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170 2014, XVII, 2 DOI: 10.15240/tul/001/2014-2-013
Introduction
A huge expansion of a competitive insurancemarket forms part of the transition process inthe former Socialist countries. New internationaltrends in securing its financial stability and newtypes of risk demand a high-powered developmentof actuarial science and its application ininsurance practice. The on-going EuropeanUnion Solvency II project, aimed at improvingthe quality of assessment of the solvency of aninsurer, places emphasis on the modelling ofrisk and on internal models for managing aninsurer's risk. This unavoidably leads to a morein depth use of the theoretical results ofactuarial science and to their further theoreticaldevelopment.
Improving the quality of premium calculationmethods is an effective factor in reducing theinsurance technical risk of an insurer. In doingso it takes into account more and more ratingfactors, thanks to which more homogeneousrating tariffs emerge allowing the possibility ofdetermining more equitable premiums. On theother hand when the insurer determines thepremium it has to contend with a smalleramount of information concerning the risk, forwhich the given tariff class is intended. If theinsurer has inadequate past data relating to thetariff class, it can make use of data from otherrisk classes which are to a given extent similar.
The insurance risk, for which there isa given tariff class, is called the individual risk.Each tariff class forms part of the wholeportfolio and thus each individual risk is part ofthe collective, i.e. portfolio, risk. On the onehand the insurer in general has at its disposala relatively large amount of statistical informationconcerning the collective risk. This informationis, however, more or less limited for pricingindividual risks, due to the wide variety of tariff
classes. An insurer usually has little statisticalinformation concerning individual risks andtherefore should use both sources of data. Thatis the fundamental idea behind credibilitytheory.
Back in 1918 Albert W. Whitney proposedthat the credible premium Pcred should bea linear combination of the individual premiumPind and the collective premium Pcol
Pcred = z . Pind + (1 – z) . Pcol (1)
where Pind is the average level of claim percontract in the given tariff class and Pcol is therespective mean for the whole insuranceportfolio. From a statistical point of view thecredible premium is the weighted arithmeticalaverage of the individual and collectivepremiums, where z is the relative weight of theindividual premium and (1 – z) that of thecollective premium. The weight z is referred toas the credibility factor. It takes values in theinterval ⟨0;1⟩ and expresses the level ofconfidence in the data for the given insuranceclass when determining the premium for thatclass. A confidence level (weight) of 0 isassigned to individual risk data which have nouse for determining the premium and a level of1 is given to those individual risk data whichcompletely suffice in order to determine thepremium for that risk class. This means that, ifz = 0, we must completely rely on the data fromcomparable risks when determining thepremium for the given class. If z = 1, then wecan completely rely on the data for the givenindividual risk.
Bühlmann established the theoreticalfoundation of modern credibility theory, presentedas distribution free credibility estimation.Bühlmann and Straub (see [3]) generalizedBühlmann’s classical credibility model. Without
MULTIDIMENSIONAL CREDIBILITY MODELAND ITS APPLICATIONViera Pacáková, Erik ·oltés, Bohdan Linda
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doubt the Bühlmann-Straub model is the mostimportant model in credibility theory. Originallyit was aimed at the determination of the crediblepremium, but after certain modifications it canalso be used to determine a reliable estimate ofclaims frequency or of the average amount ofclaim. Its importance lies in the possibility ofusing it in a wide range of applications in theactuarial practice of non-life insurance, but itcan also be used in life insurance and inreinsurance. Besides that the Bühlmann-Straubmodel represents a base for further more specificmodels, such as the hierarchical, multidimen-sional or regression credibility models. In thispaper we will consider a generalisation of theone-dimensional Bühlmann-Straub credibilitymodel to the multidimensional credibility model.
1. Generalisation of the One-Dimensional Bühlmann-StraubModel to the MultidimensionalCredibility Model
The aim of this part of article is to brieflydescribe the mathematical and statistical basisfor one-dimensional Bühlmann-Straub model(for details see [1] to [7]) and point out theanalogy of multidimensional Bühlmann-Straubmodel with the original Bühlmann-Straubmodel. This part of the article also shows thepractical use of multi-dimensional Bühlmann-Straub model in insurance practice.
Let us suppose that the portfolio ofcontracts is divided into I risk classes, for whichwe want to determine the net premium. Let theith risk class (i = 1, 2, ..., I) be characterised bythe individual risk profile ϑi, which is theoutcome of a random variable θi. To calculatethe net premium for each individual risk classwe will make use of the quantities:
Sij – the aggregate claim amount for risk i inyear j (j = 1, 2, ..., n),
wij – the number of contracts for risk i inyear j,
SijXij = –––– – the average annual level of
wij
claim for risk i in year j,which are in general known.
The one dimensional Bühlmann-Straubcredibility model stems from the followingassumptions Bühlmann, Straub [3], Herzog [6],Pacáková [8].
BS1: Conditional on the value of θi the Xij forj = 1, 2, ..., n are independent with parameters:
E[Xij|θi] = µ(θi) (2)
σ2(θi)Var[Xij|θi] = ––––– (3)
wij
BS2: The pairs (θ1, X1), (θ2, X2) ... areindependent and θ1, θ2 are independent andidentically distributed.
If the assumptions BS1 a BS2 are met, thecredible estimate of the net premium is
(4)
where the average annual level of claim percontract from the ith risk class
(5)
is the estimate of the individual premium, µ0 isthe collective premium and the credibility factorhas the form
(6)
The estimation of the so-called structuralparameters µ0, σ2 a τ2 plays a fundamental rolein credibility theory. Definitions of the structuralparameters and their interpretation are asfollows:
Structural parameter Interpretation
collective premium
the variance of theaverage annual claimamounts within a group
the inter-group varianceof the average annualclaim amounts
The credibility estimate (4), in which wereplace the unknown collective premium µ0 bythe estimate
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(7)
is the so-called homogeneous credibilityestimate with the form (see [4])
(8)
The estimate (7) of the collective premiumisthe weighted arithmetical average of theindividual premiums X
—i, where the weights are
the credibility factors zi.Assuming that there is the same number of
contracts in each risk class in each year, thecredible premium defined by formula (8) shouldcover the claims outgo for the whole portfolioand for the whole period of n previous yearsconsidered. This result is expressed in [1] bythe balance property:
(9)
For some classes of business (liability,commercial fire) a high percentage of the totalclaim amount comes from a small number ofvery large claims. In these cases a relativelysmall number of claims, which are “large”compared with the “standard” claim, have a largeweight when the premium rate is determined. Ittherefore makes sense to estimate the averageclaim amount separately for “large” and for“standard” claims. If in fact we were to calculatethe credible premium using the one-dimensionalBühlmann-Straub model separately for “large”and “standard” claims, we would not take intoaccount any dependency between them.Therefore it is more appropriate to estimate thevector of net premiums, which in this case isthe two-dimensional vector
The whole vector of net premiums for individualrisk i can be estimated by the multidimensionalcredibility model, which is a generalisation ofthe one-dimensional Bühlmann-Straub model.
Another example of the use of the multidi-mensional model would be the calculation ofthe credible premium for the separate riskclasses of a whole portfolio and the similar risk
classes of other insurers. If, therefore, aninsurer disposes of data, even from anotherinsurer, either broken-down or in total, it canuse the multidimensional credibility model.Thus, thanks to it, we can calculate thepremium for a given risk class using either pastdata for that class or past data for the whole ofthe portfolio or even historical data for the givenrisk class or for the whole portfolio of similarrisks of another insurer.
Whereas in the one-dimensionalBühlmann-Straub credibility model we estimatethe net premium µ(θi) for the risk i (i = 1, 2, ..., I),in the multidimensional Bühlmann-Straubcredibility model we estimate for risk i the wholevector of net premiums
(10)
In the one-dimensional Bühlmann-Straubcredibility model we observe the quantities Xij(the average annual claim amount for risk i inyear j) with weights wij (the number of contractsfor risk i in year j). In the multidimensionalBühlmann-Straub credibility model we observethe vector of average annual claim amounts forrisk i in year j
(11)
and the weights are the vector of the number ofcontracts for risk i in year j
(12)
The multidimensional Bühlmann-Straubcredibility model stems from the followingassumptions (for details see [2]):
MBS1: Conditional on the value of θ1 thevalues Xij for j = 1, 2, ..., n are independent witha p-dimensional vector of mean values:
(13)
and with a covariance matrix of type p × p.
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(14)
MBS2: The pairs ,
where the are indepen-dent and the θ1, θ2, ... are independent andidentically distributed.
The covariance matrix (14) is diagonal, thusit is assumed that the components of vector Xijare independent. In some situations indepen-dence is not assured. For example, if we wantto use the multidimensional credibility model toestimate the claim frequencies of larger claimsand the claim frequencies of standard claims,the assumption of their independence is notmet. The claim frequencies of large claims andof standard claims relate to the same number ofcontracts and therefore the weightsare equal. In the case of equality of the weights
we can replace the so-called standard assumption by (see [1]) the so-called alternative assumption:
In assumption MBS1 in place of thecovariance matrix (14) we can consider thecovariance matrix
(15)
where the weights for all the components of thevector Xij are equal, i.e.
The standard and alternative assumptionsdo not in fact cover all possible situations, butthey do include the majority of cases whicharise in practice. Subject to these conditions wecan generalise formulae (4) to (6), relating tothe one-dimensional credibility estimate, usingthe multidimensional credibility estimatedefined later by formulae (16) to (21).
If assumptions MBS1 a MBS2 are met, wecan express the credibility estimate of the netpremium as follows
(16)
where Bi is the individual estimate and thecollective estimate of the vector µ(θi). Zi is thecredibility matrix and it can be shown that, aftergeneralising formula (6) for the credibility factor,the following applies
(17)
It is clear that the vector of average annualclaim amounts for the ith risk class representsthe individual estimate of the vector µ(θi) andtherefore
(18)
here, for k = 1, 2, ..., p, the kth element of thevector Bi is
(19)
We can therefore express the vector Bi bythe formula
(20)
where the matrix of weights has the elements
(21)
If standard assumption (14) is met, the matrix
has elements
(22)
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If the covariance matrix of the vectormeets the alternative assumption (15), thefollowing applies
(23)
The multidimensional credibility model hasthree structural parameter vectors , S and Twhich are analogous to the structural para-meters µ0, σ2 and τ2 of the one-dimensionalBühlmann-Straub credibility model. Theirdefinitions are as follows:
The matrices of structural parameters
is the vector of collective premiums
is the covariance matrix of the vector
is the covariance matrix of the collective premium vector
We obtain the homogeneous credibility estimate
if in formula (16) we replace theunknown vector µ of the collective premiums byits estimate
(24)
Similar to the one-dimensional Bühlmann-Straub credibility model, also in the multidi-mensional model the homogeneous estimateand the individual elements of the vector
satisfy the balance property
(25)
To compare the accuracy of credibilityestimates, collective estimate and individualestimates, there are used quadratic losses. Thequadratic loss of the credible premium is zi-multiple of the quadratic loss of the individualpremium and (1 – zi)-multiple of the quadraticloss of the collective premium (see [9]). Since ziand (1 – zi) take values from interval ⟨0;1⟩, sothe quadratic loss of the credibility premium isless or at the most equal to quadratic losses ofthe collective and the individual premium.
2. Estimation of the StructuralParameters of the One-Dimensional Bühlmann-Strauband the MultidimensionalCredibility Models
In the one-dimensional Bühlmann-Straubcredibility model it is necessary to estimate, notonly the collective premium as shown informula (8), but also the structural parametersof variability σ2 and τ2. The structural parameterσ2 is the mean of the within group variances σ2(θi
). As the estimate of the variance σ2(θi) in
the ith risk class we can take the sample variance
(26)
It is possible to estimate the mean of thesevariances as the arithmetic mean. As reportedby Bühlmann and Gisler [1] the optimal weights
should be proportional to .
This expression depends on the moments up tothe fourth order. These moments are unknownand their estimation requires further structuralparameters, therefore such a procedure is notappropriate. In the special case, where Xij areconditionally on θi normally distributed, theabove mentioned expression does not dependon i. Therefore it is optimal to use the equalweights for all partial variances. On the one hand,the most of insurance data are not normallydistributed, on the other hand, weights, whichwould be better, are not known. For this reason,Bühlmann and Straub [3] suggested toestimate the structural parameter σ2 using thesimple arithmetic mean of the individual
variances :
(27)
The characteristic is an unbiased andconsistent estimate of the structural parameterσ2. A reader can find proofs of these propertiesin [1] on the page 94.
An unbiased and, provided there is nodominant risk class in the portfolio, alsoconsistent estimate of the parameter τ2 (seee.g. [7]) is given by
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(28)
where
(29)
(30)
And
(31)
Remark:If the situation arises, that is negative,
differences between the risk classes are notdemonstrable. In this case we set theparameter τ2 to 0. The credibility factors (6)then equal 1 and the credibility premiums for allthe risk classes are the same and are equal tothe average annual claim amount for the wholeportfolio (31).
So that the estimate of the structural para-meter τ2 can also be used in the situation describedin the remark, the estimate of the structuralparameter τ2 is derived from the formula
(32)
The homogeneous credibility estimate inthe multidimensional credibility model alreadyhas incorporated in it an estimate of the vectorµ in accordance with formula (24). Hence wewill concentrate only on estimation of the matrixof the structural parameters S and T.
To estimate the diagonal elements of thematrices S a T we can use analogousapproaches to those in the case of the one-dimensional Bühlmann-Straub model. Thediagonal elements of the matrix S are estimatedin a similar way to that of the parameter σ2 inthe Bühlmann-Straub model, in accordancewith formula (27) thus:
(33)
The diagonal elements of the matrix T againcan be estimated similarly to the structuralparameter τ2 in the Bühlmann-Straub model.Analogously to formulae (32) and (28) to (31)we then get
(34)
where
(35)
and
(36)
(37)
(38)
We will denote the non-diagonal element inthe kth row and the lth column of the matrix T byτkl. First we estimate τkl using the weights
thus:
(39)
Similarly we estimate τkl using the weights
:
(40)
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where
(41)
The matrix T is the covariance matrix,which means that its diagonal elements are thevariances τ2
k and its non-diagonal elements arethe covariances τkl. By taking the ratio of thecovariance τkl and the product of the standarddeviations τk . τl we get the correlation coefficient,which takes values in the interval ⟨–1; 1⟩. Onthe basis of this it must be the case that |τkl|≤τk . τl.If this condition is met, the simple arithmeticmean of the estimates (39) and (40) is anestimate of the covariance τkl. In general thefollowing estimate is proposed (see [1])
(42)
To estimate the structural parameters of theone-dimensional and also the multidimensionalBühlmann-Straub credibility model we can usestatistical methods, for which we can findprocedures in statistical programming packages.We will concentrate only on the multidimen-sional credibility model. So the kth diagonalelement (33) of the matrix S is the simplearithmetic mean of the sample variances
(43)
where
(44)
in each of the risk groups, relating to the kth
element of the estimated vector
Let us modify the elements of matrix T. Wecan rewrite the characteristics T(k) in the form
(45)
where is the sample variance of the
variable B(k) with relative weights :
(46)
We then know that in accordance withformula (45) we can express the estimate (35)of the diagonal element τ2
k of the matrix T by theformula
(47)
In the case of the estimates (39), (40) of thenon-diagonal elements τkl of the matrix T it isagain the case that
(48)
(49)
where is the sample cova-
riance between variables B(k), B(l) with relative
weights :
(50)
and is the sample covarian-
ce between variables B(k), B(l) with relative
weights :
(51)
For k = 1 we get the sample variance (46)and the sample covariance (50) (for i = 1, 2, ..., p)from the first row of the sample covariancematrix of variables B(1), B(2), ..., B(p) making use
of the relative weights . Similarly for k = 2
.
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we can read off the sample variance (46) andsample covariance (50) (for l = 1, 2, ..., p) fromthe second row of the sample covariancematrix of variables B(1), B(2), ..., B(p) by making
use of the relative weights . Analogously
we can obtain the sample variance (46) andsample covariance (50) for k = 3, 4, ..., p. Fromthese elements we can then develop the matrix
(52)
whose kth row is the kth row of the samplecovariance matrix of variables B(1), B(2), ..., B(p)
making use of the relative weights .
If we look at formula (48) and also formula(49) we see that the estimates of the non-diagonal elements of the kth row of the matrix
T are a multiple of the kth row of the
matrix SB. From formula (47) it is obvious thatthe estimate of the kth diagonal element of
the matrix T is a multiple of the diffe-
rence between the kth diagonal element of thematrix SB and the kth diagonal element of thematrix . Given that the matrix is diagonal
with elements we can define
the matrix
(53)
where ⊗ is the Hadamard matrix product andmatrix C of type p × p has elements
(54)
An estimate of the covariance matrix T isthen the matrix
(55)
If any of its diagonal elements are negative,then in accordance with formula (34) wereplace them by the value 0 and if any of itsnon-diagonal elements do not meet the
condition then in accordance with
formula (42) we replace them by the value
sgn
3. Example of ApplicationAn unnamed insurance company divides itsMTPL portfolio into eight tariff classes. For thistype of insurance the insurer has available notonly its own data, but also summary data fromother companies. Specifically for each riskclass it has available the following statistics:
X–
i – the average claim amount (in €), percontract year, for a period ofn years,
– the sample standard deviation ofthe average claim amounts for theperiod of n years,
wi• – the number of contracts for the periodof n years.
These data are set out in Table 1.The actuary has the task of setting the net
premium for the next insurance year. Given thatone expects a significant dependence betweenthe own company data and that of the otherinsurers, it is desirable to use themultidimensional credibility model to set the netpremium for each tariff class. In order to get thecredibility estimate of the net premium usingthe multidimensional model we need to carryout the calculations as shown in Table 2.
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Tab. 1: Own data and data from other insurers
Source: own
Tab. 2: Table of calculations
Source: own calculation
By taking the simple arithmetic means of
the sample variances and the simple
arithmetic means of the sample varianceswe obtain estimates of the diagonal elements ofthe matrix S
An estimate of the matrix S is therefore thematrix
To avoid the relatively lengthy calculationsto estimate matrix T we will use the adjustmentswhich we made earlier in the article (expressions
(45) to (52)). To calculate the elements ofmatrix (52) we can use several of statisticalsoftware applications. To estimate matrix T inthis article we use correlation analysisprocedure in SAS Enterprise Guide application.
First we estimate the covariance matrix ofvariables B(1) a B(2) (the values are shown inTables 1 and 2), whereby we make use of the
relative weights . Similarly we estimate the
covariance matrix of variables B(1) a B(2)
making use of the relative weights . We thus
get the covariance matrices shown in Table 3.
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Given that the resulting estimates of thediagonal elements (variances) of the matrix Tare positive and the product of their squareroots (standard deviations) is greater than theestimate of the non-diagonal element(covariance), an estimate of the matrix T isgiven by the matrix
We can use the matrices S and T toestimate the credibility matrix. In accordancewith formulae (17) and (22) we get thecredibility matrices shown in Table 4. Thecredibility matrix relating to the ith tariff classrepresents the weight for the vector of
individual premiums Bi. Its complement to theunit matrix represents the weight for the vectorof collective premiums. In accordance withformula (24) its estimate is the vector
Using the described approach of weightingthe vector of individual premiums and estimatingthe vector of collective premiums in accordancewith formula (16) we get the homogeneouscredibility estimates of the vector of netpremiums for each tariff class. These estimatesare shown in the last column of Table 4.
Tab. 3: The covariance matrix of variables B(1), B(2) using the relative weights and the covariance matrix of variables B(1), B(2) using the relative weights
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Understandably of priority interest to the insurerare the net premium rates for the next insuranceyear in respect of the individual risk classes inits own portfolio. These rates are the firstelements of the vectors for i = 1, 2, ..., 8, which are shown in the lastcolumn of Table 4. The given credibility netpremium for a particular risk class takes intoaccount not only data relating to the claimsexperience of that class for the foregoing yearsand the past data for similar risk classes, whichtogether form the insurer’s own portfolio, butalso data for similar risk classes as well as thewhole portfolio of other insurers.
Although we cannot interpret the credibilitymatrix as simply as we could the credibilityfactors in the one-dimensional credibilitymodel, we can say that the insurer when settingits premiums can the least rely on its own datafor the particular risk classes in the case of thefirst (i = 1) and last (i = 8) tariff classes, for which
the first diagonal elements (z11 = 0.317 and z11 = 0.358) of the credibility matrix Zi are thesmallest. These elements of the matrix Zireflect the fact that the first and last tariffclasses of the insurer possess the least amountof historical data and the- refore greater reliance has to be placed on thedata for the insurer’s other classes of contractor on other insurers’ data.
On the other hand when setting the pre-mium the most reliable own data (z11 = 0.898)relate to the fourth tariff class, where the insurerhas the most data For this reasonthe difference between the credibility premium
for the fourth risk class and the
corresponding individual premium B(1)4 = 78 is
negligible. On the other hand for the first andlast tariff classes the difference between thecredibility and individual premium is notnegligible. In the case of the first risk class the
Tab. 4: The credibility matrices and the net premium vectors
Source: own calculation
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credibility premium is some € 6 higher than theindividual premium and in the case of the lastclasses some € 15 lower. A graphical com-
parison of the individual, collective andcredibility premiums for each of the risk classesof the insurer’s portfolio is shown in Figure 1.
Fig. 1: A comparison of the individual, collective and credibility premiums for each of the risk classes of the insurer’s portfolio
Source: own calculation and processing in Microsoft Excel
An important property of the credibility premium is the balance property. In order to verify it wefirst rewrite formula (25) as follows
On the right-hand side of the formula wehave the average annual claim amount percontract in the portfolio for the whole n yearperiod. In the case of our example portfolioB(1)—
= € 84.290 and for the other insurers B(2)—
= € 85.658. The sum on the left-hand side iscalculated separately for the own data and theother insurers’ data in Table 5.
From Table 5 it is clear that in the portfoliowith eight tariff classes the average annualclaim amount per contract over the past n yearsis equal to the average credibility premium (ifthe number of contracts in each tariff class isunchanged). For the own portfolio the averagevalue is on the level of € 84.290 and for theother insurers’ portfolio on the level of € 85.658.
The aim of the application of the multidimen-sional Bühlmann-Straub credibility model wasto show how it is possible in determining the net
premium for the individual risk class of theheterogeneous risk portfolio use three differentsources of information from the previousinsurance period: data from the relevant riskclass, information about the claims record inother risk classes of the own portfolio and infor-mation about the claims record in risk classesof others insurance companies. As mentionedabove (without proof and with reference to thesource), the quadratic loss of the credibilitypremium in all tariff classes is smaller than thequadratic loss of the collective premium, as wellas the quadratic loss of the individual premium.Therefore credibility premium provides a betterestimate of the future costs of insurance lossesas individual or collective estimate. Moreover,this estimate satisfies the balance property.However, it must be said, that in case of thelarge number of insurance contracts the
net p
rem
ium
(in
€)
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differences between individual and crediblepremiums are negligible, as shown by thementioned application in case of risk classes 3 to 6.
Conclusion
Credibility theory belongs amongst the so-called “experience rating” techniques andserves for calculating and continuouslyupdating the premium. Similar to Bayesianestimates, the estimates gained from credibilitymodels also emerge from the a priori data froma portfolio of contracts and from individual data.As compared with Bayesian estimates thecredibility estimates of the premium arerelatively straightforward, as they do not requirethe estimation of the whole a priori distribution,but only its first two moments. Further thecredibility premiums are easy to interpret.Thanks to this credibility theory has manypossible applications in actuarial work.
Without doubt the most important model inmodern credibility theory is the Bühlmann-Straub model which can have wide applicationsand represents a base for further more specificmodels. One such model is themultidimensional model, to which we havedevoted this paper. In the paper we havementioned the generalisation of the one-dimensional Bühlmann-Straub credibility modelto the multidimensional credibility model. Givenits use is mainly in non-life insurance and
a significant proportion of the written premiumsin non-life insurance relate to MTPL we haveapplied the multidimensional credibility modelto a portfolio of such contracts. In order tominimise the calculation overhead inherent inthe multidimensional credibility model we haveadjusted the estimates of the structuralparameters so that we can use the methodsand approaches of statistical induction. Thanksto these adjustments we have been able to usein our example application the procedures inthe SAS Enterprise Guide statistical software.From the example application it ensuesamongst other things that the actuary shouldreach for the methods of credibility theory inparticular where there is a relatively smallamount of data relating to the insured risk. Atthe current time, when non-life insurers,particularly with respect to MPTL, areintroducing new tariff factors and the degree ofportfolio segmentation is increasing, a lack ofstatistical data for assessing risk is quitecommon. But not even several years’experience with a given tariff class arenecessarily sufficient to value the insurancerisk. Insurance mathematicians shouldtherefore not ignore the data from thecompanies’ own similar contracts or data fromother sources (e.g. from other insurers).Credibility models are the tool that can be usedwith such data sources when setting the netpremium. Credibility models permit the settingof a credibility premium, which in the observed
Tab. 5: Calculation table for verifying the balance property
Source: own calculation
,
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past period of n years (on the assumption of anunchanged number of contracts in each tariffclass) would cover the total claim payments forthe whole portfolio. Thanks also to this so-called balance property the credibility premiummeet the conditions for use in insurancepractice.
The paper is written in the framework of theprojects VEGA 1/0806/14 SCR calculation tocover the risks of non-life insurance inaccordance with practical needs and projectGA âR 402/09/1866 Modelling, simulation andmanagement of insurance risks.
References[1] BÜHLMANN, H., GISLER, A. A Course inCredibility Theory and its Applications. Berlin:Springer, 2005. ISBN 3540257535.[2] BÜHLMANN, H., GISLER, A., KOLLÖFFEL, D.Multidimensional Credibility applied to estimatingthe frequency of big claims. ASTIN Bulletin[online]. ETH Zürich and Winterthur InsuranceCompany, 2003. [cit. 2010-10-22]. 34 p. (PDF).Available from: http://www.actuaries.org/ASTIN/Colloquia/Berlin/Buhlmann_Gisler_Kolloffel.pdf.[3] BÜHLMANN, H., STRAUB, E. Glaubwürdigkeitfür Schadensätze. Mitteilungen der VereinigungSchweizerischer Versicherungs-mathematiker.1970, Vol. 70, pp. 111–133.[4] DANNENBURG, D.R. Basic actuarial credibilitymodels – Evaluations and extensions. Amsterdam,1996. Dissertation thesis (PhD.). TinbergenInstitute.[5] GOOVAERTS, M.J., KAAS, R., VANHEERWAARDEN, A.E., Bauwelinck, T. Effective
Actuarial Methods. Amsterdam: North-Holland,1990. ISBN 978-0444883995.[6] HERZOG, T.N. Introduction to credibility theory.3rd ed. Winsted: ACTEX Publications, 1999. ISBN978-1566983747.[7] KAAS, R., GOOVAERTS, M. J., DHAENE, J.,DENUIT, M. Modern actuarial risk theory. Boston:Kluwer academic publishers, 2001. ISBN 1-4020-2952-7.[8] PACÁKOVÁ, V. Aplikovaná poistná ‰tatistika.Bratislava: IURA Edition, 2004. ISBN 80-8078-004-8.[9] ·OLTÉS, E. Modely kredibility na v˘poãetpoistného. Bratislava: Vydavateºstvo Ekonóm,2009. ISBN 978-80-225-2798-9.
prof. RNDr. Viera Pacáková, PhD. University of Pardubice
Faculty of Economics and AdministrationInstitute of Mathematics
and Quantitative [email protected]
doc. Mgr. Erik ·oltés, PhD.University of Economics in Bratislava
Faculty of Economic InformaticsDepartment of Statistics
doc. RNDr. Bohdan Linda, PhD.University of Pardubice
Faculty of Economics and AdministrationInstitute of Mathematics
and Quantitative [email protected]
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Finance
184 2014, XVII, 2
Abstract
MULTIDIMENSIONAL CREDIBILITY MODEL AND ITS APPLICATIONViera Pacáková, Erik ·oltés, Bohdan Linda
Solvency II project places emphasis on the modelling and management of risks of the insurancecompanies. This requires further improvement in actuarial methods and their application ininsurance practice. Improving the quality of premium calculation methods is an effective factor inreducing the insurance technical risk of an insurer. Presentation the methods of premiumcalculation and its permanent updating is the aim of this article.
Credibility theory is an experience rating technique to determine premiums, claim frequenciesor claim sizes. Credibility models are based on the realistic concept of a heterogeneous insuranceportfolio. Therefore, two sources of information are used in the calculation of the credibilityestimators for the individual risk: typically little knowledge about the individual risk and quiteextensive statistical information about entire portfolio. The most important model in the credibilitytheory is Bühlmann-Straub model. This model has a wide range of possibilities to be used in praxismainly in general insurance. Besides that this model is a basis for other more specific models suchas hierarchical, multidimensional or regression credibility models. In this article we deal withgeneralisation of one-dimensional Bühlmann-Straub credibility model to the multidimensionalcredibility model. We mainly focus on estimation of so-called structural parameters and usage ofSAS Enterprise Guide application when estimating. The multidimensional Bühlmann-Straubcredibility model is applied based the real data in motor vehicle third party liability insurance.
Key Words: Credibility premium, Bühlmann-Straub model, multidimensional credibility model,estimation of structural parameters, motor vehicle third party liability insurance.
JEL Classification: C11, C13, G22.
DOI: 10.15240/tul/001/2014-2-013
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