As it has been pointed out in the introduction, some premium calculation formulas are additive for independent risks, and then the decomposition of the whole-portfolio premium into individual risks premiums is straightforward. However, sometimes a non-additive formula for pricing the whole portfolio is well justified, and then the decomposition is no more trivial. This is exactly the case of the RBC formula (and also other quantile-based formulas) derived in the previous section. This section is devoted to showing the range, interpretation and applications of some solutions to this problem.
In the case of the RBC formula decomposition means answering the question what is the share of a particular risk in the demand for capital backing the portfolio risk that in turn entails the premium. In order to solve the problem one can make use of approximations of the quantile by the normal power expansions. The most general version used in practice of the normal power formula for the quantile
of order
of the variable
reads:
Premium for the individual risk could be set on the basis of marginal cost.
This means that we look for such a price at which the insurer is indifferent whether to accept the risk or not. Calculation of the marginal cost can be based on standards of differential calculus. In order to do that, we should first write the formula explicitly in terms of a function of cumulants of first four orders:
where
,
,
,
denote partial derivatives of the function
calculated at the point
. By virtue of
additivity of cumulants for independent random variables we replace
increments
by cumulants of the additional risk
. As a result the
following formula is obtained:
Respective calculations lead to the marginal premium formula:
First two components coincide with the result obtained when the whole
premium is based on the normal approximation. Setting additionally
we obtain the premium for the case when skewness of the portfolio in
non-neglectible (making use of FC20.6 approximation), including last two
components means we regard also portfolio kurtosis (approximation based on
formula FC20.5).
For each component the problem of balancing the premium on the whole
portfolio level arises. Should all risks composing the portfolio
be charged their marginal premiums, the portfolio premium
amounts to:
Balancing correction made so as to preserve sensitivity of premium on cumulants of
order 1, 3, and 4 leads to the formula for the basic premium:
Obviously, several alternative correction rules exist. For example, in the case of the kurtosis component any expression of the form:
It seems that only in the case of the variance component
some more or less heuristic argument for the correction can be
found. The essence of the basic premium for individual risks is that it is a
basis of an open market offer. Once the cover is offered to the public,
clients decide whether to buy the cover or not. Thus the price should not
depend on how many risks out of the portfolio
have been insured before,
and how many after the risk in question. Let us imagine a particular
ordering of the basic set of
risks amended by the additional risk
in a
form of a sequence
.
Given this ordering, the respective component of the marginal cost of risk
takes the form:
We can now consider the expected value of this component, provided that each
of orderings is equally probable (as it was proposed Shapley (1953)).
However, calculations are much simpler if we assume that the share
of the aggregated variance of all risks preceeding the risk
in the total aggregate variance
is a random variable
uniformly distributed over the interval
. The error
of the simplification is neglectible as the share of each individual risk in
the total variance is small. The result:
Results presented in this section have three possible fields of application.
The first is just passive premium calculation for the whole portfolio. In
this respect several more accurate formulas exist, especially when our
information on the distribution of the variable extends its first four
cumulants.
The second application concerns pricing individual risks. In this respect it
is hard to find a better approach (apart from that based on long-run
solvability criteria, which is a matter of consideration in next sections),
which consistently links the risk relevant to the company (on the whole
portfolio level) with risk borne by an individual policy. Of course open
market offer should be based on basic valuation
, whereas the marginal cost valuation
could serve as a lower bound for contracts negotiated individually.
The third field of applications opens when a portfolio, characterized by
substantial skewness and kurtosis, is inspected in order to localize these
risks (or groups of risks), that distort the distribution of the whole
portfolio. Too high (noncompetitive) general premium level could be caused
though by few influential risks. Such localization could help in decisions
concerning underwriting limits and reinsurance program. Applying these
measures could help ``normalize'' the distribution of the variable .
Thus in the preliminary stage, when the basis for underwriting policy and reinsurance is considered, extended pricing formulas (involving higher order cumulants) should be used. Paradoxically, once the prudent underwriting and ceding policy has been elaborated, simple normal approximation suffices to price as well the portfolio as individual risks.
Clearly, such prices concern only retained portions of risk, and should be
complemented by reinsurance costs.