When the claim size distribution is exponential (or closely related to it), simple analytic results for the ruin probability in infinite time exist, see
Section 15.2. For more general claim amount distributions, e.g. heavy-tailed, the Laplace transform technique does not work and one needs
some estimates. In this section, we present 12 different well-known and not so well-known approximations. We illustrate them on a common claim size
distribution example, namely the mixture of two exponentials claims with
,
and
(see Chapter 13). Numerical comparison of the approximations is given in Section 15.4.
Cramér-Lundberg's asymptotic ruin formula for for large
is given by
In Table 15.5 the Cramér-Lundberg approximation for mixture of two exponentials claims with ,
,
and the relative
safety loading
with respect to the initial capital
is given. We see that the Cramér-Lundberg approximation underestimates the
ruin probability. Nevertheless, the results coincide quite closely with the exact values shown by Table 15.4. When the initial capital is
zero, the relative error is the biggest and exceeds
.
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0.663843 | 0.587260 | 0.359660 | 0.194858 | 0.057197 | 0.001447 |
This approximation was proposed and derived by De Vylder (1996). It requires the first three moments to be finite.
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(15.14) |
Table 15.6 shows the results of the exponential approximation for mixture of two exponentials claims with ,
,
and
the relative safety loading
with respect to the initial capital
. Comparing them with the exact values presented in Table
15.4 we see that the exponential approximation works not bad in the studied case. When the initial capital is USD
billion, the
relative error is the biggest and reaches
.
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0.747418 | 0.656048 | 0.389424 | 0.202900 | 0.055081 | 0.001102 |
The following formula, called the Lundberg approximation, comes from Grandell (2000). It requires the first three moments to be finite.
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(15.15) |
In Table 15.7 the Lundberg approximation for mixture of two exponentials claims with ,
,
and the relative safety
loading
with respect to the initial capital
is given. We see that the Lundberg approximation works worse than the exponential
one. When the initial capital is USD
billion, the relative error exceeds
.
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0.504967 | 0.495882 | 0.382790 | 0.224942 | 0.058739 | 0.000513 |
The Beekman-Bowers approximation uses the following representation of the ruin probability:
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(15.16) |
The Beekman-Bowers approximation gives rather accurate results, see Burnecki, Mista, and Weron (2004). In the exponential case it becomes the exact formula. It can be used only for distributions with finite first three moments.
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0.769231 | 0.624902 | 0.352177 | 0.186582 | 0.056260 | 0.001810 |
Table 15.8 shows the results of the Beekman-Bowers approximation for mixture of two exponentials claims with ,
,
and the relative safety loading
with respect to the initial capital
. The results justify the thesis the approximation yields quite
accurate results but when the initial capital is USD
billion, the relative error is unacceptable, reaching
, cf. the exact values in Table
15.4.
The Renyi approximation (Grandell; 2000), may be derived from (20.5.4) when we replace the gamma distribution function with an
exponential one, matching only the first moment. Hence, it can be regarded as a simplified version of the Beekman-Bowers approximation. It requires
the first two moments to be finite.
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(15.18) |
In Table 15.9 the Renyi approximation for mixture of two exponentials claims with ,
,
and the relative safety
loading
with respect to the initial capital
is given. We see that the results compared with the exact values presented in Table
15.4 are quite accurate. The accuracy ot the approximation is similar to the Beekman-Bowers approximation but when the initial capital is
USD
billion, the relative error exceeds
.
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0.769231 | 0.667738 | 0.379145 | 0.186876 | 0.045400 | 0.000651 |
The idea of this approximation is to replace the claim surplus process with the claim surplus process
with exponentially
distributed claims such that the three moments of the processes coincide, namely
for
,
see De Vylder (1978). The process
is determined by the three parameters
,
. Thus the
parameters must satisfy:
Then De Vylder's approximation is given by:
Obviously, in the exponential case the method gives the exact result. For other claim amount distributions, in order to apply the approximation, the first three moments have to exist.
Table 15.10 shows the results of the De Vylder approximation for mixture of two exponentials claims with ,
,
and the
relative safety loading
with respect to the initial capital
. The approximation gives surprisingly good results. In the considered
case the relative error is the biggest when the initial capital is zero and amounts to about
, cf. Table 15.4.
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0.668881 | 0.591446 | 0.361560 | 0.195439 | 0.057105 | 0.001424 |
The 4-moment gamma De Vylder approximation, proposed by Burnecki, Mista, and Weron (2003), is based on De Vylder's idea to replace the claim surplus process
with another one
for which the expression for
is explicit. This time we calculate the parameters of the new process with gamma
distributed claims and apply the exact formula (15.10) for the ruin probability. Let us note that the claim surplus process
with gamma claims is determined by the four parameters
, so we have to match the four
moments of
and
. We also need to assume that
to ensure that
and
, which is true for the gamma distribution. Then
When this assumption can not be fulfilled, the simpler case leads to
All in all, the 4-moment gamma De Vylder approximation is given by
In the exponential and gamma case this method gives the exact result. For other claim distributions in order to apply the approximation, the first four (or three in the simpler case) moments have to exist. Burnecki, Mista, and Weron (2003) showed numerically that the method gives a slight correction to the De Vylder approximation, which is often regarded as the best among ``simple'' approximations.
In Table 15.11 the 4-moment gamma De Vylder approximation for mixture of two exponentials claims with
,
,
(see Chapter 13) and the relative safety loading
with respect to the initial
capital
is given. The most striking impression of Table 15.11 is certainly the extremely good accuracy of the simple 4-moment gamma
De Vylder approximation for reasonable choices of the initial capital
. The relative error with respect to the exact values presented in Table
15.4 is the biggest for
and equals
.
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0.683946 | 0.595457 | 0.359879 | 0.194589 | 0.057150 | 0.001450 |
The term ``heavy traffic'' comes from queuing theory. In risk theory it means that, on the average, the premiums exceed only slightly the expected
claims. It implies that the relative safety loading is positive and small. Asmussen (2000) suggests the following approximation.
This method requires the existence of the first two moments of the claim size distribution. Numerical evidence shows that the approximation is
reasonable for the relative safety loading being and
being small or moderate, while the approximation may be far off for large
. We
also note that the approximation given by (15.21) is also known as the diffusion approximation and is further analysed and generalised to the
stable case in Chapter 16, see also Furrer, Michna, and Weron (1997).
Table 15.12 shows the results of the heavy traffic approximation for mixture of two exponentials claims with ,
,
and the relative safety loading
with respect to the initial capital
. It is clear that the accuracy of
the approximation in the considered case is extremely poor. When the initial capital is USD
billion, the relative error reaches
, cf.
Table 15.4.
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1.000000 | 0.831983 | 0.398633 | 0.158908 | 0.025252 | 0.000101 |
As for heavy traffic, the term ``light traffic'' comes from queuing theory, but has an obvious interpretation also in risk theory, namely, on the average, the premiums are much larger than the expected claims, or in other words, claims appear less frequently than expected. It implies that the relative safety loading is positive and large. We may obtain the
following asymptotic formula.
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(15.22) |
In Table 15.13 the light traffic approximation for mixture of two exponentials claims with ,
,
and the relative
safety loading
with respect to the initial capital
is given. The results are even worse than in the heavy case, only for moderate
the situation is better. The relative error dramatically increases with the initial capital.
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0.769231 | 0.303545 | 0.072163 | 0.011988 | 0.000331 | 0.000000 |
The crude idea of this approximation is to combine the heavy and light approximations (Asmussen; 2000):
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(15.23) |
The particular features of this approximation is that it is exact for the exponential distribution and asymptotically correct both in light and heavy traffic.
Table 15.14 shows the results of the heavy-light traffic approximation for mixture of two exponentials claims with ,
,
and the relative safety loading
with respect to the initial capital
. Comparing the results with Table 15.12 (heavy
traffic), Table 15.13 (light traffic) and the exact values given in Table 15.4 we see that the interpolation is promising. In
the considered case the relative error is the biggest when the initial capital is USD
billion and is over
, but usually the error is
acceptable.
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0.769231 | 0.598231 | 0.302136 | 0.137806 | 0.034061 | 0.001652 |
First, let us introduce the class of subexponential distributions
(Embrechts, Klüppelberg, and Mikosch; 1997), namely
The class contains log-normal and Weibull (for ) distributions. Moreover, all distributions with a regularly varying tail (e.g. Pareto and Burr distributions) are subexponential. For subexponential distributions we can formulate the following approximation of the ruin
probability.
If
, then the asymptotic formula for large
is given by
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(15.25) |
The approximation is considered to be inaccurate. The problem is a very slow rate of convergence as
. Even though the
approximation is asymptotically correct in the tail, one may have to go out to values of
which are unrealistically small before the fit is
reasonable. However, we will show in Section 15.4 that it is not always the case.
As the mixture of exponentials does not belong to the subexponential class we do not present a numerical example like in all previously discussed approximations.
One can use the Pollaczek-Khinchin formula (15.8) to derive explicit closed form solutions for claim amount distributions presented in
Section 15.2, see Panjer and Willmot (1992). For other distributions studied here, in order to calculate the ruin probability, the Monte Carlo
method can be applied to (15.1) and (15.7). The main problem is to simulate random variables from the density
. Only four of
the considered distributions lead to a known density: (i) for exponential claims,
is the density of the same
exponential distribution, (ii) for a mixture of exponentials claims,
is the density of the mixture of exponential distribution with the
weights
, (iii) for Pareto claims,
is
the density of the Pareto distribution with the parameters
and
, (iv) for Burr claims,
is the density of the transformed
beta distribution.
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0.769209 | 0.587917 | 0.359705 | 0.194822 | 0.057173 | 0.001445 |
For other distributions studied here we use formula (15.6) and controlled numerical integration to generate random variables (except for
the Weibull distribution,
does not even have a closed form). We note that the methodology based on the Pollaczek-Khinchin formula
works for all considered claim distributions.
The computer approximation via the Pollaczek-Khinchin formula will be called in short the Pollaczek-Khinchin approximation. Burnecki, Mista, and Weron (2004)
showed that the approximation can be chosen as the reference method for calculating the ruin probability in infinite time, see also Table
15.15 where the results of the Pollaczek-Khinchin approximation are presented for mixture of two exponentials claims with ,
,
and the relative safety loading
with respect to the initial capital
. For the Monte Carlo method purposes we
generated 100 blocks of 500000 simulations.
Table 15.16 shows which approximation can be used for a particular choice of a claim size distribution. Moreover, the necessary assumptions on the distribution parameters are presented.
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Exp. | Gamma | Wei- | Mix. | Log- | Pareto | Burr |
Method | bull | Exp. | normal | ||||
Cramér-Lundberg | + | + | - | + | - | - | - |
Exponential | + | + | + | + | + | ![]() |
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Lundberg | + | + | + | + | + | ![]() |
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Beekman- | + | + | + | + | + | ![]() |
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Renyi | + | + | + | + | + | ![]() |
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De Vylder | + | + | + | + | + | ![]() |
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4M Gamma | + | + | + | + | + | ![]() |
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Heavy Traffic | + | + | + | + | + | ![]() |
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Light Traffic | + | + | + | + | + | + | + |
Heavy-Light | + | + | + | + | + | ![]() |
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Subexponential | - | - | 0![]() ![]() ![]() |
- | + | + | + |
Pollaczek- | + | + | + | + | + | + | + |
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