Estimation of the Present Values of Life Annuities for the Different Actuarial Models

Transcription

1 h Scod Itratoal Symposum o Stochastc Modls Rlablty Egrg, Lf Scc ad Opratos Maagmt Estmato of th Prst Valus of Lf Auts for th Dffrt Actuaral Modls Gady M Koshk, Oaa V Guba omsk Stat Uvrsty Dpartmt of Appld Mathmatcs ad Cybrtcs omsk, Russa -mal kgm@maltsuru, gov7@malru Abstract h papr dals wth th problm of stmatg th actuaral prst valu of th cotuous whol lf ad -yar trm lf auts W sythsz oparamtrc stmators of ths statuss of lf auty h ma parts of thr asymptotc ma squar rrors for ths stmators ad thr lmt dstrbutos ar foud By dvduals dath momts, both paramtrc ad oparamtrc stmats ar costructd for th modls of th whol ad -yar trm lf surac h asymptotc ormalty ad ma squar covrgc of th proposd stmators ar provd h smulatos show that th mprcal ma squar rrors of lf auty stmats dcras wh th sampl sz crass Also, wh th modl dstrbuto s chagd, th oparamtrc stmats ar mor adaptabl comparso wth paramtrc stmats, ortd o th bst rsults oly for th gv dstrbutos Kywords oparamtrc stmato; lf surac; lf auty; -yar lf auts; asymptotc ormalty; bas; ma squar rror I IRODUCIO h ssc of lf auty accordac wth [, p 70] s that, from th momt t0 a dvdual oc a yar bgs to gt a crta moy, whch w tak as th ut of moy, ad paymts ar mad oly for th lftm of a dvdual It s kow that th calculato of th charactrstcs of lf auty s basd o th charactrstcs of th rspctv typ of surac hus, th avrag total cost of th prst cotuous auty s gv as (s [, p84]) A a(, whr A s a t prmum (th avrag valu of th prst valu of a sgl sum of moy th surac lftm at th ag ), s a forc of trst Lt b a dvdual ag wh th paymts bg, X b th durato of hs lf, X - b th futur lftm Itroduc th radom varabl z, > 0 () h, takg th pctato of th radom varabl z() (), th lf auty s dtrmd by th formula (s [-6]) Φ(, a ( ) M( z), ) whr M s th symbol of th pctato, S P( X > ) s a survval fucto, Φ(, t d t) I practc, oft th -yar lf auty [, p 85] s usd Hr, from th momt t 0 a dvdual yarly bgs to gt moy, whch w tak as th ut sums of moy, ad paymts ar mad oly durg yars wh h s alv It s kow that th cotuous -yar trm auty s dfd by th t prmum hus, th avrag currt valu of th cotuous -yar auty (s [, p85]) s qual to a A ( whr A s th t prmum (th avrag valu of th prst valu of a ut sum of surd th -yar lftm surac at th ag ) Df for ths cas th radom varabl, () z (, ), < (3) akg th pctato of th radom varabl z(,) (3), gt th actuaral prst valu for th -yar auty a Φ(,, ) ( ) M( z(, )), ) t Φ(,, ) d t) (4) hs work was supportd by Russa Foudato for Basc Rsarch, projct , ad th SU Compttvss Improvmt Program /6 $ IEEE DOI 009/SMRLO

2 II ESIMAIO OF AUIIES Suppos w hav a radom sampl X X of lftms of dvduals Estmat sparatly th umrator ad domator () ad (4) Substtutg th ukow fucto ) by ts o-paramtrc stmator S Ι( X > ), whr I(A) s th dcator of a vt A, w obta th followg stmators of th whol lf ad - yar trm lf auts (, ) p Φ a Ι >, X X (5) S S (,, ) p Φ a Ι < < X X S S Hr, Φ (, Φ (,, ) III p( X ) Ι( X p( X ) Ι( < X > ), MEA SQUARE ERRORS < ) Prov rsults o th proprts of th stmators (5) ad (6) Gv oly th proprts of th stmators (5) as th rsults for (6) ca b got aalogously Frst, w fd th prcpal part of th asymptotc ma squar rror (MSE) ad th covrgc rats of th bas (5) W d horm from [7], (hr t s Lmma) Itroduc th otato accordg to [7] t ( t, t,, ts ) s a s-dmsoal vctor wth compots tj tj tj ( ; X,, X ), j, s, R α α, R s th -dmsoal Euclda spac; R s R s a fucto, whr t t ( t( ),, ts ) s a s-dmsoal boudd vctor fucto; s (,) s th s-dmsoal ormally dstrbutd radom varabl wth a ma vctor μ μ( ) ( μ, μ ) ad covarac matr (); s ( H( t),, H s ), H( z) H j, j, s; z j z t (6) s th symbol of covrgc dstrbuto (wak covrgc); s th Euclda orm of a vctor Lmma Lt ) th fucto H(t) b twc dffrtabl, ad 0 ; / ) M t t Ο ( d ),,, h k,, k k ( k )/ [ ] [ ] ( ) M H( t ) H M H ( t t) ο d + (7) ot, f th formula (7) k, w obta th ma part of th bas for H ( t ), ad at k, w hav th ma part of th MSE horm If )>0 ad t) s cotuous at, th ) for th bas of (5), th followg rlato holds b( a ο( ); ) th MSE of (5) s gv by th formula ( a ) ( ) ( ) u a 3/ M a a +ο, whr ( a s dfd blow by th formula (8) Proof For th stmator a ( (5) th otato of Lmma, w hav t ( Φ (,, S ) ; d ; t ( Φ(,, )) ; (, ) Φ Φ (, a; ) H ( t ) a ; S H ) Φ(, S (, ), 0 W kow that S () s a ubasd ad cosstt stmator of ) Show that Φ (, s a ubasd stmator of th fuctoal Φ(, MΦ(,) Mp( X)( Ι X > ) Φ(,) ow, calculat th varac of Φ (,

3 X DΦ (, D Ι( X > ) ( Φ(, Φ (, ) X D{ Ι( X > ) } h rato of two ubasd stmators ca hav a bas W fd th ordr of th bas wth makg us of th rsults from [7] I vw of M( t t) 0, w obta [ ] M( a ( ) a M H( t t) M( a ( ) a ( )) ο( ) Fd th compots of th covarac matr for th statstcs t ( a { } { } ( { (, )}) { } { } D Φ (, ) Φ(, Φ (, ; D S X )( )); cov( ), Φ(, ) M S Φ M S M Φ (, ) ( )) Φ(, Usg th prvous rsult o th bas ad th covarac matr, w obta [ ] 3/ ( a 3/ u ( a ( )) M H( t t) +Ο ( ) +Ο( ), whr ( a + H p j j jp H p 3 (, ) (, ) 3 Φ Φ Φ + S S S + H + (8) Mt { s} 0, Mt { s ts}, S ts, s th as S s (0, ) horm 3 (asymptotc ormalty of H t ) ) [7-3] Lt t s{ μ, ( )} ; fucto H(z) b dffrtabl th pot, H ( μ) 0 h s s s ( H( t ) H ( μ)) H j ( μ) μ j, H j ( μ) j p j horm 4 Udr th codtos of horm ( jp H p( μ) ( a ( a (0, ( a ( ))) (9) Proof I th otato of horm, w hav s, ( a ( )), ad { Φ Φ } ( ( )) ( (, ), S ) ( (, ), )) 0, a h fucto H(z) s dffrtabl at th pot t ( Φ(,, ) ) ad Ht () 0 Cosqutly, all th codtos of horm 3 hold horm 4 s provd IV SIMULAIOS Cosdr th modl of d Movr, for whch th lftm of th dvdual X s uformly dstrbutd th trval (0,00), whr ω s th lmt ag h whol lf auty, accordac wth (), taks th form ( ω) ( ω ) + a( ( ω ) h auts ad thr stmats ar prstd Fg for th radom sampls X,,X of th szs 50,00,500, uformly dstrbutd o th trval (0, 00) Lt th forc of trst 0,0953(9,53%) ot that for ths th ffctv aual trst rat 0,(0%) horm s provd IV ASYMPOIC ORMALIY o fd th lmt dstrbuto of (3), w d th followg two thorms horm (Ctral Lmt horm th multdmsoal cas) [3, p 78-0] If t, t,, t, s a squc of dpdt ad dtcally dstrbutd s- dmsoal vctors, a)

4 Smlarly, for th -yar trm auts w hav ω ( ( a ( a G,,, 50,00,500 h calculato rsults ar gv abl III ABLE III EMPIRICAL MSES FOR DIFFERE SAMPLE SIZES G(,, ) b) It s s from abls I ad III, th qualty of stmato s mprovg wth crasg sampl sz h prst valus of th 5-yar trm lf auts for prsos of th dffrt ags, 0,0953(9,53%), ad th mothly paymt of 000 rubls, ar prstd abl IV ABLE IV PRESE VALUES OF 5-YEAR AUIIES FOR DIFFERE AGES X ( a 5 If 0, th prst valu of th 0-yar auty for a prso at ag 45 yars, 00953(953%), ad th mothly paymt of 000 rubls, s qual to c) Fgur Dpdc of th whol auty ad ts stmat o th ag by th sampl szs a) 50; b) 00; c) 500 W wll charactrz th qualty of th stmats by th mprcal MSE 00 ( a( a G (, ) h calculato rsults ar gv abl I ABLE I EMPIRICAL MSES FOR DIFFERE SAMPLE SIZES G(, ) h prst valus of th whol lf auts for prsos of th dffrt ags, 0,0953(9,53%), ad th mothly paymt of 000 rubls, ar prstd abl II ABLE II PRESE VALUES OF AUIIES FOR DIFFERE AGES a ( a (00953) V COCLUSIO h papr dals wth th problm of stmatg th prst valus of th cotuous whol lf ad -yar trm lf auts W prov asymptotc proprts of th proposd stmators ubasdss, cosstcy ad ormalty Also, w foud th ma parts of th asymptotc MSEs of ths stmators h smulato rsults for th modl of d Movr show that th qualty of valuato by th crtra G (,, s mprovg wth crasg sampl sz ot that o ca obta th mprovd stmators of lf auts (5) ad (6) chagg th mprcal survval fuctos th domators of (5) ad (6) by th smooth mprcal survval fuctos (cf [4-0]) REFERECES [] GI Fal, Mathmatcal Foudatos of th hory of Lf Isurac ad Pso Schms Moscow Akl Publ, 00 (I Russa) [] GM Koshk ad Ya Lopukh, Estmato of t prmums th modls of log trm lf surac, Rvw of Appld ad Idustral Mathmatcs, vol 0,, pp 35-39, 003 (I Russa) [3] GM Koshk, Itroducto to Mathmatcs of Lf Isurac omsk omsk Stat Uvrsty Publ, 004 (I Russa) [4] OV Guba ad GM Koshk, Estmato of th actuaral prst valu of th whol cotuous lf auty, omsk Stat Uvrsty Joural of Cotrol ad Computr Scc, (30), pp 38-43, 05 (I Russa)

5 [5] Bowrs, H Grbr, J Hckma, D Jos, ad C sbtt, Actuaral Mathmatcs Socty of Actuars, Itasca, III, 986 [6] H Grbr, Lf Isurac Mathmatcs, 3rd d Sprgr-Vrlag, w York, 997 [7] GM Koshk, Approach to th vstgato of fuctoals of codtoal dstrbutos udr statstcal ucrtaty, Automato ad Rmot Cotrol, vol 39, 8, pp 4-5, 978 [8] GM Koshk, Asymptotc proprts of fuctos of statstcs ad thr applcato to oparamtrc stmato, Automato ad Rmot Cotrol, vol 5, 3, pp , March 990 [9] GM Koshk, Stabl stmato of ratos of radom fuctos from prmtal data, Russa Physcs Joural, vol 36, 0, pp , 993 [0] GM Koshk, Dvato momts of th substtuto stmator ad ts pcws smooth appromatos, Sbra Mathmatcal Joural, vol 40, 3, pp 55-57, 999 [] A Ktava ad G Koshk, Sm-rcursv krl stmato of fuctos of dsty fuctoals ad thr drvatvs, IFAC Procdgs Volums (IFAC-PaprsOl), 9 (PAR ), pp 43-48, 007 [] I Fuks, ad G Koshk, Smooth rcurrt stmato of multvarat rlablty fucto, h Itratoal Cofrc o Iformato ad AV Ktava ad GM Koshk, Rcurrt oparamtrc stmato of fuctos from fuctoals of multdmsoal dsty ad thr drvatvs, Automato ad Rmot Cotrol, vol 70, 3, pp , March 009 [3] A Dobrovdov, G Koshk, ad V Vaslv, o-paramtrc Stat Spac Modls Hbr, U 8403, USA Kdrck Prss, Ic 0 [4] AV Ktava ad GM Koshk, Stabl multdmsoal tsty fucto Stabl oparamtrc stmato wth rfd covrgc rat, Automato ad Rmot Cotrol, vol 58, 5, pp , 997 [5] VA Vaal ad GM Koshk, oparamtrc stmato of th hazard rat fucto ad ts drvatv, Russa Physcs Joural, vol 4, 3, pp , 999 [6] VA Vaal ad GM Koshk, Krl oparamtrc stmato of th hazard rat fucto ad ts drvatvs, 3rd Itratoal Russa-Kora Symposum o Scc ad chology (Korus 99) ovosbrsk ovosbrsk Stat chcal Uvrsty Procdgs vol, pp , 999 [7] GM Koshk, Smooth stmators of th rlablty fuctos for orstorabl lmts, Russa Physcs Joural, vol 57, 5, pp 67-68, 04 [8] GM Koshk, Smooth rcurrt stmato of th rlablty fucto, Russa Physcs Joural, vol 58, 7, 05, prss [9] I Fuks, ad G Koshk, Smooth rcurrt stmato of multvarat rlablty fucto, h Itratoal Cofrc o Iformato ad Dgtal chologs 05 (ID 05), 7 9 July 05 Zla, Slovaka, pp 84-89, 05 [0] IL Fuks ad GM Koshk, Smooth stmato of multvarat rlablty fucto, Procdgs of th Itratoal Workshop Appld Mthods of Statstcal Aalyss oparamtrc Approach (AMSA 05), 4-9 Sptmbr 05 ovosbrsk SU publshr, Russa, pp 8-9,