PREMIUMS CALCULATION FOR LIFE INSURANCE


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1 ls of the Uiversity of etroşi, Ecoomics, 2(3), 202, REIUS CLCULTIO FOR LIFE ISURCE RE, RI GÎRBCI * BSTRCT: The pper presets the techiques d the formuls used o itertiol prctice for estblishig the premiums for life policy. The formuls re geerlly bsed o series of idictors med mortlity idictors which mily poit out the isured survivl probbility, the deth probbility d life epectcy t certi ge. I determied, usig cse study, the uique et premium, the ul et premium for survivl isurce, whole life isurce d mied life isurce. KEY WORS: premium; life isurce; mortlity tbles; commuttio umbers. JEL CLSSIFICTIO: G22; G32; G35.. ITROUCTIO cturil sciece ppered from the eighteeth cetury through combitio of the iterest rte with mortlity tbles, cturil term ppers oce with the society "The Equitble" i 762, d the first who fouded the cturil professio ws Willim org, who held this post t this society (ghelche, 2006). The ctury hs the most importt role i clcultig isurce premiums, clled so life isurce mthemtici" (Ciurel, 2000) with the followig mjor tsks: determiig the isurce premium, determiig mthemticl reserves, the surreder vlues of cpitliztio products, ew products developmet, d others. The mi elemets uderlyig the clcultio of isurce premiums re mortlity tbles, geerl helth of the popultio, ge d se of the isured, cotrct period, the level of sums isured to be pid i deth cse or t isurce cotrct mturity if it is edowmets isurce, iterest rte derived from the ivestmets of premiums, the epeses relted to the issuce of the isurce cotrct, the isurce compy's profit. This techique of determiig the isurce premiums d mthemticl reserves i life isurce is prticulrly importt. Correct determitio of premiums * h Cdidte, Uiversity of Criov, Romi, h Cdidte, Uiversity of Criov, Romi
2 98 red,.; Gîrbci,. is bsed o observce of some cler priciples (Ciurel, 2000): isurce premiums must be dequte, which mes tht for group of cotrcts, the moey collected from policyholders, plus the iterest ered from the ivestmet of these mouts, shll be sufficiet to py ll promised mouts d cover the isurce compy epeses; isurce premiums must be equitble, tht risk must cosider ech perso isured; isurce premiums should ot be ecessive compred to the isured sums. I will lyze the price of three types of isurce policy: survivl isurce, whole life isurce, mied life isurce. Survivl isurce is chrcterized by the fct tht the isurer udertkes to py the isured sum whe the cotrct epires provided tht the isured is live. Sum isured cosists from isurce premiums pid by the isured durig the isurce period, ccumulted d cpitlized i vrious wys by the isurer. So, the isured receive the isured sum oly if, o epiry of the cotrct, is live, otherwise "the isurer is cosidered relesed from tht service which is stted i the isurce cotrct, hvig o obligtio to the heirs of the isured" (ăuleţiu, 2007) Survivl isurce is ot ttrctive, so does ot stimulte svig. Whole life isurce ims to protect the isured gist the risk of deth, future risk, sure, but ucerti s time. whole life isurce hs limited durtio d requires tht isurer pys to the beeficiry the sum isured if the isured's deth occurred durig the period of the cotrct. Isted if the isured survives whe the cotrct epires, the the isurer is relieved of y obligtio to the isured, "isurce premiums re set ccordig to the chrcteristics of ech cotrct bsed o cturil clcultios (ledru & rmeu, 2003), whole life isurce is isurce protectio gist the risk determied d ot svigs isurce. ied life isurce is chrcterized by the fct tht "the isurer offers product tht covers both risks through sigle cotrct, survivl d deth." (Bde & Ioescu, 200). Coverge i isurce cotrct the two ltertive risks ot remove their cotrdictory chrcter. It seems tht the isured is protected i both cses. I fct to the deth of the isured, isurce beeficiry shll receive the sum isured d if the isured survives, he beefits of the sum isured. If the isured is protected for both risks, he pys the isurce premium relted to both risk of deth d survivl, cumultively, the isurer ctully justified by the requiremet to chieve blce betwee premiums ered d beefits icurred. 2. CSE STUIES COCERIG THE ET UIQUE REIU I studied the uique et premium for survivl isurce. These types of isurce hve the followig techicl chrcteristics: isurce period deoted by, epressed i yers; isured sum, deoted by S; Risks covered by such isurce my be: if the isured to ge, survives to ge + yers (= ed isurce), the isurce compy pys to this perso the sum S t the ed of the isurce period; if the isured to ge does ot survives to ge + yers, the the isurce compy does ot py the sum S, d o premium refud;
3 remiums Clcultio for Life Isurce 99 The isured sum c be pid by the isurer: lump sum settlemet or pymet i driblets. The uique et premium for survivl isurce, the isured sum pid with lump sum settlemet, is clculted s: E l v () l l Where: E  the uique et premium with uitry sum isured; l  umber of survivors to ge ;  period of isurce, yers; v  life ctuliztio fctor. We both deomitor d umertor weightig of v d we get: l v v l v l E l v lv The reltio l. v =,(  re commuttio umbers) so the: (2) l (3) Whe the isured sum S is other th uitry, the premium is clculted: S S (4) l For emple, we clculte the et premium to be pid by yers old perso, who buy survivl isurce policy, period cotrct 20 yers, with sum isured of 0000 c.u., the iterest rte is 0%. From mortlity tbles we hve: 50 =76; 76 =5403; =0000 =325,8 c.u Besides survivl isurce, the most populr d the chepest re cosidered the whole life isurce. The isurer will py to the isurce beeficiry certi mout of moey t the isured perso's deth. epedig o cotrct period these isurce c be:  uity isurce where the premium is deoted by ; term isurce where the premium is deoted by. The first type of isurce supposes the isurer obligtio to py to the isurce beeficiry sum of moey upo the deth of the isured perso fter y period whe deth hppes. The pymet of sum isured c be t the ed of yer whe the isured died or immeditely fter deth. I clcultig the uique et premium for whole life isurce uity, with pymet of sum isured t the ed of yer of isured deth re tke ito ccout: the isurer suppose tht ll persos l of ge buy whole life isurce idefiite period. From the mortlity tble we kow tht durig the first yer of cotrct, will die d people d will remi live l +, so the isurer will py d c.u. :
4 200 red,.; Gîrbci,. Sice pymet will be mde t the ed of deth yer, the preset vlue of pymets i the first yer will be d v, i the secod yer d + v 2, i the third yer d + v 3, etc., i this cse the preset vlue of totl pymets i  + yers re: 2. d vd v d v (5) 2 dvd v. d v (6) l To simplify the clcultios we multiply both deomitor d umertor with v d l.v is deoted by d d v + = C : C C. C t C t (7) The umertor is clculted d represet (  re commuttio umbers): (8) S S (9) Whe the isured sum S is other th uitry, the premium is clculted ccordig to rel. (9). If the pymet of sum isured is mde immeditely fter isured deth the isurce premium is deoted by (,  re commuttio umbers, i  iterest rte): i (0) I will eemplify these types of premiums usig the followig dt: yers old perso, who buys uity isurce policy, sum isured 0000 c.u., ul iterest rte is 0%.  the sum isured will be received t the yer ed of deth: 328 = 0, 0607, = 607 c.u.; the sum isured will be pid immeditely fter deth:  0, 0,060,048 = 0,0628, = 628 lei
5 remiums Clcultio for Life Isurce 20 The secod type of cotrct, term isurce, ssumes the obligtio to py sum of moey to isurce beeficiry fter the isured deth, if his deth occurs withi the period specified i the isurce cotrct. The resoig behid the premium clcultio is the sme s the uity isurce. : () Whe the isured sum is other th uitry: = S : S (2) Usig the dtes bove for emple, if the isured would hve bought isurce policy, cotrct period  5 yers: :5 0,00962, = 96,2 lei et, I determie the uique et premium for mied life isurce, deoted by : d is clculted by ddig et uique premiums of survivl isurce d whole life isurce: Whe the isured sum is other th uitry: : (3) = S (4) I will eemplify usig the followig dt: yers old perso, cotrct period  5 yers, sum isured 0000 c.u., ul iterest rte is of 0% ,6224, =6224; :5 3. CSE STUIES COCERIG THE ET UL REIU The cotrct period for life isurce is for severl yers d usig the et uique premium is etremely rre, becuse it requires too much ficil effort for isured, therefore c be used istlmets throughout ll cotrct period, or shorter period. We tlk bout et ul isurce premium of survivl isurce, et ul isurce premium of whole life isurce d ul et premium of mied life isurce. I cse of survivl isurce we hve two possibilities:
6 202 red,.; Gîrbci,.  if the pymet period of et ul premium coicide with the cotrct period (m = ), the ul et premium for perso ge, which requires survivl isurce for period of yers is: (5) : : : : (6) Where: : the et ul premium; : the et uique premium for survivl isurce,  the et uique premium relted to ticipted uity : isurce for specified period,,  re commuttio umbers.  if the pymet period of et ul premium is less th the cotrct period (m < ), the ul et premium for perso ge, which requires survivl isurce for period of yers is: : : (7) : m : (8) : m I clculted the et ul premium tht should be pid by yers old perso, which requires survivl isurce, cotrct period 5 yers, sum isured 0000 c.u., ul iterest rte 0%: m =, :5 = 0,475 =475 c.u.; m <, if the isured wts to py the premiums i 3 yers: : 0,2244 =2244 c.u.; Reltig to whole life isurce I preseted the et ul premium for:  uity isurce where the premium is deoted by ;  term isurce where the premium is deoted by ; For the first type of isurce we hve two situtios:. if the pymet period of et ul premium coicide with the cotrct period, m = : (9) (20) Where:  the et uique premium for uity isurce,  the et uique premium relted to ticipted uity isurce,,  re commuttio umbers 2. if the pymet period of et ul premium is less th the cotrct period m < : ȧ.
7 remiums Clcultio for Life Isurce 203 (2) m m (22) For the secod type of isurce we hve, lso, two situtios:. m = : : (23) : (24) : Where: : :  the et uique premium for term isurce (whole life isurce), the et uique premium relted to ticipted term isurce;,  re commuttio umbers 2. m< : : (25) (26) m : m I clculted the et ul premium tht should be pid by yers old perso, which requires whole life isurce, sum isured 0000 c.u., ul iterest rte 0%:. uity isurce  m= 328 = 0,0058, =58 c.u.; m <, the isured wishes to py the premium i 0 yers: 328 0,0090 =90 c.u.; term isurce  m= = :5 0,0023 =23 c.u.; m < the isured wishes to py the premium i 3 yers: :5 0,0035 =35 c.u.; The et ul premium for mied life isurce is clculted by ddig et ul premiums of survivl isurce d whole life isurce: : : : : : (27) : (28) : :
8 204 red,.; Gîrbci,. whe m < the formul becme: : (29) I clculted the et ul premium tht should be pid by yers old perso, which requires mied life isurce, sum isured 0000 c.u., ul iterest rte is 0%:  m = = 5, :5 = :5 :5 =0,475+0,0023=0,498, =498 c.u.;  m < = :5 0, 2279 =2279 c.u COCLUSIOS 33 The bove et premiums preseted gurtees oly cover obligtios to the isured, isurce pymets, whe hppe the isured risk, but i order to cover costs of coductig isurce opertios, the isurers dd dditio to et premium thus obtiig gross premium, pid by the isured whe purchsig the isurce policy. Isurer creted o ccout of premiums received from policyholders, fud which is cpitlized s bk deposits or other ivestmets i ecoomic cycle d produce icomes tht stisfy isurer s obligtios to policyholders, the isured sum pymet. Isurce premiums must be dequte, equitble, should ot be ecessive compred to the isured sums, so tht show hw importt is the ctury ctivity. The resercher s lysis show tht i ll developed coutries i the world, mles hve higher overll rtes of mortlity th femles, so for premium clcultio must be tke ito discussio other importt fctors: icludig whether or ot perso smokes, their ge d mritl sttus, where they live d their lifestyle i geerl. The sme thig we c sy bout femle life epectcy eceeds tht of mles tody, but this hs bee the cse i these coutries for my yers. REFERECES: []. ledru, F.; rmeu,. (2003) sigurări de buuri şi persoe, Editur Ecoomică, Bucureşti [2]. rmeu,. (2008) etode şi tehici de sigurre, Editur Uiversităţii Româo Britice, Bucureşti [3]. ghelche, C.V. (2006) liză cturilă î sigurări, Editur Ecoomică, Bucureşti [4]. Bde,.; Ioescu, L. (200) sigurările de persoe și reflectre lor î cotbilitte, Editur Ecoomică, București [5]. Ciurel, V. (2000) sigurări şi resigurări:bordări teoretice și prctici iterţiole, Editur llbeck, Bucureşti [6]. ăuleţiu,.c. (2007) sigurări comercile, Editur Risoprit, Cluj poc m
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