Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 Poducts of the Second Pilla Pension Jana Špiková Depatent of Quantitative Methods and Infoation Systes, Faculty of Econoics, Matej Bel Univesity Tajovského 10, 975 90 Banská Bystica, Slovakia e-ail: jana.spikova@ub.sk Abstact: The second pilla pension guaantees an additional incoe fo pensiones in thei etieent. It is the ain suppot fo the fist pilla o the state pension. The second pilla pension is inheitable duing the savings peiod. The pension contact is a life insuance poduct on the basis of which annuity payouts ae ade to pensiones. This eans that the ageed pension payents ae guaanteed to pensiones until the end of thei life. Recently the attention focuses ostly on pension funds which have diffeent isk levels and stategies, but only a sall attention is paid to the calculation of futue pensions. Theefoe, we offe seveal life insuance poducts which would be necessay to conside in the pecise calculation of futue pension annuity payouts. Keywods: second pilla pension; annuity; odelling; gende 1 Intoduction Deogaphic tends - the low bith ate, the decease in woking-age population and especially the apid inceased life and health epectancy eans that thee will be less oney fo pensiones in the futue. It is clea that the state pension alone is not enough if a an wants to aintain his/he cuent living standad. Recently in ost counties the funding of pensions is based on thee pillas: the 1 st pilla - the copulsoy, pay-as-you-go, old age state pension; the 2 nd pilla - the suppleentay, often funding-based, occupational pension and the 3 d pilla - individual savings. The goal of the fist pension pilla o state pension is to guaantee that all pensiones eceive a pension on which they can subsist. The state pension insuance is financed though egula payents of the pension insuance peius to the State Social Insuance Copany which ae edistibuted in the fo of the pension annuity fo pensiones. The aounts of the payouts ade to pensiones fo the fist pension pilla ae oe o less the sae and only slightly affected by ealie salaies and the aount 17
J. Špiková Poducts of The Second Pilla Pension of social ta paid. The social ta paid by the woking population is the souce of state pension insuance. The 2 nd pilla pension guaantees additional incoe fo pensiones in thei etieent. It is the ain suppot fo the fist pilla o the state pension. The second pension pilla is inheitable duing the savings peiod. The pivate pension savings ae adinisteed by pivate Pension Fund anageent copanies. A client of such a copany contibutes to a pivate savings account, while the savings the popety of this client. If the case of death of the client, all oney which was not used to epay the pension annuity becoes a subject-atte of the inheitance as a pat of a decedent s estate o ay be tansfeed to the peson appointed by the client in the contact on etieent pension savings. All Pension Fund anageent copanies ust establish thee kinds of pension funds based on vaious levels of the financial isk consevative pension fund, balanced pension fund and a so-called gowth pension fund. Accoding to law, people have the ight to stat withdawing the oney they have saved in the second pension pilla when they attain etieent age. The aount of a pension payout depends on the peiod and fequency of payouts and the nube of units accuulated. The thid pilla is epesented by the voluntay additional insuance, which is designed to keep a high standad of living afte eaching the etieent age. The thid pilla includes, fo instance, an additional pension insuance, life insuance, investent to the utual funds. The client fully contols the aount and the pupose of the contibutions. Recently the attention focuses ostly on pension funds which have diffeent isk levels and stategies, but only a sall attention is paid to the calculation of futue pensions. Theefoe, this pape offes thee life insuance poducts which would be necessay to conside in the pecise calculation of futue pension annuity payouts. Moeove, this pape bings analysis of the ipact of a ban on the use of gende in insuance, with special stess on the 2 nd pilla pension annuity, accoding to the equieents of the Euopean Cout of Justice. This pape is oganized as follows. Section 2 pesents the peliinaies. In Section 3 we intoduce odels of thee insuance poducts which ae suitable on an evaluation of pension annuities fo the 2 nd pilla pension. Section 4 discusses the ipact of diffeent input paaetes on the size of onthly pension annuities. The last pat contains the conclusions. Poducts of the 2 nd pilla pension wee pogaed by MS Office Ecel 2010 VBA with using life tables obtained fo web page of the Statistical Office of the Slovak Republic (Motality tables, 2012). 18
Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 2 Peliinaies Pension contact is the life insuance poduct on the basis of which annuity payouts ae ade to pensiones. As a basic poduct of the second pilla pension we assue a single whole life pension insuance. This eans that the ageed pension payents ae guaanteed to pensione until the end of his/he life. To odeling the poducts of the second pilla pension we can use ultiple appoaches. One of the is the deteinistic. At the beginning we ecall the basic notations which ae used on an evaluation of entioned poducts. They ae: i - a technical inteest ate; q - an accuulated facto, q 1 i ; - a discounting facto, 1/ q ; - a nube of paid, esp. paid out annuities within one yea; y - a stat saving age; - a etieent age; n - a nube of yeas of saving on pesonal pension account, n y ; A - an yealy annuity duing the peiod of saving; l - a nube of living at age ; d - a nube of deaths at age ; - a aiu age to which a peson can live to see (egading used life tables is hee 100 ; p - a pogaed withdawal fo an accuulated value at the beginning of etieent tie (in pecent); z - a quotient of the -thly paid out pensions within t yeas (in pecent); t - a nube of yeas of payent of a quotient of the cetain pensions. We eind so-called coutation functions which facilitate full actuaial odeling. The fist one is D l (1) and it eans a nube of living at age discounted to the date of bith. 19
J. Špiková Poducts of The Second Pilla Pension Coutation function follows N epesents a su of D up to age and is given as i1 N D i. (2) The following functions elated to the nube of deaths. A function C epesses the nube of deaths at age discounted to the date of bith, and is given by 1 C. (3) d And finally, coutation function is given as follows M C i i1 M epesents a su of C up to age and. (4) In ou odel we need also othe financial and actuaial functions. These functions ae usually evaluated fo intege ages and tes, assuing that cash flows ae payable annually. Howeve, annuities ae vey often paid oe fequently than annually, naely onthly, quately, but also sei-yealy. We can find in vaious souces well known foulas on evaluating of cetain and epected annuities which ae paid oe fequently than annually. Hee, we eind only those which we need fo ou odeling. The accuulated value of tepoay cetain annuities of 1 / onetay unit s which ae payable in advance duing n yeas is as follows n 1 n 1 1 q s n q. (5) 1 q 1 The pesent value of tepoay cetain annuities of which ae payable in advance duing t yeas is as follows 1 / onetay unit a t 1 t 1 1 q a t q. (6) 1 q 1 In ou life insuance poducts we odel as a single peiu fo whole pension insuance an accuulated aount S as follows S A s. (7) n 20
Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 A net single peiu of the yealy whole life annuities in advance in the size of the one onetay unit fo an -aged insued is given by N a, (8) D And a coesponding pesent value of -thly paid annuities a is the epected pesent value of the whole annuity of one pe yea, payable to the enty aged with 1 / at the beginning of each -thly peiod pe yea and it is given as follows 1 a a. (9) 2 By A is denoted a net single peiu fo whole life insuance in the case of death fo -aged peson with a policy su of the one of onetay unit, and is given by M A (10) D At this point we ust eflect that the accuulated aount of savings actually epesents a single goss peiu and theefoe we cetainly will calculate the costs associated with the payent of pensions. We assue initial costs which ae usually calculated as a pecentage of the accuulated aount, and usually ae paid at the beginning of the insuance; adinistative costs and collection costs associated with the payent duing of etieent which insuance copanies calculated as a poile of the yealy pension annuity. 3 Poducts of the Second Pilla Pension In this pat we intoduce thee basic insuance poducts which could be used on an evaluation of pension annuities fo the 2 nd pilla pension. Obseve, the thid poduct epesents a genealization of two pevious ones, and pactically on a pogaing of the all poducts it is sufficient to poga just the thid one. Moeove, we intoduce foulas fo evaluation of the coesponding povisions which insues ust ceate unde the laws of individual counties. 21
J. Špiková Poducts of The Second Pilla Pension 3.1 Poduct 1 The fist Poduct 1 assues that the entie accuulated aount will be spent on pensions without esidue. A net pension annuity NP 1 paid out -thly pe yea is given by S NP 1 a (11) and a coesponding goss pension annuity 1 is as follows S 1 1. a 1 The following foulas epess the level of povisions at the beginning of -th yea of the payent of pensions. Foula (13) epesses a net povision NV at the beginning of -th yea and is given by NV 1 1 (12) NP a. (13) 1 A goss povision GV 1 at the beginning of -th yea is given by GV a 1. (14) 1 1 Obseve, in this case an insuance copany ust constitute povisions on costs which will need in the futue. 3.2 Poduct 2 The second poduct is chaacteized in that it allows the client to choose at the beginning of etieent p pecent of the saved aount as the fist highe pension. A net peiu NP S 1 p NP 2 in this case we can evaluate as follows. (15) a 2 So, as in the pevious poduct, in this insue ust calculate individual costs, which is epessed by the foula S 1 p 2 a 1 which epesses -thly pe yea paid out goss peiu 2. (16) 22
Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 Foulas (17), (18) epess coesponding povisions at the beginning of -th yea of the payent of pensions. A net povision NV 2 is as follows NV NP a 2 2 (17) and a goss povision GV2 GV by a 1. (18) 2 2 3.3 Poduct 3 The thid poduct is a ost geneal possibility of payent of pensions. In addition to the fist highe incoe p pecent of saving aount, this poduct also iplies that in case of a pensione death at the tie of payent of the pension, to suvivos o authoized pesons will eceive the guaanteed quotient z pecent of his/he oiginal pension duing t yeas. A net -thly pension annuity S 1 p NP 3 we can evaluated by NP 3 (19) a z a A t and a coesponding goss -thly pension annuity 3 as follows S 1 p 3. a z a t A 1 The net and goss povisions NV 3, GV 3 at the beginning of -th yea of the payent of pensions ae given as follows 3 3 NV NP a, z A a (21) t GV 3 a 1. z A a (22) t 3 (20) 4 The Ipact of the Individual Input Paaetes on the Pension As can be seen fo peceding foulas, the aount of futue pensions is influenced by a nube of input paaetes. 23
J. Špiková Poducts of The Second Pilla Pension Let conside fo illustation 25-yea-old peson (unise) who saving fo etieent 30 euos onthly up to 65 his/he age. When he/she will be 65-yea-old, he/she can buy onthly whole life pension annuity in the fo of the single goss peiu. Moeove, let conside a possibility of the choice of the fist highe pension 20 pecent of the saving aount and the cetain quotient 50 pecent of the onthly pension annuity duing net 5 yeas. Individual peius and povisions (fo the fist ten yeas) ae at the Table 1. Individual costs ae calculated as follows: 5%, 3, 1. Coesponding onthly peius consequently fo ou thee poducts ae 12 1 148.75 euos, 12 2 117.43euos, 12 3 104.74 euos, see also Table 1 and Figue 1. Table 1 Net and goss onthly peius and povisions accoding to individual poducts Accuulated Net Monthly Pension (euo) Goss Monthly Peiu (euo) Su (euo) Poduct 1 Poduct 2 Poduct 3 Poduct 1 Poduct 2 Poduct 3 24 592,27 157,2 125,76 112,17 148,75 117,43 104,74 Yea Net povisions Goss povisions of the Pension Poduct 1 Poduct 2 Poduct 3 Poduct 1 Poduct 2 Poduct 3 0 24 592,27 19 673,81 19 673,81 23 362,65 18 444,20 18 444,20 1 23 734,94 18 987,95 19 097,19 22 548,19 17 801,20 17 903,62 2 22 879,49 18 303,59 18 521,84 21 735,52 17 159,62 17 364,22 3 22 022,02 17 617,61 17 945,12 20 920,92 16 516,51 16 823,55 4 21 135,96 16 908,77 17 349,17 20 079,16 15 851,97 16 264,85 5 20 238,34 16 190,68 16 745,46 19 226,43 15 178,76 15 698,87 6 19 338,17 15 470,53 16 140,02 18 371,26 14 503,63 15 131,27 7 18 462,51 14 770,01 15 551,07 17 539,38 13 846,88 14 579,13 8 17 589,05 14 071,24 14 963,59 16 709,59 13 191,78 14 028,37 9 16 722,38 13 377,90 14 380,69 15 886,26 12 541,78 13 481,90 10 15 872,17 12 697,73 13 808,86 15 078,56 11 904,12 12 945,80 Figue 1 illustates the developent of the technical povisions fo ou entioned pensione in individual yeas of etieent accoding to offeed insuance poducts fo the 2 nd pilla pension. The developent of povisions is obvious. Obseve, appoiately fo 83 yeas the povisions fo the thid poduct ae highe than fo othes. This is due to the fact that the pobability of death fo that age is high and the copanies have to ceate highe povisions fo the payent of cetain pensions. Fo the sae eason, the povisions fo the second poduct ae highe copaed to the fist one. 24
Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 Figue 1 Goss povisions at -th yea of the eceiving of the pension These goss peius ae lowe by 6.21 % in aveage against to pensions if we do not assue costs of an insuance copany. On the Figue 2 you can see the developent of onthly pension annuities fo ou pensione accoding to etieent age. Figue 2 Monthly pension annuities accoding to etieent age With the sae conditions and with diffeent costs in the size 1 %, 2, 1 goss onthly peius ae lowe by 1.46 % in aveage against to the pensions if we do not assue costs. 25
J. Špiková Poducts of The Second Pilla Pension Figue 3 shows how uch onthly pension annuities would incease if the costs 5 %, 3, 1 decease to the level 1 %, 2, 1. It is clea that incease of the onthly pensions annuities fo the two fist poducts ae the sae. Figue 3 Incease of the onthly pension by educing costs On the Figue 4 ae illustated onthly pension annuities with espect to unise life tables and life tables fo both gendes, too. Figue 4 Monthly pension annuities accoding to etieent age Accoding to equieents of ECJ all insuance copanies ust fo 21 st Decebe 2012 to use on evaluation of peiu in all life insuance poducts unise life tables. In the calculation by unise life tables onthly pension annuities fo en deceasing by 11% in aveage and fo woen inceasing by 7% in aveage. 26
Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 Conclusions In this pape we offeed poducts which could be used by insuance copanies fo futue pensiones. Howeve, thee eist anothe possibilities on an etension of poducts fo the 2 nd pilla pension. In ou net investigation in this aea we want to study poducts which ae based on so-called joint annuities. In this case we can use deteinistic appoach, too. Howeve, in this case we assue independence of the joint lives. The copula odels becae an inceasingly popula tool fo odeling dependencies between joint lives. In actuaial studies, it is often of inteest to eaine the joint otality patten of goups of oe than a single individual. This goup could be, fo eaple, a husband and wife, a faily with childen, o twins. Intuitively, pais of individuals ehibit dependence in otality because they shae coon isk factos. of joint suvival analysis and paticula pobles of two associated lives. It howeve ust be undestood that even substantial developent of second and thid pilla pensions will pobably not be sufficient to copensate both longe life-epectancy and a ising popotion of people ove 65 yeas. With good health epectancy it is not only possible but also essential to plan fo fleible etension of woking life. Pension funds ust encouage and facilitate this etension which will also benefit the insuance secto wokfoce. Acknowledgeents This pape is based upon wok suppoted by the Poject: Mobility - Enhancing Reseach, Science and Education at Matej Bel Univesity, ITMS code: 26110230082, unde the Opeational Poga Education co-financed by the Euopean Social Fund. Refeences [1] BOOTH, P. et al.: Moden Actuaial Theoy and Pactice. Chapan & Hall/CRC, 2000 [2] DICKSON, D. C. M. et al.: Actuaial Matheatics fo Life Contingent Risks. New Yok: Cabidge Univesity Pess, 2009. [3] GERBER, H. U. et al.: Life Insuance Matheatics. Geany: Belin Heidelbeg, New Yok : Spinge-Velag, 1997. [4] MACDONALD, A. S. Eule-Maclauine Epansion and Woolhouse's Foula. 2011. http://peso.univ-ennes1.f/athu.chapentie/euleact.pdf. [cit. 3 June 2011] [5] MIGRATION INFORMATION CENTRUM. 2013. http://ic.io.sk/en/social-issues/social-secuity/75-dochodkove-poisteniena-slovensku-syste.htl [cit. 15 June 2013] 27
J. Špiková Poducts of The Second Pilla Pension [6] MORTALITY TABLES. 2012. http://potal.statistics.sk/showdoc.do?docid=2464 [cit. 10 Mach 2012]. [7] SEKEROVÁ, V., BILÍKOVÁ, M.: Insuance Matheatics (in Slovak). Batislava: Ekonó, 2007. [8] ŠPIRKOVÁ, J., URBANÍKOVÁ, M.: Actuaial Matheatics - Life Insuance (in Slovak). Batislava: Iua Edition, spol s. o., 2012. [9] OXERA: The ipact of a ban on the use of gende in insuance. http://www.slaspo.sk/tp/asset_cache/link/0000033890/11207%20oeastudy-on- gende-use-in-insuance.pdf. [cit.14 Mach 2012]. [10] URBANÍKOVÁ, M.,VACULÍKOVÁ, Ľ.: Actuaial atheatics (in Slovak). Batislava: Slovak Univesity of Technology, 2006. [11] URBANÍKOVÁ, M.: Financial atheatics (in Slovak). Nita: Constantine The Philosophe Univesity, 2008 28