the finite time De Vylder approximation for mixture of two exponentials claims with ,
,
with respect to the initial capital
and the time horizon
.
The relative safety loading
. We see that the approximation gives even better results than the corrected
diffusion one, cf. the reference values presented in Table 15.18.
Table 15.23 shows which approximation can be used for each claim size distribution. Moreover, the necessary assumptions on the distribution parameters are presented.