This section is devoted to extending the decision problem considered in previous sections by allowing for reinsurance. Then the pricing obey the form:
No matter which particular objective function is chosen, the space of
possible subdivisions of the variable has to be reduced somehow. One of the
most important cases is when the variable
has a compound Poisson
distribution, and the excess of loss reinsurance is chosen. Denoting by
the
number of claims, we could define for each claim amount
,
its subdivision into the truncated loss
and the excess of loss
and then define variables representing subdivision of the whole portfolio:
Assuming that capital of the insurer is not flexible, and that the current
amount of capital is smaller than the amount
necessary to accept solely the whole portfolio, we could simply find such
value of
, for which
. In the case when the
current amount of capital is in excess, it is still relevant to assess such
portion of the capital, which should serve as a protection for insurance
operations. The excess of capital over this amount can be treated
separately, as being free of prudence requirements when investment decisions
are undertaken.
It is more interesting to assume that the amount of capital is flexible, and to choose
the retention limit to minimize the total premium
given parameters
,
, and
. The objective
function reflects the aim of maximizing competitiveness of the company. If
the resulting premium (after being charged by respective cost loadings) is
lower than that acceptable by the market, we can revise assumptions. Revised
problem could consist in maximizing expected rate of return given the
premium level and parameters
and
. This would mean
getting higher risk premium than that offered by the capital market.
Reasonable solutions could be expected in the case when reinsurance premium
formula
contains loadings proportional
primarily to the expected value, and its sensitivity to the variance (more so as to skewness and kurtosis) is small. This could be expected as a
result of transaction costs on the one hand, and larger capital assets of
reinsurers on the other. Also the possibility to diversify risk on the
world-wide scale work in the same direction, increasing transaction costs and
at the same time reducing the reinsurer's exposure to risk.
Example 2
Aggregate loss has a compound Poisson distribution with truncated-Pareto severity distribution, with cdf given for
by the formula:
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For expository purposes we take the following values of parameters:
Problem consists in choosing the retention limit
that minimize the total premium
.
Solution.
First step is to express moments of first four orders of variables
and
as functions of parameters
and the real variable
. Expected value of the
truncated-Pareto variable with parameters
equals by definition:
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Similar calculations made for moments of higher order yield the recursive
equation:
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|||
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The above formulas could serve to calculate raw moments as well of the variable
as the variable
, provided we replace
by
. Having calculated moments for both variables
and
already, we make use of the
relation:
The second step is to express cumulants of both variables
and
as a product of the parameter
and respective raw moments of variables
and
. Finally, both components
and
of the total premium are expressed as a function of parameters
and
the decision parameter
. Now the search of
such a value of
that minimizes the total premium
is
a quite feasible numerical task. Optimal retention level and related minimal
premium entail optimal amount of capital
.
The problem described in Example 2 has been solved in several different variants of assumptions on parameters.
Variants 1-5 consist in minimization of the total premium, in variant 1 the parameters are
. In variants 2, 3, 4 and 5 value of one of parameters
is modified and in variant 6 there is no reinsurance,
are as in variant 1. Variant 7
consists in maximization of
where
are as in variant 1, and premium loading equals
. Results are presented in Table 20.1.
Optimization variants |
Quantile approx. method for
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Retention Limit ![]() |
RBC |
Loading
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|
V.1:
(basic) |
FC20.6 | 114.5 | 386.6 | 4.11% | |
FC20.5 | 106.5 | 385.2 | 4.13% | ||
V.2:
|
FC20.6 | 114.5 | 483.3 | 4.11% | |
FC20.5 | 106.5 | 481.5 | 4.13% | ||
V.3:
|
FC20.6 | 129.7 | 382.3 | 4.03% | |
FC20.5 | 134.7 | 382.1 | 4.01% | ||
V.4:
|
FC20.6 | 79.8 | 373.3 | 4.03% | |
FC20.5 | 76.3 | 372.1 | 4.03% | ||
V.5:
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FC20.6 | 95.5 | 380.0 | 4.05% | |
FC20.5 | 90.9 | 379.0 | 4.06% | ||
V.6:
(no reinsurance) |
FC20.6 | 500.0 | 446.6 | 4.47% | |
FC20.5 | 500.0 | 475.1 | 4.75% | ||
V.7: |
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FC20.6 | 106.0 | 372.3 | 4.47% |
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FC20.5 | 99.6 | 371.5 | 4.47% |
Reinsurance reduces the required level of RBC, which coincides either with
premium reduction (compare variant 1 and 6) or with increase of the expected
rate of return (compare variant 7 and 6). Reinsurance also reduces
difference between results obtained on the basis of two different
approximation methods (FC20.6 and FC20.5). In variant 6 (no reinsurance) the
difference is quite large, which is caused by the fairly long right tail of
the distribution of the variable .
Comparison of variants 2 and 1 confirms that the choice of a smaller expected rate of return (given substitution rate) automatically raises the need for capital, leaving the premium level unchanged (and therefore also the optimal retention level).
Comparison of variants 3 and 1 shows that admission of greater loss
probability
causes reduction of premium, which coincides with
substantial reduction of the need for reinsurance cover, and slight
reduction in the need for capital. It is worthwhile to notice that
replacement of
by
entails reversing the relation of results obtained by two approximation methods.
Formula FC20.5 leads to smaller retention limits when safety standard is high (small
, and to larger retention limits when safety
standard is relaxed (large
).
Comparison of variants 4 and 5 with variant 1 illustrates the obvious rule that it does pay off to reduce retention limits when reinsurance is cheap, and to increase them when reinsurance is expensive.
It could happen in practice that pricing rules applied by reinsurers differ
by lines of business. When the portfolio
consists of
business lines, for which the market offers reinsurance cover priced on the
basis of different formulas
, ...,
, the natural generalization of the
problem lies in minimization of the premium (or maximization of the rate
made by choosing
retention limits
, ...,
, for
each of business lines separately. Separation of business lines makes it
feasible to assume different severity distributions, too.