19.3 Premiums Under Deductibles for Given Loss Distributions
In the preceding section we showed a relation between the pure risk premium under several deductibles and a limited expected value function. Now, we
use the relation to present formulae for premiums in the case of deductibles for a number of loss distributions often used in non-life actuarial
practice, see Burnecki, Nowicka-Zagrajek, and Weron (2004). To this end we apply the formulae for levf for different distributions given in Chapter 13.
The log-normal, Pareto, Burr, Weibull, gamma, and mixture of two exponential distributions are typical candidates when looking for a suitable analytic
distribution, which fits the observed data well, see Aebi, Embrechts, and Mikosch (1992), Burnecki, Kukla, and Weron (2000), Embrechts, Klüppelberg, and Mikosch (1997), Mikosch (1997), Panjer and Willmot (1992),
and Chapter 13. In the log-normal and Burr case the premium formulae will be illustrated on a real-life example, namely on the fire loss
data, already analysed in Chapter 13. For illustrative purposes, we assume that the total amount of risk
simply follows one of the
fitted distributions, whereas in practice, in the individual and collective risk model framework (see Chapter 18), in order to obtain an
annual premium under a per occurrence deductible we would have to multiply the premium by a number of policies and mean number of losses per year,
respectively, since in the individual risk model
provided that the claim amount variables are identically distributed,
and in the collective risk model
19.3.1 Log-normal Loss Distribution
Consider a random variable
which has the normal distribution. Let
. The distribution of
is called the log-normal distribution
and its distribution function is given by
where
,
and
is the standard normal distribution function, see Chapter 13. For
the log-normal distribution the following formulae hold:
(a) franchise deductible premium
(b) fixed amount deductible premium
(c) proportional deductible premium
(d) limited proportional deductible premium
(e) disappearing deductible premium
We now illustrate the above formulae using the Danish fire loss data. We study the log-normal loss distribution with parameters
and
, which best fitted the data. Figure 19.6 depicts the premium under franchise and fixed amount deductibles in the
log-normal case.
Figure 19.6:
The premium under the franchise deductible (thick blue line) and fixed amount deductible (thin red line). The
log-normal case.
|
Figure 19.7 shows the effect of parameters
,
, and
of the limited proportional deductible. Clearly,
is a decreasing function of these parameters.
Figure 19.7:
The premium under the limited proportional deductible with respect to the parameter
. The thick blue solid line
represents the premium for
and
DKK, the thin blue solid line for
and
DKK, the dashed red line
for
and
million DKK, and the dotted red line for
and
million DKK. The log-normal case.
|
Finally, Figure 19.8 depicts the influence of parameters
and
of the disappearing deductible. Markedly,
is a decreasing function of the parameters and we can observe that the effect of increasing
is rather minor.
Figure 19.8:
The premium under the disappearing deductible with respect to the parameter
. The thick blue line represents the
premium for
DKK and the thin red line the premium for
DKK. The log-normal case.
|
19.3.2 Pareto Loss Distribution
The Pareto distribution function is defined by
where
, see Chapter 13. The expectation of the Pareto distribution exists only for
.
For the Pareto distribution with
the following formulae hold:
(a) franchise deductible premium
(b) fixed amount deductible premium
(c) proportional deductible premium
(d) limited proportional deductible premium
(e) disappearing deductible premium
19.3.3 Burr Loss Distribution
Experience has shown that the Pareto formula is often an appropriate model for the claim size distribution, particularly where exceptionally large
claims may occur. However, there is sometimes a need to find heavy tailed distributions which offer greater flexibility than the Pareto law. Such
flexibility is provided by the Burr distribution which distribution function is given by
where
,
,
,
, see Chapter 13. Its mean
exists only for
. For the Burr distribution with
the following formulae hold:
(a) franchise deductible premium
(b) fixed amount deductible premium
(c) proportional deductible premium
(d) limited proportional deductible premium
(e) disappearing deductible premium
where the functions
and
are defined as:
and
In order to illustrate the preceding formulae we consider the fire loss data. analysed in Chapter 13. The analysis showed that the losses
can be well modelled by the Burr distribution with parameters
,
and
. Figure 19.9
depicts the premium under franchise and fixed amount deductibles for the Burr loss distribution.
Figure 19.9:
The premium under the franchise deductible (thick blue line) and fixed amount deductible (thin red line). The Burr
case.
|
In Figure 19.10 the influence of the parameters
,
, and
of the limited proportional deductible is illustrated. Figure
19.11 shows the effect of the parameters
and
of the disappearing deductible.
Figure 19.10:
The premium under the limited proportional deductible with respect to the parameter
. The thick solid blue line
represents the premium for
and
DKK, the thin solid blue line for
and
DKK, the dashed red line
for
and
million DKK, and the dotted red line for
and
million DKK. The Burr case.
|
Figure 19.11:
The premium under the disappearing deductible with respect to the parameter
. The thick blue line represents the
premium for
DKK and the thin red line the premium for
DKK. The Burr case.
|
19.3.4 Weibull Loss Distribution
Another frequently used analytic claim size distribution is the Weibull distribution which is defined by
where
, see Chapter 13.
For the Weibull distribution the following formulae hold:
(a) franchise deductible premium
(b) fixed amount deductible premium
(c) proportional deductible premium
(d) limited proportional deductible premium
(e) disappearing deductible premium
where the incomplete gamma function
is defined as
19.3.5 Gamma Loss Distribution
All four presented above distributions suffer from some mathematical drawbacks such as lack of a closed form representation for the Laplace transform
and nonexistence of the moment generating function. The gamma distribution given by
for
does not have these drawbacks, see Chapter 13. For the gamma distribution the following
formulae hold:
(a) franchise deductible premium
(b) fixed amount deductible premium
(c) proportional deductible premium
(d) limited proportional deductible premium
(e) disappearing deductible premium
19.3.6 Mixture of Two Exponentials Loss Distribution
The mixture of two exponentials distribution function is defined by
where
and
, see Chapter 13.
For the mixture of exponentials distribution the following formulae hold:
(a) franchise deductible premium
(b) fixed amount deductible premium
(c) proportional deductible premium
(d) limited proportional deductible premium
(e) disappearing deductible premium