**Insurance Abstract**
A premium for stop loss insurance for a fleet of vehicles is calculated
as a stop loss premium for an assumed loss distribution having only
losses with a value of one of zero and maximum individual loss.
The stop loss premium is calculated based on a loss frequency, a
maximum individual loss, and a deductible. The loss frequency is
calculated by dividing an expected total loss by the maximum individual
loss. Subsets of the fleet of vehicles are associated with different
treaty durations. For each treaty duration a stop loss premium is
calculated for the fleet of vehicles. Subsequently, for each treaty
duration a premium is calculated for the subset of the fleet of
vehicles associated with the treaty duration by weighting the stop
loss premium, calculated for the fleet of vehicles, with the number
of vehicles in the subset. Without having to store and process complex
distributions of individual losses of the fleet of vehicles, a worst-case
premium for stop loss insurance for the fleet of vehicles can be
calculated. Repetitive steps used in the prior art for discretizing
and processing distributions of individual losses can be eliminated,
and thus, processing time and processing power can be reduced.
**Insurance Claims**
What is claimed is:
1. A computer-implemented method for calculating a premium for
stop loss insurance for a fleet of vehicles, the method including:
determining an expected total loss for the fleet of vehicles; storing
in a computer a maximum individual loss equivalent to a cost of
a most expensive vehicle of the fleet; calculating by the computer
a loss frequency by dividing the expected total loss by the maximum
individual loss; storing in the computer a deductible payable by
an insurance holder; and calculating by the computer the premium
based on the loss frequency, the maximum individual loss, and the
deductible as a stop loss premium for an assumed loss distribution
having only losses with a value of one of zero and maximum individual
loss.
2. The method according to claim 1, wherein the method further
includes associating in the computer subsets of the fleet of vehicles
with different treaty durations; wherein for each treaty duration
a separate premium is calculated by the computer for the subset
of the fleet of vehicles associated with the treaty duration; and
wherein the premium for the fleet of vehicles is calculated by the
computer by aggregating the separate premiums.
3. The method according to claim 1, wherein the method further
includes associating in the computer subsets of the fleet of vehicles
with different treaty durations; wherein for each treaty duration
a stop loss premium is calculated by the computer for the fleet
of vehicles; and wherein for each treaty duration a premium is calculated
by the computer for the subset of the fleet of vehicles associated
with the treaty duration by weighting the stop loss premium, calculated
for the fleet of vehicles, with the number of vehicles in the subset.
4. The method according to claim 1, wherein the method further
includes associating in the computer subsets of the fleet of vehicles
with different treaty durations; wherein for each treaty duration
a duration-dependent loss frequency is calculated by the computer
by dividing an expected total loss for the treaty duration by the
maximum individual loss; wherein a stop loss premium is calculated
by the computer for the fleet of vehicles for each treaty duration
based on the duration-dependent loss frequency, the maximum individual
loss, and a deductible assigned to the treaty duration, the stop
loss premium being calculated for an assumed loss distribution having
only losses with a value of one of zero and maximum individual loss;
and wherein for each treaty duration a premium is calculated by
the computer for the subset of the fleet of vehicles associated
with the treaty duration by dividing the stop loss premium for the
treaty duration by the total number of vehicles in the fleet and
by the treaty duration, and by multiplying the stop loss premium
for the treaty duration with the number of vehicles in the subset.
5. The method according to claim 1, wherein the method further
includes calculating by the computer stop loss premiums for the
fleet of vehicles for different treaty durations; wherein for each
treaty duration a stop loss premium per vehicle is calculated by
the computer by dividing the stop loss premium, calculated for the
treaty duration and for the fleet of vehicles, with the number of
vehicles in the fleet; wherein the method further includes associating
in the computer subsets of the fleet of vehicles with the different
treaty durations; and wherein for each treaty duration a premium
is calculated by the computer for the subset of the fleet of vehicles
associated with the treaty duration by multiplying the stop loss
premium per vehicle, calculated for the respective treaty duration,
with the number of vehicles in the respective subset.
6. The method according to claim 1, wherein the method further
includes associating in the computer subsets of the fleet of vehicles
with different treaty durations; wherein for each treaty duration
a duration-dependent loss frequency is calculated by the computer
by dividing an expected total loss for the treaty duration by the
maximum individual loss, the expected total loss for a multi-year
treaty duration being calculated by adding an expected total loss
for each year included in the multi-year treaty.
7. The method according to claim 6, wherein an expected total loss
for a first year of a multi-year treaty is calculated by the computer
by multiplying an expected number of incidents, expected in the
first year, with an average individual loss amount for an incident
involving one of the vehicles; wherein an expected total loss for
one of the years after the first year of the multi-year treaty is
calculated by the computer by multiplying an expected total loss
of a preceding year with an index; and wherein an expected total
loss for the multi-year treaty is calculated by the computer by
aggregating expected total losses for years included in the multi-year
treaty.
8. The method according to claim 1, wherein the method further
includes storing in the computer a maximum total insurance coverage,
and calculating by the computer a premium excess based on the loss
frequency, the maximum individual loss, and the maximum total insurance
coverage as a stop loss premium for an assumed loss distribution
having only losses with a value of one of zero and maximum individual
loss; and wherein calculating the premium includes subtracting at
least a defined part of the premium excess from the premium.
9. The method according to claim 1, wherein the method further
includes calculating by the computer the premium for defined values
of the deductible and producing by the computer a graphical representation
showing the premium as a function of the defined values of the deductible;
and wherein the deductible payable by the insurance holder is selected
by an insurance holder based on the graphical representation.
10. The method according to claim 1, wherein determining the expected
total loss includes entering and storing risk factors in the computer
and calculating by the computer the expected total loss based on
the risk factors; and wherein the method further includes producing
by the computer a graphical representation showing the premium as
a function of the risk factors.
11. The method according to claim 1, wherein the method further
includes calculating by the computer the premium for defined values
of the expected number of incidents, and producing by the computer
a graphical representation showing the premium as a function of
the defined values of the expected number of incidents.
12. The method according to claim 1, wherein determining the expected
total loss includes storing in the computer an expected number of
incidents involving one of the vehicles, storing in the computer
an expected average individual loss amount for an incident involving
one of the vehicles, and calculating by the computer the expected
total loss by multiplying the expected number of incidents with
the expected average individual loss amount.
13. Computer program product comprising computer program code means
for controlling one or more processors of a computer, such that
the computer determines an expected total loss for a fleet of vehicles
to be insured by stop loss insurance; that the computer stores a
maximum individual loss equivalent to a cost of a most expensive
vehicle of the fleet; that the computer calculates a loss frequency
by dividing the expected total loss by the maximum individual loss;
that the computer stores a deductible payable by an insurance holder;
and that the computer calculates a premium for the insurance based
on the loss frequency, the maximum individual loss, and the deductible
as a stop loss premium for an assumed loss distribution having only
losses with a value of one of zero and maximum individual loss.
14. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer associates subsets
of the fleet of vehicles with different treaty durations; that the
computer calculates for each treaty duration a separate premium
for the subset of the fleet of vehicles associated with the treaty
duration; and that the computer calculates the premium for the fleet
of vehicles by aggregating the separate premiums.
15. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer associates subsets
of the fleet of vehicles with different treaty durations; that the
computer calculates for each treaty duration a stop loss premium
for the fleet of vehicles; and that the computer calculates for
each treaty duration a premium for the subset of the fleet of vehicles
associated with the treaty duration by weighting the stop loss premium,
calculated for the fleet of vehicles, with the number of vehicles
in the subset.
16. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer associates subsets
of the fleet of vehicles with different treaty durations; that the
computer calculates for each treaty duration a duration-dependent
loss frequency by dividing an expected total loss for the treaty
duration by the maximum individual loss; that the computer calculates
a stop loss premium for the fleet of vehicles for each treaty duration
based on the duration-dependent loss frequency, the maximum individual
loss, and a deductible assigned to the treaty duration, the stop
loss premium being calculated for an assumed loss distribution having
only losses with a value of one of zero and maximum individual loss;
and that the computer calculates for each treaty duration a premium
for the subset of the fleet of vehicles associated with the treaty
duration by dividing the stop loss premium for the treaty duration
by the total number of vehicles in the fleet and by the treaty duration,
and by multiplying the stop loss premium for the treaty duration
with the number of vehicles in the subset.
17. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer calculates for
different treaty durations a stop loss premium for the fleet of
vehicles; that the computer calculates for each treaty duration
a stop loss premium per vehicle by dividing the stop loss premium,
calculated for the treaty duration and for the fleet of vehicles,
with the number of vehicles in the fleet; that the computer associates
subsets of the fleet of vehicles with the different treaty durations;
and that the computer calculates for each treaty duration a stop
loss premium for the subset of the fleet of vehicles, associated
with the treaty duration, by multiplying the stop loss premium per
vehicle, calculated for the respective treaty duration, with the
number of vehicles in the respective subset.
18. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer associates subsets
of the fleet of vehicles with different treaty durations; that the
computer calculates for each treaty duration a duration-dependent
loss frequency by dividing an expected total loss for the treaty
duration by the maximum individual loss, the expected total loss
for a multi-year treaty duration being calculated by adding an expected
total loss for each year included in the multi-year treaty.
19. The Computer program product according to claim 18, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer calculates an
expected total loss for a first year of a multi-year treaty by multiplying
an expected number of incidents, expected in the first year, with
an average individual loss amount for an incident involving one
of the vehicles; that the computer calculates an expected total
loss for one of the years after the first year of the multi-year
treaty by multiplying an expected total loss of a preceding year
with an index; and that the computer calculates an expected total
loss for the multi-year treaty by aggregating expected total losses
for years included in the multi-year treaty.
20. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer stores a maximum
total insurance coverage; that the computer calculates a premium
excess based on the loss frequency, the maximum individual loss,
and the maximum total insurance coverage as a stop loss premium
for an assumed loss distribution having only losses with a value
of one of zero and maximum individual loss; and that the computer
calculates the premium by subtracting at least a defined part of
the premium excess from the premium.
21. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer calculates the
premium for defined values of the deductible and produces a graphical
representation showing the premium as a function of the defined
values of the deductible.
22. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer receives and stores
risk factors; that the computer calculates the expected total loss
based on the risk factors; and that the computer produces a graphical
representation showing the premium as a function of the risk factors.
23. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer calculates the
premium for defined values of the expected number of incidents and
produces a graphical representation showing the premium as a function
of the defined values of the expected number of incidents.
24. The Computer program product according to claim 13, comprising
further computer program code means for controlling the one or more
processors of the computer such that the computer stores an expected
number of incidents involving one of the vehicles; that the computer
stores an expected average individual loss amount for an incident
involving one of the vehicles; and that the computer calculates
the expected total loss by multiplying the expected number of incidents
with the expected average individual loss amount.
25. A computer-based data processing system for calculating a premium
for stop loss insurance for a fleet of vehicles, the system including:
means for determining an expected total loss for the fleet of vehicles;
means for storing a maximum individual loss equivalent to a cost
of a most expensive vehicle of the fleet; means for calculating
a loss frequency by dividing the expected total loss by the maximum
individual loss; means for storing a deductible payable by an insurance
holder; and means for calculating the premium based on the loss
frequency, the maximum individual loss, and the deductible as a
stop loss premium for an assumed loss distribution having only losses
with a value of one of zero and maximum individual loss.
26. The system according to claim 25, further including means for
associating subsets of the fleet of vehicles with different treaty
durations; means for calculating for each treaty duration a separate
premium for the subset of the fleet of vehicles associated with
the treaty duration; and means for calculating the premium for the
fleet of vehicles by aggregating the separate premiums.
27. The system according to claim 25, further including means for
associating subsets of the fleet of vehicles with different treaty
durations; means for calculating for each treaty duration a stop
loss premium for the fleet of vehicles; and means for calculating
for each treaty duration a premium for the subset of the fleet of
vehicles associated with the treaty duration by weighting the stop
loss premium, calculated for the fleet of vehicles, with the number
of vehicles in the subset.
28. The system according to claim 25, further including means for
associating subsets of the fleet of vehicles with different treaty
durations; means for calculating for each treaty duration a duration-dependent
loss frequency by dividing an expected total loss for the treaty
duration by the maximum individual loss; means for calculating a
stop loss premium for the fleet of vehicles for each treaty duration
based on the duration-dependent loss frequency, the maximum individual
loss, and a deductible assigned to the treaty duration, the stop
loss premium being calculated for an assumed loss distribution having
only losses with a value of one of zero and maximum individual loss;
and means for calculating for each treaty duration a premium for
the subset of the fleet of vehicles associated with the treaty duration
by dividing the stop loss premium for the treaty duration by the
total number of vehicles in the fleet and by the treaty duration,
and by multiplying the stop loss premium for the treaty duration
with the number of vehicles in the subset.
29. The system according to claim 25, further including means for
calculating for different treaty durations stop loss premiums for
the fleet of vehicles; means for calculating for each treaty duration
a stop loss premium per vehicle by dividing the stop loss premium,
calculated for the treaty duration and for the fleet of vehicles,
with the number of vehicles in the fleet; means for associating
subsets of the fleet of vehicles with the different treaty durations;
and means for calculating for each treaty duration a stop loss premium
for the subset of the fleet of vehicles associated with the treaty
duration by multiplying the stop loss premium per vehicle, calculated
for the respective treaty duration, with the number of vehicles
in the respective subset.
30. The system according to claim 25, further including means for
associating subsets of the fleet of vehicles with different treaty
durations; means for calculating for each treaty duration a duration-dependent
loss by dividing an expected total loss for the treaty duration
by the maximum individual loss, the expected total loss for a multi-year
treaty duration being calculated by adding an expected total loss
for each year included in the multi-year treaty.
31. The system according to claim 30, further including means for
calculating an expected total loss for a first year of a multi-year
treaty by multiplying an expected number of incidents expected in
the first year with an average individual loss amount for an incident
involving one of the vehicles; means for calculating an expected
total loss for one of the years after the first year of the multi-year
treaty by multiplying an expected total loss of a preceding year
with an index; and means for calculating an expected total loss
for the multi-year treaty by aggregating expected total losses for
years included in the multi-year treaty.
32. The system according to claim 25, further including means for
storing a maximum total insurance coverage; means for calculating
a premium excess based on the loss frequency, the maximum individual
loss, and the maximum total insurance coverage as a stop loss premium
for an assumed loss distribution having only losses with a value
of one of zero and maximum individual loss; and means for calculating
the premium by subtracting at least a defined part of the premium
excess from the premium.
33. The system according to claim 25, further including means for
calculating the premium for defined values of the deductible; and
means for producing a graphical representation showing the premium
as a function of the defined values of the deductible.
34. The system according to claim 25, further including means for
receiving and storing risk factors; means for calculating the expected
total loss based on the risk factors; and means for producing a
graphical representation showing the premium as a function of the
risk factors.
35. The system according to claim 25, further including means for
calculating the premium for defined values of the expected number
of incidents; and means for producing a graphical representation
showing the premium as a function of the defined values of the expected
number of incidents.
36. The system according to claim 25, further including means for
storing an expected number of incidents involving one of the vehicles;
means for storing an expected average individual loss amount for
an incident involving one of the vehicles; and means for calculating
the expected total loss by multiplying the expected number of incidents
with the expected average individual loss amount.
**Insurance Description**
FIELD OF THE INVENTION
[0001] The present invention relates to a computer-implemented
method and devices for calculating an insurance premium. Specifically,
the present invention relates to a computer-implemented method,
a computer program product, and a computer-based data processing
system for calculating a premium for stop loss insurance for a fleet
of vehicles.
BACKGROUND OF THE INVENTION
[0002] Estimating the loss potential and pricing of a treaty is
central to the underwriting process. Usually, pricing methods work
with `static` input (distributions) to yield a `static` premium.
In certain cases, however, input parameters may not be known with
sufficient certainty (e.g. loss experience) or can be subject to
change (treaty conditions). In these cases, it is important to determine
the sensitivity of the premium (or expected loss) to changes in
input parameters, e.g. deductible.
[0003] In long term renting of vehicles, typically, fleet operators
rent to individuals or companies for a duration of 1 to 5 years.
By outsourcing its fleet to a specialized provider, a company can
expect to save considerable costs. In consequence, the business
sees a greater increase over the past years. For example, in Spain
about 7% of all newly licensed vehicles belong to this category.
Growing by 22% in 2001, the number of renting vehicles in Spain
reached 265,000 vehicles in 2002 (statistics from the Asociaci6n
Espanola de Renting). Fleet operators take over all administration
of the vehicles, including agreements with service providers, e.g.
garages and insurers. Regarding motor hull damages, fleet operators
may be willing to retain some financial risk, but seek balance sheet
protection through insurance instruments more typical of Reinsurance
than Insurance.
[0004] For example, renting of vehicles is shifting the demand
for motor hull insurance in the Spanish market. Instead of standard,
per vehicle insurance handled by insurers, a balance sheet protection
is sought, which is better achieved by Reinsurance instruments.
[0005] What is missing are a method and tools suitable for estimating
efficiently and flexibly the loss potential and pricing of an insurance
treaty for fleets of vehicles.
SUMMARY OF THE INVENTION
[0006] It is an object of this invention to provide an improved
computer-implemented method, an improved computer program product,
and an improved computer-based data processing system for calculating
a premium for stop loss insurance for a fleet of vehicles; particularly,
a premium for stop loss reinsurance for the fleet of vehicles.
[0007] According to the present invention, the above-mentioned
objects are particularly achieved in that for calculating a premium
for stop loss insurance for a fleet of vehicles, particularly, a
premium for stop loss reinsurance for the fleet of vehicles, an
expected total loss for the fleet of vehicles is determined, a maximum
individual loss, equivalent to a cost of a most expensive vehicle
of the fleet, is stored in a computer, a loss frequency is calculated
by the computer by dividing the expected total loss by the maximum
individual loss, a deductible, payable by an insurance holder, is
stored in the computer, and the premium is calculated by the computer
based on the loss frequency, the maximum individual loss, and the
deductible, as a stop loss premium for an assumed loss distribution
having only losses with a value of one of zero and maximum individual
loss. Generally, if the probability distribution of individual losses
is known for the fleet of vehicles, the stop loss premium can be
calculated. For probability distributions having the same maximum
individual loss and the same average individual loss or aggregated
total loss, respectively, Gagliardi and Straub have shown that a
probability distribution having only individual losses with a value
of either zero or the maximum individual loss is the worst case
probability distribution resulting in the highest stop loss premium
[Gagliardi and Straub (1974): "Eine obere Grenze fur Stop-Loss-Prmien",
Mitteilungen der Vereinigung schweizerischer Versicherungs-mathematiker
1974, volume 2, pages 215 to 221]. Consequently, a worst case or
upper bound stop loss premium can be calculated for an assumed loss
distribution having only losses with a value of either zero or the
maximum individual loss. For that purpose, the (assumed) loss frequency
is calculated by dividing the expected total loss by the maximum
individual loss. Therefore, without having to know and without having
to store and process complex distributions of individual losses
of the fleet of vehicles, a worst case (and thus safe) premium for
stop loss insurance for the fleet of vehicles can be calculated
based solely on the expected total loss, the maximum individual
loss, and a deductible payable by the insurance holder. Consequently,
for calculating the premium, repetitive steps used in the prior
art for discretizing and processing distributions of individual
losses can be eliminated, and thus, processing time and processing
power can be reduced. Furthermore, memory space used in the prior
art for storing distributions of individual losses, for storing
discretized distributions of individual losses, and for storing
intermediate processing results can be saved. Incorporating the
Gagliardi/Straub method for calculating a premium for stop loss
insurance for a fleet of vehicles according to the present invention
reduces processing time, and thus, makes it possible to reduce operating
time for negotiating with a client from several hours to a few minutes.
[0008] In a preferred embodiment, subsets of the fleet of vehicles
are associated in the computer with different treaty durations.
For each treaty duration, a separate premium is calculated by the
computer for the subset of the fleet of vehicles associated with
the treaty duration. Subsequently, the premium for the fleet of
vehicles is calculated by the computer by aggregating the separate
premiums.
[0009] Preferably, for each treaty duration, a stop loss premium
is calculated by the computer for the fleet of vehicles. Moreover,
for each treaty duration, a premium is calculated by the computer
for the subset of the fleet of vehicles associated with the treaty
duration by weighting the stop loss premium, calculated for the
fleet of vehicles, with the number of vehicles in the subset. Thus,
as discussed above in the context of calculating the premium for
stop loss insurance for the fleet of vehicles, the premium can be
calculated efficiently for fleets of vehicles having subsets associated
with different treaty durations. There is no need for storing or
processing distributions of individual losses. In addition to the
expected total loss, the maximum individual loss, and the deductible,
only the number of vehicles in the different subsets must be known
for calculating the premium for stop loss insurance for the whole
fleet of vehicles.
[0010] Preferably, for each treaty duration, a duration-dependent
loss frequency is calculated by the computer by dividing an expected
total loss for the treaty duration by the maximum individual loss.
Moreover, based on the duration-dependent loss frequency, the maximum
individual loss, and a deductible assigned to the treaty duration,
a stop loss premium is calculated by the computer for the fleet
of vehicles for each treaty duration. The stop loss premium is calculated
by the computer for an assumed loss distribution having only losses
with a value of one of zero and maximum individual loss. For each
treaty duration, a premium is calculated by the computer for the
subset of the fleet of vehicles associated with the treaty duration
by dividing the stop loss premium for the treaty duration by the
total number of vehicles in the fleet and by the treaty duration,
and by multiplying the stop loss premium for the treaty duration
with the number of vehicles in the subset. In addition to the above-stated
advantages, different deductibles can be specified for the different
treaty durations, thus enabling insurance holders to define different
scenarios for short term and long-term risks.
[0011] In an embodiment, stop loss premiums for the fleet of vehicles
are calculated by the computer for different treaty durations. For
each treaty duration, the computer calculates a stop loss premium
per vehicle by dividing the stop loss premium, calculated for the
treaty duration and for the fleet of vehicles, with the number of
vehicles in the fleet. In the computer, subsets of the fleet of
vehicles are associated with the different treaty durations. For
each treaty duration, a premium is calculated by the computer for
the subset of the fleet of vehicles associated with the treaty duration
by multiplying the stop loss premium per vehicle, calculated for
the respective treaty duration, with the number of vehicles in the
respective subset. Stop loss premiums per vehicle for each treaty
duration can be calculated at a time when the portfolio distribution,
i.e. the number of vehicles of the fleet associated with the different
treaty durations, is not known yet, for example at the time when
the contract of the stop loss insurance is prepared. At a later
time, when the portfolio distribution is known, the premium for
each treaty duration can be calculated by multiplying the stop loss
premium per vehicle for the respective treaty duration with the
number of vehicles associated with the respective treaty duration.
Consequently, it is possible for an insurance holder and/or for
an insurance provider to calculate easily the premium for each treaty
duration (and through aggregation the premium for the fleet of vehicles)
as an estimate for an expected portfolio distribution or as a very
accurate approximation for a known portfolio distribution.
[0012] Preferably, for each treaty duration, a duration-dependent
loss frequency is calculated by the computer by dividing an expected
total loss for the treaty duration by the maximum individual loss.
The expected total loss for a multi-year treaty duration is calculated
by the computer by adding an expected total loss for each year included
in the multi-year treaty.
[0013] Preferably, an expected total loss for a first year of a
multi-year treaty is calculated by the computer by multiplying an
expected number of incidents, expected in the first year, with an
average individual loss amount for an incident involving one of
the vehicles. An expected total loss for one of the years after
the first year of the multi-year treaty is calculated by the computer
by multiplying an expected total loss of a preceding year with an
index. Finally, an expected total loss for the multi-year treaty
is calculated by the computer by aggregating expected total losses
for years included in the multi-year treaty. Time-dependent indexing
of the expected total loss has the advantage that monetary inflation,
on one hand, and age-dependent devaluation of a vehicle, on the
other hand, can be considered for multi-year treaties.
[0014] In an embodiment, a maximum total insurance coverage is
stored in the computer and, based on the loss frequency, the maximum
individual loss, and the maximum total insurance coverage, a premium
excess is calculated by the computer as a stop loss premium for
an assumed loss distribution having only losses with a value of
one of zero and maximum individual loss. At least a defined part
of the premium excess is subtracted by the computer from the premium.
Calculating and subtracting the premium excess from the premium
has the advantage that the premium is not charged for losses exceeding
the maximum total insurance coverage, i.e. for losses not covered
by the insurance.
[0015] In an embodiment, the premium is calculated by the computer
for defined values of the deductible and a graphical representation
is produced by the computer, showing the premium as a function of
the defined values of the deductible. The deductible payable by
the insurance holder is selected based on the graphical representation.
Illustrating the premium for the stop loss insurance as a function
of the deductible makes it possible for the insurance holder to
specify a deductible, knowing the corresponding premium, or vice
versa.
[0016] In an embodiment, determining the expected total loss includes
entering 5 and storing in the computer risk factors and calculating
by the computer the expected total loss based on the risk factors.
Moreover, a graphical representation is produced by the computer,
showing the premium as a function of the risk factors. Typically,
risk factors have a direct influence on the number of incidents
and/or on the individual loss amount, and thus, on the expected
total loss. For example, a geographical area where vehicles are
frequently stolen represents a quantifiable risk factor, having
a direct influence on the expected number of incidents and on the
expected total loss. Illustrating the premium as a function of risk
factors has the advantage that the influence of risk factors on
the premium, as well as the impact of reducing specific risk factors,
can be illustrated to the insurance holder.
[0017] In an embodiment, the premium is calculated by the computer
for defined values of the expected number of incidents and a graphical
representation is produced by the computer, showing the premium
as a function of the defined values of the expected number of incidents.
Illustrating the premium as a function of the expected number of
incidents has the advantage that the influence of the expected number
of incidents on the premium, as well as the impact of reducing the
expected number of incidents, can be illustrated to the insurance
holder.
[0018] Preferably, determining the expected total loss includes
storing in the computer an expected number of incidents involving
one of the vehicles, storing in the computer an expected average
individual loss amount for an incident involving one of the vehicles,
and calculating by the computer the expected total loss by multiplying
the expected number of incidents with the expected average individual
loss amount.
[0019] In addition to a computer-implemented method and a computer-based
data processing system for calculating a premium for stop loss insurance
for a fleet of vehicles, the present invention also relates to a
computer program product including computer program code means for
controlling one or more processors of a computer, particularly,
a computer program product including a computer readable medium
containing therein the computer program code means.
BRIEF DESCRIPTION OF THE DRAWINGS
[0020] The present invention will be explained in more detail,
by way of example, with reference to the drawings in which:
[0021] FIG. 1 shows an example of a time sequence of incidents
having individual loss amounts and a chart illustrating the corresponding
stop loss cover.
[0022] FIG. 2 shows block diagram illustrating schematically an
exemplary configuration of a computer-based data processing system
for practicing embodiments of the present invention, said configuration
comprising a computer with a processor and memory.
[0023] FIG. 3 shows a block diagram illustrating schematically
the interdependencies between various variables and a premium for
stop loss insurance.
[0024] FIG. 4 shows a block diagram illustrating schematically
an exemplary configuration of programmed software modules for practicing
embodiments of the present invention.
[0025] FIG. 5 shows a block diagram illustrating schematically
an exemplary configuration of data flow and processing for practicing
embodiments of the present invention for calculating a premium for
stop loss insurance for a fleet of vehicles.
[0026] FIG. 6 shows a block diagram illustrating schematically
an exemplary configuration of data flow and processing for practicing
embodiments of the present invention for calculating a premium for
stop loss insurance for a fleet of vehicles, defined subsets of
the fleet being associated with different treaty durations.
[0027] FIG. 6b shows a block diagram illustrating schematically
an exemplary configuration of data flow and processing for practicing
embodiments of the present invention for calculating a premium for
stop loss insurance for a fleet of vehicles, stop loss premiums
being calculated per vehicle for different treaty durations.
[0028] FIG. 7 shows a graph illustrating the premium for stop loss
insurance as a function of the deductible.
[0029] FIG. 8 shows a graph illustrating the premium for stop loss
insurance as a function of the frequency of incidents.
[0030] FIG. 9 shows a graph illustrating the premium for stop loss
insurance as a function of the number of vehicles.
[0031] FIG. 10 shows a graph illustrating the premium for stop
loss insurance as a function of a defined percentage of robberies
actually observed.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0032] In FIG. 1, reference numerals 11, 12, 13, 14, and 15 refer
to individual loss amounts caused by a corresponding time sequence
of incidents, for example vehicle accidents or vehicle thefts. Reference
numerals 11', 12', 13', 14', and 15' refer to the individual loss
amounts arranged vertically to illustrate the aggregation of the
individual loss amounts over time. Reference numeral 16 relates
to a deductible, having a value of 115% in the illustrated example.
The deductible 16 defines the portion of the aggregated individual
loss amounts 11', 12', 13', 14', 15' that is to be paid by an insurance
holder. Reference numeral 18 relates to a stop loss cover, i.e.
a range of the aggregated individual loss amounts 11', 12', 13',
14', 15' for which insurance coverage is provided. As is illustrated
in FIG. 1, the stop loss cover 18 is limited by a maximum insurance
coverage 17 (also referred to as exit point), having a value of
150% in the illustrated example. The insurer does not cover any
aggregated loss exceeding the exit point.
[0033] In FIG. 2, reference numeral 2 refers to a computer-based
data processing system, particularly a computer such as a personal
computer. As is illustrated schematically, computer 2 includes a
display 24, at least one processor 21, memory 22 for storing data
and programs, as well as a computer program product 23. The computer
program product 23 comprises computer program code for controlling
processor 21 so that the computer 2 executes various functions described
below in more detail with reference to FIGS. 3, 4, 5 and 6. Particularly,
the computer program product 23 comprises computer program code
for calculating a premium for stop loss insurance for a fleet of
vehicles. The computer program code is stored in a computer readable
medium, either in memory integrated in computer 2 or on a data carrier
that can be inserted into computer 2. The computer 2 is connected
via communication link 27 to printer 25.
[0034] In FIG. 3, illustrated are the interdependencies between
various variables and the premium 31 for stop loss insurance. The
premium 31 is determined by pricing module 32. The pricing module
32 determines the premium 31 based on pricing parameters 33 and
treaty conditions 34. The pricing parameters are influenced by loss
components 35. The pricing parameters 33 include the average cost
per incident (i.e. the average individual loss amount), the incident
frequency (e.g. the number of incidents per year), the number of
vehicles in the fleet to be insured, a portfolio distribution, and
an index, preferably a loss inflation index. For a portfolio including
multiple treaties having different treaty durations (i.e. multi-year
treaties), the portfolio distribution indicates the number of vehicles
of the fleet associated with each treaty. In Table 1, an example
of a portfolio distribution is shown for different treaties having
individual treaty durations of one, two, three, four, or five years,
respectively. The treaty conditions 34 include a maximum individual
loss amount, i.e. the maximum single loss that is equivalent to
the most expensive vehicle in the fleet. The treaty conditions 34
also include information about the treaty structure. The information
about the treaty structure includes a deductible, payable by the
insurance holder, and a maximum insurance coverage (exit point).
It is possible to associate and store different deductibles and/or
exit points for different treaty durations. The loss components
include information about a client's loss experience. The loss experience
includes the number of losses or incidents by type of loss or incident
(e.g. theft of the vehicle), date of loss or incident, place of
loss or incident (e.g. type of place, such as highway, inner city,
or suburbs; and/or geographical location, including information
such as country, state/province, and city). Each loss or incident
also includes a unique identifier and a detailed description of
the incident, for example a description of an accident.
1 TABLE 1 Treaty Distribution Distribution Duration (percentage)
(numbers) 1 Year 6% 192 2 Years 24% 708 3 Years 40% 1,200 4 Years
26% 792 5 Years 4% 108
[0035] One skilled in the art will understand that the computer
program code, included in the computer program product 23, may be
implemented as one program application, as multiple separate program
application modules or as program extension modules for conventional
spreadsheet applications, such as Microsoft Excel, for example.
In FIG. 4, an exemplary configuration of programmed software modules
for practicing embodiments of the present invention is illustrated.
As illustrated in FIG. 4, computer 2 includes a main program module
41, an expected loss calculation module 42, a treaty module 43,
a pricing module 44, a calculate rate module 45, a control module
46, as well as a visualization module 47.
[0036] The main program module 41 is responsible for receiving
and storing input parameters needed for calculating the premium
for stop loss insurance for a fleet of vehicles. The input parameters
include the average cost per incident, the incident frequency, the
number of vehicles to be insured, the portfolio distribution, the
index (e.g. the loss inflation index), the treaty structure, and
the maximum individual loss. It is also possible to have the average
cost per incident and the incident frequency calculated based on
loss experience information and/or risk factors, as will be explained
later in more detail.
[0037] The expected loss calculation module 42 calculates the expected
total loss by multiplying the average cost per incident (expected
average individual loss amount for an incident involving one of
the vehicles) with the incident frequency (expected yearly number
of incidents involving one of the vehicles). Furthermore, for fleets
having subsets of vehicles associated with different treaty durations,
the expected loss calculation module 42 calculates the expected
total loss for treaty durations of one, two, three, four and five
years, for example. For multi-year treaties, the expected loss for
the years after the first year is calculated by multiplying the
expected total loss for the preceding year with an index. Preferably,
the index is a loss inflation index. The expected total loss for
multi-year treaties having treaty durations of two, three, four,
and five years, is calculated by aggregating the expected total
losses for the years included in the respective multi-year treaty.
[0038] The treaty module 43 is responsible for applying the treaty
conditions to calculations and simulations.
[0039] The pricing module 44 is used to analyze the loss experience.
Particularly, the pricing module 44 is used to determine the average
cost per incident and the incident frequency based on loss experience
information and/or risk factors. Most input parameters, for example
the average individual loss amount, are better described by a distribution
rather than a fixed value. In the present invention the Monte Carlo
method is used for risk calculation, allowing a user to determine
the probability level of a result. The pricing module 44 invokes
the expected loss calculation module 42, the treaty module 43, and
the calculate rate module 45 for calculating a premium for the stop
loss insurance for the fleet of vehicles. The pricing module 44
is also configured to provide reverse pricing for determining treaty
parameters based on a set total premium. For example, if a client
is willing to allocate a defined total sum for the premium, key
parameters of the treaty, such as the deductible, are calculated
for the specified premium.
[0040] Using the Gagliardi/Straub method (or Gagliardi method for
short), the calculate rate module 45 calculates a premium for stop
loss insurance for a fleet of vehicles or for a defined subset of
the fleet, respectively, as will be explained in more detail with
reference to FIGS. 5 and 6.
[0041] The control module 46 can also be used to analyze the loss
experience. Particularly, the control module 46 is used for sensitivity
analysis and simulations, i.e. for assessing how the premium changes
if specific input parameters are changed. For example, specific
input parameters and risk factors are selectable and for a selected
input parameter or risk factor, the premium is calculated for different
values of the selected input parameter or risk factor. These simulated
results are illustrated graphically on display 24 or on a report
26 printed on printer 25. Results are simulated based on the interdependencies
of certain parameters.
[0042] The visualization module 47 is responsible for visualizing
selected information in graphical form. For example, the visualization
module 47 displays graphs of simulated scenarios showing the impact
of different variables (e.g. input parameters or risk factors) on
the premium. In FIGS. 7, 8, and 9, examples of graphs are shown,
illustrating the premium for stop loss insurance as a function of
the deductible, as a function of the frequency of incidents, or
as a function of the number of vehicles, respectively. In FIG. 10,
an example of a graph is shown, illustrating the premium for stop
loss insurance as a function of a risk factor. Particularly, FIG.
10 illustrates the premium for stop loss insurance as a function
of a defined percentage of robberies (theft of vehicles) actually
observed. Preferably, the visual images are displayed on display
24 in a graphical interface. The visual images refresh automatically
when one or more of the input parameters are changed. The visual
images can also be reproduced on a report 26 printed on printer
24.
[0043] As is illustrated in FIG. 5, the expected total loss 51
is calculated in block 50. The expected total loss 51 is calculated
by multiplying the expected average individual loss m by the expected
incident frequency .lambda.. The expected average individual loss
m and/or the expected incident frequency .lambda. are entered into
computer 2 or calculated in block 504. Block 504 analyzes the loss
experience 502 and calculates the expected average individual loss
m and the expected incident frequency A based on the loss experience
502 and the number of vehicles 501. In addition, Block 504 uses
risk factors 503 to calculate the expected average individual loss
m and the expected incident frequency .lambda..
[0044] In block 55, according to Gagliardi/Straub, an assumed loss
frequency .LAMBDA. is calculated by dividing the expected total
loss 51 by the maximum individual loss M.
[0045] The maximum individual M loss, the deductible d, as well
as the maximum insurance coverage (exit point) x are values entered
and stored in computer 2. The maximum individual loss M and the
loss frequency .LAMBDA. are passed to block 552. The deductible
d is passed to block 553 and the maximum insurance coverage (exit
point) x is passed to block 554.
[0046] In block 552, according to Gagliardi/Straub, a stop loss
premium P.sub.d is calculated for the deductible d based on the
loss frequency .LAMBDA., the maximum individual loss M and the deductible
d. The stop loss premium P.sub.d is calculated according to formula
(1) for an assumed loss distribution having only losses with either
a value of zero or maximum individual loss M, wherein k=Integer(d/M).
1 P d = M ( 1 - j = 0 k - 1 e - j j ! ) - d ( 1 - i = 0 k e - i
i ! ) ( 1 )
[0047] Furthermore, in block 552, according to Gagliardi/Straub,
a stop loss premium P.sub.x is calculated for the maximum insurance
coverage (exit point) x based on the loss frequency A, the maximum
individual loss M and the exit point x. The stop loss premium P.sub.x
is calculated according to formula (2) for an assumed loss distribution
having only losses with either a value of zero or maximum individual
loss M, wherein k=Integer(x/M). 2 P x = M ( 1 - j = 0 k - 1 e -
j j ! ) - x ( 1 - i = 0 k e - i i ! ) ( 2 )
[0048] An example of a computer program function for calculating
stop loss premiums P.sub.d and P.sub.x according to formulas (1)
or (2), respectively, is shown in Table 2.
2 TABLE 2 Public Function STOPLOSS (exp_loss As Double, max_loss
As Double, prio As Double) Dim frequency As Double Dim sum_a As
Double Dim sum_b As Double Dim k As Integer Dim j As Integer Dim
i As Integer k = Int(prio / max_loss) frequency = exp_loss / max_loss
p_i = Exp(-1 * frequency) p_i = Exp(-1 * frequency) sum_a = p_i
sum_b = p_i For j = 1 To (k - 1) p_j = p_j * (frequency / j) sum_a
= sum_a + p_j Next j For i = 1 To k p_i = p_i * (frequency / i)
sum_b = sum_b + p_i Next i STOPLOSS = frequency * max_loss * (1
- sum_a) - prio * (1 - sum_b) End Function
[0049] Finally, in block 552, the premium P for the stop loss insurance
for the fleet of vehicles is calculated according to formula (3).
The factor c (0.ltoreq.c.ltoreq.1) should correct for the fact that
a subtraction of two upper limits for the stop loss premium is not
necessarily itself an upper limit for the layer.
P=P.sub.d-c.multidot.Px (3)
[0050] In FIG. 6, calculation of the premium for stop loss insurance
is illustrated for a fleet of vehicles having subsets of the fleet
associated with different treaty durations. In the example illustrated
in FIG. 6, the treaties have durations of one, two, three, four,
and five years. However, in FIG. 6, only calculations for the multi-year
treaties having two and five years are explicitly shown; the multi-year
treaties having a duration of three and four years are indicated
symbolically only by periods (" . . . ").
[0051] The expected total loss for the first year 63 is calculated
in block 60. Block 60 corresponds to block 50 described above with
reference to FIG. 5.
[0052] For multi-year treaties, the expected total losses are each
calculated by adding the aggregated losses expected for years included
in the treaty duration. The aggregated losses expected for years
after the first year are calculated by indexing the expected total
loss for the first year 63, i.e. by multiplying the expected total
loss for the first year 63 with an index, preferably an inflation
index. For example, in block 61, the expected total loss 65 is calculated
for the multi-year treaty having duration of two years (i.e. the
two year treaty). The expected total loss for the two-year treaty
65 is calculated by adding the expected total loss for the first
year 63 and the expected aggregated loss for the second year. The
expected aggregated loss for the second year is calculated by indexing
the expected total loss for the first year 63. In block 62, the
expected total loss 67 is calculated for the five-year treaty. The
expected total loss for the five-year treaty 67 is calculated by
adding the expected total loss for the first year 63 and the expected
aggregated losses for the second, the third, the fourth, and the
fifth year.
[0053] As is illustrated in FIG. 6, the same maximum individual
loss amount M is used for the one-year treaty as well as for the
multi-year treaties. However, different deductibles d.sub.1, d.sub.2,
d.sub.5 can be entered and stored in computer 2 for each of the
treaties. Moreover, it is also possible to enter and store different
maximum insurance coverage values (exit points) x.sub.1, x.sub.2,
x.sub.5 for each of the treaties.
[0054] In block 64, the stop loss the premium for the full fleet
of vehicles is calculated according to Gagliardi/Straub for the
one-year treaty. Block 64 corresponds to block 55 described above
with reference to FIG. 5. Block 64 calculates the premium for the
one-year treaty for the full fleet based on the expected total loss
for the first year 63, the maximum individual loss amount M, the
deductible d.sub.1 for the one-year treaty, and the maximum insurance
coverage (exit point) x.sub.1 for the one-year treaty.
[0055] For multi-year treaties, the stop loss premiums for the
full fleet of vehicles are calculated according to Gagliardi/Straub
based on the respective expected total loss calculated for the respective
treaty. For the multi-year treaties, the stop loss premiums for
the full fleet of vehicles are calculated according to Gagliardi/Straub
based on the deductible d.sub.2, d.sub.5 and the maximum insurance
coverage (exit point) x.sub.2, x.sub.5 assigned to the respective
treaty. For example, in block 66, the stop loss premium for the
full fleet of vehicles is calculated for the two-year treaty based
on the expected total loss for the two-year treaty 65, the maximum
individual loss amount M, the deductible d.sub.2 for the two-year
treaty, and the maximum insurance coverage (exit point) x.sub.2
for the two-year treaty. In block 68, the stop loss the premium
for the full fleet of vehicles is calculated for the five-year treaty
based on the expected total loss for the five-year treaty 67, the
maximum individual loss amount M, the deductible d.sub.5 for the
five-year treaty, and the maximum insurance coverage (exit point)
x.sub.5 for the five-year treaty.
[0056] In block 691, the stop loss premiums 641, 661, 681 calculated
for the different treaty durations for the full fleet of vehicles
are weighted by the actual number of vehicles in the respective
subset associated with the treaty duration. For that purpose, the
portfolio distribution 69 is passed to block 691. Moreover, the
stop loss premiums 641, 661, 681 calculated for the multi-year treaties
are converted into yearly rates. For example, in block 642, the
premium for the stop loss insurance for the one-year treaty 643
is calculated. In block 642, the premium for the one-year treaty
for the full fleet 641 is divided by the number of vehicles 501
of the fleet and multiplied by the number of vehicles in the subset
associated with the one-year treaty. In block 662, the yearly premium
for the stop loss insurance for the two-year treaty 663 is calculated.
In block 662, the premium for the two-year treaty for the full fleet
661 is divided by the number of vehicles 501 of the fleet, multiplied
by the number of vehicles in the subset associated with the two-year
treaty, and divided by the two-year duration. In block 682, the
yearly premium for the stop loss insurance for the five-year treaty
683 is calculated. In block 682, the premium for the five-year treaty
for the full fleet 681 is divided by the number of vehicles 501
of the fleet, multiplied by the number of vehicles in the subset
associated with the five-year treaty, and divided by the five-year
duration.
[0057] The total yearly premium for stop loss insurance for the
full fleet is calculated by aggregating the yearly premiums 643,
663, 683 for the stop loss insurance for the different treaty durations.
[0058] Since renting firms are usually start-up companies, most
input values are only approximately known, so rather than calculating
only a fixed premium, the present invention determines the impact
of a parameter on the price (premium) of the insurance. This often
leads to adaptations in the treaty. For example, a reasonable upper
limit for the loss per vehicle can be determined and included in
the price of the insurance. Also, other high impact parameters can
be monitored and/or simulated.
[0059] In FIG. 6b, for a fleet of vehicles having subsets of the
fleet associated with different treaty durations, the calculation
of stop loss premiums per vehicle for each treaty duration is illustrated.
In block 692, the stop loss premiums 641, 661, 681 calculated for
the different treaty durations for the full fleet of vehicles are
divided by the number of vehicles in the fleet. For example, in
block 644, the stop loss premium per vehicle for the one-year treaty
is calculated and stored; in block 664, the stop loss premium per
vehicle for the two-year treaty is calculated and stored; and in
block 684, the stop loss premium per vehicle for the five-year treaty
is calculated and stored. Once the portfolio distribution 69 is
known (or provided as an estimate) and passed to block 692, the
premiums for the stop loss insurance for the different treaties
are calculated in block 692. For example, the premium for the stop
loss insurance for the one-year treaty 645 is calculated by multiplying
the stored stop loss premium per vehicle for the one-year treaty
644 with the number of vehicles associated with the one-year treaty.
The premium for the stop loss insurance for the two-year treaty
665 is calculated by multiplying the stored stop loss premium per
vehicle for the two-year treaty 664 with the number of vehicles
associated with the two-year treaty. The premium for the stop loss
insurance for the five-year treaty 685 is calculated by multiplying
the stored stop loss premium per vehicle for the five-year treaty
684 with the number of vehicles associated with the five-year treaty.
[0060] Typically, the precise portfolio distribution is known only
after the beginning of the stop loss insurance. Consequently, the
calculated premium for stop loss insurance may be too high or too
low, if the portfolio distribution was not estimated correctly at
the beginning of the insurance contract. For example, an average
individual loss of 1,000, an expected incident frequency of 10%,
a number of vehicles of 5,000, a maximum individual loss of 100,000,
an assumed percentage of 80% of the fleet associated with a one-year
treaty, and an assumed percentage of 20% of the fleet associated
with a two-year treaty, results an expected total loss for the one-year
treaty of 80%.multidot.5,000.multidot.10%.multidot.- 1,000=400,000
and an expected total loss for the two-year treaty of 20%.multidot.5,000.multidot.10%.multidot.1,000=100,000
(in two years 200,000). Assuming an 115% stop loss deductible of
the expected total loss (600,000) of 690,000, the precise premium
of the stop loss insurance, calculated according to the method described
herein, is 60941. However, if the portfolio distribution turns out
to have a percentage of 20% of the fleet associated with the one-year
treaty and an percentage of 80% of the fleet associated with the
two-year treaty, the precise premium of the stop loss insurance
would be 6,089 (about 10%) higher (the value calculated for the
assumed portfolio distribution is too low). In our example, the
stop loss premium per vehicle for the one-year treaty is 11.79;
the stop loss premium per vehicle for the two-year treaty is 13.65.
For a portfolio distribution with a percentage of 80% of the fleet
associated with the one-year treaty and a percentage of 20% of the
fleet associated with the two-year treaty, the premium for the stop
loss insurance is 5,000.multidot.80%.multidot.11.79+5000.multidot.20%.multidot-
.13.65=60,810. For a portfolio distribution with a percentage of
20% of the fleet associated with the one-year treaty and a percentage
of 80% of the fleet associated with the two-year treaty, the premium
for the stop loss insurance is 5,000.multidot.20%.multidot.11.79+5000.multidot.80%.mul-
tidot.13.65=66,390. In both cases, the difference to the precise
premium for stop loss insurance is negligibly small. In Table 3,
the difference between the approximation, based on the stop loss
premium per vehicle, and the precise calculation of the premium
for the stop loss insurance is listed for different portfolio distributions.
3TABLE 3 Approx- imation in % of Precise precise premium premium
Percentage Percentage for stop for of one-year of two-year Stop
loss loss Approx- stop loss treaties treaties deductible insurance
imation insurance 0 100 1150000 68253 68253 100 10 90 1092500 66787
67321 101 20 80 1035000 67030 66389 99 30 70 977500 66142 65456
99 40 60 920000 65256 64524 99 50 50 862500 64968 63592 98 60 40
805000 62819 62660 100 70 30 747500 63146 61728 98 80 20 690000
60941 60795 100 90 10 632500 60512 59863 99 100 0 575000 58931 58931
100
[0061] As can be seen in Table 3, calculating the premium of the
stop loss insurance from the stop loss premiums per vehicle, calculated
for individual treaty durations, provides a very good approximation
to the precise calculation of the premium of the stop loss insurance
with known portfolio distribution.
[0062] In order to proof that (U+V).sup.+.ltoreq.U.sup.++V.sup.+
(inequation 1) is true for random variables U and V, the following
three cases must be reviewed: (a) U+V.ltoreq.0; (b) U+U<0; and
(c) U>0, V>0.
[0063] Let us assume that X.sub.1 and X.sub.2 are two expected
losses, that P.sub.1 and P.sub.2 are the respective stop loss deductibles,
and that 0.ltoreq.a.ltoreq.1.
[0064] a.multidot.X.sub.1+(1-a).multidot.X.sub.2 is a weighted
expected loss.
[0065] a.multidot.P.sub.1+(1-a).multidot.P.sub.2 is a weighted
stop loss deductible.
[0066] It is: (a.multidot.X.sub.1+(1-a).multidot.X.sub.2-[a.multidot.P.sub-
.1+(1-a).multidot.P.sub.2]).sup.+=(a.multidot.[X.sub.1-P.sub.1]+(1-a).mult-
idot.[X.sub.2-P.sub.2]).sup.+.
[0067] If one sets U=a.multidot.(X.sub.1-P.sub.1) and V=a.multidot.(X.sub.2-P.sub.2),
then, according to inequation (1), the expression above is .ltoreq.a.multidot.(X.sub.1-P.sub.1).sup.++(1-a).mult-
idot.(X.sub.2-P.sub.2).sup.+.
[0068] If on both sides of the inequation the expected value is
formed, inequation (2) follows as indicated below:
E{(a.multidot.X.sub.1+(1-a).multidot.X.sub.2-[a.multidot.P.sub.1+(1-a).mul-
tidot.P.sub.2]).sup.+}.ltoreq.a.multidot.E([X.sub.1-P.sub.1].sup.+)+(1-a).-
multidot.E([X.sub.2-P.sub.2].sup.+).
[0069] The left side of inequation (2) is the stop loss premium
of the weighted expected loss; the right side of inequation (2)
is the weighted stop loss premium of the individual expected losses.
[0070] However, in the method for calculating the premium for stop
loss insurance according to the present invention (incorporating
the Gagliardi/Straub method), one is not dealing with weighted values
of expected losses X.sub.1 and X.sub.2, but the Poisson distributed
number of losses are weighted, whereas the maximum values of the
losses remain unchanged. Therefore, in Table 3, approximations are
not always higher than the precise value but often lower. However,
for practical purposes, the differences are insignificant. |