Exposure Modelling in Property Reinsurance

Abstract

Exposure curves play significant role in modelling of property per risk excess of loss non-proportionalreinsurance contracts, especially in the situations when not enough historical data is availablefor applying experience-based methods or if the underlying exposure changed significantly.The paper deals only with the first loss scale (FLS) approach which is frequently used in Europe. Analternative approach is based on ISO´s PSOLD methodology which is typical for the U.S. The firstresearch into FLS approach was done by Ruth E. Salzmann in 1963 and some further curves havebeen developed since that time, however, their availability is limited. According to the authors´knowledge only limited number of articles were published on this topic and no comprehensivepublication which would describe the methodology to a larger extent exists. The paper providesa comprehensive description of the FLS exposure rating methodology, aims to summarise bothhistorical and latest developments in this area and also includes various authors´ own practicalconsiderations. The theory is illustrated on numerical examples.

Full text

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Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
EXPOSURE MODELLING INPROPERTY REINSURANCE
Jan Hrevuš,1Luboš Marek*
Abstract
Exposure curves play significant role inmodelling ofproperty per risk excess ofloss non-proportional
reinsurance contracts, especially inthe situations when not enough historical data is available
for applying experience-based methods or if the underlying exposure changed significantly.
Thepaper deals only with the first loss scale (FLS) approach which is frequently used inEurope.
An alternative approach is based onISO´s PSOLD methodology which is typical fortheU.S. Thefirst
research intoFLS approach was done by Ruth E. Salzmann in1963 and some further curves have
been developed since that time, however, their availability is limited. According totheauthors´
knowledge only limited number ofarticles were published onthis topic and no comprehensive
publicationwhich would describe themethodology toalarger extent exists. Thepaper provides
acomprehensive description oftheFLS exposure rating methodology, aims tosummarise both
historical and latest developments inthis area and also includes various authors´ own practical
considerations. Thetheory is illustrated onnumerical examples.
Keywords: reinsurance, exposure, property, non-proportional, excess ofloss
JEL Classification: G22, C58, C15
1. Introduction
The exposure curves play very significant role in pricing of property per risk excess of loss
non-proportional reinsurance treaties, especially in cases where not enough historical losses
are available or the underlying exposure changed significantly. The first research on this
topic was undertaken by Ruth E. Salzmann (see Salzmann, 1963), where a cumulative
loss cost distribution by percentage of insured value was introduced and it was shown that
for homogeneous groups of risks such distribution was stable. The research was based
on the claims data of Insurance Company of North America (INA) and the fire losses incurred
in 1960 as of May 31, 1961 were analysed (due to their short-tail nature it could have been
assumed that the losses were already fully developed at that time). The Salzmann´s curves
were reviewed latter by S. Ludwig (see Ludwig, 1991) and the methodology was applied
on more recent data including various practical suggestions.
* Jan Hrevuš, VIG Re zajistovna, a.s., Czech Republic (hrevus@volny.cz);
Luboš Marek, Faculty of Informatic and Statistic, University of Economics in Prague, Prague,
Czech Republic (marek@vse.cz).
This paper was written with the support of the Czech Science Foundation project No. P402/12/G097
“DYME – Dynamic Models in Economics”.
ARTICLES

130 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
Some further exposure curves exist and are used in practice. The so called Gasser
curves also known as Swiss Re curves are one of the most popular among European
reinsurers. These curves are suitable for wide scope of property portfolios. On the Swiss
Re curves wider related theory was demonstrated including deeper statistical description
by S. Bernegger (Bernegger, 1997) where also MBBEFD1 distribution class for degree of da-
mage was introduced. The exposure curve directly expresses the percentage of the retained
amount by the cedant from the total gross loss.
Although the curves are widely used, they are all based on very old data and according
to authors´ knowledge no recent, comprehensive and publicly available research on this
topic exists. White (2005) provides a list of curves widely accepted by practitioners. The list
complemented by the authors´ experience is provided here:
1.1 Lloyds curves
The curves were historically used by London market and are based on very old data with
unknown origin. The underlying data might have been marine data or even fire losses from
WWII period. The curves vary neither by size of the risks nor by line of business.
1.2 Salzmann curves
The curves were developed in Salzmann (1963), were based on homeowners data and vary
by construction (frame/brick) and by fire protection (protected/unprotected) class, i.e. four
different Salzmann curves are available. The analysis was based only on building (i.e. no
content etc.) fire losses from the early 1960s. Salzmann constructed the curves in order
to demonstrate the methodology and meant them just as an example, she did not recommend
them for further use.
1.3 Ludwig curves
Ludwig (Ludwig, 1991) based his curves on Hartford Insurance Group homeowners and
commercial losses for accident years 1984–1988. The curves vary by construction/
protection class for homeowners and occupancy class for commercial losses, however,
the commercial book analysed was very small and it is discussable if the derived curves would
be also representative for large accounts. The curves also include all property covers and
perils which were: wind losses, other property causes loss, and 1989 Hurricane Hugo losses.
1.4 Swiss Re curves
The original Swiss Re curves, also called Gasser curves, were developed by Peter Gasser
based on the data of “Fire statistics of the Swiss Association of Cantonal Fire Insurance
Institutions” for the years 1959–1967. The curves are widely used by European reinsurers
and some recommendations for their application are available (Guggisberg, 2004). Due
to their importance the curves will be described in further sections in more detailed way.
1 Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac

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1.5 Other reinsurers´ curves
Some further curves like Munich or Skandia are used by the reinsurance market, however,
their origins are not publicly documented.
1.6 MBBEFD curves
Bernegger (1997) built a new theoretical concept how any exposure curve can be described
by two parameters function which is based on MBBEFD distribution of degree of damage.
Again due to its importance the concept will be dealt in latter chapters.
1.7 Distributionfunction, density functionand estimates ofthese
function.
In our text, we often give a definition of distribution function, density function and estimates
of these functions. We specify only intervals in which these functions are different from
0 or 1. Other intervals are not interesting.
2. Basics ofExposure Rating
The exposure rating technique is often used in non-proportional per risk reinsurance
modelling when only limited loss history of the reinsured is available or the exposure
of the underlying portfolio significantly changed. This usually concerns the cases when an
excess of loss reinsurance treaty for a new portfolio or treaty layers with high retentions
are modelled. The exposure rating methodology is based on the so called risk profiles and
the exposure curves are derived from degree of damage distributions which will be described
in latter sections. It is also important to mention, that any perception of the words “rating”
or “pricing” in this paper relates only to the so-called “risk reinsurance premium” which
equals to estimated mean loss to reinsurance treaty layer, i.e. no profit, volatility or other
loadings will be applied. Further, the paper follows the notation as stated in Daykin (1995),
i.e. stochastic variables are denoted by bold letters, e.g. X, non-stochastic variables
(e.g. monetary variables) are denoted by capital letters, e.g. M. Further, ratio variables are
denoted by small letters, corresponding to the capital letter of that variable from which
they are derived (again, if ratio variable is stochastic, than it is denoted by small bold letter
(e.g. x for degree of damage).
The risk profile is a snapshot of portfolio (usually all risks come from the same line
of business) where risks are allocated to bands by their value of risk (this can be e.g. maximum
possible loss MPL2, sum insured or EML3). Each profile further includes information about
2 MPL is according to Guggisberg (2004) defined as “the maximum possible loss that occurs if all
conceivable negative – and thus even improbable – circumstances accumulate in particularly
unfortunate way”.
3 EML is according to Guggisberg (2004) defined as “estimated maximum loss, or the largest loss that
could occur under normal conditions of operation, use and loss prevention (e.g. intervention of fire
brigades, operation of fixed extinguishing systems) in the building in question, whereby any exception-
al circumstances (accident or unforeseen event) which could significantly alter the risk are ignored”.

132 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
the number of risks and gross premium. Let´s denote the value of i-th risk as Vi, total gross
premium for h-th band as Ph and the number of risks in h-th band as risk Nh, then the example
of risk profile typically provided for renewal purposes is shown on Table 1.
Table 1 | Example ofRisk Profile forProperty Portfolio
hBands ofV risk NhPh
10 500,000 528,008 166,891,508
2500,001 1,000,000 30,864 30,758,545
31,000,001 1,500,000 6,325 11,412,224
41,500,001 2,000,000 2,939 6,632,123
52,000,001 2,500,000 1,329 3,814,369
62,500,001 3,000,000 1,188 4,115,279
73,000,001 3,500,000 789 2,742,529
83,500,001 4,000,000 484 2,007,070
94,000,001 4,500,000 485 2,038,237
10 4,500,001 5,000,000 276 1,090,990
11 5,000,001 6,000,000 554 2,432,686
12 6,000,001 7,000,000 463 2,368,648
Note: All figures inEUR, fictive insurance company.
Source: Own calculations.
The idea behind the exposure rating is to estimate the expected gross loss per band and
by applying the exposure curves to estimate a split of the total expected gross loss between
the reinsurer and the cedant. Let´s denote the total gross loss, which is random variable,
arising from all risks from h-th band as Zh,
1
0;
h
risk N
hh
i
i
V


Z. The exposure curve G(m)
directly expresses the percentage of the retained amount by the cedant from the total gross
loss and the curve can be expressed for any i-th risk as
,
,
()
() ()
i ced
i ced
i
E
Gm r E

Z
Z, (1)
where m denotes the relative retention as percentage of value of risk (normalised retention)
defined as m = min(M / Vi ; 1). Further, M denotes a retention of given excess of loss treaty
layer and E(Zi.ced ) denotes the mean claim amount which occurred on i-th risk and only its
portion which is retained by cedant after impact of excess of loss reinsurance. The formu-
la (1) can be also generalised for h-th band as

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()
() ()
h
ced
h
E
Gm E
Z
Z, (2)
where ()
h
ced
EZdenotes mean retained claim amount by the cedant from all losses originating
from all risks allocated to h-th risk profile band (assuming unlimited reinsurance layer
capacity). Some of the curves commonly used by European reinsurers are demonstrated
on Figure 1.
2.1 Exposure curves and which curve tochoose
The theoretical background for construction of the curves will be described further,
however, at this stage some thoughts how to select the appropriate curve will be discussed.
The exposure curve is derived from distribution of degree of damage which is random
variable and will be further denoted as ,0;1xx and defined as
V
X
x, (3)
where X denotes a random variable describing gross amount (i.e. before application
of reinsurance) of individual claim and V denotes value of the respective risk on which
the given claim occurs.
Figure 1 | Total, Swiss Re and Lloyd´s Exposure Curves
Source: Guggisberg (2004), Bernegger (1997), own calculations
rced
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 m
Total Loss
Swiss 1
Swiss 2
Swiss 3
Swiss 4
Lloyd's

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It will be shown in further sections that the curves tending more towards to the diagonal
are suitable for portfolios characterised by total or higher degree of damage. Typical
example can be fire insurance for residential property risks where the fire usually causes
substantial loss related to the value of respective risk. For such type of risks Swiss 1 curve
would be suitable.
On the other hand, the more the curve runs to the outer area, the less probable the total
loss is and such portfolios are characterised rather by partial losses (lower average degree
of damage). Typical example for such risk would be fire industrial insurance. The industrial
complexes are usually well equipped by various types of fire protection measures and it can
be well expected that the fire loss would be relatively low compared to the value of given
risk. According to Guggisberg (2004) following aspects when selecting the curve must be
considered.
Perils covered intheportfolio
Fire typically causes more damage to an individual building than windstorm which in many
cases damages just the roofs. Gas explosion can completely destroy the whole building,
losses caused by floods and earthquakes are dependent on their strengths.
Class ofrisk
The gunpowder factories are more exposed to total loss than for example airport buildings
with good fire protection.
Size ofrisk
Fire often causes only partial damage to large buildings or industrial plants, whereas small
risks are more exposed to suffer total loss. The good indicator how to measure the size
(value) of the risks is by sums insured or MPL.
All the aspects above must be considered simultaneously and useful rough guide
for choice of the exposure curve is provided in Table 2.
Table 2 | Choice ofExposure Curve by Peril and Class ofRisk From
Curve tends
towards thediagonal
Curve runs
inthemiddle area
Curve runs
intheouter area
Evaluationbasis EML MPL SI
Fire
Risks with poor fire
protection
Risks with average fire
protection
Risks with good, above-
average fire protection
Personal lines Commercial lines Industrial lines
Farm building Industrial building Administrative building
Windstorm – Radio tower Office building
Hurricane Radio tower Office building –
Source: Guggisberg (2004)

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2.2 Steps inFLS exposure rating process
After receiving the risk profile (e.g. as shown in Table 1) the procedure of exposure rating
can be conducted in the following steps:
1. Setting mean h
Vfor each band either as mid of the h-th band or if aggregated value
of risk per band or per individual risks is provided then
1
h
risk N
h
i
hi
h
risk
V
VN

. (4)
2. Setting mean normalised retention h
m for each band h as
min ; 1
h
h
M
mV



. (5)
3. Choice of appropriate exposure curve for each band.
4. For each band calculating value of exposure curve function ()
h
Gm .
5. Estimating mean gross loss per band.f
As only gross premiums per band are usually provided, then the estimated mean total
gross loss per band shall be calculated as
ˆˆ
()
hhh
EPZq
, (6)
where
ˆh
q denotes estimated gross loss ratio for h-th band. This information can be
obtained directly from the client or based on market experience.
6. Estimating ceded loss (reinsurer´s share) per band from


ˆˆ
()1 ()
hhh
re
EGmEZZ
. (7)
7. Considering only one layer with retention M and unlimited capacity, then the esti-
mated mean total ceded loss into the layer can be expressed as
1
ˆˆ
() ()
H
h
re re
h
EE


ZZ
. (8)
for all bands h = {1, 2, ..., H}.
Assuming enough reinsurance capacity is obtained by the cedant, i.e. the highest
value of any risk is within the treaty capacity, then in case of more layers with
corresponding retentions denoted as ()sM, s = {1, 2, ..., S} the formula (9) can be
expressed for the h-th band and s-th layer as




(1) ()
()
()
ˆ() for
ˆ() ˆ
1 ( ) for .
sh sh h
sh
re sh h
GmGmE sS
E
GmE sS


Z
Z
Z (9)
Further, the mean aggregated loss ceded to the s-th layer can be expressed as
() ()
1
ˆˆ
() ()
H
ssh
re re
h
EE


ZZ
. (10)

136 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
2.3 Numerical example
The example in Table 3 follows the risk profile from Table 1 for retention of M = 1 Mio.
EUR, and treaty limit of L = 6 Mio. EUR. Unlimited number of free reinstatements is
further considered for simplification, the estimated loss ratio ˆh
q is flat for all h and amounts
to ˆh
q = 45%, the selected exposure curve is Swiss 44. Following all the steps above,
the gross premium for all risks is P = 236 Mio. EUR and corresponding estimated gross
loss ˆ( ) 106 Mio. EUREZ. The estimated mean ceded loss (i.e. risk reinsurance premium)
for selected parameters would be ˆ( ) 2.9 Mio. EUR
re
EZ which leads to the risk reinsurance
rate of 1.23% (results generously rounded, for more precise results please see Table 3).
Table 3 | Exposure Rating Example
hBands ofV risk NhPh(Z)E
ˆh
m()
h
Gm (Z )
h
re
E
ˆ
10 500,000 528,008 166,891,508 75,101,179 1.00 1.00 0
2500,001 1,000,000 30,864 30,758,545 13,841,345 1.00 1.00 0
31,0 00 ,0 01 1,500,00 0 6 ,325 11,412, 224 5,135, 501 0. 80 0 .95 231,553
41,500,001 2,000,000 2,939 6,632,123 2,984,455 0.57 0.89 336,341
52,000,001 2,500,000 1,329 3,814,369 1,716,466 0.44 0.84 281,016
62,500,001 3,000,000 1,188 4,115,279 1,851,875 0.36 0.80 375,824
73,000,001 3,500,000 789 2,742,529 1,234,138 0.31 0.77 286,311
83,500,001 4,000,000 484 2,007,070 903,181 0.27 0.74 233,651
94,000,001 4,500,000 485 2,038,237 917,207 0.24 0.72 258,066
10 4,500,001 5,000,000 276 1,090,990 490,946 0.21 0.69 150,688
11 5,000,001 6,000,000 554 2,432,686 1,094,709 0.18 0.66 368,139
12 6,000,001 7,000,000 463 2,368,648 1,065,892 0.15 0.63 395,192
Sum – – 573,704 236,304,208 106,336,894 – – 2,916,780
Source: Own calculations
The advantage of the basic exposure rating approach as described above is that it
can be relatively easily applied. However, the whole procedure allows significant degree
of subjectivity when choosing appropriate exposure curve and loss ratio. Further, basic
application of this method does not provide any characteristics of volatility or the loss
distribution, these problems are solved in the following sections.
4 Assumption for using one exposure curve made just for illustrative purposes. The curve is suitable
for portfolios with higher values of risk (commercial or industrial portfolios) and in practice it would
be used only for higher bands.

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2.4 Theoretical background
This section is further inly based on Bernegger (1997) where two-parameter family
of analytical functions for modelling exposure curves and loss distributions was introduced.
Despite the year of publishing (1997), it is probably still the most developed theoretical
concept describing exposure curves background and also the concept gives more freedom
to underwriters and pricing actuaries to define their own desired exposure curves as until
that time only few discrete curves were available in practice.
Besides to already in previous sections introduced random variable degree of damage x
and deterministic ratio of normalized retention denoted as m, the distribution function F(x)
is defined on interval 0, 1 5 and according to Bernegger (1997) its limited expected value
function is defined as

() min ,Lm E m

x. (11)
Further, the expected retained loss by the cedant is defined as () ()
ced
EVLmX and
the expected ceded loss as

 () (1)()
re
EVLLmX. The ratio of losses retained by cedant
is given by the relative limited expected value function () ()/(1)Gm Lm L which is also
known as exposure curve function and is expressed as






00
1
0
11
()
() (1) ( )
1
mm
Fx dx Fx dx
Lm
Gm LEx
Fx dx

 



. (12)
According to Bernegger (1997), the distribution function of normalized loss F(x) can be
derived from

1()
() ()
Fm
Gm Ex (13)
and with F(0) = 0 and G´(0) = 1/E(x) is the distribution function of degree of damage has
the form
11
() ()
101.
(0)
x
Fx Gx x
G






(14)
The expected degree of damage has the form
1
() (0)
EG

x (15)
and the probability of total loss is further defined as
(1)
(1) (1)()
(0)
G
pP G E
G

  

xx
. (16)
The properties of exposure curves are that they are concave and increasing functions
on the interval 0; 1 with G(0) = 0 and G(1) = 1 and also 0(1)()1PE xx
.
5 Any underinsurance is not assumed, i.e. incurred loss amount cannot be higher than
the corresponding value of risk.

138 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
2.5 Example ofbasic approach toconstructing piecewise linear
exposure curve
In order to demonstrate how exposure curve can be determined 20 claims were simulated from
lognormal distribution, where ~ (10.6 ; 3.6)LNX whose parameters correspond to mean
gross claim of E(X) = 250,000 and standard deviation of gross claim ( ) 1, 500, 000Var X
(all further monetary amounts are in EUR if not stated differently). The corresponding degree
of damage values were simulated independently from MBBEFD distribution with Swiss 36
curve parameters, the simulated values are shown in Table 4 and the empirical distribution
function of x on Figure 2. Further, from formula (13) and Table 4 the following ratio is received




00
11
() , 0
ˆ0.0979
()
mm
Fx dx Fx dx
Gm m
E

 

x. (17)
Table 4 | Exposure Rating Example
iX
ixiVi
170,184 0 .012 5,62 2,140
2365,443 0.062 5,872,593
3106,710 0.005 21,912,266
45,697 0.015 387,735
52,343,391 0.029 82,108,438
618,091 0.144 125,4 68
7139,658 0.102 1,365,561
833,822 0.062 545,521
95,838 0.309 18,904
10 39,327 0.023 1,679,923
11 348,997 0.072 4,816,840
12 13, 347 0.0 05 2 ,432,933
13 24 4 ,1 4 8 0 .0 4 2 5, 8 0 1,111
14 6,012 0.014 421,864
15 53,340 0.005 10,938,014
16 9,807 0.023 432,773
17 44,890 0.005 9,603,076
18 13, 03 0 0. 020 651, 36 4
19 324 0.007 45,265
20 63,227 1.000 63,227
ˆ()EX 196,264 0.098 7,742,251
()
^
Var x 504,197 0.218 17,846,246
Source: Own calculations
6 All parameters for probabilistic distributions of loss severity and degree of loss selected
for illustrative purposed, use of MBBEFD distribution will be discussed latter.

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The amount of 9.79% in denominator of formula (18) is the estimated mean degree
of damage and is represented by the area above the CDF curve on Figure 2, whilst e.g.
for m = 7% the numerator is represented by the area above the curve and bounded
by the dotted curve on its right side. For the relative retention of m = 7% the value of res-
pective exposure curve is


0.07
0
10.034
(0.07) 0.347
0.0979 0.0979
Fx dx
G



. (18)
Figure 2 | Empirical DistributionFunctionF(x)
Source: Own calculations
Remark 1: This EDF is not continuous, of course. It is step function in general. But it
is very difficult to show the steps in software. Therefore, this Figure 2 is a simplified view
only.
As m ∊ 0; 1, the values of corresponding piecewise linear exposure curve for all m are
determined in Table 5 and also shown graphically on Figure 3.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
F(x)

140 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
Table 5 | Calculated Values ofExposure Curve
mG(m)
< 0.0000; 0.0047) (1/0.0979)(0 + m)
< 0.0047; 0.0048) (1/0.0979)(0.0002 + 0.95m)
< 0.0048; 0.0049) (1/0.0979)(0.0005 + 0.90m)
< 0.0049; 0.0055) (1/0.0979)(0.0007 + 0.85m)
< 0.0048; 0.0072) (1/0.0979)(0.0010 + 0.80m)
< 0.0048; 0.0125) (1/0.0979)(0.0014 + 0.75m)
< 0.0048; 0.0142) (1/0.0979)(0.0020 + 0.70m)
< 0.0048; 0.0147) (1/0.0979)(0.0027 + 0.65m)
< 0.0048; 0.0200) (1/0.0979)(0.0034 + 0.60m)
< 0.0048; 0.0227) (1/0.0979)(0.0044 + 0.55m)
< 0.0048; 0.0234) (1/0.0979)(0.0056 + 0.50m)
< 0.0048; 0.0285) (1/0.0979)(0.0067 + 0.45m)
< 0.0048; 0.0421) (1/0.0979)(0.0082 + 0.40m)
< 0.0048; 0.0620) (1/0.0979)(0.0103 + 0.35m)
< 0.0048; 0.0622) (1/0.0979)(0.0134 + 0. 30m)
< 0.0048; 0.0725) (1/0.0979)(0.0165 + 0.25m)
< 0.0048; 0.1023) (1/0.0979)(0.0201 + 0.20m)
< 0.0048; 0.1442) (1/0.0979)(0.0252 + 0.15m)
< 0.0048; 0.3088) (1/0.0979)(0.0324 + 0.10m)
< 0.0048; 1.0000) (1/0.0979)(0.0479 + 0.05m)
Source: Own calculations, based onvalues inTable 4.
Figure 3 | Empirical DistributionFunctionG(m)
Source: Own calculations
Because we work with EDF, theRemark 1 is valid forthis EDF, too.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
G(m)
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% m

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3. MBBEFD DistributionApproach
Although the previously discussed approach is continuous, practical implementation without
further theoretical extension allows only construction of piecewise linear function which when
based only on limited number of observations is not always suitable for further calculations.
Bernegger (1997) came with an idea of implementing MBBEFD distribution class also
into insurance risk theory and proved it to be very appropriate for modelling probability
distribution of degree of damage on the interval x ∊ 0; 1. His approach is widely used by
European reinsurance actuaries.
In his paper Bernegger described parameters b ∊ 0, ∞), g ∊ 1, ∞) and normalised
retention m ∊ 0; 1 two parameters exposure curve in a form of


 

,
10
ln 1 ( 1) 11
ln
()
111
1
11
ln 101 11.
ln
bg
m
m
mgb
gm bg>
g
Gm
bbg g >
b
g b gb b
bb> b bg g>
gb


 


















 


(19)
Based on MBBEFD distributed degree of damage x and corresponding distribution func-
tion for degree of damage of
1
11
01(10)
1
1111
() 1( 1)
1111
1
110111.
(1) (1 )
x
x
x
xgb
xbg>
Fx gx
bxbgg>
bxb>bbg g>
gb gb



  







    


(20)
Further, density function of degree of damage is defined as

142 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683


2
1
2
1
010
111
1( 1)
()
ln( ) 1 1
(1)( 1)ln() 01 11
(1) (1 )
x
x
x
gb
gbg>
gx
fx
bb bg g>
bg bb b> b bg g >
gb gb


 











 


. (21)
and mean degree of damage is from Bernegger (1997) defined as

110
ln( ) 11
1
1
11
(0) 11
ln( ) ln( )
ln( )(1 ) 01 11
ln( )(1 )
gb
gbg>
g
Ebg
Gbg g >
bgg
gb b b> b bg g>
bgb
 











 


x (22)
and the total loss probability as
(1) 1
P( 1) (0)
G
pGg

 

x (23)
3.1 Exposure curve fitting
We know type of distribution, but we do not know the exact parameters values. It is
necessary to estimate parameters. Several ways how to estimate the b and g parameters
exist. Bernegger (1997) recommends methods of moments (MM), whilst use of maximum
likelihood method (ML) or least squares method on either empirical exposure curve or
distribution function of degree of damage F(x) might be also possible and will be further
tested.
Exposure curve fitting based onestimations of ˆ()Exand ˆ
P (MM)
Bernegger (1997) showed that for each given pair of functional p = 1 / q and E(x) fulfilling
condition

01pE x only one distribution function belonging to the MBBEFD
class exists. The parameter p can be computed directly from the formula (23) and the other
parameter b by solving the equations from (22) or iteratively for general case

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
ln( )(1 )
ln( )(1 )
gb b
Ebgb


x. (24)
This method will be further denoted as MM7.
Exposure curve fitting based onmaximum likelihood method (ML)
As the distribution of x is known, the maximum likelihood method for estimating the un-
known parameters b, g is also a suitable option. The likelihood function which needs to be
maximised has (from formula (21)) for the general case the following form:


1
2
1
11
(1)( 1)ln()
( , ) b,g
(1) (1 )
i
i
x
NN
ix
ii
bg bb
lbg f x
gb gb






 (25)
and its logarithm

1
2
1
1
(1)( 1)ln()
ln ( , ) ln
(1) (1 )
i
i
x
N
x
i
bg bb
lbg
gb gb





. (26)
The further step in this method would be finding maximum of ln / (xi | b, g) by solving
the set of equations:
ln ( , ) 0
ln ( , ) 0.
lbg
b
lbg
g




(27)
Unfortunately, the set of equations (27) cannot be solved analytically and unknown
parameters have to be calculated iteratively from formulas (25) or (26).
Exposure curve based onfitting thedistributionfunctionF(x) by non-linear
regression
Once the empirical distribution of degree of damage is obtained, t is possible to fit theoretical
distribution to empirical distribution one as described in the formula (21) and estimate
parameters b, g. The exposure curve is further derived by applying the estimated parameters
to formula (20). Advantage of this method is that the MBBEFD distribution parameters are
directly obtained.
Direct exposure curve fitting (DF MBBEFD)
Further alternative how to estimate the unknown parameters of MBBEFD is direct fit
of the function (20) based on the empirical exposure curves. Such approach gives very good
results and also allows for further stochastic modelling as the probabilistic distribution can
be fully parameterised.
7 MM from method of moments, however, use of such name is not exact in this case as p is not any
moment.

144 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
Direct exposure curve fitting by polynomial function(DF PL)
This is a task from non-linear regression and two parameters exposure curve in the following
form can be fitted:

2
12
2
,1
() for 0 1;0 () 1Gm mmm m Gm



 
(28)
Such curve gives a very good fit of the empirical exposure curve. Various approaches
as described above were compared on the example from the Figure 3 and are visualised
on the Figure 4. For parameterisation of all the curves below the solver included in MS Excel
2007 was used, namely the Newton iterative method to minimise the squared residuals or
to maximise the logarithm of likelihood function8.
Figure 4 | ComparisonofVarious Approaches toFitting theExposure Curves
Source: Own calculations
8 For parameterisation of the curves by the non-linear regression it would be better to use appropriate
techniques, like calculating the first estimations of parameters by Marquardt or Steepest Descendent
methods which are not so sensitive on the choice of initial parameters. These results can be further
used as input estimations for Gauss-Newton method which provides more accurate estimations.
The selected approach using MS Excel provides enough accuracy for this work as it is not the aim
to focus on non-linear regression techniques. Many details about non-linear regression can be found
e.g. in Härdle (1995).
G(m)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
G(m) Empirical G(m) MM G(m) ML G(m) Fitted F(x)
G(m) DF MBBEFD G(m) DF PF G(m) Swiss
m

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Table 6 | ComparisonofVarious Approaches toFitting theExposure Curves
Empirical MM ML Fitted F(x) DF MBBEFD DF PL Swiss 3
G´(0) 10.219 10.219 12.744 18. 041 15.685 n .a . 11.471
G´(1) 0.511 0.511 0.151 0.005 0.456 n.a. 0.375
p
ˆ0.050 0.050 0.012 0.000 0.029 n.a. 0.033
ˆ()Ex 0.098 0.098 0.078 0.055 0.064 n.a. 0.087
b
ˆn.a. 15.963 0.284 0.002 13.394 n.a. 3.669
g
ˆ20.000 20.000 84.630 3,724.619 34.375 n.a. 30.569
RSS n.a. 0.101 0.983 4.677 0.008 0.025 n.a.
Source: Own calculations9
As a common practise method for measuring the fit empirical and theoretical distri-
bution the residual sum of squares criterion is applied



2
1
() ()
n
Empirical i Fitted i
i
RSS G m G m . (29)
All the three methods MM, DF PF and DF MBBEFD provided very good results.
As the probabilistic loss distribution in analytical form cannot be derived from the DF
PL method, the method of moments would be the recommended option to get initial
estimates of unknown parameters and further DF MBBEFD method would be applied.
The parameters obtained from MM were also used as initial parameters for the method
of maximum likelihood. The method based on fitting the distribution function F(x) was
the least accurate. For comparison, also Swiss 3 curve is included in Figure 4 as initially
the observations were generated from its probabilistic distribution.
3.2 Swiss Re exposure curves as special case
Already before Bernegger (1997) published his paper the so called Swiss Re exposure
curves existed and were widely used. He also parameterized these curves with help
of MBBEFD distributions and after evaluating the estimations of parameters b, g he
formulated the dependency between the total loss probability p = 1/g and expected degree
of damage E(x). In the next step he defined a new sub-class of the one-parameter MBBEFD
exposure curves as:
,
() ()
cc
cbg
Gm G m
(30)
9 Method called DF PL is based on different parameterisation of exposure curve according
to the function described by formula (29), therefore only RSS is displayed. Based on this
parameterisation the estimated values of parameters are 1
ˆ0.060

 and 2
ˆ0.388

.

146 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
with
3.1 0.15(1 )
(0.78 0.12 )
()
()
cc
c
cc
c
bbce
ggce



 . (31)
For c = 0 the curve corresponds to the total loss distribution (diagonal on Figure1), the curves
defined by c = {1.5; 2.0; 3.0; 4.0} correspond to Swiss Re curves and c = 5 corresponds
to Lloyd´s curve used for heavy industrial business. The Table 7 shows the overview
of Swiss Re curves and recommendation for their use.
Table 7 | Overview ofSwiss Re Exposure Curves and their Parameters under MBBEFD
Exp.
Curve cb g P
(%)
E(x)
(%) Scope of application Basis Size of Risk
(CHF, 2004)
Size of Risk
(EUR, 2013)
n.a. n.a. n.a. n.a. n.a. OPC business
interruption MPL – –
Swiss 1 1.5 12.648 4.221 23.69 34.855 Personal lines SI < 400,000 340,000
n.a. n.a. n.a. n.a. n.a. OPC property damage/
business interruption MPL – –
n.a. n.a. n.a. n.a. n.a. OPC property damage MPL – –
Swiss 2 2.0 9.025 7.691 13.00 22.609 Commercial lines
(small- scale) SI < 1,000,000 850,000
Swiss 3 3. 0 3.6 69 30.569 3.27 8.718 Commercial lines
(medi um-s cale) SI < 2,000,000 1,700,000
3.1 3.299 35.559 2.81 7.891 Captive business
interruption MPL – –
3.4 2. 354 56.781 1.76 5.836
Captive property
damage/business
interruption
MPL – –
3.8 1.439 109.596 0.91 3.895 Captive property
damage MPL – –
Swiss 4 4.0 1.105 154.470 0.65 3.185 Industrial and large
commercial ? MPL > 2,000,000 1,700,000
Lloyd´s 5.0 0.247 992.275 0.10 1.215 Industry Top
location ––
up
to 8 n.a. n.a. n.a. n. a.
Large-scale industry/
multinational
companies
MPL – –
Source: Own calculations, Guggisberg (2004), Swiss Federal Statistical Office (2013)
The original Swiss Re curves, also called Gasser curves, were developed by Peter
Gasser based on the data of “Fire statistics of the Swiss Association of Cantonal Fire
Insurance Institutions” for the years 1959–1967 and due to the fact that they are based
on relative degree of damage (i.e. do not develop by inflation over years as it can be easily
assumed the same inflation for both values of risks and corresponding claim amounts
applies). The “Scope of application” column in Table 7 serves as an indication which
exposure curve should be used for which band from risk profile. Also the size of risk gives

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a good indication as in many cases the risk profiles are not separated by personal lines,
commercial lines, etc. The original size of risk indicator suggested by Guggisberg (2004)
was given in Swiss Francs (CHF) and in 2004 values (prices), therefore an alternative
in EUR and 2013 values was calculated and rounded generously (tens of thousands EUR).
For this recalculation Swiss consumer prices index and rate of exchange 0.81 EUR / 1
CHF were used.
What is also important to mention is that various CDFs of degrees of damage do not
smoothly converge to the cumulative probability at the point 1 at their maximums (examples
are shown for Swiss Re curves on Figure 5) and this is why there is a different form
of function F(x) for x < 1 and for x = 1 in formula (21), the CDF is not continuous function.
Figure 5 | CDFs ComparisonofDegree ofDamage forSwiss Re Curves
Note: F(x) forx<1.
Source: Own calculations, Guggisberg (2004), Bernegger (1997)
It can be further shown that for general cases of parameters b, g and for x ∊ ∞)
(i.e. without further restrictions10 for x as introduced in formula (21)) and where x is
MBBEFD distributed, the distribution function F(x), would not have maximum and
the CDF would converge to
10 By definition of exposure curves for CDF of degree of damage F(x = 1) = 1 applies (formula (20)),
therefore sharp step (discontinuity) in the tail of F(x) can be observed.
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
F(x)
x
Swiss1 Swiss2 Swiss3 Swiss4

148 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
1
lim ( ) 1 1
x
b
Fx gb


. (32)
It is also interesting to show some dependencies of selected characteristics on the para-
meter c, brief comparison is shown in Table 8. For each exposure curve 1,000,000
simulations of degree of damage were ran and various statistics on their basis calculated.
When comparing mean degree of damage for each curve from simulated sample in Table 8
and for theoretical values in Table 7 the results are very close and 1 Mio. simulations seem
to be enough. Also Var (x) is decreasing with increasing c parameter.
All distributions of Swiss Re curves are positively (right) skewed and the higher the c
parameter the more the right tail of the distribution is stretched from the centre. The moment
method for calculating the skewness was used as it is more suitable for resampling11.
Kurtosis measures how “peaked” the probabilistic distribution is and again moment method
of calculation was used and the higher the c parameter the higher kurtosis can be expected.
Table 8 | ComparisonofMBBEFD Distributions Characteristics ofx forSwiss Re Curves Parameters
 Swiss1 Swiss2 Swiss3 Swiss4
ˆ()Ex 0.34842 0.22580 0.08741 0.03161
^()Var x 0.15 6 5 4 0. 11119 0 . 0 4 0 5 6 0. 01110
Skewness 0.86352 1.63042 3.63288 7.02407
Kurtosis 1.98132 4.06 347 15.81750 58. 48 595
Source: Own calculations, Bernegger (1997)
4. Deriving Loss Severity Distributionand Frequency from MBBEFD
Exposure Curves
Once it is decided which exposure curve should be used for reinsurance modelling and its
parameters b, g are known, for any further stochastic modelling number of losses lossN and
the respective individual loss amounts X need to be simulated.
The following theory is based on selected h-th risk profile band, however, it can be
also used for any i-th single risk, where in the following formulas Vi and RPi would be used
instead of h
Vand ˆ()
h
EZrespectively.
From the collective risk model and from (16) the expected number of gross losses can
be expressed as

()/()
() ()(0)
()
hhh
loss
hh
hh h
EEE
EE
G
VE V



NZX
ZZ
x
(33)
and more general for losses exceeding given threshold U as
11 R software also offers Fisher method which is based on unbiased sample variance.

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Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683

>>
,
h
hh
loss loss
hhh
h
hh
UU
EE
VVV
EU
G
VV












X
Nx N
Z
(34)
where the derivative of exposure curve as function of normalised retention m is defined
in Bernegger (1997) as

1
110
111
ln( )(1 ( 1) )
() ln( ) 11
1
ln( )(1 ) 01 1 1.
ln( ) ( 1) (1 )
m
m
gb
gbg>
ggm
Gm bb bg g >
b
bgb b> b bg g>
gb g b gb

 










 



(35)
Further, for each threshold U the mean excess function can be calculated as

()1
>(>)
h
h
hh
hh
loss
U
EG
V
EUU
EU







Z
XX NX
(36)
and the conditional mean individual loss can be expressed as




1
>>
>
h
h
hh h h
hh
loss
U
EG
V
EU UEUUU
EU







Z
XX X X
NX
. (37)
Once the expected number of losses for each threshold is computed, the CDF for size
of the individual loss F(X) can be calculated iteratively. As it is not always suitable to derive
the CDF for all losses Sometimes only large losses are of interest. Let´s denote 0U as basis
threshold and elements of the vector

01 1
, , ..., D
UU U

U will define the elements
of vector

01 1
, , ..., D
XX X

X for which respective percentiles should be calculated
in D − 1 iterations. The conditional CDF of size of the loss in d-th iteration can be then
expressed as




0
0
1,
hh
loss d
hh
dd hh
loss
E>U
FX U >U
E>U
NX
X
NX
(38)
where d ∊ {0, 1, ..., D – 1}.

150 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
An alternative option of computing the number of expected claims above threshold
exists. This method is also iterative and in each iteration it is based on creation of artificial
layer with limit of one unit (e.g. 1 EUR, 1 CZK etc.) in excess of each dU, i.e. in each d-th
iteration an artificial layer with limit L = 1 ad retention M = dU is created. By applying
the exposure curve parameters G(m) and calculating the ceded loss into each of the artificial
layers according to formula (10) the expected number of claims above threshold is obtained
as from each claim 1 EUR or CZK is ceded. More formally the expected number of claims
above threshold can be written as

,1xs
()( >)
d
hhh
re U loss d
EEUZNX
, (39)
where ,1xs
()
d
h
re U
EZ denotes mean ceded loss to layer with limit of L = 1 and M = dU. The cal-
culation of F(X) is then identical to the previous method, i.e. according to the formula (39).
Once distributions of degree of damage and frequency are parameterised (for frequency
Poisson distribution will be assumed) the gross losses for each band can be simulated.
For simulating the losses from severity model the inverse transform method (also known
as probability integral transform method) can be applied. Assuming the degree of damage
distribution parameters are known and for each i-th loss x ~ MBBEFD(b,g) (i.e. all losses
have identical distribution of degree of damage) and r ~ uniform, than
1()F
xr
. (40)
Further, as size of any i-th loss Xi = xi . Vi , where value of risk Vi is deterministic and further
assuming that for all risks from h-th band Vi
h = h
V then the random loss for h-th band can
be expressed as
1() h
FV

Xr
. (41)
From (21), (33) and (32) the gross individual loss for h-th band is then defined as
11* 1
1
1 log 1
11
1
1.
1
h
b
h
bgb b
Vggb
b
Vgb
 






 













rr
X
r
(42)
5. Case Study
For the more complex and more realistic illustration of property exposure rating the risk
profile from Table 1 was analysed and for various bands various Swiss Re curves were
chosen as shown in Table 9 below.

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Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
Table 9 | Selected Swiss Re Curves by Risk Profile Bands
hBands ofV Exposure Curve
10 500,000 Swiss 1
2500,001 1,000,000 Swiss 2
31,000,001 1,500,000 Swiss 3
4–12 1,500,001 7,000,000 Swiss 4
Source: Own calculations
Further, for each of the bands of the risk profile the expected annual numbers of losses
were estimated and their amounts used as Poisson distribution parameters ˆh

for each h-th
band. The estimated ˆh

were calculated from formula (34) and their values are shown
in Table 10.
Table 10 | Expected Number ofLosses per Band
hBands ofV()
h
Gm λh
10 500,000 2.8690 861.8756
2500,001 1,000,000 4.4230 81.6270
31,000,001 1,500,000 11.4706 47.1257
41,500,001 2,000,000 31.3952 53.5415
52,000,001 2,500,000 31.3952 23.9506
62,500,001 3,000,000 31.3952 21.1418
73,000,001 3,500,000 31.3952 11.9219
83,500,001 4,000,000 31.3952 7.5615
94,000,001 4,500,000 31.3952 6.7755
10 4,500,001 5,000,000 31.3952 3.2449
11 5,000,001 6,000,000 31.3952 6.2488
12 6,000,001 7,000,000 31.3952 5.1483
Source: Own calculations
Further, for each h-th band 100,000 years were simulated, the claims were simulated
with threshold of 0U = 0, i.e. the simulations generated both attritional and large losses.
The reinsurance structure applied was the same as in the previously demonstrated example
ˆ

152 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
in Table 3, i.e. limit L = 6Mio., retention M = 1Mio. and unlimited number of free
reinstatements were applied. The main aim of non-proportional excess of loss reinsurance
is to protect a cedant against large single claims and decrease the volatility of retained
losses, the results based on Monte Carlo simulations are shown in Table 11. Based
on simulations the total estimated mean losses retained by cedant were slightly reduced
by reinsurance, this is due to the obvious fact valid in insurance business that the majority
of claims come from attritional losses, whilst only large losses are meant to be subject
of excess of loss reinsurance. It can be also seen in Table 1 that only losses from limited
number (2.6%) of risks in the portfolio can theoretically trigger the retention. The volatility
measured by sample standard deviation is again reduced due to the reinsurance (from 6.5
Mio. for gross loss to 5.4 Mio. for net loss). Table 11 also demonstrates, how volatile such
structure for reinsurer is, coefficient of variation of ceded losses amounts to 79.86% which
is relatively high, however, not unusual level in such type of reinsurance.
Table 11 | Selected Results ofExposure Rating forTotal Loss
Gross Ceded Net
Numbers ofsimulations 100,000 100,000 100,000
Mean 106,33 0,654 3,080,670 103, 249,98 4
Standard deviation 6,469,931 2,4 60,130 5,439,38 6
Coefficient ofvariation 6.08% 79.86% 5.27%
Minimum 82,339,861 0 82,339,861
Maximum 137, 4 8 7, 5 0 4 21,19 5, 537 12 8 , 7 8 4 , 5 02
VaR 0.9 114,784,490 6,500,002 110,301,821
VaR 0.99 122 ,441,24 0 10, 87 7,0 98 116, 2 37,14 0
VaR 0.995 124,324,079 12,054,375 117,634,710
VaR 0.996 124,900 , 576 12,381,433 118 , 07 7,675
Note: Estimations based onMonte Carlo simulations.
Source: Own calculations
Some of the estimated percentiles which are used in practice were further added
into Table 11 (denoted as VaR from “Value at Risk”). The 99.5-th percentile is often
used for Solvency II purposes, whilst the 99.6-th percentile has been traditionally used
by reinsurance buyers. When such rule should be applied in deciding about the number
of reinstatements the cedant needs, the 99.6-th percentile of ceded loss is slightly above
12 Mio. which would lead to recommendation to buy one or maximum two reinstatements
(from 99.5-th percentile of ceded loss in Table 11 the 1st reinstatement would be fully
exhausted only with 0.5% probability, which very low). For illustration the cumulative

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distribution functions based on Monte Carlo simulations are shown on Figure 6, the limited
effect of such type of reinsurance on mean retained loss is clearly visible.
Figure 6 | CDFs ComparisonofGross, Ceded and Net Loss
Source: Own calculations
6. Conclusion
Exposure rating is a powerful alternative to traditional modelling approaches based
on historical claims data and when such historical data of the cedant is not available it is
the only possible modelling approach which can be used. Such exposure based approach is
also very useful for situations when a reinsurance buyer changed its underwriting approach
and historical claims are not any longer representative for future projections.
Unfortunately, only limited number of papers on the topic of exposure rating exist and
each of the papers deals with slightly different issue. This paper provides comprehensive
description of the methodology and can serve as a guide how to construct exposure curves
and how to apply the methodology for the purpose of reinsurance modelling. All the methods
are further demonstrated on numerical examples.
One of the weaknesses of the property exposure rating methodology is a very high
degree of subjectivity when choosing an appropriate exposure curve. The choice of the curve
always requires an in-depth knowledge of the analysed portfolio. Further, the curves are
often based on very old data, although they are resistant to inflation, the insurance covers
(products) have changed historically.
1.0
0.8
0.6
0.4
0.2
0.0
0.0 30 60 90 120 150
F(z)
Z(in Mio.)
–––– Gross –––– Ceded ––– Net

154 Prague Economic Papers, 2019, 28(2), 129–154, https://doi.org/10.18267/j.pep.683
Very often a combined risk profile is provided which includes all homeowners,
commercial and also large industrial risks, in such situation it is very important to decide
by value of risks which curve should be used for which band. Sometimes it is also helpful
to blend the exposure curves and apply the combined curve on some band of risk profile
which includes combination of various types of risks. Further, for various bands might
be applied various estimated loss ratios, unfortunately, not enough information is usually
provided by the cedant to make such assumptions.
References
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Daykin, C. D., Pentikäinen, T., Pesonen, M. (1995). Practical Risk Theory forActuaries. Suffolk:
Chapman& Hall, ISBN 978-0412428500.
Guggisberg, D. (2004). Exposure Rating. Zurich: Swiss Re.
Härdle, W. (1995). Applied Nonparametric Regression. Cambridge: Cambridge University Press.
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Salzmann, E. R. (1963). Rating by Layer ofInsurance. Proceedings oftheCasualty Actuarial
Society, 50, 15–26, https://www.casact.org/pubs/proceed/proceed63/1963.pdf
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