Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 Mahemaica Moeig of Life Isurace Poicies Ui M. Kuub eparme of Mahemaics, Uiversiy of haka, Bagaesh kuubu9@gmai.com Isam M. Rafiqu eparme of Bakig, Uiversiy of haka, Bagaesh Rehma Tasima eparme of Mahemaics, merica Ieraioa Uiversiy Bagaesh (IUB) asima.rehma@homai.com Moa Rabira Nah (orrespoig auhor) Mahemaics iscipie, Khua Uiversiy, Bagaesh E-mai: rmoa7@yahoo.com bsrac: The Life Isurace ompay cacuaes he poicy price wih ie o recover caims o be pai a amiisraive cos a o make profi. The cos of isurace is eermie usig Moraiy Tabe cacuae by cuaries. The isurace compaies receive premiums from he poicy ower a ives hem o creae a poo of moey from which o pay caims a fiace he isurace compay s operaios. Raes charge for ife isurace icrease wih isure s age because saisicay peope are more ikey o ie as hey ge oer. I his paper, we iscusse abou iffere isurace poicies icuig epeses. We have aso iscusse abou he aua premium raes of boh eowme pa a hree-payme pas. We have use mahemaica programmig o cacuae he premium raes. Keywors: Isurace, Premium, Eowme ssurace, Life uiies.. Iroucio cuaria Sciece appies mahemaica a saisica mehos o fiace, isurace paricuary o risk assessme. cuaria Mahemaics eas wih he mahemaics of uceraiy a risk (Bowers, Gerber, Hickma, Joes a Nesbi 986). Some key areas where acuaria mahemaics is pricipay appie are moraiy suy, fiacia risk, risk a rui heory, creibiiy ec. Basic ieas from cacuus, iear agebra, umerica aaysis, saisics, mahemaica programmig a ecoomics wi appear as a buiig bock i moes of isurace sysem (ii, Moi a Joshi 00). isurace sysem is a mechaism for reucig he averse fiacia impac of raom eves ha preves he fufime of reasoabe epecaios, i.e. Isurace is esige o proec agais serious fiacia reversas ha may resu from raom eves iruig o he pas of iiviuas. The face amou of he poicy is ormay he amou pai whe he poicy maures, ahough poicies ca provie greaer or esser amous. The poicy maures whe he isure ies or reaches a specifie age. The mos commo reaso o buy a ife isurace poicy is o proec he fiacia ieres of he ower of he poicy i he eve of he isure s emise (Ui 999). Life isurace is mus for a a sury who have famiy o ook afer. Nowihsaig his, he ee is o equa for a. For rich peope i may be a uury bu for ow icome commuiy i is a mus. Ye he aer secios of he sociey are o covice ha hey ee i. The Goverme a No Goverme Orgaizaios ee. 308
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 The Life Isurace ompay cacuaes he poicy price wih ie o recover caims o be pai a amiisraive cos a o make profi (i 004). The cos of isurace is eermie usig Moraiy Tabe cacuae by cuaries. The isurace compaies receive he premiums from he poicy ower a ives hem o creae a poo of moey from which o pay caims, a fiace he isurace compay s operaios (am 993). Raes charge for ife isurace icreases wih isure s age because saisicay peope are more ikey o ie as hey ge oer. Sice averse seecio ca have a egaive impac o he fiacia resus of he isurer, he isurer ivesigaes each propose isure begiig wih he appicaio which becomes a par of he poicy (Ziam a Brow 005). Life isurace is mus for a a sury who have famiy o ook afer. Nowihsaig his, he ee is o equa for a. For rich peope i may be a uury bu for ow icome commuiy i is a mus. Ye he aer secios of he sociey are o covice ha hey ee i. The Goverme a No Goverme Orgaizaios ee. Lack of isurace ca coribue o iequaiy i he sociey as whoe a which has a irec effec o he ecoomic growh of ay coury. Life isurace compaies, seig ages, NGOs a he Goverme shou ake his issue ogeher as a chaege equay. However cosierig he fac ha umber of peope wih ow icome far ecees hose wih higher icomes, herefore ow premium ife isurace poicies wih o fris ca aso be remueraive o he seig ages a he Life Isurace compaies, as ow margis of profi wou be more ha offse by he high voumes of poicies so (hauhury 994). Nee i such awareess shou be creae i boh sies, ife isurace compaies, heir seig ages a he cocere sraa of peope wih ow icomes. Here is where he reguaory boies, Goverme a No Goverme Orgaizaio ca pay impora roes. I his paper, we iscusse abou iffere isurace poicies icuig epeses. We iscusse abou he aua premium raes of boh eowme pa a hree-payme pas. We have use mahemaica programmig o cacuae he premium raes.. Mahemaica Formuaios Whoe Life ssurace The essece of Whoe Life ssurace is ha i provies for he payme of he face amou upo he eah of he isure regaress of whe he eah occurs. This is oe of he simpes forms of ife assurace (Hafiz, Isam, a howury 995). The vaue of whoe ife assurace of payabe o a perso age i.e. he prese vaue of he assurace is eoe by, a is give by Iroucig ommuaio Fucios 3 3 309
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 30 M Temporary or Term ssurace Term ife isurace furishes ife isurace proecio for a imie umber of years, he face amou of he poicy beig payabe oy if eah occurs urig he sipuae erm a ohig beig pai i case of surviva. The vaue of year erm assurace of o he ife of a perso age is eoe by :. The umber over iicaes ha i orer for he sum assure becomig payabe he saus () mus come o a e before he saus. Thus he epressio for he prese vaue of his assurace of payabe o eah urig -year erm is give by : = 3 ; : = Iroucig ommuiy Fucio : : } { } { 0 M M Pure Eowme ssurace -year pure eowme provies for payme a he e of he h year if a oy if he isure survives a eas years from he ime of poicy issue The vaue of -year pure eowme assurace of o he ife of a perso age, is eoe by : is give by : = Eowme ssurace - year eowme isurace provies for a amou o be payabe eiher foowig he eah of isure or upo he surviva of he isure o he e of he -year erm, whichever occur firs. This is a combiaio of pure eowme a emporary assurace. The prese vaue of he assurace of uer his pa is eoe by :.
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 3 : = : + : { } ) ( : = M M Life uiies ife auiy is a series of paymes mae coiuousy or a equa iervas whie a give ife survives. I may be emporary, ha is, imie o a give erm of years, or i may be payabe for he whoe of ife. The payme iervas may commece immeiaey or, aeraivey, he auiy may be eferre. uiy ue osier ives. Sice he paymes are o be mae a he begiig of each year, ives wi receive firs payme a he prese ime. ä = Iroucig he commuaio fucios, we have ä = N Temporary uiies emporary ife auiy is a series of paymes mae a reguar iervas o a perso urig his ife ime for a specifie perio, each payme beig mae a he e of each year of ife urig years. The prese vaue of such auiy is eoe by a :. Thus a : a : N N Temporary Life uiies ue If isea a he e of he year, he paymes are mae a he begiig of each year, he series of paymes are kow as emporary ife auiy ue for years. The prese vaue of emporary ife auiy ue of o a perso age X is eoe by ä :, a is vaue is give by ä : ) ( ) (
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 ä : 0 N N 3. Mahemaics of Premiums Ne premiums for ssurace Pas The e premiums are obaie by iviig he prese vaue of beefis by he prese vaue of premiums. Prese vaue of various assurace pas aso represes he sige premium o be pai a he begiig of a corac o secure he beefis uer he assurace pa. Whoe Life ssurace Le P be he aua premium for a whoe ife assurace of o he ife age. Uer his pa he premium is payabe hroughou he ife ime of he assure. The vaue of he of he premium wou herefore, be equa o P ä. We aso kow ha he vaue of he whoe ife sum assure of is. Therefore, we ge P ä = P = M / N Temporary ssurace Uer his pa ife assure age wi pay he eve aua premium for years. The vaue of emporary assurace of o a ife age, is P : ä :. Hece P : P : ä : = M N M N Year Eowme ssurace : P : a he begiig of each poicy year :. The prese vaue of he premium is The vaue of a -year eowme assurace o of he ife age is :. The prese vaue of he premiums is P : ä :. Hece P : ä : : P : M M N N Isurace Moes Icuig Epeses 3
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 more reaisic view of he isurace busiess icues provisio for epeses. The profi for he compay ca aso be icue here as a epese. The commo meho use for he eermiaio of he epeses oae premium is a moificaio of he equivaece pricipe. ccorig o he moifie equivaece pricipe he gross premium P is se o ha o he poicy issue ae he acuaria prese vaue of he beefi pus epeses is equa o he acuaria prese vaue of he premium icome. The premium is usuay cosiere o be cosa. Uer hese assumpios i is fairy easy o wrie a formua o eermie P. Three eemes which is o be ake io cosieraio whie esigig a prouc a pricig he prouc, i.e. o cacuae he premium are: ) Rae of moraiy ) Epeses icurre by ife isurace busiess 3) Rae of reur o ivesme. Prouc esig s per ar. 39 of he isurace rue 958, he imiaio of epeses of maageme (icuig commissio a ay oher remueraio for procreaio of busiess) i ay caear year is a amou o eceeig 90% of he s year premium a 5% of reewa premium for a ife isurace compay whose year of operaio are 0 years or more a erms of he isurace poicy o ess ha years. ua premium of a eowme pa We cacuae he aua Premium of a prouc which provies beefi of Tk.000 o surviva up o mauriy a Tk.000 o eah before mauriy. This ype of pa is cae eowme pa. If we cosier he erm of he poicy o be years a we wa o cacuae he aua premium for a perso age (), if P is he aua premium he, aue of eah beefi is 000 aue of surviva beefi is 000 : : Hece he prese vaue of he premium is Pä :. osierig he epeses foowig he rue of isurace ac we have Pä : =000 : +000 +.75P +.5 Pä : : 000( M M P 0.85( N N ) ) 0.75.......................................... () We use Mahemaica Program for equaio (). We obai a poyomia for a he commuaio fucio usig Newo's Forwar Ierpoaio meho (Bure a Faires 003). The aua premium abe per Tk.000 for a isurace poicy of erm 5 years usig mahemaica program is give beow. Tabe for 5 years pa ge Premiums ge Premiums 0 64.49 4 66.60 64.68 4 66.605 33
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 64.508 43 66.97 3 64.436 44 67.398 4 64.49 45 67.87 5 64.449 46 68.465 6 64.47 47 68.989 7 64.493 48 69.670 8 64.58 49 70.344 9 64.549 50 7.44 30 64.588 5 7.04 3 64.64 5 7.85 3 64.703 53 74.55 33 64.783 54 75.366 34 64.878 55 76.77 35 65.00 56 78.37 36 65.7 57 80.885 37 65.300 58 8.67 38 65.494 59 84.53 39 65.696 60 86.39 40 65.966 The aua premium Tabe per Tk.000 of a ife isurace poicy of erm 0 is give beow which is aso obaie by usig mahemaica programmig. Tabe for 0 years pa ge Premiums ge Premiums 0 43.605 4 46.97 43.783 4 46.685 43.69 43 47.9 3 43.640 44 47.698 4 43.646 45 48.500 5 43.676 46 48.966 6 43.74 47 49.60 7 43.75 48 50.655 8 43.799 49 5.483 9 43.855 50 5.496 30 43.99 5 53.74 3 44.007 5 55.57 3 44.6 53 56.38 33 44.39 54 58.35 34 44.369 55 59.697 35 44.546 56 6. 36 44.74 57 63.40 37 44.976 58 67.054 38 45.03 59 75.998 39 45.53 60 74.447 40 45.867 The foowig curve (Fig. ) shows he variaio of premiums wih respec o he age for a 5 years a 0 years isurace poicy. 34
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 premium 80 5 years 70 60 50 0 years 40 30 40 50 ge Figure. ariaio of premiums wih respec o he age for 5 years a 0 years isurace poicy ua premium for a hree-payme pa I hree paymes pa surviva beefi is give a 3 sages of he oa erm of he poicy. If he erm of he poicy is years he we may cosier ha 5% of he sum assure is provie afer he epiry of 4 years, 5% of he sum assure is provie afer he epiry 8 years a fiay 50% of he sum assure is provie a he e of he erm i.e. afer years as surviva beefi. So he Mahemaica formuaio for a hree payme pa, where he basic sum assure is Tk 000 usig he commuaio fucio M,, N for years a for a perso age is: aue of surviva beefi is 000 : The prese vaue of he premium is Pä :. osierig he epeses foowig he rue of isurace ac we have, 000( M M ) 50 50 500 P /3 /3 0.85( N N ) 0.75 The aua premium abe per Tk.000 of a hree-payme pa for a erm of years usig mahemaica program is give beow. Tabe for years hree-payme pa ge Premiums ge Premiums 0 96.36 4 98.98 96.605 4 99.46 96.473 43 99.99 3 96.393 44 00.50 4 96.387 45 0.58 5 96.43 46 0.8 6 96.443 47 0.6 7 96.474 48 03.480 8 96.50 49 04.479 9 96.555 50 05.38 30 96.65 5 06.63 3 96.689 5 07.808 3 96.783 53 09.474 35
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 33 96.896 54 0.75 34 97.035 55.30 35 97.99 56 4.560 36 97.398 57 6.386 37 97.69 58 8.56 38 97.905 59.045 39 98.08 60 4.95 40 98.573 Tabe for 5 years hree-payme pa 36
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 ge Premiums ge Premiums 0 74.6 4 77.489 74.43 4 78.03 74.3 43 78.596 3 74.59 44 79.50 4 74.66 45 79.993 5 74.30 46 80.87 6 74.344 47 8.637 7 74.39 48 8.67 8 74.447 49 83.678 9 74.57 50 85.047 30 74.60 5 86.79 3 74.708 5 87.477 3 74.835 53 89.33 33 74.989 54 90.87 34 75.69 55 9.86 35 75.386 56 94.86 36 75.66 57 97.964 37 75.96 58 99.906 38 76.43 59 03.77 39 76.599 60 06.09 40 77.0 The foowig curve (Fig. ) shows he variaio of premiums wih respec o he age for a years a 5 years hree-payme pa. Premiums 0 0 00 years years 90 80 5 years 30 40 50 ge Figure. ariaio of premiums wih respec o he age for years a 5 years isurace poicy Premium Tabe of merica Life Isurace ompay (ico) ge years 5 years 3PPP ge years 3PPP 5 years 3PPP 3PPP 0 00.80 80.70 35 0.0 8.60 00.80 80.80 36 0.50 8.90 00.80 80.80 37 0.80 83.0 37
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 3 00.90 80.90 38 03.0 83.60 4 00.90 80.90 39 03.40 84.00 5 00.90 8.00 40 03.80 84.40 6 0.00 8.0 4 04.30 85.00 7 0.0 8.0 4 04.90 85.60 8 0.0 8.30 43 05.50 86.30 9 0.0 8.40 44 06.0 87.0 30 0.30 8.50 45 06.90 87.90 3 0.50 8.70 46 07.60 88.80 3 0.60 8.90 47 08.50 89.70 33 0.80 8.0 48 09.40 90.70 34 0.00 8.30 49 0.40 9.80 We have see ha he premium rae charge by LIO, a iffere ife isurace compaies ike ea Life Isurace, Naioa Life Isurace ec. are higher ha wha we have cacuae usig mahemaica program. This may be ue o he foowig reasos: ) Sice we have go ay iformaio abou he cacuaio of he premium raes from iffere eisig compay. ) The rae of ieres assume by me i he premium raes cacuaio is higher ha wha have bee assume by ge 37 74.7 57.93 55 00.00 93.0 hese compaies. 3) Epeses oae i he premium eermiaio formua are higher ha wha have bee aowe i he isurace rue. 4) combiaio of boh he above reasos. 4. ocusio 5 years Eowme pa 0 years Eowme pa I his paper, we have presee how oe ca appy mahemaica programs o cacuae he aua premiums of various isurace poicies. I is very ifficu o ge he age specific premium raes bu by coig mahemaica ge 5 years Eowme pa 0 years Eowme pa 0 70.99 53.85 38 74.63 58.56 7.07 53.95 39 75.5 59.9 7.6 54.07 40 75.73 60. 3 7.6 54.6 4 76.38 6.04 4 7.37 54.3 4 77. 6.05 5 7.48 54.47 43 77.93 63.5 6 7.60 54.79 44 78.85 64.34 7 7.7 54.79 45 79.90 65.65 8 7.83 54.98 46 8.0 67. 9 7.97 55.9 47 8.46 68.80 30 7.3 55.4 48 83.98 70.74 3 7.3 55.65 49 85.66 7.98 3 7.5 55.93 50 87.5 75.57 33 7.76 56.4 5 89.55 78.47 34 73.04 56.58 5 9.8 8.67 35 73.37 56.96 53 94.9 85.7 36 73.75 57.40 54 97.00 88.98 38
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 program we ca easiy ge he premium raes for iffere isurace poicies for a perso age () a for a erm of he poicy of years i a cusomize way. I is fou ha ife epecacy of he isure popuaio is more ha he acua popuaio, ha meas isurace compaies are chargig more premium raes ha wha hey shou charge. I his paper, we have iscusse how o evauae he premium of iffere assurace pas such as whoe ife assurace, emporary assurace, eowme assurace ec. We have cacuae he premium for iffere ife isurace poicies ike eowme assurace pa, hree payme pa, si payme pa, weve payme pa, a micro ife isurace poicy usig Mahemaica Program. he begiig iffere commuaio fucio has bee evauae which are furher use o cacuae he premium of a perso age () for a isurace poicy of erm years, usig Newo s Forwar ierpoaio meho. The hese fucios are use o evauae he premium of a perso usig Mahemaica Program. We have cacuae he aua premium for a - year eowme assurace of he ife age (), where he basic sum assure is Tk.000. We aso cacuae he aua premium for a hree payme pa. The basic sum assure for a hese poicy is TK.000. The we have give a abuar form of premium raes for hese poicies a we have aso compare i wih he premium raes of some eisig compaies ike merica Life Isurace ompay, ea Life Isurace ompay a Popuar Life Isurace ompay ec. We have fou ha he premium raes of hese compaies are higher ha ha we have compue, a we have came o a cocusio ha hese variaio i he premium raes migh occur because of he foowig reasos. ) The rae of ieres assume by us i he premium raes cacuaio is higher ha wha have bee assume by hese compaies. ) Epeses oae i he premium eermiaio formua are higher ha wha have bee aowe i he isurace rue. 3) Or a combiaio of boh he above reasos. Three-payme pa is a very popuar ife isurace pa. I a hree-payme Life Isurace Pa of erm years he isurer pays premium afer every 4 years. O he coiuaio of hree payme pa we have propose si-payme pa a weve-payme pa. We have see ha cusomer wi be more ierese o buy a sipayme pa raher ha buyig a hree-payme, o he oher ha cusomer wi be more ierese i buyig a weve-payme pa raher ha buyig a si-payme pa. This is because if he erm of he poicy is years he i a si-payme pa cusomer wi ge some par of his sum assure a he e of every year whie o he oher ha i a hree-payme pa he cusomer wi ge some par of his sum assure afer every four years. Simiary i a weve-payme pa for a poicy of erm years a cusomer wi ge some par of his sum assure afer every oe year. Hece he cusomer wi be more arace owars a weve-payme pa. he same i wi be easier for he compay o covice peope o buy a si-payme pa raher ha o buy a hree-payme pa a o buy a weve-payme pa raher ha buyig a si-payme. We have aso cacuae premium raes for micro isurace poicies for ow cass popuaio of he coury. Here we have cosiere he basic sum assure o be 6000 a we have ower he epeses. gai sice i is easier for he poor peope o give premium mohy hece we have cacuae he premium raes mohy raher ha auay as we have cacuae for oher poicies. We have come io a cocusio ha hese compaies are chargig more premium raes ha wha hey shou acuay charge i.e. he isurace compaies are earig more profis ha usua. Refereces 39
Europea Joura of Busiess a Maageme ISSN -905 (Paper) ISSN -839 (Oie) o 3, No.4, 0 Bowers, Gerber, Hickma, Joes a Nesbi, (986), cuaria Mahemaics, Pubishe by he Sociey of cuaries, Loo. Bure, R. L. a Faires,. J., (003), Numerica aysis, 7h Eiio, New York. ii, S.P. Moi,.S. a Joshi, R.. (00), Mahemaica Basis of Life ssurace, Iia. Ui, M.S., (999), Iroucio o cuaria a Fiacia Mahemaics, s Eiio. Ziam, P a Brow, R. I., (005), Theory a Probems of Mahemaics of Fiace, Eiio. i, K. M. M., (004), Povery eviaio a ee for Muua Micro Isurace for he Poor, Isurace Joura; oume: 55, Juy. Hafiz G..S, Isam, S a howury, J.., (995), Life Isurace Busiess i Bagaesh: evauaio. Isurace Joura; oume: 47. am, S., (993), Rura Isurace i Iia-is iverse Uses, Isurace Joura; oume: 44, Jue. hauhury, S.., (994), Life Isurace i Bagaesh- Isurer s iew, Isurace Joura; oume: 46, Jue. 30
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