# Actuarial Models for Valuation of Critical Illness Insurance Products

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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 o-iformative 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 data-based 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 o-iformative 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 oe-year 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 41-50, 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. 8-Fig. 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:

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