20.5 Ruin Probability Criterion when the Initial Capital is Given
Presuming long-run horizon for premium calculation we turn back to ruin theory.
Our aim is now to obtain such a level of premium for the portfolio yielding
each year the aggregate loss
, which results from a presumed
level of ruin probability
and initial capital
. This is done by inverting various approximate formulae for the probability of ruin. Information requirements of different methods are emphasized. Special attention is paid also to the problem of decomposition of the whole portfolio premium.
20.5.1 Approximation Based on Lundberg Inequality
This is a simplest (and crude) approximation method, simply assuming replacement of the true function
by:
At first we obtain the approximation
of the desired level of the
adjustment coefficient
:
In the next step we make use of the definition of the adjustment coefficient
for the portfolio:
to obtain directly the premium formula:
where
denotes the cumulant generating function. The result is well known as the exponential premium formula. It possesses several
desirable properties - not only that it is derivable from ruin theory.
First of all, by the virtue of properties of the cumulant generating function, it is additive for independent risks. So there is no need to distinguish between marginal and basic premiums for individual risks. By the same reason the formula does not reflect the cross-sectional diversification effect when the portfolio is composed of large number of risks, each of them being small. The formula can be practically applied once we replace the adjustment coefficient
by its approximation
.
Under certain conditions we could rely on truncating higher order terms in the expansion of the cumulant generating function:
 |
(20.9) |
and use for the purpose of individual risk pricing the formula (where higher
order terms are truncated as well):
 |
(20.10) |
Some insight into the nature of the long-run criteria for premium calculation could be gained by re-arrangement of the formula (20.9). At first we could express the initial capital in units
of standard deviation of the aggregate loss:
. Now the adjustment coefficient could be expressed as:
and premium formula (20.9) as:
 |
(20.11) |
where in the brackets appear only unit-less figures, that form together the
pricing formula for the standardized risk
. Let us notice that the contribution of higher order terms in the expansion is neglectible when initial capital is large enough. The above phenomenon could be interpreted as a result of risk diversification in time (as opposed to cross-sectional risk diversification). Provided the initial capital is large, the ruin (if it happens at all) will rather appear as a result of aggregation of poor results over many periods of time. However, given the skewness and kurtosis of one-year increment of the risk process, the sum of increments over
periods has skewness of order
, kurtosis of order
etc. Hence the larger initial capital, the smaller importance of the difference between the distribution of the yearly increment and the normal distribution. In a way this is how the diversification of risk in time works (as opposed to cross-sectional diversification). In the case of a cross-sectional diversification the assumption of mutual independency of risks plays the crucial role. Analogously, diversification of risk in time works effectively when subsequent increments of the risk process are independent.
20.5.2 ``Zero'' Approximation
The ``zero'' approximation is a kind of naive approximation, assuming replacement of the function
by:
where
denotes the relative security loading, which means that
. The ``zero'' approximation is applicable to the case of Poisson
claim arrivals (as opposed to Lundberg inequality, which is applicable under more general assumptions). Relying on ``zero'' approximation leads to the
system of two equations:
The system could be solved by assuming at first:
and next by executing iterations:
that under reasonable circumstances converge quite quickly to the solution
, which
allows applying formula (20.9) for the whole portfolio and formula (20.10) for
individual risks, provided the coefficient
is replaced by its approximation
.
20.5.3 Cramér-Lundberg Approximation
Premium calculation could also be based on the Cramér-Lundberg
approximation. In this case the problem can be reduced also to the
system of equations (three this time):
where
and
denote respectively the
first order derivative of the moment generating function and the expectation of the severity
distribution. Solution of the system in respect of unknowns
,
and
requires now a bit more complex calculations.
Obtained result
could be used then to replace
in formulas (20.9) and
(20.10). The method is applicable to the case of Poisson claim arrivals.
Moreover, severity distribution has to be known in this case. It can be
expected that the method will produce accurate results for large
.
20.5.4 Beekman-Bowers Approximation
This method is often recommended as the one which produces
relatively accurate approximations, especially for moderate amounts of
initial capital. The problem consists in solving the system of three equations:
where
denotes the cdf of the gamma distribution with parameters
, and
denotes the raw moment of order
of the severity distribution. Last two equations arise from equating moments of the gamma distribution to conditional moments of the maximal loss distribution (provided the maximal loss is positive). Solving this system of equation is a bit cumbersome, as it involves multiple numerical evaluations of the cdf of the gamma distribution. The admissible solution exists provided
, that is always satisfied for arbitrary severity distribution with support on the positive part of the axis. Denoting the solution for the unknown
by
, we can write the latter as a function:
and obtain the whole portfolio premium from the equation:
Formally, application of the method requires only moments of first three orders of the severity distribution to be finite. However, the problem arises when we wish to price individual risks. Then we have to know the moment generating function of the severity distribution, and it should obey conditions for adjustment coefficient to exist. If this is a case, we can replace the coefficient
of the equation:
by its approximation
, and thus obtain the approximation
of the adjustment coefficient
. It allows calculating premiums
according to formulas (20.9) and (20.10). It
is easy to verify that there is no danger of contradiction, as both formulas
for the premium
produce the same result
.
20.5.5 Diffusion Approximation
This approximation method requires the scarcest information. It suffices to know the first two moments of the increment of the risk process, to invert the formula:
where:
in order to obtain the premium:
that again is easily decomposable for individual risks. The formula is equivalent to the exponential formula (20.9), where all terms except the first two are omitted.
20.5.6 De Vylder Approximation
The method requires information on moments of the first three orders of the increment of the risk process. According to the method, ruin probability could be expressed as:
where for simplicity the abbreviated notation
is used. Setting
equal to
and rearranging the equation we obtain another form of it:
that can be solved numerically in respect of
, to yield as a result premium formula:
which again is directly decomposable.
When the analytic solution is needed, we can make some further simplifications. Namely, the equation entangling the unknown coefficient
could be transformed to a simplified form on the basis of the following approximation:
Provided the error of omission of higher order terms is small, we obtain the approximation:
The error of the above solution is small, provided the initial capital
is several times greater than the product
. Under this condition we obtain the explicit (approximated) premium formula:
where the star symbolizes the simplification made. Applying now the method of linear approximation of marginal cost
presented in Section 20.3 yields the result:
The reader can verify that the formula
is additive for
independent risks, and so it can serve for marginal as well as for basic
valuation.
20.5.7 Subexponential Approximation
This method applies to the classical model (Poisson claim arrivals) with
thick-tailed severity distribution. More precisely, when the severity cdf
possesses the finite expectation
, then the integrated tail distribution
cdf
(interpreted as the cdf of the variable
, being the ``ladder height'' of the claim surplus process) is defined as follows:
Assuming now that the latter distribution is subexponential (see Chapter 15), we could obtain (applying the Pollaczek-Khinchin formula) the approximation, which should work for large values of initial capital:
The extended study of consequences of thick-tailed severity distributions can
be found in Embrechts, Klüppelberg, and Mikosch (1997).
20.5.8 Panjer Approximation
The Pollaczek-Khinchin formula could be also used in combination with the
Panjer recursion algorithm, to produce quite accurate (at the cost of
time-consuming calculations) answers in the case of the classical model
(Poisson claim arrivals). The method consists of two basic steps. In the first
step the integrated tail distribution
is calculated and discretized.
Once this step is executed, we have a distribution of a variable
(discretized version of the ``ladder height''
):
The second step is based on the fact that the maximal loss
has a compound geometric distribution. Thus the distribution of the discretized version
of the variable
is obtained by making use of the Panjer recursion formula:
and for
:
where
.
Iterations should be stopped when for some
the cumulated probability
exceeds for the first time the predetermined value
. The approximated value of the capital
at which the ruin
probability attains the value
could be set then on the basis of
interpolation, taking into account that the ruin probability function is
approximately exponential:
Calculations should be repeated for different values of
in order to
find such value
, at which the resulting capital
approaches the predetermined value of capital
. Then the
resulting premium is given by the formula:
It should be noted, that it is only the second step of calculations which has
to be repeated many times under the search procedure, as the distribution of
the variable
remains the same for various values of
being
tested. The advantage of the method is that the range of the approximation
error is under control, as it is a simple consequence of the width of the
discretization interval
and the discretization method used. The disadvantage
already mentioned is a time-consuming algorithm. Moreover, the method produces
only numerical results, and therefore, provides no rule for decomposition of
the whole portfolio premium for individual risk premiums. Nevertheless, the
method could be used to obtain quite accurate approximations, and
thus, a reference point to estimate approximation errors produced by
simpler methods.
All approximation methods presented in this section are more
or less standard, and more detailed information on them can be found in any
actuarial textbook, as for example in ``Actuarial Mathematics'' by
Bowers et al. (1986, 1997). More advanced analysis can be found in a book ``Ruin
probabilities'' by Asmussen (2000) and numerical comparison of this and other approximations are given in Chapter 15.