Let denote a non-negative random variable describing the size
of claim (risk, loss) with the distribution function
. Moreover, we assume that the
expected value
, the variance
and the moment generating function
exist.
The simplest premium (calculation principle) is called pure risk
premium and it is equal to the expectation of claim size variable:
![]() |
(18.1) |
This premium is often applied in life and some mass lines of business in non-life insurance. As it is known from the ruin theory, the pure risk premium without any kind of loading is insufficient since, in the long run, the ruin is inevitable even in the case of substantial (though finite) initial reserves. Nevertheless, the pure risk premium can be - and still is - of practical use because, for one thing, in practice the planning horizon is always limited, and for another, there are indirect ways of loading a premium, e.g. by neglecting interest earnings (Straub; 1988).
The future claims cost may be different from its expected value
and the estimator
drawn from past
may be different from the true
. To reflect this fact, the insurer can impose the risk loading on the pure risk premium.
The pure risk premium with safety (security) loading given by
![]() |
(18.2) |
The pure risk premium and the premium with safety loading are
sometimes criticised because they do not depend on the degree of
fluctuation of . Thus, two other rules have been proposed. The
first one, denoted here by
and given by
The other one, denoted here by and given by
The rules described so far are sometimes called ``empirical'' or ``pragmatic''. Another approach employs the notion of utility (Straub; 1988). The so-called zero utility principle states that the premium for a risk
should be calculated such that the expected utility is (at least) equal to the zero utility. This principle yields a technical minimum premium in the sense that the risk
should not be accepted at a premium below
. In the trivial case zero utility premium equals the pure risk premium. A more interesting case is the exponential utility which leads to a premium, denoted here by
and called the exponential premium, given by
![]() |
(18.5) |
Another interesting approach to the problem of premium calculations is the quantile premium, denoted here by
, is given by
![]() |
(18.6) |