In this section premium calculation is considered under predetermined ruin probability and predetermined rate of dividend, with reinsurance included. At first the example involving fixed dividend is presented.
Example 5
We assume (as in Example 2), that the aggregate loss has a compound Poisson distribution with expected number of claims
, and with severity distribution being truncated-Pareto distribution
with parameters
. We assume also that the excess of each loss over
the limit
is ceded to the reinsurer using the same pricing formula:
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Solution.
Risk process can be written now as:
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and in the version based on the Beekman-Bowers approximation method take a form:
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Moments of the first three orders of the variable
as well
as cumulants of variables
and
are
calculated the same way as in Example 2. All these characteristics are
functions of parameters
and the
decision variable
.
Variants of minimization problems | Method of approx. of the ruin probability |
Retention limit ![]() |
Initial capital ![]() |
Loading
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V.1:
(basic) |
BB | 184.2 | 416.6 | 4.17% |
dV | 185.2 | 416.3 | 4.16% | |
V.2:
|
BB | 179.5 | 408.2 | 4.25% |
dV | 180.5 | 407.9 | 4.25% | |
V.3:
|
BB | 150.1 | 463.3 | 4.65% |
dV | 156.3 | 461.7 | 4.63% | |
V.4:
|
BB | 126.1 | 406.2 | 4.13% |
dV | 127.1 | 406.0 | 4.13% | |
V.5:
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BB | 139.7 | 409.0 | 4.13% |
dV | 140.5 | 408.8 | 4.13% | |
V.6:
(no reinsurance) |
BB | 500.0 | 442.9 | 4.25% |
dV | 500.0 | 442.7 | 4.25% |
Results of numerical optimization are reported in Table 20.2.
In the basic variant of the problem, parameters has been set on the level
. In
variant 6 the value
is assumed, so as this variant represents the
lack of reinsurance. Variants 2, 3, 4 and 5 differ from the basic wariant by
the value of one of parameters
. In
variant 2 the dividend rate
has been increased so as to obtain the same
level of premium, than it is obtained in variant 6. Results could be
summarized as follows:
In the next example assumptions are almost the same as in Example 5, except that the fixed dividend is replaced by the dividend dependent on financial result by the same manner, as in Example 4.
Example 6
Assumptions on the aggregate loss are the same as in Example 5: compound Poisson truncated-Pareto distribution with parameters
. Assumptions concerning available
reinsurance (excess of loss over
, pricing formulas characterized by parameters
and
) are also the same. Dividend is defined as in
Example 4, with a suitable correction due to reinsurance allowed:
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Now the problem lies in choosing capital , risk-sharing parameter
and retention limit
so as to minimize premium
under the restriction
, and predetermined values of parameters characterizing the distribution
, parameters
characterizing reinsurance costs
and
parameters characterizing profitability and safety
.
Solution.
Under the predetermined values of decision variables
and remaining parameters the risk process has a form:
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The problem differs from that presented in Example 4 by two factors:
variable
is not gamma distributed, and the premium
is now replaced by the constant
.
However, variable
could be approximated by the shifted
gamma distribution with parameters
chosen so as to match moments of order 1, 2, and 3 of the original
variable
. Suitable calculations lead to the definition of
the variable
, that approximates the original variable
:
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where the variable has a gamma
distribution, and the constant
equals
. Thus we could express moments of the variable
as functions of parameters
exactly this way, as it is done with respect to variable
and parameters
in Example 4. It
suffices in turn to approximate ruin probability with the De Vylder method:
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where
and
,
and where the expected value of dividend
satisfies the
restriction:
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Hence it is clear that the problem of minimization of premium under
restrictions
,
,
,
,
and predetermined values of parameters
is in essence analogous to the problem
presented in Example 4, and differs only in details. The set of decision
variables
in Example 4 is now extended by
additional variable
, and the variable
is only
approximately gamma distributed.
Results are presented in Table 20.3. In all variants predetermined values of
parameters
are the same. In variant
1 (basic) the ruin probability
is assumed, and reinsurance is
allowed. Variant 2 differs from the basic one by higher safety standard
(
, whereas variant 3 differs by lack of reinsurance. In each
variant three slightly different versions of the problem have been solved.
Version A is a simplified one, assuming fixed dividend rate
, so that
. Consequently the minimization of the premium is conducted with respect to
only. In fact, the results from Table 20.2 are quoted for this version. Versions B and C assume minimization with respect to
.
Version B plays a role of a basic version, where premium
is minimized under the expected rate of dividend
. In version C such a rate of dividend
has been chosen, that leads (through minimization) to the same premium level, as obtained
previously in version A. So two alternative effects of the consent of
shareholders to participate in risk could be observed. Effect in terms of
reduction of premium (expected rate of dividend remaining unchanged) is
observed when we compare version B and A. Effect in terms of increase of
the expected rate of dividend (premium being fixed) is observed when
versions C and A are compared. Results could be summarized as follows.
In each of three variants, the consent of shareholders for risk participation allows for substantial reduction of premium (loading reduced by about 20%).
It is interesting that shareholder's consent to participate in risk allows for much more radical reduction of premium than reinsurance. It results from the fact that reinsurance costs have been explicitly involved in optimization, whereas the ``costs of the shareholder's consent to participate in risk'' have not been accounted for.
Variant of assumptions |
Version of
problem |
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M |
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V.1:
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A | 5% | 185.2 | 416.3 | 4.16% | - | 0 |
B | 5% | 189.3 | 406.0 | 3.35% | 41.7% | 5.02% | |
C | 8.54% | 143.6 | 305.2 | 4.16% | 48.6% | 8.14% | |
V.2:
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A | 5% | 156.3 | 461.7 | 4.63% | - | 0 |
B | 5% | 157.0 | 447.5 | 3.67% | 44.4% | 4.94% | |
C | 8.96% | 122.9 | 329.7 | 4.63% | 52.3% | 8.29% | |
V.3:
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A | 5% | 500.0 | 442.7 | 4.25% | - | 0 |
B | 5% | 500.0 | 429.8 | 3.45% | 42.0% | 4.70% | |
C | 8.09% | 500.0 | 340.0 | 4.25% | 48.2% | 7.15% |
Comparison of versions C with versions A in each variant of the problem
allows us to see the outcome (increase of expected rate of dividend) of the
shareholder's consent to share risk. In the last column of the table the
(relative) standard deviation
of dividends is
reported; it could serve as a measure of ``cost'' at which the outcome, in
terms of the increment of the expected dividend rate, is obtained.
Comparing versions B and C in variants 1 and 2 we could observe effects of
the increment in the expected rate of dividend. Apart from the obvious effect on
premium increase, also the reduction of capital could be observed (cost of
capital is higher), and at the same time retention limits are reduced. Also
the sharing parameter increases, as well as the (relative)
standard deviation of dividends
.
Comparing variants 1 and 2 (in all versions A, B, and C) we notice the
substantial increase of the premium as an effect of higher safety standard
(smaller . Also the amount of capital needed increases and the retention
limit is reduced. At the same time a slight increase of sharing parameter
is observed (versions B and C).