Methods for Calculating Life Insurance Rates

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1 World Appled Scences Journal 5 (4): , 03 ISSN IDOSI Pulcatons, 03 DOI: 0.589/dos.wasj Methods for Calculatng Lfe Insurance Rates Madna Movsarovna Magomadova Chechen State Unversty, Grozny, Russa Sumtted: Aug 7, 03; Accepted: Oct, 03; Pulshed: Oct 6, 03 Astract: Ths artcle analyzes the mechansm for modelng the proalty of survval and death for a populaton, wth a vew to determnng the attrutes of mortalty gven dfferent age groups; t looks nto methods of calculatng lfe nsurance rates for short-term and long-term nsurance and attempts to assess the posslty of determnng rsks for nsurance companes under dfferent types of nsurance. The artcle also presents the general workflow of the lfe nsurance system. The author has conducted a theoretcal analyss of the mortalty process y usng the models developed y A. de Movre (a person s lfespan s dstruted evenly over the nterval (0, ), where the parameter s the lmtng age people don t normally lve past), Gompertz (some parameters are determned ased on statstcal data for a certan populaton) and Makeham (the force of mortalty s calculated usng the µ A + Be formula). The author has eamned the practcal applcaton of all the methods used n the study wth respect to dfferent types of nsurance: t s customary for an nsurance company to enter nto a large numer of contracts; ndvdual clams are ndependent quanttes; the long-term lfe nsurance model s characterzed y that n these cases, one takes nto account changes n the value of money occurrng over tme; the eponental smoothng method s used for short-term and long-term forecasts and s predcated on the mean-weghted value of sales over a certan numer of perods passed. Key words: Model Proalty Insurance event Mathematcal epectaton Mortalty Survval functon Rsk INTRODUCTION Changes n the value of money over tme are counted n; The demographc stuaton n a regon governs the Changes n the value of money over tme are not net cost of lfe nsurance: the lower the populaton s counted n []. age-related mortalty level, the lower the rsk rate and vce versa. Of great sgnfcance n mplementng lfe nsurance We can dspense wth money value changes for short n a regon s the demographc stuaton stalty factor, nsurance polcy terms. for the nvaralty of age-related mortalty ndcators n For ths reason, lfe epectancy models that do not tme lets one realze the major prncple ehnd determnng take nto account changes n the value of money over tme lfe nsurance rates rate stalty wth no detrment to the are called short-term lfe nsurance models []. materal nterests of the polcyholder and the nsurer. Otherwse, when changes n the value of money over When t comes to lfe nsurance, of nterest s tme are taken nto account n calculatons, we are dealng modelng the survval and death proalty wth a vew to wth long-term lfe nsurance models. detectng the attrutes of mortalty for dfferent age To a commercal nsurance company, compared wth groups. one n the Islamc nsurance system called takaful [3], of Ths area s known as actuaral mathematcs and s mportance s not an ndvdual clam or A dsursement characterzed y the use of the proalty theory, on that clam, ut the total sze of payments to mathematcal statstcs and mathematcal modelng polcyholders, who fle ndvdual clams,.e. the total rsk methods []. whch should not eceed the company s captal. Lfe nsurance models are nomnally susumed nto Otherwse, there s a proalty the company can go two large groups wheren, respectvely: ankrupt. Correspondng Author: Magomadova, Chechen State Unversty, Sharpov, 3 street, Grozny, Russa. 653

2 World Appl. Sc. J., 5 (4): , 03 The assessment of ths proalty s crucal to the operaton of a commercal nsurance company. The Man Part: The general workflow of the lfe nsurance system. The sze of dsursements on an nsurance contract s of a fortutous character, whch means that the sze of payments on all contracts s lkewse a chance quantty. The sze of payments s lmted y the nsurance fund whch s formed of premum payments. Based on that, the cumulatve nsurance sze s vared over a certan nterval, the upper lmt whereof equals the sze of all dsursements on all contracts taken together. Therefore, the calculaton of the premum payment T s of great mportance to commercal nsurers. If the proalty of an nsurance event q occurrng s known ahead of tme (ased on former eperence, y analogy, etc.), then theoretcally, wthout takng nto account all the other factors (ncludng the tme factor), the sze of the premum T s calculated usng the followng formula [4]. T S.q where S s an nsurance dsursement; q s the proalty of an nsurance event occurrng. The equaton aove just llustrates the prncple of the fnancal equvalency of polcyholder and nsurer olgatons. Suppose that T s the sze of the premum and q n s the proalty of an nsurance event occurrng (for nstance, the death of a polcyholder n n years after the nsurance polcy comes nto effect). If an nsurance event occurs wthn the frst year of nsurance, the nsurer wll get an amount T (assume that the premum s pad at the egnnng of the year), whle f ths event occurs wthn the second year, the nsurer wll get an amount equalng T, etc. The mathematcal epectaton of ths row of premums s T.q +.T.q +...+n n Although the resultng quantty sums up all the polcyholder s payments, gven the credlty of these havng een pad, when summng the correspondng quanttes we do not take nto account the fact that the premums are pad at dfferent ponts n tme. Wth ths factor n mnd (through dscountng the payment amounts), we fnd the mathematcal epectaton of the up-to-the-mnute cost (actuaral cost) of payments to e ( n ) ( ) E A T q v q v v q v v q n where v /( + ) s a dscount multpler; s a percentage rate. We could admt that t s pad at the end of the year the nsurance event occurred n. Then the mathematcal epectaton of the dsursement wthn the frst year wll e Sq, the second year - Sq, etc. The mathematcal epectaton nclusve of the dsursement tme factor (actuaral cost) s n E S S v q+ v q v q n Based on the prncple of the equvalence of polcyholder and nsurer olgatons, we get an equaton E(S)E(A) whch enales us to derve the sought value of the net premum T. The calculaton of net and gross premums s ased on the nsurance company s model more specfcally, on dervng the total nsurance amount on all nsurance contracts and the credlty of nsurance events occurrng. To ensure a 00% guarantee that the sze of net premums wll eceed the sze of dsursements, the nsurer has to create an nsurance fund the sze of the cumulatve nsurance amount. In ths case, the premum wll equal the nsurance amount. The nsurer determnes for tself the sze of ts acceptale rsk, whch mathematcally can e epressed as the followng n equaton ( < ) ( ) P S P y P S P where P s the proalty; y s a safety guarantee estalshed y the nsurer; st S s a dsursement on the contract; st P s a premum on the contract; s the upper lmt of the safety guarantee. 654

3 World Appl. Sc. J., 5 (4): , 03 The ratonale ehnd these n equatons s that the proalty that the sze of all premums wll eceed the sze of all polcyholder payments should e determned upfront. Accordng to the Lyapunov theorem, nsurance events and nsurance dsursements are dstruted accordng to the normal law [4]. If the chance quantty dstruton law s determned, the aove n equaton s solved easly. Frstly, the proalty that the contnuous quantty X wll get a value that elongs to the nterval (A, ) equals the defnte ntegral of the frequency of the dstruton over the nterval from A to, namely Ðà ( < Ð< ) f d f e a Secondly, the normal dstruton functon equals ( a) where s the mathematcal epectaton of a chance quantty; y s the mean square devaton. Then fð X à Ô P S P Ô where s the Laplace functon. Based on the fnancal equvalence prncple, the sought sze of the net premum can e epressed as the product of the nsurance amount and the net rate epressed as a percentage. The major part of the rate s the net rate whch epresses the cost of the nsurance rsk and ensures the coverng of losses [4]. u S Tn 00 where T s the net rate; u s the sze of the net rate for 00 ru.; S s the sze of a asc nsurance amount. The rate s the gross rate. Tn T f where T s the gross rate; T s the net rate; f s the load fracton n the gross rate. The major part of the rate s the net rate whch epresses the cost of the nsurance rsk and ensures the coverng of losses. The load fracton f can e calculated ased on accountng data. R f + K + V Ï where R s epenses; s the sze of premums collected on ths type ofnsurance; K s the percentage of commsson pad to agents n effectng ths type of nsurance; V s a part of proft n the gross rate the nsurer looks to get on ths type of nsurance. The ass of any actuaral calculatons s the study of the mortalty process of any gven populaton, determnaton of certan patterns and further calculaton of rates. In the nsurance system, the mortalty process s mostly modeled ased on certan analytcal laws,.e. the theoretcal analyss of the state of a populaton over a long perod of tme was conducted [5]. Wth ths theoretcal analyss of mortalty processes, the prmary and smplfed study of real-lfe stuatons normally employs standard models that let us derve major patterns. Besdes, the mortalty process s well appromated y the analytcal laws consdered elow. One of the founders of the credlty theory, A. de Movre, suggested n 79 [6] holdng that a person s lfespan s dstruted evenly over the nterval (0, ) where the parameter s the lmtng age people normally don t lve past. For ths model, wth 0 < < f( ) s we get 655

4 World Appl. Sc. J., 5 (4): , 03 Tale :Input data Name of parameter Age (n years) Lmt (n years) Numer of dsursements per year Dsursement n case of death Proalty Use The prmary and most crucal parameter that helps determne correspondng values n the mortalty tale Used n calculatng the cost of rent:postnumerando and pronumerando Used n calculatng the cost of rent: postnumerando and pronumerando Based on the value of ths parameter, the cost of rates s calculated The parameter tes the mechansms of actuaral mathematcs and the proalty theory together: relyng on the value of ths parameter, the polcyholder and the nsurer take a decson Tale : Input data Name of parameter Net premum for short-term and long-term nsurance Gross premum for short-term and long-term nsurance Load Use The rate that epresses the cost of the nsurance rsk and ensures the coverng of losses The rate at whch an nsurance contract s entered nto Determnes what part of the gross premum s determned y the nsurer s ependtures assocated wth the mplementaton of the nsurance process, attracton of new clents, etc. Tale 3: Calculaton of the man ndcators of the moralty tale Indcator Formula Parameters Numer of people who survved untl a certan age l l 0. S() l 0 s the ntal populaton; S() s the survval functon; s the age (the tale s formed for any age, ) Numer of people who ded whle transtng from age to age + d l-l+ l s the numer of people of age ; l+ s the numer of people of age + The proalty of dyng at age, wthout survvng untl age + q -d d s the numer of people who ded whle transtng from age to age + The proalty of survvng untl a certan age P - q q s the proalty of dyng at age, wthout survvng untl age + f s In the Gompertz model, the force of mortalty s calculated va the formula f Be s Later on, n 860 [6], Makeham suggested determnng the force of mortalty usng the formula A + Be where the parameter A takes nto account the occurrence of nsurance events and the summand E takes nto account the effect of age on mortalty. where > > 0 are some parameters determned ased on statstcal data for a certan populaton eng studed. The survval functon s u e s ep udu ep Be du ep B du and the deaths curve s For the Makeham model, the survval functon s B u s ep ( A + Be ) du ep A ( e ) 0 and the deaths curve s ( e ) B f s A + Be ep A e f s B ep B We have a mamum n the pont ln ln B The models presented are used for the theoretcal analyss of the populaton mortalty process. In Tale, one can see the ntal data for the algorthms presented elow. The most commonly used algorthm for solvng actuaral mathematcs prolems works the followng way. 656

5 World Appl. Sc. J., 5 (4): , 03 For convenence, we take the ntal aggregate l 0 to equal l we have ES N EX VarS N VarX We wll further model up the lfe epectancy of an aggregate of people over the net 00 years and wll e takng down the tmes of death. The modelng wll e mplemented ased on the analytcal laws consdered aove. Let s calculate the commutaton numers: ( ) D l + where s the current age; N M w Dj j w Cj j ( ) C d + + l s the percentage rate; l s the numer of people who were stll alve y age. Mortalty tales are used n calculatng the sze of the nsurance rate [7]. Let s take a look at the algorthm for calculatng short-term nsurance rates. We ll take the sze of an nsurance premum as the unt of measurement for amounts of money. In ths case, dsursements on the -th contract X ³ take the values 0 and wth the proalty ( q) and q respectvely. We get EX q 0+ q q EX q 0 + q q VarX EX EX q q Now for the mean value and summary dsursements S X X n Usng Gaussan appromaton for the centered and scaled quantty of summary dsursements, we ll epress the proalty of a dsursement y a tradtonal nsurance company as follows: S ES u ES u ES P( S u) P Ô VarS VarS VarS We fnd those that satsfy the value of the quntle of the normal dstruton, where p s the proalty. p We get the equaton u ES Ô VarS Then u p VarS + ES p We ve otaned the value of a net premum for 00 rules. Let s fnd the sze of the net premum u S Tn 00 where S s the amount the clent wants to get n the event of hs/her death, however paradocal that may sound. We fnd the value of the gross premum Tn T f where f s the load the sze of whch each nsurer determnes for tself. As a result, the algorthm for calculatng short-term nsurance rates was otaned. When t comes to long-term nsurance, the remanng term of lfe s characterzed y a constant force of mortalty [8]. 657

6 World Appl. Sc. J., 5 (4): , 03 Let s calculate the net premum: t t0 A v f () t dt 0 where f(t) s the densty of the remanng term of lfe. Snce the force of ntensty s gven, we fnd the survval functon usng the formula t S t e and the mortalty functon f(t) f () t e Then the net premum s ( + ) t A e dt 0 Z + t VarZ EZ EZ VarZ A N A T n ( ) + 00 ( ) T T f + The up-to-the-mnute sze of dsursements on an ndvdual contract can e otaned through susttutng for. Consequently Now we can calculate the relatve nsurance mark-up: Let s determne the sze of rates: As a result, the algorthm for calculatng long-term nsurance rates was otaned. Tale 4: Structure of ntal data for algorthmc models Name of parameter Value entered Age (n years) 5 Insurance polcy term (n years) 5 Lmt (n years) 0 Numer of dsursements per year 6 Insurance amount Proalty of dsursement 0,89 Let s consder the practcal applcaton of the aove methodology usng a specfc eample (Tale 4). We ll frst model up the lfe epectancy parameters, whch s an alternatve method for otanng data needed n calculatng an nsurance company s rates. We ll use the Gompertz law for the modelng; the ntal populaton s people [9]. We ll model the mortalty process through appromaton y the law chosen. Parameters () s(0) 0, e s Åõð 0, , , l , ,3364 d l l , , ð q 0, , , , Thus and so on. The numer of those who ded whle transtng from age to age + s determned through the formula We ll determne the proalty of dyng at age wthout survvng untl age + the followng way: The proalty of survvng untl a certan age s determned as follows: 0 0 The commutaton numers are calculated the smlar way 0,09 e 0,

7 World Appl. Sc. J., 5 (4): , 03 D 99803,3364 ( + 0, 09474) 93, 4 98 N Dj 04898, j + C 96, , , We ll calculate short-term nsurance rates. The numer of nsurance contracts s N450; the nsurance amount s py. The proalty of death takng place wthn a year s q 0, st In ths case, dsursements on the contract X take the values 0 and wth the proalty ( q) and q respectvely. Consequently, EX q 0 + q q 0, VarX EX EX q q 0, 0486 Now for the mean value and summary dsursements S X X n Tn T 3, , 08 ðóá. 00 The gross rate equals 3467, ,85 ðóá. 0, We can calculate long-term lfe nsurance rates the smlar way. CONCLUSION We shouldn t, nevertheless, forget that multple scentfc studes reveal the need to keep track of polcyholders gender n determnng lfe nsurance rates. Furthermore, mortalty n men depends on the socal, economc, poltcal stuaton n the country [0]. Mortalty n women s a relatvely steady ndcator, thanks to the attrutes of women s psychology, ehavor, eng less stress-prone compared wth men. The resdence factor (whether he/she lves n the country or a town/cty) doesn t have a tangle effect on the proalty of death we can gnore t n uldng nsurance rates. we have ES N EX 300 0,003, VarS N VarX 300 0,003,75434 Usng Gaussan appromaton for the centered and scaled sze of summary dsursements, we ll epress the proalty of a dsursement y the company as follows: S ES u ES u, P( S u) P Ô VarS VarS, If we need the proalty to equal 0,89, the quantty u,809556, has to equal 0,87 (whch satsfes the value of the quantle of the normal dstruton),.e. è 0,87, , , the sze of the nsurance amount for 00 rules. For ru. the net rate wll e Inferences: Thus, the use of mortalty tales n calculatng net lfe nsurance rates leads to a decrease n the net cost of nsurance for the most actve group of polcyholders. We get a far layout of the rsk of ncurrng materal losses assocated wth a person s lfe epectancy (etween all polcyholders); the consstency of nsurance operatons s oosted; the relalty of nsurance companes ncreases; reducng the cost of nsurance for the most proale polcyholders enales nsurance companes to rng n more customers, whch rngs the nsurance rates down for all polcyholders. Let s consder the practcal applcaton of the aove methods for dfferent types of nsurance:. It s customary for an nsurance company to enter nto a large numer of contracts. Therefore, what s mportant to t s not an ndvdual clam and effectng a dsursement on that clam ut the total sze of dsursements, S, across all ndvdual clams. Let s assume that the company has entered nto N nsurance contracts and s ndvdual clams on these contracts. Then 659

8 World Appl. Sc. J., 5 (4): , 03 n S k k We ll call the amount S the summary rsk. If the summary rsk S doesn t eceed the company s captal, then the company wll succeed n fulfllng ts olgatons efore ts clents. But f S > u, the company won t e ale to dsurse all ndvdual clams. In ths case, the company could go ankrupt. Thus, the proalty (R) of the company gong ankrupt can e epressed va the formula R P{ S > u} In other words, the proalty (R) of the company gong ankrupt equals the value of the addtonal functon of the dstruton of the summary rsk n the pont u, where u s the company s captal. Accordngly, the proalty of the company not gong ankrupt equals the value of the functon of the dstruton of the summary rsk n the pont u. The calculaton of these proaltes s crucal to the company, for t s on the ass of these calculatons that the company makes ts mportant decsons. We ll assume that ndvdual clams are ndependent quanttes. The numer of polcyholders nsured wth a company s normally very hgh. Therefore, there s always a need to calculate the dstruton of the amount of a large numer of deposts. It s clear that manual calculaton methods usng generatng functons are not helpful here. Consequently, t s practcally mpossle to effect an accurate calculaton of the attrutes of the summary rsk of the sum of a large numer of summands. In a stuaton of ths knd, we can resort to the lmt theorems of the proalty theory. The ratonale ehnd these theorems s that gven some qute general condtons, the functon of the dstruton of the sum of ndependent chance quanttes at N 8 overlaps wth the functon of the dstruton of some specfc chance quantty. Ths enales us to employ ths specfc dstruton functon n solvng practcal prolems, nstead of usng the functon of the dstruton of the sum, whch s hard to compute. Note that the error resultng from ths susttuton s qute small and does satsfy practcal requrements wth respect to the accuracy of calculatons. For the further calculaton of the attrutes of the summary rsk on short-term polces we use the Posson theorem: P{ a } In other words, the captal of the company u (provded that the proalty of the company not gong ankrupt s not less than 95%) must satsfy the condton P P N k k u 0,95 And accordng to the Posson theorem { u} 0,95 Numerc calculatons are done usng the tale of quntles of the level 0,95 of the Posson dstruton. Eample: There are N 3000 ndvduals aged 38 and N 000 ndvduals aged 8 nsured wth the company. The company pays the polcyholder s hers a dsursement to the tune of rules n the event of hs/her death wthn one year and pays nothng f he/she lves for over one year after enterng nto the contract. Usng the Posson method, we ll calculate the sze of the nsurance dsursement whch guarantees that the proalty of the company gong ankrupt won t eceed 5%. Soluton: Let s make the sze of the nsurance dsursement our unt. Usng the lfe epectancy tales, we fnd q 0, q 0,00 8 Therefore, accordng to the Posson theorem, the summary rsk from polcyholders aged 38 (8) can e consdered as a chance quantty dstruted y the Posson law wth the parameter N q 38 N q 8 660

9 World Appl. Sc. J., 5 (4): , 03 Then, the company s summary rsk s the Posson chance quantty wth the parameter P P + 0 Consequently, the proalty of the company not gong ankrupt appromately equals { u} where s the Posson chance quantty wth the parameter 0; u s the company s captal. The prolem says t has to e { u} 0,95 From ths condton we have u 0,95 where 0,95 s the quntle of the level 0,95 of the Posson dstruton wth the parameter 0. Usng the quntle tale, we fnd that u 5 0,95 The net premum for the frst polcyholder group equals + q38 0,003 and for the second group + q8 0, 00 Thus, through net premums the company wll get the amount , , 00 0 The remanng amount needed wll e consttuted y the nsurance mark-up. Snce the nsurance mark-up consttutes 50% of the sze of net premums, the company has to ncrease net premums y 50%. Consequently, for the frst polcyholder group the sze of a premum wll e 0,003,5 0,0045 Aand for the second group 0,00,5 0,005 The long-term nsurance model s characterzed y that n ths case we count n changes n the value of money over tme. Wth full lfe nsurance, there s no uncertanty n the fact of presentng a clam y the polcyholder: the clam wll e fled for sure the moment the polcyholder passes away. There s no fortuty to the sze of the clam ether: the company has to pay the polcyholder s hers the amount, as s requred y the contract. Ths s what sets full nsurance apart from short-term types of nsurance. It s clear that the sze of the nsurance premum p has to e much hgher than the sze of the nsurance dsursement. In ths regard, the queston then arses: how does the company manage to come up wth the money for the dsursement of clams? Indeed, wth full lfe nsurance, all polcyholders hers wll fle clams wth the company for sure. The thng s the company receves the nsurance premum p at the moment the contract s eng entered nto and t pays the dsursement a lot later. Wthn ths perod of tme (whch equals the remanng term of lfe T() of the polcyholder, where s hs/her age) the nsurance premum makes the company proft and grows va the followng y the amount T pe where s the ntensty, %. Thus, the company s proft from enterng one full nsurance contract s T ( ) pe In order to get the amount at the moment of the polcyholder s death (.e. n T() years after the concluson of the contract), the nsurance company, accordng to the formula for changes n the value of money over tme, has to get from hm/her the amount T z : e at the moment the contract s eng entered nto. 66

10 ( + a) ( a + ) a + ( + a)( a + ) World Appl. Sc. J., 5 (4): , 03 The amount z s a chance quantty that epresses the e Q s the forecast polcy sales volume wthn the up-to-date (.e. as of the moment the contract s eng current perod; entered nto) sze of the future nsurance dsursement. The average sze p0of ths quantty s desgnated y s the smoothng constant; the nsurance company as the net premum. Q t s the sales volume over the perod t; Numerc calculatons are done usng the Erlang Q t - s the smoothed sales volume for the perod t -. model. The smoothng constant s taken y an analyst Eample: Let s assume that lfe epectancy s descred nteractvely wthn the nterval from 0 to. Its value s usng the Erlang model and the average lfe epectancy s low when there s lttle change n sales and tends to 80 years. when there are strong fluctuatons on the nsurance We ll calculate the net premum for polcyholders market. aged 0 and 30, the ntensty equalng 0%. Eample: We need to calculate the forecast value of Soluton: The average lfe epectancy n the Erlang model possle short-term polcy sales n May 0, f n Aprl equals A. 0 the company entered nto nsurance contracts and n March forecast sales consttuted 9 contracts. Then Soluton: The forecast value s calculated va the formula a a + + a A + Q Qt + ( ) Qt the value A 40 0, we get À À ,0933 0,086 We can demonstrate that the functon of the dstruton F z(t) of the up-to-date sze of the future nsurance dsursement z, under full lfe nsurance, s epressed va the followng formula (usng the Erlang model): ln t a t Fz t t, t () ( + a), ( 0,) The eponental smoothng method s used for shortterm and long-term forecasts and s predcated on the mean-weghted value of sales over a certan numer of perods passed. The forecast value s calculated va the formula ( ) t Q Qt + Q where Q s the forecast volume of polcy sales for May 0; s the smoothng constant; Q t s the sales volume for Aprl 0; Q t - s the forecast sales volume for March 0. As we can see, the value of the smoothng constant s low ecause there s lttle change n sales over the gven perod. Then Q 0, + ( 0,) 9 9, 9 REFERENCES. Gerer, H., 995. Lfe Insurance Mathematcs. M., Mr, nd 995. (In Englsh: Sprnger-Verlag, ed.). Chetyrkn, E.M., 000. Fnancal Mathematcs: Tetook M.: Delo, pp: Sddq, M.N., Insurance n an Islamc Economy. - London: Islamc Foundaton. 4. Bowers, N., H. Gerer and D. Jones, 00. Actuaral Mathematcs. M.: Yanus, pp: Fyodorova, T.A., 008, Insurance. M.: Economst, pp: Skpper, H. and K. Black, 999. Lfe and Health Insurance. Prentce Hall ISBN ; pp: Shkhov, A.K., 00. Insurance Law. M.: Yursprudentsya, pp: 8. 66

11 World Appl. Sc. J., 5 (4): , Gantenen, M. and M. Mata, 008, Swss Annutes 0. Khan, M. and A. Mrakhor, 990. Economc and Lfe Insurance: Secure Returns, Asset Protecton Development and Cultural Change, 38:. and Prvacy. Wley ISBN ; 9. Laronov, V.G., S.N. Selevanov and N.A. Kozhurova, Insurance. M.: URAO, 008. pp:

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

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