MULTIPLE LIFE INSURANCE PENSION CALCULATION *

Transcription

1 ULIPLE LIFE INSURANCE PENSION CALCULAION * SANISŁA HEILPERN Universi of Economics Dermen of Sisics Komndors rocł Polnd emil: snislheilern@uerocl Absrc he conribuion is devoed o he deenden mulile life insurnce of mrried coule A more relisic ssumion of deenden lifeime of mrried coule is invesiged s disinc from clssicl roch hich ssumes he indeenden lives o models: firs is bsed on he rov chin nd second uses he couls minl Archimeden re sudied he curil vlues of hree cses of ensions: ido s n-er join-life nd n-er ls survivl nnuiies re clculed in hese models he differences beeen he vlues of ensions in he indeenden model nd model bsed on he deendences re invesiged using he emiricl d from Polnd he resuls ill be comre ih he resuls obined in he uhor s erlier invesigions Ke ords: mulile life insurnce coul rov model ension DOI: 56/mse2472 Inroducion e ill sud he mulile life insurnce concerns ih he mrried coule in order o evlue he remiums of conrcs Clssicl curil heor conneced ih he mulile life insurnce ssumes he indeendence for he remining lifeimes (Boers e l 986; Frees Crriere Vldez 996) Bu i is no relisic ssumion In he rel life he souses m be eosed o he sme riss nd he heir lifeimes re ofen lile deenden bu deenden e lso m observe so-clled he broen her sndrome In he er e invesige o models lloing he deendence of lifeime of souses Firs is bsed on he rov chin nd second uses he couls e derive he vlues of hree nnuiies: he ido s he n-er join-life nd n-er ls survivl nnuiies e sud he imc of deendences on he vlues of hese nnuiies he er is bsed on he Denui s e l (2) er he uhors sudied in i he siuion in Belgium Heilern (2) ried o l he mehods from his er in Polish cse bsed on he d from 22 No e coninue his or nd use he ne d from Polish Cenrl Sisicl Office o 2 he im of his er is sud he imc deendences on he vlue of bove hree ensions he differences of he vlues of hese nnuiies beeen he vrins bsed on indeendence nd deendence re clculed * he rojec s funded b he Nionl Science Cenre lloced on he bsis of decision No DEC-23/9/B/HS4/49 4

2 2 Generl ssumions nd noions No e inroduce he generl noion nd ssumion conneced ih his subjec Le nd be he remining lifeimes of -er-old mn nd -er-old omn ing vlues in [ ] nd [ ] he (res ge of he mn (res omn) nd (res ) he disribuion funcion e cn lso derive he survivl robbili nd survivl funcion using force of morli ) denoes he difference beeen he border of re given b formuls: ) : similr If e n o sud he deendence of rndom vribles e sds () nd he deh robbili of ife re obined in he e mus no heir join disribuion he robbili of join-life sus surviving o ime is given b formul: nd ls-survivl sus m{ ) } ) he rndom vribles re osiive udrn deendence (PQD) hen (Lehmnn 966; Dhene Goovers 997) ) e cn see h if he lifeimes re PQD hen e obin 3 Pensions No e resen hree ensions conneced ih he mulile life insurnce of souses Firs e sud he ido's ension: = here v v nd v = ( + ξ) - be he discoun fcor conneced ih he nnul effecive re ξ he mens srs ih he husbnd's deh nd ermining ih he deh of his ife in his cse Second ensions re he n-er join-life survivl nnuiies described b formul: n ; n v hird n-er ls survivl nnuiies is eul n ; v 5

3 In ls o nnuiies he s $ he end of he ers s long boh or eiher souse survives hen he lifeimes re indeenden e denoe hese ension b smbols: nd hen he lifeimes ; re PQD hen e obin he folloing relion beeen hese ension ih resec he indeenden cse: nd ; ; n ; ; n e see h hen e ssume he indeendence e cn overesime or underesime he vlue of he nnui he second cse occurs hen e n o comue he vlue of he n- er join-life survivl nnuiies ; 4 rov model In his secion e invesige he rov model bsed on sionr rov chin I is n recied ool for he clculion of life coningencies funcions nd ensions (see olhuis Vn Hoec 986; Norberg 989) his rov chin hve four ses nd he forces of morliies μij ij = 2 3 in his cse (see fig ) μ husbnd nd ife live husbnd ded ife ded 2 μ3 6 μ2 μ23 husbnd nd ife ded 3 Figure he sce of ses of rov model Source: Denui e l 2 e denoe b smbol ij( he rnsiion robbiliies his is he condiionl robbili h he coule is in se j ime s given h i s in se i ime he forces of morli μij() from he se i o se j ime is done b formul ij( ) ( ) lim ij he rnsiion robbiliies ij( cn be reresened b he forces of morliies in he folloing (Denui e l 2): s ( e ( ( 2( ) du s ii( e i3( du ( ( u du ( s ( i i ii here i = 2 If e no he robbiliies of sing se e cn comue he join nd mrginl survivl funcions of rndom vribles nd using he formul (Denui e l 2):

4 ( ( ) ( s (2) ( ) ( 2( s ) s P ) ( ) ( ) ( 2 ( ) ( ) ( P ) he lifeimes nd re indeenden iff μ() = μ23() μ2() = μ3() nd if μ() < μ23() μ2() < μ3() hen he re PQD (Norberg 989) In our er e use for fied ges of husbnd nd ife he folloing simlifing ssumion done b Denui e l (2): μ() = ( α) μ23() = ( + α23) (3) μ2() = ( α2) μ23() = ( + α3) hese formuls lin he rov forces of morli μij() nd he mrginl lifeime forces of morli nd using he consns αij So e cn comue he join survivl funcion: 2 ( ) e ( 2( du e see h if e n o use his model in rcice e mus esime he coefficiens α α2 nd e obin he mrginl survivl funcions nd from he survivl life bles (Heilern 2) e esime he rmeers α nd α2 using he Nelson-Alen esimor bsed on he cumulive funcion (Jones 997; Denui e l 2): ij( ) ij( ds he Nelson-Alen esimor minimizes he sum of sured differences beeen he incremens ΔΩij nd heir esimor ie he semen: Using (3) e obin 2 ij ij ( ) ij ( ) d ( ) d ( ) d ( ) ln 2( ) d ( 2) d ( 2) ln he esimors of he coefficiens α α2 re soluions of he bove oimizion roblem: 2 2 ( )ln 2( )ln 2 2 (4) (ln ) (ln ) here he esimor i (Denui e l 2) is eul o L i( ) i( ) (ln Li ( ) ln Li ( )) (5) L ( ) L ( ) i i 2 7

5 he smbol L ( ) (res L ( ) 2 ) mens number of -er-old husbnds (res ive ding during fied er eg 2 L() (res L2()) is number of -er-old husbnds (res ive 2 nd L( + ) (res L2( + )) is number of ( + )-er-old husbnds (res ive 22 5 Coul model 5 Bsic definiion nd roeries e cn describe he deenden srucure of join lifeimes using coul Coul is lin beeen he join nd mrginl disribuions (Genes ck 986; Nelsen 999): C( ) ) Bu in our nlsis e need more he join nd mrginl survivl funcions hn cumulive disribuion funcions e use he survivl coul C * o his end: * C ( ) ) he funcion C * is coul oo nd i sisfies he folloing relion (Nelsen 999): C * (u v) = u + v + C( u v) he robbiliies cn be comued using he survivl coul C * in he folloing : * ) C ( ) For he indeenden rndom vribles he corresonding coul es he simle form: CI(u v) = uv nd for he sric osiive C nd he sric negive C deendence e hve: C(u v) = min{u v} C(u v) = m{u + v } hese ereme couls sisf he folloing relion: C(u v) C(u v) C(u v) (6) for ever coul C (Nelsen 999) Using he relion (6) e obin he folloing ineuliies: m{ F( ) F 2( 2) } F( 2) min{ F( ) F 2( 2)} for ever join nd mrginl survivl funcions he lef nd righ sides of hese ineuliies re clled he Freche bounds So e cn esime he join-life sus surviving o ime : nd ls-survivl sus m{ } min{ min{ } m{ } he bove relions le us esime he ensions nd e obin (Denui e l 2): min m min m min m here ; ; ; min v v } min{ } ; ; ; m v v } m{ 8

6 min n v n } ; min n v min{ n } ; m m{ v ( n min{ n } ; m n v m{ n } ; ( In Heilern s er (Heilern 2) he deenden srucure of join lifeimes s described b he Archimeden coul I is simle coul done b formul (Nelsen 999): C(u v) = φ - (φ( + φ(v)) here φ: [ ] R+ is decresing funcion clled generor sisfing condiion φ() = Archimeden couls form he fmilies of couls chrcerized b some rmeer his rmeer described he degree of deendence he Kendll s coefficien of correlion τ is done b formul ( ) 4 d '( ) 52 Coul selecion No e resen he mehod of selecion of coul bes fi o he d (Genes Rives 993; Heilern 27) e resriced ourselves o Archimeden couls onl his mehods roceeds in four ses: i) se he fmilies of Archimeden couls ii) esime Kendll s τ coefficien of correlion bsed on he emiric d iii) selec he coul conneced ih his Kendll s τ from ever fmil iv) choose oiml coul using some crierion e cn use he crierion bsed on he on he Kendll funcion (Genes nd Rives 993): ( ) K C ( ) F( ) ) ( ) hen e choose he coul minimized he disnce beeen emiricl Kn() nd heoreicl KC() Kendll s funcions: Sn Kn( ) KC ( ) dkc ( ) Denui e l (2) colleced he ges deh of 533 coules buried in o cemeeries in Brussels he used his d o selec he Archimeden coul describing he deenden srucure of he join lifeimes ( ) of souses nd used he crierion bsed on he Kendll s funcion he selec Gumbel coul Cα(u v) = e( (( ln α + ( ln v) α ) /α ) α ih rmeer α = 5 using his d Heilern (2) used he de n = 36 from o cemeeries in rocł nd he crierion bsed on he Kendll s funcion oo He oo ino ccoun he Clon Gumbel Frn nd AH fmilies of couls he AH coul Cα(u v) = uv/( α( ( v)) ih he rmeer α = 5879 roved o be he bes coul in his cse he comued he vlues of ensions nd he comred hem ih he vlues obined under indeenden ssumion 2 9

7 he couls Cα obined in hese ers re conneced ih he survivl robbili of join lifeimes ( ) If e n o obin robbili e mus comue he folloing condiionl survivl robbili C ( ) i ) C ( ) 6 Emle In his secion e resen he resuls of he invesigion of he souses in Polnd e use he rov model nd he d from Polish Cenrl Sisicl Office from 2 here s he Polish Generl Census in 2 nd he d re more deil in his er So e cn obin he vlues Li() bu Li( + ) s he number of ( + )-er-old husbnds (res ive 22 is uninble e cn esime he semen ΔL = L( + ) L() s he difference beeen he number of -er-old men geing mrried during 2 nd he sum of he number of -er-old mrried men ding during 2 -er-old mrried men hose ife died during 2 nd -er-old mrried men geing divorced during 2 hen L( + ) = L() + ΔL e obin he vlue of L( + ) in he similr hese d ere groued in he 5-er clsses So he ere evenl disribued over he one er eriods he effecive re ξ = 3 Using (5) nd (4) e obin he folloing vlues of he rmeers: α = 257 α2 = 29 Heilern (2) conduced he similr sud bsed on he d from 22 nd obin he vlues α = 76 nd α2 = 55 he rmeers obined in invesigion using he d from Belgium in 99 (Denui e l 2) re eul o α = 929 nd α2 = 27 he relive vlues of he ido's ension hen he souse re in he sme ge ie = for minimum indeenden nd mimum cses received ord rov model (he ension for rov model is eul o ) re given in ble e see h if he rov model is ruh hen he indo's ension hen e ssume indeenden lifeimes is overesime his overesime is eul verge 2% nd i increses ih ge e obin he similr resuls for Freche bounds bu he errors re bigger riculrl for uer bound ble he relive vlues of ido ension ord rov model rov min indenden m Source: on elborion ble 2 conins he vlues of he ensions ;n hen = = 5 for differen vlues of n e see h he indeenden cse underesimes he ruh ension in his cse Bu he errors re smller hn in he cse of idos ension 2

8 ble 2 he relive vlues of ension ord rov model ; n rov indeende m min Source: on elborion References BOERS N L GERBER H U HICKAN J JONES D A NESBI C J 986 Acuril hemics Isc Illinois: he Socie of Acuries DENUI DHAENE J Le BAILLY de ILLEGHE C EGHE S 2 esuring he imc of deendence mong insured lifelenghs In Belgin Acuril Bullein 2 vol () DHAENE J GOOVAERS J 997 On he deendenc of ris in he individul life model In Insurnce: hemics nd Economics 997 vol FREES E CARRIERE J F VALDEZ E 996 Annui vluion ih deenden morli In he Journl of Ris nd Insurnce 996 vol GENES C ACKAY R J 986 he jo of coule: bivrie disribuions ih uniform mrgins In he Americn Sisicin 986 vol GENES C RIVES L-P 993 Sisicl inference rocedures for bivrie Archimeden couls In JASA 993 vol HEILPERN S 27 Funcje łączące rocł: rocł Universi of Economics 27 8 HEILPERN S 2 zncznie ielości ren zleżnch gruoch ubezieczenich n żcie In Prce Nuoe UE rocł 2 vol JONES B L 997 ehods for he nlsis of CCRC d In Norh Americn Acuril Journl 997 vol 4-54 LEHANN E L 966 Some conces of deendence In Annls of hemicl Sisics 966 vol NELSEN R B 999 An Inroducion o Couls Ne Yor: Sringer NORBERG R 989 Acuril nlsis of deenden lives In Bullein de l'associion Suisse des Acuries 989 vol POLISH CENRAL SAISICAL OFFICE: [ci ] sgovl 4 OLHUIS H VAN HOECK L 986 Sochsic models for life coningencies In Insurnce: hemics nd Economics 986 vol