Actuarial Models for Valuation of Critical Illness Insurance Products


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1 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 Actuarial Models for Valuatio of Critical Illess Isurace Products P. Jidrová, V. Pacáková Abstract Critical illess isurace (CII or critical illess cover is a isurace product, where the isurer is cotracted to typically make a lump sum cash paymet if the policyholder is diagosed with oe of the critical illesses listed i the isurace policy. The schedule of isured illesses varies betwee isurace compaies. The basis for the valuatio of each isurace product, ot excludig CII products, is the kowledge of probability of isurace evet specified i the policy. This article aims to explai ad apply methods of classical ad ayesia statistical iferece how to estimate the probability of critical evet diagoses i the Slovak isurace compaies, specifically for the me ad wome ad for various age groups. The estimated evet probabilities are subsequetly used for settig risk premiums i the homogeeous groups by sex ad age. The idividual risk model has bee used for calculatio of premiums. Data submitted by the Decree No. 0/008 to the Natioal ak of Slovakia from Slovak isurace compaies givig exposure to the critical illess risk have bee used for all calculatios i the article. Keywords ayesia estimatio, biomial/beta model, evet probability, idividual risk model, premium. I. INTRODUCTION HE critical illess isurace (CII first came to the scee T i South Africa early i the 1980s uder the ame of Dread Disease Isurace. However, before this, i the USA, Japa ad Israel some life isurace policies were exteded to cover cacer. CII has bee very popular i the UK. Although CII policies have bee issued sice the 1980s i the UK, the umber of policies icreased dramatically i the early 1990s. Curretly critical illess isurace is commo product of may isurace compaies aroud the world, although these isurace products vary i umber ad set up of diseases, they cover [1]. CII covers pay a isurace beefit if the isured perso suffers a serious coditio, depedig o the defiitios stipulated i the policy wordig, such as cacer, heart attack, P. Jidrová is with Istitute of Mathematics ad Quatitative Methods, Faculty of Ecoomics ad Admiistratio, Uiversity of Pardubice, Pardubice, Studetská 84, Pardubice, Czech Republic ( V. Pacáková is with Istitute of Mathematics ad Quatitative Methods, Faculty of Ecoomics ad Admiistratio, Uiversity of Pardubice, Pardubice, Studetská 84, Pardubice, Czech Republic ( stroke, coroary artery (bypass surgery or kidey failure. The umber of diseases covered varies cosiderably depedig o the market ad provider cocered. This kid of isurace products covers agaist the fiacial cosequeces of the serious coditio. People affected are give fiacial support to eable them to better maage their chaged circumstaces of life. Isurace products differ i their specificatios ad i premiums. I their creatio there is ecessary kowledge about the probabilities of claims that are covered by critical illess policy. These probabilities eed to kow for differet homogeeous groups of cliets. ecause critical illess cover oly idemifies the isured whe a dread disease is diagosed, umber of CII evets i homogeeous group of isured persos has biomial distributio i(;θ with parameters ad θ, where θ is probability of evet, which is which the diagosis is a critical evet uder the policy coditios. Article ivestigates the classical ad the ayesia estimators of the parameter θ of biomial distributio usig quadratic loss fuctio for homogeeous groups of isured persos ad presets usig of these estimators i premium calculatio. II. THE POINT ESTIMATION OF EVENT PROAILITY The classical approach to poit estimatio treats parameters as somethig fixed but ukow. The method of maximum likelihood provides estimators which are usually quite satisfactory. They have the desirable properties of beig cosistet, asymptotically efficiet for large samples uder quite geeral coditios. So the maximum likelihood method is the most frequetly used. The priciple of maximum likelihood tells us that we should use as our estimate that value which maximises the likelihood of the observed evet [], [3]. The essetial differece i the ayesia approach to iferece is that parameters are treated as radom variables ad therefore they have probability distributios. Suppose x x, x,..., x is a radom sample from a ( 1 populatio specified by desity fuctio f ( x /θ ad it is required to estimate parameterθ. y [4], [5], [6] prior iformatio about θ that we have before collectio of ay data is the prior distributio f ( θ which is probability desity fuctio or probability mass fuctio. The iformatio about θ provided by the sample data x ( x1, x,..., x is cotaied i the likelihood ISSN:
2 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 ( x / f ( x i / θ f θ (1 i 1 ayes theorem combies this iformatio with the f θ i the form iformatio cotaied i ( f ( ( x θ f ( θ f Θ θ / x ( f ( x θ f ( θ dθ θ which determies the posterior distributio. A useful way of expressig the posterior desity is to use proportioality. We ca write ( θ x f ( x θ f ( θ f / / (3 or simply posterior likelihood * prior. The posterior distributio cotais all available iformatio about θ ad therefore should be used for makig decisios, estimates or ifereces. The followig procedure for ayesia estimatio of the biomial parameter θ is explaied for example i [7] [11]. For estimatio of a biomial probability θ from a sigle observatio X with the prior distributio of θ beig beta with parameters α ad β, we will ivestigate the form of the posterior distributio of θ. Prior beta desity fuctio by assumptio ad omittig the costat is α ( θ θ ( 1 θ 1 β 1 f, 0 < θ < 1. Note that the uiform distributio o (0, 1 is a special case of the beta distributio with α 1 ad β 1. This correspods to the oiformative case. Omittig the costat likelihood is give by f x ( x / θ θ ( 1 θ x, (4 x 0,1,..., (5 where is umber of idepedet trials (i our case umber of policies ad x is umber of evets. y (3 we get the posterior desity of θ i the form x x α 1 β 1 α + x 1 β + x 1 ( / x θ ( 1 θ θ ( 1 θ θ ( 1 θ f θ (6 Apart of the appropriate costat it is the posterior beta desity fuctio of θ with ew parameters α α + x (7 β ' β + x (8 The ayesia estimator of θ, give the sample data x ( x1, x,..., x is the loss fuctio g (x, which miimizes the expected loss with respect to the posterior distributio [10]. There is oe very commoly used loss fuctio, called quadratic or squared loss. The quadratic loss is defied by ( g( x ; θ [ g( x θ ] L (9 y miimizig the quadratic loss the ayesia estimator of θ ca be expressed as the mea of this posterior distributio as follows: α + x α + x θ (10 ( α + x + ( β + x α + β + We ca rewrite the ayesia estimator of θ i the form of credibility formula by [1], [1] or [13]: x θ ( Z + 1 Z µ (11 where factor credibility Z ca be expressed as Z α + β + (1 ad μ is the mea of the prior beta distributio expressed as α µ (13 α + β We ote that as (umber of isurace policies icreases, the weight Z attachig to the databased estimator icreases ad the weight attachig to the prior mea correspodigly decreases. I practice may be some situatios i which there is o prior kowledge. The we use a oiformative prior. For example if θ is a biomial probability ad we have o prior iformatio at all about θ, the a prior distributio which is uiform o iterval (0, 1 would seem appropriate. I case this case the prior distributio is the beta distributio with parameters α 1, β 1 ad it leads to the prior estimate θ 0.5. If parameter θ is probability of diagosis critical illess, this prior estimator fortuately highly overstates real value of this probability. To elimiate this drawback, istead of iterval (0, 1 for prior estimate of probability θ eed to propose more realistic iterval i which we assume a uiform prior distributio. Such iterval ad the algorithm for its use i ayesia estimatio of the biomial probability of radom evet θ was published i articles [14] ad [11]. The proposed procedure is as follows: We set the iterval ( θ,θ mi max, i which we suppose to get ISSN:
3 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 a better estimate. We deote by the symbol s the mea of prior beta distributio, which is the cetre of this iterval: x mi + x s max. We mark as θ the more distat boudary from the value of of the iterval ( θ,θ. mi max Calculate the allowable error as h θ 0 s. We calculate q accordig to the formula θ ( 1 θ0 ( 1 θ 0 q. (14 h θ0 0 We estimate the parameters α, β of the prior beta distributio as follows: α qs, β q qs. (15 III. THE INDIVIDUAL RISK MODEL Oe of the key quatities of iterest to a isurace compay is the total amout to be paid out o a particular portfolio of policies over a fixed time iterval, such as a accoutig period. Oe short term model is the idividual risk model, where we cosider the portfolio to cosist of a fixed umber,, of idepedet policies, or idividual risks. We will model aggregate claims from the portfolio as the sum of claims from idividual risks, hece the ame idividual risk model. Aother short term model is the collective risk model [4], [7], [15], [16], [18]. Here we model successive claims arisig from the portfolio as idepedet, idetically distributed radom variables X, X, 1..., X N, ad we igore which policy gives rise to which claim. The umber of claims i the fixed time period is a radom variable, N, say, which is assumed to be idepedet of X i. The total claim amout (or aggregate claims is modelled as a radom variable give by S X X + X N (16 The distributio of S is a example of a compoud distributio. We will cosider aggregate claims whe N has the biomial distributio, ad so S has the compoud biomial distributio. I this case expressios of mea ad variace of S are by [4], [15]: E D ( S πm1 (17 ( S πm π (18 m1 where π is the parameter of biomial distributio of the variable N ad m1 ad m are the momets of X i about zero. I idividual risk model we deote the aggregate claim from the portfolio by S. We ow write S Y1 + Y Y (19 where Y deotes the claim amout uder the th risk ad deotes the umber of risks. It is possible that some risks will ot give rise to claims. Thus, some of the observed values of Y, 1,,..., may be 0. For each risk, we make the followig assumptios: 1 the umber of claims from the th risk, N, is either 0 or 1, the probability of a claim from the th risk is q. We assume a situatio where there is a maximum of oe claim from each policy. This case icludes also risk of critical illess diagosis i oeyear term policies. If a claim occurs uder the th risk, we deote the claim amout by the radom variable X. Let F ( x, µ a σ deote the distributio fuctio, mea ad variace of X respectively. Assumptios 1 ad say that N i( 1; q. Thus, the distributio of Y is compoud biomial, with idividual claims distributed as X. We ca immediately write dow from (17 ad (18 that ( Y q E µ (0 ( Y q ( σ + µ q µ q σ + q ( q D µ (1 1 Tha S is the sum of idepedet compoud biomial distributed radom variables ad it is easy to fid the mea ad variace of S : E ( S E Y E( Y q µ ( D Y D Y [ q σ + q 1 q µ ] (3 ( ( ( D S For calculatio of D( S we have used the assumptio that idividual risks are idepedet. ecause of Y are geerally ot idetically distributed, there is o geeral result that tells us the distributio of such a sum. We ca state this distributio oly whe the compoud biomial variables are idetically distributed, as well as idepedet. I a special case, whe Y, 1,,..., is a sequece of idetically distributed, as well as idepedet radom ISSN:
4 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 variables, the by cetral limit theorem we ca approximate distributio of S by ormal distributio. The we set the risk premium as the 95. percetile of the ormal distributio with parameters µ E(, ( S σ D S (4 IV. SOURCE OF DATA Let θ is ukow probability of diagosis a critical illess. To estimate this probability for Slovak isurace market we have foud the the data about the umber of claims x ad risk exposure i the years from dataset of Natioal ak of Slovakia [17]. Data coverig the period were submitted to the Natioal ak of Slovakia based its Decree No. 0/008 o submittig of actuarial data ad statistical data of isurace compay ad brach of a foreig isurace compay, o the basis of which it started to gather statistical data about isured people from isurace udertakigs i 009. The data were gathered i classificatio accordig to geder, age ad thirtee isurace risks. Amog them there are also the critical illess risks. illess diagosis for me by four age categories The small umber of isured persos i the early observed years is the cause of large variability i the estimates. A small umbers of isured me ad wome i the age group over 50 years are the cause of large fluctuatios i θ estimates by the relative umber of claims. I Figures 1 ad we ca observe sigificat differeces i the relative umbers of diagoses of CI disease i differet age categories. I priciple, these relative umbers grow with age. While i groups of wome uder 30 years ad of wome i the age rage years are large differeces i estimated probabilities especially i the first years of time series, the relative umbers of diseases i the same age groups of me are almost idetical. Positive feature is the decreasig tedecy of occurrece of the disease amog wome aged 4150, but maily i the age rage years. V. RESULTS AND DISCUSSION Let θ is ukow probability of diagoses some critical illess. To estimate this probability we have foud the data about the umber of claims x ad risk exposure i the years from dataset of Natioal ak of Slovakia [17]. The radom variable X the umber of claims durig the year is biomial distributed ad maximum likelihood estimatios we get simply by formula x est θ (5 Maximum likelihood estimators of the probabilities of critical illess diagosis of isured me ad wome i the Slovak Republic based o data of Natioal ak of Slovakia [17] preset the Figures 1 ad. Fig. 1 Maximum likelihood estimates of probabilities of critical Fig. Maximum likelihood estimates of probabilities of critical illess diagosis for wome by four age categories The shortcomigs of the maximum likelihood estimates of probabilities of critical illess diagosis i homogeeous groups of isured persos by age ad sex, maily due to the small umber of data, we try to remove usig ayesia estimatios. Accordig to the publicatios [19], [0] we selected the itervals ( θ,θ for prior estimatio of θ for each mi max homogeeous group of policies by sex ad age categories. For example it is iterval ( ; 0.00 for prior estimate of parameter θ for group of me less tha 30 years. Followig the procedure described above by [14] we have obtaied the prior estimate of the probability θ of critical illess diagosis i the year I calculatio by (14 we used the umber of the populatio i Slovak republic i the year Procedure for obtaiig the parameters of prior beta distributio for category of me less tha 30 years is as follows: 1. θ mi ; θmax s ISSN:
5 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, θ h q ( ( α β , ' The parameters α ad β of the posterior gamma distributio for each of the subsequet years we have estimated by relatios (7 ad (8, usig as the prior parameters α, β their ayesia estimates from the previous year. Table I Updated ayesia estimatio of critical illess probabilities for me less tha or 30 years Year x x/ α β θ Source: ow calculatios based o data with prior e( ; distributio i years we have used for permaetly updated ayesia estimates of the probability of diagosis of a critical illess accordig to the formula (10 by miimizig the quadratic loss. The results of the procedure for me uder or 30 years are summarized i Table I ad the aalogous results for wome i the same age category cotai Table II. Table II Updated ayesia estimatio of critical illess probabilities for wome less tha or 30 years Year x x/ α β θ Source: ow calculatios based o data with prior e( ; Fig. 4 Maximum likelihood ad ayesia estimatios of critical illess probabilities for wome less the or 30 Fig. 3 Maximum likelihood ad ayesia estimatios of critical illess probabilities for me less the or 30 Permaetly updated parameters of the posterior beta Maximum likelihood estimates x/ ad ayesia estimates θ from Table I i successive years are show i Fig. 3 ad from Table II i Fig 4. We ca see that the ayesia estimates are ot so strog affected by radomess as a maximum likelihood estimates, because ayesia estimates cotai also a priori iformatio from the previous years. Therefore the ayesia estimates are more suitable for actuarial calculatios i compariso with maximum likelihood estimates. ISSN:
6 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 Graphical comparisos of the maximum likelihood ad ayesia estimates of the probabilities of a critical illess diagosis for me of other age categories have show Fig. 5 Fig. 7. Aalogous comparisos of the maximum likelihood estimates by x ad ayesia estimates θ of critical illess for wome of differet age groups show the Fig. 8Fig. 10. Fig. 5 Maximum likelihood ad ayesia estimatios of critical illess probabilities for me above 30 ad less the or 40 Fig. 8 Maximum likelihood ad ayesia estimatios of critical illess probabilities for wome above 30 ad less the or 40 Fig. 6 Maximum likelihood ad ayesia estimatios of critical illess probabilities for me above 40 ad less the or 50 Fig. 9 Maximum likelihood ad ayesia estimatios of critical illess probabilities for wome above 40 ad less the or 50 Fig. 7 Maximum likelihood ad ayesia estimatios of critical illess probabilities for me above 50 Fig. 10 Maximum likelihood ad ayesia estimatios of critical illess probabilities for wome above 50 ISSN:
7 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 Table III Compariso of ayesia estimatios of critical illess probabilities for me ad wome above 30 ad less the or 40 Year θ  me θ  wome Source: ow calculatios probabilities for me ad wome above 40 ad less the or 50 Year θ  me θ  wome Source: ow calculatios Fig. 11 Graphical compariso of ayesia estimatios by sex from Table III As we ca see from Table III ad Fig. 11 i the age group above 30 ad less the or 40 the differeces betwee estimated probabilities of critical illess are sigificatly higher i the group of isured wome i compariso with isured me, particularly i the early years of the referece period. Due to the decreasig tred i the values of the probability of critical disease of wome, these differeces i the later years of moitorig cosiderably reduced. I the age group the sigificat differeces of estimated probabilities of critical illess for me ad wome have bee also gradually reduced i the later years, but the reaso was ot oly a decreasig tred of estimated values of these probabilities for wome, but also the icreasig tred for me. Table IV Compariso of ayesia estimatios of critical illess Fig. 1 Graphical compariso of ayesia estimatios by sex from Table IV Table V Compariso of ayesia estimatios of critical illess probabilities for me ad wome above 50 Year θ  me 51+ θ  wome Source: ow calculatios ISSN:
8 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 Total Source: ow calculatios Table VIII Risk premiums for homogeeous groups of wome by age RP/ Wome Θ est 011 E(Y D(Y perso Total E+11 Source: ow calculatios Fig. 13 Graphical compariso of ayesia estimatios by sex from Table IV Oly i the highest age category of isured persos above 50 years the estimated probabilities of critical illess there are higher for me tha for wome. ayesia estimates we use to determie the risk premiums i homogeeous groups accordig to age ad sex of the isured persos. For this aim we eed to estimate the umber of isured persos i 011 by extrapolatio the tred of the time series. To determie the tred of the time series of isured persos i each of the homogeeous groups i the period we used the statistical package Statgraphics Ceturio. Procedure Compariso of Alterative Models allows to choose the most appropriate tred fuctio ad procedure Forecast provides predicted value for the comig year. The quality of the selected model expresses the R squared measure. Table VI The results of predictio the umber of isured persos i year 011 for homogeous group by sex ad age Group Model Predicted R Values 011 squared Me, 30 Square rooty % Me, Multipliative % Me, Multipliative % Me, above 50 Multiplicative % Wome, 30 Double reciprocal % Wome, Square rooty % Wome, Square rooty % Wome, above 50 Multiplicative % Source: ow calculatios Table VII Risk premiums for homogeeous groups of me by age RP/ Me Θ est 011 E(Y D(Y perso For the calculatio of risk premiums i homogeeous groups of isured persos by sex ad age we used the formulas (0(4. Results we ca see i tables VII ad VIII. VI. CONCLUSION ayesia estimatio theory provides methods for permaetly updated estimates of the evet probability for each comig year i isurace compay. ayesia approach combie prior iformatio that are kow before collected of ay data ad iformatio provided by the sample data, which are i our case umber of cocluded isurace cotracts ad umber of claims i previous years. Probabilities of the claims which are the subect of isurace cotracts are ecessary to kow for isurace compay especially whe calculatig premiums for ext year. The isurace compay ca correctly determie premiums oly if use correctly estimates probabilities of claims. This article is both theoretical ad practical demostratio of permaetly updated ayesia estimates of evet probability which i this case is critical illess. This procedure has of course geeral use ad provides better estimates of probabilities as maximum likelihood method. The maximum likelihood estimate is assiged to period which has already expired, while ayesia estimate of evet probability is for ext period. This is udoubtedly advatage for premium calculatio. The possibility to express ayesia estimate of biomial probability i the form of credibility formulas by expressio (10 allow easy applicatio of this theory i isurace practice. The weakest poit of ayesia estimatio is the choice of parameters of prior distributio ad the associated a priori estimate of the parameter. Article also presets a algorithm to improve the a priori estimates. REFERENCES [1] P. Jidrová, V. Pacáková, ayesia Probability Models for Critical Illess Isurace. Proceedigs of the 014 Iteratioal Coferece o Mathematical Models ad Methods i Applied Scieces (MMAS '14, Sait Petersburg, Septembre 3 5, 014, pp [] P. H. Garthwaite, P. H., I. T.Jollifee,. Joes, Statistical Iferece (d Editio, Oxford Uiversity Press, 00. [3] D. R. Cox, Priciples of Statistical Iferece, Cambridge Uiversity Press, 006. ISSN:
9 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 [4] V. Pacáková, Aplikovaá poistá štatistika (Applied Isurace Statistics, ratislava: Iura Editio, 004. [5] Y. K. Tse, Nolife Actuarial Models, Cambridge: Uiversity Press, 009. [6] H. R. Waters, A Itroductio to Credibility Theory, Lodo ad Ediburgh: Istitute of Actuaries ad Faculty of Actuaries, [7] P. J. olad, Statistical ad Probabilistic Methods i Actuarial Sciece, Lodo: Chapma&Hall/CRC, 007. [8] P. Jidrová, Quatificatio of Risk i Critical Illess Isurace. I: Coferece proceedigs from 9th iteratioal scietific coferece Fiacial Maagemet of Firms ad Fiacial Istitutios, VŠ Ostrava, 013. pp [9] V. Pacáková, ayesia Estimatios i Isurace Theory ad Practice, Proceedig of the 14th WSEAS Iteratioal Coferece o Mathematical ad Computatioal Methods i Sciece ad Egieerig (MACMESE'1, Sliema, Malta, September 79, 01, pp [10] V. Pacáková, Credibility models for permaetly updated estimates i isurace. Iteratioal Joural of Mathematical Models ad Methods i Applied Scieces, Issue 3, Volume 7, 013, pp [11] V. Pacáková, E. Kotlebová, ayesia Estimatio of Evet Probability i Accidet Isurace. I: Europea Fiacial Systems 014. Proceedigs of the 11th Iteratioal Scietific Coferece, ro: Masaryk Uiversity, 014, pp [1] J. Gogola, Spôsob permaete úpravy výšky poistého v eživotom poisteí (Method for Permaet Adustmets of Premium i NoLife Isurace. E+M Ecoomics ad Maagemet, No. 4/013, 013, pp [13]. Lida, J. Kubaová, Credibility Premium Calculatio i Motor ThirdParty Liability Isurace, Proceedigs of the 14th WSEAS Iteratioal Coferece o Mathematical ad Computatioal Methods i Sciece ad Egieerig (MACMESE'1, Sliema, Malta, September 79, 01, pp [14] E. Kotlebová, I. Láska, Využitie bayesovského prístupu pri odhade podielu a možosti eho aplikácie v ekoomicke praxi (Use of ayesia approach i estimatig the proportio ad its possible applicatios i the ecoomic practice. Sloveská štatistika a demografia (Slovak Statistics ad Demography, Volume 4, No, 014, pp [15] D. C. M. Dickso, H. R. Waters, Risk Models, Ediburgh: Heriot/Watt Uiversity, [16] R. J. Gray, S. M. Pitts, Risk Modellig i Geeral Isurace. Cambridge: Cambridge Uiversity Press, 01. [17] Natioal ak of Slovakia (01. Data Disclosure accordig to Directive 004/113/EC: Uderlyig data_1999_010.xls. [Olie]. Available: [18] M. Käärik, M. Umblea, O claim size fittig ad rough estimatio of risk premiums based o Estoia traffic isurace example, Iteratioal Joural of Mathematical Models ad Methods i Applied Scieces, Issue 1, Volume 5, 011, pp [19] Cotiuous Mortality Ivestigatio Committee (010. Workig Paper 43, 50, 5, 58CMI critical illess diagosis rates for accelerated busiess, Istitute of Actuaries ad Faculty of Actuaries. [Olie]. Available: [0] Material of Muich Re Group: Critical Illess Isurace, 001, pp : [Olie]. Available: She was workig at Departmet of Statistics, Faculty of Ecoomic Iformatics, Uiversity of Ecoomics i ratislava sice 1970 to Jauary 011. At the preset she has bee workig at Faculty of Ecoomics ad Admiistratio i Pardubice sice 005. She has bee cocetrated o actuarial sciece ad maagemet of fiacial risks sice 1994 i coectio with the actuarial educatio i the Slovak ad Czech Republic ad she has bee achieved cosiderable results i this area. Mgr. Pavla Jidrová, Ph.D. graduated from Mathematical aalysis at the Faculty of Sciece of Palacky Uiversity i Olomouc i She lectures mathematics, isurace ad fiacial mathematics i brach of study of Isurace egieerig. She fiished her dissertatio thesis which deals with problems of aggregatio ad disaggregatio i ecoomics ad mathematics. Prof. RNDr. Viera Pacáková, Ph.D. graduated i Ecoometrics (1970 at Comeius Uiversity i ratislava, RNDr. i Probability ad Mathematical Statistics at the same uiversity, degree Ph.D at Uiversity of Ecoomics i ratislava i 1986, associate prof. i Quatitative Methods i Ecoomics i 1998 ad professor i Ecoometrics ad Operatio Research at Uiversity of Ecoomics i ratislava i 006. ISSN:
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