Some Quantitative Aspects of Life Annuities in Czech Republic

Transcription

1 Some Quaniaive Aspecs of Life Annuiies in Czech Republic Tomas Cipra The conribuion deals wih some quaniaive aspecs of life annuiies when applied in he Czech Republic. In paricular, he generaion Life Tables and he gender problem for he pensions in he Czech Republic are discussed. One deals also wih he financial susainabiliy of life annuiies and wih he annuiy puzzle. Keywords: Generaion Life Tables, unise Life Tables, susainabiliy of life annuiies, annuiy puzzle.. Inroducion The pension reform is discussed in he Czech Republic since he demographic siuaion becomes criical. Besides he qualiaive aspecs here are quaniaive quesions ha should be solved in his cone: () Generaion Life Tables (GLT): In he pension cone one should apply special Life Tables wih projecions disinguishing among paricular generaions (cohors). Moreover, hese Tables should be adjused in a special way o be suiable for pensions (see Secion 2). (2) Unise Life Tables (ULT): Due o various ani-discriminaion laws concerning he insurance he ULT should be consruced for he Czech populaion (see Secion 3) (3) The susainabiliy of pensions: One invesigaes (under he given olerance and wih he given invesmen efficiency) which regular annuiy spending (spending rae) is admissible for a given pension accoun. For his purpose special pension models may be used which are presened here and which are applied numerically o pension sysems in he Czech Republic o provide imporan conclusions (see Secion 4). (4) Annuiy puzzle: In spie of he fac ha he annuiizaion hrough life annuiies or pensions can be an effecive insrumen how o ensure economically seniors, he low demand on life annuiies conradics his resul (see Secion 5). 2. Generaion Life Tables for Pension Sysem The Life Tables applied for he pension sysem and pension funds in he Czech Republic should respec wo facs ha are ypical jus for pensions: () decreasing rend of he raes of moraliy in ime and (2) selecion of pension porfolios. A possible soluion is o consruc he Generaion (or Cohor) Life Tables (GLT) wih projecions disinguishing among paricular generaions (cohors), see Cipra (998). Moreover, hese Tables are adjused in a special way o be suiable for pensions. Tomas Cipra, Charles Universiy of Prague, Faculy of Mahemaics and Physics, Dep. of Saisics, Czech Republic, cipra@karlin.mff.cuni.cz

2 In he Period (or Curren) Life Tables he probabiliy of deah q (and he probabiliy of survival) depends only on he age ha is looked upon as he curren age in he year for which he LT were consruced. Therefore e.g. q 65 from he LT for he Czech Republic in 2 concerns he male generaion 945. However, one uses his value approimaively as he probabiliy of deah in he age also for oher male cohors. Beer soluions consiss in τ applying he GLT where he probabiliy of deah q depends also on he year of birh (herefore he probabiliy discussed above should be denoed as q ). In addiion, his principle mus be respeced also for muliyear probabiliies, e.g p65 p65 p66 p67 p68 p69 The applicaion of he GLT can be recommended jus for he Czech Republic wih a high longeviy risk in he pension sysems. Figures a b show numbers of survivors l and l y for male and female generaions 935, 955 a 975 according o he GLT for he Czech Republic consruced by Cipra (998). These GLT respec anoher imporan fac, namely he seleciviy of annuiy porfolios (i is obvious since he life annuiies are preferred by persons wihou healh problems). Fig. a. Numbers of survivors l for male generaions 935, 955 a 975 according o he GLT for he Czech Republic l(935) l(955) l(975) Source: Cipra (998) Fig. b. Numbers of survivors l y for female generaions 935, 955 a 975 according o he GLT for he Czech Republic

3 ly(935) ly(955) ly(975) Source: Cipra (998) If applying he GLT he differences are no negligible from he financial poin of view. Tab. conains he monhly paymens for life annuiies paid off from he age 6 achieved in various years if he pension accoun in he age 6 is CZK. The hree alernaive calculaion formulas are / 2e 6 (where e 6 is he life epecancy in he age 6) or / 2ä 6 (4 %) (where ä 6 (4 %) is he annuiy facor in he age 6 for he ineres rae 4 % p.a.) or / 2ä 6 (6 %) (where ä 6 (6 %) is he annuiy facor in he age 6 for he ineres rae 6 % p.a.). E.g. for he ineres rae 4 % p.a. and for he male generaion 2 in he age 6 he monhly annuiy paymen is 5 58 CZK. On he oher hand, if applying he classical (non-generaion) LT he monhly annuiy paymen would be subsanially higher, namely 7 CZK. Tab.. Annual paymen for life annuiy using Generaion and Curren LF Monhly paymen(in CZK) for life annuiy (he pension accoun of CZK in age 6) Pension Male in age 6 by means of: Female in age 6 by means of: from year: e 6 = ä 6 (%) ä 6 (4%) ä 6 (6%) e 6 = ä 6 (%) ä 6 (4%) ä 6 (6%) Curren LT Source: Cipra (998)

4 3. UNISEX LIFE TABLES FOR PENSION SYSTEM This Secion can be looked upon as a reacion o he movemen agains he discriminaion using he gender as a calculaion parameer in he life insurance and pension sysems. The Unise Life Tables (ULT) presened here (see also Smeana and Cipra (25)) have been consruced applying he same mehodology as he Czech Saisical Office for he gender depending LT. The comparison of he ne regular annual premiums for he deferred life annuiies is added. Fig. 2 (he probabiliies of deah q according o he ULT for he Czech Republic in 23) confirms he epecaion ha he unise moraliy is approimaely an average of he male and female moraliy. Fig. 2. Probabiliies of deah q according o he ULT for he Czech Republic in 23, ,,, ln(q ) males ln(q) muži ln(q y) females ln(qy) ženy ln(q z) unise ln(qz) unise,, Source:Smeana a Cipra (25) Tables 2a 2b presen single and level premium for deferred life annuiies wih annual paymens of 2 CZK for he Czech Republic in 23 (he echnical ineres rae is 2.4 % p. a.). The Tables demonsrae he fac ha he analysis of longeviy risk mus be modified if respecing he gender effec. Tab. 2a. Single premium for deferred life annuiies wih annual paymens of 2 CZK for he Czech Republic in 23 (2.4 % p. a.) Age Defermen Unise (U) Males (M) Females (F) Raio U/M (%) Raio U/F (%) Raio M/F (%) ,88 % 8,58 % 7,4 % ,48 % 8,88 % 7,9 % ,97 % 82,26 % 7,55 % ,72 % 83,6 % 73,3 % Source:Smeana a Cipra (25) Tab. 2b. Level annual premium for deferred life annuiies wih annual paymens of 2 CZK for he Czech Republic in 23 (2.4 % p. a.) Age Defermen Unise (U) Males (M) Females (F) Raio U/M (%) Raio U/F (%) Raio M/F (%)

5 ,8 % 82,36 % 7,74 % ,33 % 82,72 % 72,35 % ,62 % 83,23 % 73,25 % ,4 % 84,2 % 74,84 % Source:Smeana a Cipra (25) 4. SUSTAINABILITY OF LIFE ANNUITIES If a pension sysem is based on he fund principle hen a quesion appears how much o save annually during he accumulaion phase and how much o spend annually during he decumulaion (annuiy) phase. In view of many random aspecs he bes approach is similar o he one applied in modern finance, namely he Value-a-Risk (VaR, see e.g. Cipra (2)), when one calculaes he highes loss which can occur wih a given probabiliy (olerance). In he pension cone i mus be modified o he probabiliy ha he reired person will no be ruined before he momen of deah (such a probabiliy can be called he probabiliy of susainable pension) while is complemen o is he probabiliy of ruin (or he probabiliy of unsusainable pension). In he pracice i is closely conneced wih he pension planning (pension consulans speak someimes on pension risk managemen). For simpliciy le us consider only so called DC pension plans (Defined Conribuion) where he conribuions o he sysem are defined in advance (frequenly by a percenage of he paricipan s salary). In he age of reiremen (e.g. in 65) here is an accumulaed capial w on he paricipan s accoun which will be decumulaed by corresponding annual pension paymens. In paricular, he invesmen risk is fully on he side of he paricipans of he pension plan (no on he side of he pension provider). Two aspecs of random characer should be a leas considered: () Randomness of ineres raes r() for invesmen of he capial from he paricipan s accoun (originally in he age of reiremen i holds S = w bu his sum is gradually reduced due o pension paymens). This ype or randomness is in he modern finance usually modeled by means of geomeric Brown moion: S S (, ) B B e S e, () where μ is he drif and is he volailiy. In paricular, S is log-normally disribued ln S ~ N(ln S 2, ) (2) wih he mean value 2 2 E( S ) S e S e (3) and he median value M( S ) S e. (4)

6 (2) Randomness of he fuure lifeime T of an individual aged can be modeled in he simples case by he eponenial law of moraliy: s p e d ep, (5) where is he insananeous force of moraliy. In paricular, he epeced remaining lifeime (he life epecancy) a age is e T ) ( E (6) and he median remaining lifeime a age is T M ln(2) ) (. (7) If we combine he models () and (5) we ll obain he presen value PV of he sandard pension (which pay uni annual paymens in coninuous ime) as random variable T B PV ) ( d e. (8) Finally one can show (see e.g. Dufresne (99), Milevsky (997), Milevsky (26)) ha he probabiliy of ruin (he probabiliy of unsusainable pension) w w PV T B ) ( d e P P (9) (w > is he sum on he paricipan s accoun a he age of reiremen ) can be approimaed as y y y z z z w PV w /w d ep ) ( d ep ) ( ~ P ) (, () where 2, () and () is he gamma funcion

7 z d Moreover, he mean presen value E(PV ) can be approimaed as ( ) z e z. (2) E ( PV ) ~. (3) 2 The previous formulas enable o perform he corresponding calculaions for pension plans in he Czech Republic. For his purpose one mus use boh financial and longeviy (or equivalenly moraliy) daa: (a) Financial daa: For he purpose of he invesmen formula () one could use he so called echnical ineres rae 2.5 % used for insurance calculaions according o he insurance legislaive in he Czech Republic in year 22 (correcly speaking, i is he upper limi in he case of long-erm insurance producs in he commercial life insurance including life annuiies). The beer sraegy is o es more scenarios boh for he drif μ and he volailiy using he echnical ineres rae 2.5 % as one of possibiliies. (b) Longeviy daa: We have used Life Tables (LT) for males and females in he Czech Republic in 2. From he epeced remaining lifeime (he life epecancy) e a paricular ages given in hese LT one esimaes according o he formula (6) very simply he parameers (see Tab. 3). One could obain beer esimaes of longeviy applying Gomperz-Makeham law of moraliy or even Lee-Carer approach (insead of our eponenial law of moraliy) bu he eponenial approach seems o be for our calculaions sufficien. Tab. 3. The insananeous forces of moraliy of he eponenial law of moraliy for males and females in he Czech Republic in 2 Males CZ 2 Females CZ 2 and y e λ e y λ y Tab. 4. The probabiliy of ruin (i.e. he probabiliy of unsusainable pension in per cen) for various reiremen ages and spending raes: for μ = % and σ = 5 % Males CZ 2 for μ = %, σ = 5 % : probabiliy of unsusainable pension (%) e λ Spending rae / w: % 4. %.8 % 2.2 % 3. % 4.9 % 52.2 % 6.5 % 69.5 % 76.2 % % 2.7 % 7.5 % 4.4 % 22.9 % 32. % 4.3 % 5. % 58.2 % 65.5 % %.7 % 4.8 % 9.7 % 5.9 % 23. % 3.5 % 38.2 % 45.7 % 52.8 % %. % 2.9 % 5.9 %. % 5. % 2.6 % 26.6 % 32.7 % 38.9 % %.5 %.5 % 3. % 5.5 % 8.4 % 2. % 5.9 % 2.3 % 24.8 %

8 %.2 %.6 %.4 % 2.5 % 4. % 5.9 % 8. %.6 % 3.4 % %. %.2 %.5 %. %.6 % 2.4 % 3.4 % 4.5 % 5.9 % Females CZ 2 for μ = %, σ = 5 % : probabiliy of unsusainable pension (%) y e y λ Spending rae / w: % 6. % 5.6 % 27.9 % 4.9 % 53.2 % 64. % 72.9 % 8. % 85.5 % % 4.2 %.2 % 2.9 % 3.8 % 42.9 % 53.3 % 62.6 % 7.6 % 77.2 % % 2.7 % 7.5 % 4.5 % 22.9 % 32. % 4.3 % 5.2 % 58.3 % 65.5 % %.5 % 4.5 % 9. % 4.9 % 2.7 % 29. % 36.4 % 43.8 % 5.8 % %.8 % 2.3 % 4.9 % 8.4 % 2.6 % 7.6 % 22.9 % 28.5 % 34.2 % %.3 %. % 2.2 % 3.9 % 6. % 8.8 %.9 % 5.3 % 9. % %. %.4 %.8 %.5 % 2.4 % 3.5 % 4.9 % 6.6 % 8.4 % Tab. 5. The probabiliy of ruin (i.e. he probabiliy of unsusainable pension in per cen) for various reiremen ages and spending raes: for μ = 5 % and σ = % Males CZ 2 for μ = 5 %, σ = % : probabiliy of unsusainable pension (%) e λ Spending rae / w: %.6 % 2.2 % 5.5 %.4 % 6.7 % 24. % 3.9 % 4. % 47.9 % %.4 %.7 % 4.2 % 8. % 3. % 9. % 25.6 % 32.7 % 39.8 % %.3 %.3 % 3. % 5.9 % 9.7 % 4.3 % 9.6 % 25.4 % 3.4 % %.2 %.9 % 2. % 4. % 6.7 %. % 3.9 % 8.3 % 23. % %. %.5 %.3 % 2.4 % 4. % 6.2 % 8.7 %.7 % 4.9 % %. %.3 %.7 %.3 % 2.2 % 3.3 % 4.7 % 6.4 % 8.4 % %. %. %.3 %.6 %. %.5 % 2.2 % 3. % 3.9 % Females CZ 2 for μ = 5 %, σ = % : probabiliy of unsusainable pension (%) y e y λ Spending rae / w: %.7 % 2.8 % 7. % 3.2 % 2. % 29.8 % 39. % 48. % 56.5 % %.6 % 2.3 % 5.6 %.6 % 7. % 24.5 % 32.5 % 4.7 % 48.7 % %.4 %.7 % 4.2 % 8. % 3. % 9. % 25.7 % 32.7 % 39.8 % %.3 %.2 % 2.9 % 5.6 % 9.2 % 3.6 % 8.7 % 24.3 % 3. % %.2 %.7 %.8 % 3.5 % 5.8 % 8.7 % 2. % 6. % 2.3 % %. %.4 %. %.8 % 3. % 4.7 % 6.7 % 9. %.6 % %. %.2 %.4 %.8 %.4 % 2. % 3. % 4. % 5.5 % In Tables 4 5 we have calculaed according o (9) he probabiliy of ruin (i.e. he probabiliy of unsusainable pension) for various reiremen ages 55, 6,, 85 in dependence on he spending raes /w (e.g. he spending rae.6 means ha he given pension accoun of CZK (Czech crowns) will pay 6 CZK annually, i.e. 5 CZK monhly); he calculaions are performed separaely for males and females and for various values of invesmen drifs and volailiies, namely μ = % and σ = 5 % in Tab. 4, μ = 5 % and σ = % in Tab. 5.

9 Tables 6 7 regard he problem of unsusainable pensions in an opposie way which seems o be more pracical: hey conain (again separaely for males and females in he Czech Republic in 2) he maimal spending raes /w ha are admissible wih a given olerance (i.e. wih a given probabiliy of ruin which is ye oleraed); he calculaions are performed for various values of invesmen drifs μ = %,.5 %, %,.5 %, 2 %, 2.5 %, 3 %, 4 %,, 7 %, only for single value of invesmen volailiy σ = 5 % and for various values of olerance, namely % in Tab. 6 and % in Tab. 7: Tab. 6. The maimal spending raes wih olerance % Males CZ 2 for σ = 5 % : maimal spending rae /w (%) wih olerance of % : e λ Drif μ: %.5 % %.5 % 2 % 2.5 % 3 % 4 % 5 % 6 % 7 % %.23 %.2 %.387 %.583 %.786 %.998 % 2.44 % 2.99 % % 3.97 % %.22 %.397 %.582 %.774 %.973 % 2.79 % 2.69 % 3.6 % % 4.24 % %.483 %.659 %.84 % 2.3 % % % % % % 4.25 % %.855 % 2.3 % 2.2 % % % % 3.86 % 3.67 % 4.44 % % % % 2.69 % % 2.97 % 3.56 % % % 4.43 % 4.56 % 4.99 % % % 3.52 % % % 4.6 % % % 5.6 % 5.46 % % % % 5.3 % 5.25 % % % % 6.2 % % % % Females CZ 2 for σ = 5 % : maimal spending rae /w (%) wih olerance of % y e y λ Drif μ: %.5 % %.5 % 2 % 2.5 % 3 % 4 % 5 % 6 % 7 % %.853 %.3 %.22 %.42 %.629 %.847 % 2.35 % 2.79 % % % %.6 %.83 %.37 %.566 %.77 %.982 % % % % % %.22 %.396 %.58 %.773 %.972 % 2.78 % 2.68 % 3.6 % % 4.23 % %.53 %.76 %.888 % 2.76 % 2.27 % 2.47 % % 3.32 % % 4.24 % % 2.5 % 2.9 % % % % % % % 4.84 % % % 2.8 % % 3.62 % % % 3.76 % 4.2 % % 4.99 % 5.33 % % 4.53 % % 4.52 % 4.68 % 4.86 % 5.44 % 5.47 % % 6.89 % % Tab. 7. The maimal spending raes wih olerance % Males CZ 2 for σ = 5 % : maimal spending rae /w (%) wih olerance of % e λ Drif μ: %.5 % %.5 % 2 % 2.5 % 3 % 4 % 5 % 6 % 7 % % % % 3.24 % % % 4.98 % % % % 7.3 % % 3.86 % % 3.7 % 4.3 % % % % 6.43 % 6.74 % 7.45 % % % 4.58 % 4.37 % 4.69 % 5.2 % % 6. % % % 8.6 % % % % 5.39 % % % % 6.92 % % % % % 6.53 % 6.46 % % 7.86 % 7.43 % % % 9.23 % %.358% % % % 9.79 % 9.39 % 9.75 %.2 %.658 %.33 %.955 % 2.63% % % % 2.89 % 3.2 % 3.53 % % % 5.92 % % 6.379%

10 Females CZ 2 for σ = 5 % : maimal spending rae /w (%) wih olerance of % y e y λ Drif μ: %.5 % %.5 % 2 % 2.5 % 3 % 4 % 5 % 6 % 7 % % 2.55 % % % 3. % % % % 5.75 % % 6.63 % % % % 3.7 % % 3.82 % 4.55 % % % % % % 3.83 % % 3.78 % 4.28 % % % % 6.4 % % % % % 4.77 % 4.49 % 4.89 % 5.3 % % 6.8 % % % 8.73 % % 5.93 % 5.4 % 5.73 % 6.29 % % % 7.32 % % % % % 7.4 % 7.42 % % 8.35 % 8.35 % % 9.3 % 9.96 %.68 %.283 % %.496 %.83 %.2 %.423 %.736 % 2.5 % % % % 4.68 % The previous resuls provide very ineresing hins for fuure paricipans of pension plans, financial consulans, sales represenaives of commercial insurance companies and managemens of pension funds: E.g. under a conservaive invesmen sraegy wih μ = % and σ = 5 % a male wih reiremen age of 65 and spending rae.6 (i.e. 6 CZK annually from a pension accoun of CZK) faces he unsusainable pension wih probabiliy 23. % (naurally for a female of he same age his probabiliy is higher, namely 32. %, see Tab. 4). In oher words each fifh male and each hird female are ruined before deah according o his scenario. However, if he invesmen drif increases han (even for higher invesmen volailiy) he ruin probabiliy decreases significanly, e.g. for μ = 5 % and σ = % he ruin probabiliies are only 9.7 % for males and 3. % for females (see Tab. 5), i.e. only each enh male and each eighh female are ruined. Similarly, under he olerance of %, reiremen age 65 and conservaive invesmen sraegy wih μ = % and σ = 5 % a male may spend from his pension accoun of CZK no more han 6 59 CZK annually (i.e. 383 CZK monhly) and a female of he same age no more han 3 96 CZK (i.e. 63 CZK monhly, see Tab. 6). These values are surprisingly low bu he reason is very simple, namely he high level of safey wih olerance of only %. For higher invesmen drif μ = 2.5 % ceeris paribus he maimal annual wihdrawals are CZK for males and 9 72 CZK for females. If he olerance of % is sufficien (see Tab. 7) hen under conservaive invesmen parameers μ = % and σ = 5 % he male can draw 4 58 CZK annually (i.e CZK monhly) and he female CZK annually (i.e CZK monhly). For higher invesmen drif μ = 2.5 % (which corresponds o he siuaion in he Czech Republic) ceeris paribus he male can spend 5 2 CZK annually (i.e CZK monhly) and he female CZK annually (i.e CZK monhly). 5. ANNUITY PUZZLE The annuiizaion is a powerful insrumen o ensure economically one s old age. On he oher hand, alhough a maimum annuiizaion should be he bes scenario for scenarios a common pracice preferring he immediae consumpion is quie differen. In order o make his paper more pracical we ll ry in conclusion o lis some argumens answering his srange annuiy puzzle. Impac of social benefis

11 If individuals can rely on he eisence of social benefis (e.g. he fla pension in UK) hen hey mus do decision how much hey provide from heir savings o raise hese benefis. On he oher hand, he eisence of such savings is closely relaed o volume of incomes during working life. Possible invesmens o risky asses and pension deferral The invesmen models discussed in his e do no involve he possibiliy o inves o asses ha are more risky han bonds (le s summarize such asses under he denoaion socks ). In his cone some auhors (see e.g. Milevsky (998)) sugges various pension sraegies where he iniial capial W is in he firs insance invesed o socks, and he annuiizaion is posponed o an opimal age several years afer he pension age. Since such an opimal age uses o be much higher han he legislaive pension age (in paricular for females), i delivers anoher argumen why life annuiies are no so frequen in pracice. Le s demonsrae he previous consideraion by means of a simple eample. Le an average annual yield of bonds be 2 % and of socks 9 %, he risk margin for socks being 4 %. I means ha a raional individual invess he corresponding savings o life annuiies only if he annual yield from such an invesmen is a leas 5 %. Denoing he yearly surviving probabiliy in he given age as p hen he invesmen o annuiies will be obviously advanageous only if i holds r p,2,5, (4) p where (in addiion o he bond yield of r = 2 %) he moraliy drag due o he inequaliy p < is also effecive. Alhough he moraliy drag is bundled wih he risk of deah his risk is usually much lower han he risk of socks. According o he Life Tables for males and females in he Czech Republic in 2, he inequaliy (4) occurs since age 68 for males and since age 75 for females. According o he pension sraegy described above one should defer he annuiizaion ill hese ages invesing o riskier asses before (in pracice i means ha seniors should reain heir money longer in producive pension funds). Impac of consumpion disribuion during decumulaion period () Seniors need irregularly higher amouns for invesmens relaed o heir age (e.g. he insallaion of lif) or relaed o heir sae of healh (e.g. he implans of hip joins uncovered by he healh insurance). Such financial flows are no cerainly smooh and due o limied access o credis for seniors hey can be very demanding as o he liquidiy of seniors asses. (2) Consumpion preferences change wih age. In senior ages heir composiion ransfer from culure, gasronomy and ravelling o day care cenres, healh visiors and similar epenses. (3) Moreover, he irregular characer of seniors spending is muliplied by he randomness and uncerainy of he corresponding changes: i is cerain ha he majoriy of seniors will have healh problems bu heir iming and relevance is unknown in advance. (4) There are suiable producs of commercial insurance which are suggesed o cover he ypical social and healh problems of seniors. E.g. he long-erm care (LTC) producs (popular in Germany, Swizerland and UK) cover he disabiliy o perform aciviies of daily living (ADL). However, such commercial producs are usually very epensive and hey are accessible only for rich families.

12 Impac of aniselecion The impac of aniselecion concerning life annuiies can be very srong. Such an aniselecion follows from he simple fac ha he purchase of a life annuiy is reasonable only if he individual s life epecancy is long. I can have ineresing consequences: () If govermen pensions are guaraneed hen an individual wih subjecively shor life epecancy will no choose any annuiizaion of his or her savings due o he convicion ha he insiuional cover of pension is sufficien. (2) In he case of moneyed seniors, i is difficul o disinguish wheher heir pension behavior is influenced by heir wealh or wheher hey ake advanage of he aniselecion. Impac of markeing If he pensions producs offer rained sales represenaives he presenaion of he produc can play an imporan role including unepeced psychological effecs, e.g. differen feelings of saisfacion epressed by cliens of defined benefi (DB) and defined conribuion (DC) pension plans. 6. CONCLUSIONS The paper has concenraed on several imporan aspecs ha mus be aken ino accoun when consrucing producs burdened by longeviy risk: () he generaion longeviy, (2) he gender longeviy, (3) he susainabiliy of life annuiies, (4) he annuiy puzzle. I has been shown numerically ha hese aspecs really play a very imporan role for he pensions in he Czech Republic. Acknowledgemen: This paper was wrien wih he suppor of he Czech Science Foundaion No. P42/2/G97 DYME Dynamic Models in Economics REFERENCES Cannon, E., Tonks, I. (28). Annuiy Marke. Oford Universiy Press. Cipra, T. (998): Cohor life ables for pension insurance and pension funds in Czech Republic. Pojisné rozpravy, 3, 3-57 (in Czech). Cipra, T. (2): Financial and Insurance Formulas. Heidelberg, Dordrech, London, New York: Physica-Verlag / Springer. Dufresne, D. (99): The disribuion of a perpeuiy wih applicaions o risk heory and pension funding. Scandinavian Acuarial Journal, 9, Milevsky, M. A. (997). The presen value of a sochasic perpeuiy and he Gamma disribuion. Insurance: Mahemaics and Economics, 2, Milevsky, M. A. (26). The Calculus of Reiremen Income. Cambridge Universiy Press. OECD (2). Pensions a a Glance 2: Reiremen-income Sysems in OECD and G2 Counries. OECD Publishing. Smeana, P., Cipra, T. (25): Unise life-ables for he Czech Republic and consequences for insurance calculaions. Pojisný obzor, 82, -7 (in Czech).