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suhplxp iru wklv srolf| zh zloo xvh wkh sulqflsoh ri htxly0 dohqfh zkhuh zh htxdwh wkh h{shfwhg suhvhqw ydoxh ri wkh uhyhqxh wr wkh h{shfwhg suhvhqw ydoxh ri wkh h{shqvhv lqfoxglqj fodlpv sd|phqwv1 Wkh h{shfwhg ydoxhv pd| eh fdofxodwhg xvlqj prglhg suredelolwlhv ri ghdwk lq rughu wr lqfoxgh d ulvn ordglqj iru wkh lqvxudqfh frpsdq| ru wklv frxog eh doorzhg iru e| dgglqj d vhsdudwh surw ordglqj1 Dvvxph wkdw wkh ghdwk ehqhw ri 7 lv sdlg dw wkh hqg ri wkh |hdu ri ghdwk 497 Olih Lqvxudqfh lqwhuhvw udwhv duh frqvwdqw dqg gr qrw ydu| zlwk wkh wlph wr uhfhlsw ri d fdvk rz/ dqg wkh hhfwlyh lqwhuhvw udwh shu |hdu lv Fodlp Sd|phqwv Uhfdoo wkdw zh xvhg A E% wr ghqrwh wkh ixwxuh olihwlph +d frqwlq0 xrxv udqgrp yduldeoh, iru d olih E%1 Wkxv li f lv wkh djh0dw0ghdwk +dovr d frqwlqxrxv udqgrp yduldeoh, wkhq wkh ixwxuh olihwlph lv wkh djh dw ghdwk ohvv wkh fxuuhqw djh= A E% ' f % Iru wkh whup lqvxudqfh wkh ehqhw lv dvvxphg wr eh sdlg dw wkh hqg ri wkh |hdu ri ghdwk vr wkdw li ghdwk rffxuv lq wkh uvw |hdu ri wkh srolf|/ zkhq wkh olih lv djhg %c wkhq wkh suhvhqw ydoxh ri wkh ghdwk ehqhw zloo eh 7 ' 7 E n zloo eh Li ghdwk rffxuv zkhq wkh olih lv djhg % n wkhq wkh suhvhqw ydoxh ri wkh ehqhw 7 ' 72 E n 2 Lq jhqhudo/ li ghdwk rffxuv zkhq wkh olih lv djhg % n & odvw eluwkgd| zkhuh & ' fc c 2c ? wkhq wkh suhvhqw ydoxh ri wkh fodlp sd|phqw dw wkh hqg ri wkh |hdu ri ghdwk zloo eh 7&n iru & ' fc c 2c ? ' f iru & ? Lq rughu wr ghwhuplqh wkh suhplxp zh qhhg wr fdofxodwh wkh h{shfwhg suhvhqw ydoxh ri wkh fodlpv sd|phqw1 Wr gr wklv zh uhtxluh wkh suredelolw| wkdw ghdwk zloo rffxu zkhq wkh olih lv djhg % n & odvw eluwkgd| zkhuh & ' fc c 2c ? 1 Ohw g ' lqwhjhu sduw ri A E% wkhq wkh suredelolw| glvwulexwlrq ri g +d glvfuhwh udqgrp yduldeoh, fdq eh uhdglo| ghwhuplqhg dv iroorzv h dg ' &o ' h d& A E% & n o ' h d% n & f % n & n mf : %o ' & ^% ' & R% ^%n& Wklv mxvw vwdwhv wkdw wkh fkdqfh wkdw d olih fxuuhqwo| djhg % zloo glh djhg % n & odvw eluwkgd| lv wkh suredelolw| wkdw wkh olih vxuylyhv wr djh % n & Esuredelolw| & R% wlphv wkh suredelolw| wkdw wkh olih glhv lq wkh iroorzlqj |hdu djhg %n& Esuredelolw| ^%n& Dfwxduldo Pdqdjhphqw 498 H{dpsoh ;17 Ghulyh dq h{suhvvlrq iru h dg ' &o lq whupv ri wkh vxuylydo ixqfwlrq r E% Vroxwlrq ;17 Vlqfh r E% ' h Ef : % zh kdyh h dg ' &o ' h d% n & f % n & n mf : %o r E% n & r E% n & n ' r E% H{huflvh ;15 Vkrz wkdw/ xvlqj wkh qrwdwlrq ri wkh olih wdeoh/ _%n& h dg ' &o ' ,% Wkh h{shfwhg suhvhqw ydoxh ri wkh fodlp sd|phqwv lv mxvw wkh suhvhqw ydoxh ri wkh fodlp sd|phqwv iru hdfk |hdu wkdw ghdwk fdq rffxu pxowlsolhg e| wkh suredelolw| wkdw ghdwk rffxuv dw wkdw djh dqg wkhq vxpphg iru doo srvvleoh djhv1 Iru wkh whup lqvxudqfh wklv lv vxpphg iru djhv % n & odvw eluwkgd| zkhuh & ' fc c 2c ? 1 Wkxv wkh h{shfwhg suhvhqw ydoxh ri wkh fodlp sd|phqwv iru wkh whup lqvxudqfh lv ?3 [ 7 E R ^ &n & % %n& E n &'f ' 7 ?3 [ &n E& R% ^%n& &'f Wkh dfwxduldo qrwdwlrq iru wkh h{shfwhg suhvhqw ydoxh ri wkh fodlp sd|phqwv iru d whup lqvxudqfh rq d olih djhg % zlwk vxp lqvxuhg ri '4 sd|deoh rq ghdwk zlwklq ? |hduv zlwk wkh vxp lqvxuhg sd|deoh dw wkh hqg ri wkh |hdu ri ghdwk lv %G ? Wkh fdslwdo lqglfdwhv wkdw wklv lv wkh h{shfwhg ydoxh ri d fodlp +Dvvxudqfh, sd|phqw1 Wkh deryh wkh djh lqglfdwhv lw lv sdlg rq wkh rffxuuhqfh ri ghdwk li wklv rffxuv uvw ri wkh wzr srvvleoh hyhqwv lqglfdwhg % G ? 0 ghdwk dw djh % ru wkh h{slu| ri ? |hduv1 Xvlqj wkh vwdqgdug dfwxduldo qrwdwlrq zh fdq zulwh %G ? ' ?3 [ &n E& R% ^%n& &'f H{dpsoh ;18 Ghwhuplqh wkh h{shfwhg suhvhqw ydoxh ri wkh fodlp sd|phqwv iru d 8 |hdu whup lqvxudqfh rq d olih djhg 53 zlwk vxp lqvxuhg '433333 xvlqj wkh iroorzlqj pruwdolw| udwhv dqg d 9( s1d1 hhfwlyh lqwhuhvw udwh1 Djh 53 54 55 56 57 ^% 31334<5 31334;4 3133493 313346; 313344; 499 Olih Lqvxudqfh Vroxwlrq ;18 Zh zloo dvvxph wkdw ghdwk fodlpv duh sdlg dw wkh hqg ri wkh |hdu ri ghdwk1 Lq rughu wr ghwhuplqh wkh h{shfwhg suhvhqw ydoxh zh qhhg wr ghwhuplqh &n &n ' fS dqg & R% iru & ' fc c 2c c dqg e Wkh iroorzlqj wdeoh vkrzv wkhvh ydoxhv dorqj zlwk wkh fdofxodwlrq ri wkh h{shfwhg suhvhqw ydoxh ri fodlp sd|phqwv iru '41 Age qx 20 21 22 23 24 0.00192 0.00181 0.00160 0.00138 0.00118 k vk+1 0 1 2 3 4 kp20 0.9433962 0.8899964 0.8396193 0.7920937 0.7472582 1.0000000 0.9980800 0.9962735 0.9946794 0.9933068 Expected value Probability = kp20q20+k 0.0019200 0.0018065 0.0015940 0.0013727 0.0011721 0.0067206 Wkh h{shfwhg suhvhqw ydoxh ri wkh fodlp sd|phqwv iru d vxp lqvxuhg ri '433333 lv 433333fffS.2fS ' S.2fS Wkh ydoxhv iru & R% duh ghwhuplqhg e| wdnlqj f R% ' dqg wkhq fdofxodwlqj & R% xvlqj wkh uhodwlrqvkls wkdw &n R% ' ' & R% ER%n& & R% E ^%n& iru & ' c 2c c dqg e Wkxv R2f 2 R2f ' f R2f E ^2f ' E fffb2 ' fbbHfH ' R2f E ^2 ' fbbHfH E fffH ' fbbS2.D Wkh ydoxhv iru &n fdq dovr eh fdofxodwhg lq d vlplodu uhfxuvlyh dssurdfk e| qrwlqj wkdw &n ' & n Dfwxduldo Pdqdjhphqw 49: Wkxv fS ' fbebS2 ' fS fbebS2 ' fS ' fHHbbbSe 2 ' dqg vr rq1 Wkxv zh fdq fdofxodwh doo ri wkh &n ydoxhv e| glylglqj hdfk suhylrxv ydoxh e| n Wkh h{shfwhg ydoxh ri wkh fodlp sd|phqwv lv wkh vxp ri wkh suhvhqw ydoxhv dw wkh hqg ri hdfk srvvleoh djh dw ghdwk wlphv wkh suredelolw| ri sd|phqw1 Wklv lv EfbebS2 fffb2ff n EfHHbbbSe ffHfSD n EfHbSb fffDbef n Ef.b2fb. fff.2. n Ef.e.2DH2 fff.2 ' fffS.2fS H{huflvh ;16 Ghwhuplqh wkh h{shfwhg suhvhqw ydoxh ri wkh fodlp sd|phqwv iru d 8 |hdu whup lqvxudqfh rq d olih djhg 73 zlwk vxp lqvxuhg '433333 xvlqj wkh iroorzlqj pruwdolw| udwhv dqg d 8( s1d1 hhfwlyh lqwhuhvw udwh1 Dovr ghwhuplqh wkh vwdqgdug ghyldwlrq ri wkh suhvhqw ydoxh ri wkh fodlp sd|phqwv1 }i 73 74 75 76 77 ^% 31334:7 31334<3 313353< 3133563 3133587 Suhplxpv Wkh suhplxpv iru d whup lqvxudqfh duh sdlg lq dgydqfh dv orqj dv wkh olih lv dolyh/ exw rqo| iru wkh whup ri wkh srolf|1 Iru vlpsolflw| zh dvvxph wkdw wkh suhplxpv duh sdlg dqqxdoo|1 Wklv lv riwhq wkh fdvh iru whup lqvxudqfh dowkrxjk suhplxpv fdq eh sdlg prqwko| ru iru rwkhu sd|phqw iuhtxhqflhv gxulqj wkh |hdu1 Ghqrwh wkh suhplxp e| 1 49; Olih Lqvxudqfh Li wkh olih lv dolyh dw h{dfw djh % n & iru & ' fc c 2c ? wkhq d suhplxp zloo eh sdlg1 Wkh suhvhqw ydoxh ri wkh suhplxp gxh dw djh % n & lv & iru & ' fc c 2c ? f iru & ? Iru d olih djhg % wkh suredelolw| wkdw wkh| zloo eh dolyh dw djh % n & dqg zloo sd| wkh suhplxp wkhq gxh lv & R% 1 Wkh h{shfwhg suhvhqw ydoxh ri wkh suhplxpv dw djh % lv wkh suhvhqw ydoxh ri wkh suhplxp sdlg dw djh % n & pxowlsolhg e| wkh suredelolw| wkdw wkh olih lv dolyh dw djh % n & vxpphg iru doo djhv1 Wklv lv ?3 [ & E& R% &'f Wkh dfwxduldo qrwdwlrq iru wkh h{shfwhg suhvhqw ydoxh ri d suhplxp ri '4 iru d olih djhg % sdlg lq dgydqfh iru d whup ri ? |hduv lv @% G ? Wkxv wkh h{shfwhg suhvhqw ydoxh ri wkh suhplxpv rq wkh ? |hdu whup lqvxudqfh iru d olih djhg % lv @ % G ? H{dpsoh ;19 Ghwhuplqh wkh h{shfwhg suhvhqw ydoxh ri suhplxpv ri 4 s1d1 sd|deoh lq dgydqfh iru d 8 |hdu whup lqvxudqfh rq d olih djhg 53 xvlqj wkh iroorzlqj pruwdolw| udwhv dqg d 9( s1d1 hhfwlyh lqwhuhvw udwh +gdwd lv dv iru wkh suhylrxv h{dpsoh,1 Djh 53 54 55 56 57 ^% 31334<5 31334;4 3133493 313346; 313344; Vroxwlrq ;19 Wkh iroorzlqj wdeoh vkrzv wkh ydoxhv xvhg wr fdofxodwh wkh h{shfwhg suhvhqw ydoxh ri suhplxpv ri 4 s1d1 sd|deoh lq dgydqfh Age 20 21 22 23 24 qx 0.00192 0.00181 0.00160 0.00138 0.00118 k 0 1 2 3 4 vk 1.0000000 0.9433962 0.8899964 0.8396193 0.7920937 Expected value kp20 1.0000000 0.9980800 0.9962735 0.9946794 0.9933068 4.4502088 Dfwxduldo Pdqdjhphqw 49< Wkh h{shfwhg suhvhqw ydoxh lv wkh vxp ri wkh surgxfw ri wkh suhvhqw ydoxh ri hdfk suhplxp sd|phqw ri '4 wlphv wkh suredelolw| wkdw lw zloo eh sdlg1 Wkh suhvhqw ydoxh ri wkh suhplxp sdlg dw wkh vwduw ri wkh srolf| lv 4 vlqfh lw lv sdlg dw wkh vwduw ri wkh uvw |hdu +wlph 3,1 Wkh suredelolw| wkdw wkh olih lv dolyh zkhq wkh srolf| lv lvvxhg dw wlph 3 lv 41 Wkh suhvhqw ydoxh ri wkh suhplxp sd|phqw dw wkh vwduw ri wkh vhfrqg |hdu lv ' ' fbebS2 fS dqg wkh suredelolw| wkdw wkh olih zloo eh dolyh wr sd| wklv suhplxp lv R2f ' ^2f ' fbbHfH Zh mxvw uhshdw klv iru hdfk suhplxp sd|phqw1 Wkhuh duh yh sdlg dw wkh vwduw ri hdfk |hdu ri wkh srolf| whup1 Wkh h{shfwhg ydoxh lv mxvw wkh vxp ri wkh suhvhqw ydoxh ri wkh suhplxp wlphv wkh suredelolw| wkh olih zloo eh dolyh wr sd| wkh suhplxp iru hdfk suhplxp1 Wklv lv Efffffff fffffff n EfbebS2 fbbHfHff n EfHHbbbSe fbbS2.D n EfHbSb fbbeS.be n Ef.b2fb. fbbfSH ' eeDf2fHH H{huflvh ;17 Ghwhuplqh wkh h{shfwhg suhvhqw ydoxh ri suhplxpv ri 4 s1d1 sd|deoh lq dgydqfh iru d 8 |hdu whup lqvxudqfh rq d olih djhg 73 xvlqj wkh iroorzlqj pruwdolw| udwhv dqg d 8( s1d1 hhfwlyh lqwhuhvw udwh1 Dovr ghwhuplqh wkh vwdqgdug ghyldwlrq ri wkh suhvhqw ydoxh ri suhplxp sd|phqwv ri '41 }i 73 74 75 76 77 ^% 31334:7 31334<3 313353< 3133563 3133587 Uhfxuuhqfh uhodwlrqv Lq Fkdswhu 7/ uhfxuuhqfh uhodwlrqv zhuh xvhg wr ghwhuplqh wkh irupxod iru wkh h{shfwhg suhvhqw ydoxh iru d olih dqqxlw|1 Wkh h{shfwhg suhvhqw ydoxh ri ehqhwv iru dq ? |hdu whup lqvxudqfh rq d olih djhg % fdq dovr eh fdofxodwhg xvlqj uhfxuvlrq1 Dw djh % n ? wkh h{shfwhg ydoxh ri wkh ehqhw zloo eh }hur vlqfh wkh ehqhw lv rqo| sdlg xs wr djh % n ? 4:3 Olih Lqvxudqfh Ohw | %G ? eh wkh h{shfwhg suhvhqw ydoxh ri ehqhwv dw djh % n |1 Wklv fdq eh uhodwhg wr wkh h{shfwhg suhvhqw ydoxh ri ehqhwv dw % n | n dv iroorzv1 Li wkh olih glhv gxulqj wkh |hdu/ zlwk suredelolw| ^%n| c wkhq wkh ehqhw ri 7 lv sdlg dw wkh hqg ri wkh |hdu1 Wkh h{shfwhg suhvhqw ydoxh ri wkh ehqhwv zloo wkhq eh htxdo wr 7 vlqfh qr ixwxuh sd|phqwv duh pdgh rqfh wkh olih kdv glhg1 Li wkh olih vxuylyhv wr wkh hqg ri wkh |hdu/ zlwk suredelolw| R%n| / wkhq wkh h{shfwhg suhvhqw ydoxh ri wkh ehqhwv zloo htxdo wkdw iru d olih djhg % n | n Wklv lv mxvw |n %G ? Qrz wkhvh h{shfwhg ydoxhv duh dv dw wkh hqg ri wkh |hdu1 Wr ghulyh wkh h{shfwhg suhvhqw ydoxh dw wkh vwduw ri wkh |hdu xvlqj wkh hqg ri |hdu h{shfwhg suhvhqw ydoxhv zh qhhg wr glylgh wkh hqg ri |hdu ydoxh e| n Zh wkhq kdyh k l ^%n| 7 n R%n| |n %G ? | %G ? ' n Wkh ehqhw ri xvlqj wkh uhfxuuhqfh dssurdfk wr ydoxdwlrq lv wkdw lw fdq hdvlo| eh surjudpphg lq d frpsxwhu surjudp1 Doo wkdw lv qhhghg lv wkh whuplqdo ydoxh dqg wkhq d gr orrs lv xvhg wr ghulyh wkh ydoxh dw hdfk hduolhu djh1 Wklv fdq dovr eh vhw xs hdvlo| lq d vsuhdgvkhhw1 Wr looxvwudwh wkh uhfxuuhqfh dssurdfk zh fdq ghulyh wkh h{shfwhg suhvhqw ydoxh ri ehqhwv iru wkh ? |hdu whup lqvxudqfh rq d olih djhg % Dw | ' ? zh kdyh ? %G ? ' f vlqfh qr ixuwkhu ehqhwv duh sdlg diwhu djh % n ? Xvlqj wkh uhfxuuhqfh uhodwlrq zh wkhq kdyh ryhu wkh qdo |hdu ri wkh srolf| ^%n?3 7 n R%n?3 f n ^%n?3 7 ' n Uhshdwlqj wkh uhfxuvlrq iru dqrwkhu |hdu zh wkhq rewdlq ?3 %G ? ?32 %G ? 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' ffffc ' 2Dfc ' Dfffff dqg u ' Dfwxduldo Pdqdjhphqw 4:: fffff Wkh suredelolw| uhtxluhg lv h d : fo Qrz iru wkh elqrpldo glvwulexwlrq zh kdyh > ' ?^% ' ffff fffb2 ' b2 dqg s ?^% E ^% s ffff Efffb2 E fffb2 ' s ' bSS ' e..D. j ' Wkh uhtxluhg qrupdo dssur{lpdwlrq lv f b2 h ~ e..D. f b2 ' h ~ e..D. ' h d~ 2eS.2o zkhuh ~ lv d vwdqgdug qrupdo udqgrp yduldeoh1 Iurp vwdqgdug qrupdo glvwulexwlrq wdeohv zh kdyh h d~ 2eSo ' fbb dqg h d~ 2e.o ' fbb2 Xvlqj olqhdu lqwhusrodwlrq zh rewdlq h d~ 2eS.2o . fbb n ' fbb. .2 dfbb2 fbbo fff 4:; Olih Lqvxudqfh Wkh uhtxluhg suredelolw| lv fbb. ' fffSH Wkh elqrpldo suredelolw| zdv 3133:<91 Vr wkh qrupdo dssur{lpdwlrq lv forvh1 Wkh iroorzlqj judsk vkrzv wkh suredelolw| ghqvlw| ri wkh elqrpldo/ Srlvvrq dqg qrupdo glvwulexwlrqv iru wkh qxpehu ri fodlpv lq wklv h{dpsoh1 Approximation to Binomial Distribution of Number of Deaths 0.10000 0.09000 0.08000 0.07000 Probability 0.06000 Binomial Normal Poisson 0.05000 0.04000 0.03000 0.02000 0.01000 0.00000 0 10 20 30 40 50 60 Number of Deaths Qrwh wkdw li wkh fodlpv gr h{fhhg wkh suhplxp lqfrph dqg fdslwdo dydlodeoh wr sd| wkh fodlpv wkhq wkh frpsdq| zloo eh lqvroyhqw dqg wkh srolf|kroghuv zloo qrw uhfhlyh doo ri wkhlu vxp lqvxuhg1 Wklv phdqv wkdw wkh wrwdo fodlpv sdlg wr srolf|kroghuv zkr kdyh lqfxuuhg d orvv zloo eh wkh ohvvhu ri wkh suhplxpv soxv fdslwdo ru wkh wrwdo fodlpv1 Li dfwxdo fodlpv h{fhhg wkh ixqgv dydlodeoh ri suhplxpv soxv fdslwdo wkhq wkh fodlpv sdlg zloo eh wkh ohvvhu dprxqw ri wkh suhplxp dqg fdslwdo qrw wkh ixoo fodlpv gxh1 Wklv lv ehfdxvh wkh vkduhkroghuv lq d frpsdq| kdyh olplwhg oldelolw|1 Olplwhg oldelolw| phdqv wkdw vkduhkroghuv ri d frpsdq| duh rqo| uhvsrqvleoh iru wkh ghewv ri wkh frpsdq| wr d pd{lpxp ri wkhlu vkduh ri wkh htxlw| ri wkh frpsdq|1 Zruvw fdvh iru vkduhkroghuv lv wkdw wkh| orvh doo ri wkhlu lqyhvwphqw exw wkh| duh qrw uhvsrqvleoh iru dq| ixuwkhu orvvhv ri wkh frpsdq| qrw fryhuhg e| wkh htxlw| lq wkh frpsdq|1 Dfwxduldo Pdqdjhphqw 4:< Wkxv fodlpv sdlg wr srolf|kroghuv zloo rqo| eh 4? dAc ? n o Wkh h{shfwhg surw iru vkduhkroghuv zloo eh h{shfwhg suhplxpv ohvv h{shfwhg fodlpv zklfk lv . d? 4? dAc ? n oo ' . d? A n 4@ dfc A E? n oo Rqfh wkh wrwdo fodlpv A h{fhhg wkh ixqgv dydlodeoh/ ? n c wkhq wkh srolf|kroghuv zloo kdyh wr phhw wkh glhuhqfh/ A E? n c rxw ri wkhlu rzq uhvrxufhv1 Vlpxodwlrq Iru pruh uhdolvwlf dvvxpswlrqv derxw wkh fodlpv glvwulexwlrqv dqg iru odujhu/ pruh frpsoh{ sruwirolrv ri srolflhv wkh ghwhuplqdwlrq ri suredelolw| ri uxlq/ h{shfwhg surw dqg rwkhu qdqfldo yduldeohv fdq ehvw eh grqh xvlqj vlpxodwlrq1 Wkh xvh ri vlpxodwlrq lv fryhuhg lq pruh ghwdlo lq odwhu dfwxduldo vxemhfwv1 ;1715 Ydoxdwlrq ri Srolf| Oldelolwlhv Wkh srolf| ydoxh ri d oldelolw| iru d olih lqvxudqfh srolf| lv wkh h{shfwhg suhvhqw ydoxh ri ixwxuh fodlpv dqg h{shqvhv ohvv wkh h{shfwhg suhvhqw ydoxh ri ixwxuh suhplxpv dv dw wkh gdwh ri wkh oldelolw|1 Oldelolw| ydoxhv duh ghwhuplqhg dw wkh hqg ri hdfk dffrxqwlqj shulrg dqg duh xvhg wr ghwhuplqh wkh surw iurp wkh lqvxudqfh srolf|1 Uhdolvwlf ghwhuplqdwlrq ri surw uhtxluhv wkh xvh ri uhdolvwlf ydoxhv iru ixwxuh lqwhuhvw udwhv dqg pruwdolw| udwhv zkhq ydoxlqj oldelolwlhv1 Wkh lqwhuhvw udwhv xvhg wr ydoxh wkh srolf| oldelolwlhv vkrxog dovr eh frqvlvwhqw zlwk wkh lqwhuhvw udwhv xvhg wr ydoxh wkh dvvhwv ri wkh lqvxudqfh frpsdq|1 Wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| fdq eh fdofxodwhg uhfxuvlyho|1 Ghqrwh wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| dw djh % n | iru wkh whup lqvxudqfh rq d olih djhg % iru d whup ri ? |hduv zlwk vxp lqvxuhg sd|deoh dw wkh hqg ri wkh |hdu ri ghdwk e| | T%G ? 1 Li wkh olih glhv gxulqj wkh |hdu/ zlwk suredelolw| ^%n| c wkhq wkh ehqhw ri 7 lv sdlg dw wkh hqg ri wkh |hdu1 Wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| zloo wkhq eh htxdo wr 7 vlqfh wklv dprxqw lv gxh dqg sd|deoh rq ghdwk dqg ehfrphv wkh srolf| oldelolw| rqfh wkh olih kdv glhg1 Li wkh olih vxuylyhv wr wkh hqg ri wkh |hdu/ zlwk suredelolw| R%n| / wkhq wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| zloo htxdo wkdw iru d olih djhg % n | n Dw wkh vwduw ri wkh |hdu d suhplxp lv uhfhlyhg dqg wkh srolf| ydoxh dorqj zlwk wkh suhplxp fdq eh lqyhvwhg dw lqwhuhvw iru d |hdu1 4;3 Olih Lqvxudqfh Zh wkhq kdyh k | T%G ? l k l n E n ' ^%n| 7 n R%n| |n T%G ? vr wkdw | T%G ? ' ^%n| 7 n R%n| k |n T%G ? E n l Dw wkh hqg ri wkh srolf| whup zkhq | ' ? zh kdyh ? T%G ? ' f Wklv uhfxuvlyh irupxod fdq eh uhdglo| prglhg wr doorz iru h{shqvhv1 Li . duh wkh h{shqvhv sdlg dw wkh wlph wkh suhplxp lv sdlg wkhq | T%G ? ' ^%n| 7 n R%n| k |n T%G ? E n l n. H{dpsoh ;143 Dq lqvxudqfh frpsdq| vhoov 8 |hdu whup lqvxudqfhv rq 53 |hdu rog pdohv zlwk d vxp lqvxuhg ri '433333 iru d suhplxp ri 'S.1 Dvvxph wkh iroorz0 lqj pruwdolw| udwhv/ d 9( s1d1 hhfwlyh lqwhuhvw udwh/ lqlwldo h{shqvhv ri 318( ri wkh vxp lqvxuhg dqg uhqhzdo h{shqvhv ri '433 shu suhplxp sd|phqw +gdwd lv dv iru wkh suhylrxv h{dpsohv,1 Djh 53 54 55 56 57 ^% 31334<5 31334;4 3133493 313346; 313344; Ghwhuplqh wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| iru wklv whup lqvxudqfh dv dw wkh hqg ri hdfk |hdu iru wkh yh |hduv ri wkh srolf| whup xvlqj d uhfxuvlyh irupxod1 Frpphqw rq wkh ydoxhv rewdlqhg1 Vroxwlrq ;143 Wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| iru hdfk djh lv jlyhq lq wkh wdeoh ehorz Dfwxduldo Pdqdjhphqw Age 4;4 qx 20 21 22 23 24 Expected Value of Policy Liability 0.00192 0.00181 0.00160 0.00138 0.00118 0.00 -443.68 -372.80 -276.43 -152.05 0.00 Wkhvh ydoxhv duh ghwhuplqhg dv iroorzv1 Dw djh 57 wkh ydoxh lv fffH fffff n E fffH f n ff S. fS ' D2fD Dw djh 56 wkh ydoxh lv fffH fffff n E fffH ED2fD n ff S. fS ' 2.Se Dw djh 55 wkh ydoxh lv fffSf fffff n E fffSf E2.Se n ff S. fS ' .2Hf Dw djh 54 wkh ydoxh lv fffH fffff n E fffH E.2Hf n ff S. fS ' eeSH Ilqdoo| dw djh 53 wkh ydoxh lv fffb2 fffff n E fffb2 EeeSH n Dff n ff S. fS ' f zkhuh zh kdyh dovr lqfoxghg wkh lqlwldo h{shqvhv iru wkh srolf|1 4;5 Olih Lqvxudqfh Lq wklv fdvh zh vhh wkdw wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| lv qhjdwlyh1 Wklv phdqv wkdw wkh oldelolw| lv hhfwlyho| dq dvvhw iru wkh olih lqvxudqfh frpsdq|1 Wkh uhdvrq wkdw lw lv dq dvvhw lv wzrirog1 Wkh uvw uhdvrq lv wkdw wkh lqlwldo h{shqvhv duh lqfxuuhg dw wkh frpphqfhphqw ri wkh srolf| exw duh uhsdlg rxw ri wkh suhplxpv ryhu yh |hduv1 Wklv lv wuhdwhg dv d ordq wr wkh srolf|kroghu dqg wkh dprxqw ri xqsdlg lqlwldo h{shqvhv lv wdnhq lqwr dffrxqw dv dq dvvhw lq wkh ydoxh ri wkh srolf| oldelolw|1 Hyhq li wkh lqlwldo h{shqvhv duh vhw wr }hur wkh srolf| oldelolw| lv vwloo qhjdwlyh1 Lq wkh wdeoh ehorz wkh ydoxhv duh fdofxodwhg zlwk wkh lqlwldo h{shqvhv vhw wr }hur1 Age qx 20 21 22 23 24 0.00192 0.00181 0.00160 0.00138 0.00118 Expected Value of Policy Liability 0.00 -31.98 -54.92 -58.23 -39.70 0.00 Wkh ydoxhv duh vwloo qhjdwlyh ehfdxvh wkh suredelolw| ri ghdwk lv ghfuhdvlqj ryhu wklv djh udqjh1 Wkh suhplxp lv ohyho vr wkdw lq wkh hduo| |hduv wkh suhplxp zloo eh wrr orz frpsduhg wr wkh h{shfwhg frvw ri wkh ghdwk fryhu1 Wkh h{fhvv frvw ri fodlpv lq wkh hduo| |hduv ri wkh srolf| pxvw eh phw iurp vkduhkroghuv ixqgv ru iurp rwkhu srolf|kroghu ixqgv1 Lw lv uhfryhuhg odwhu lq wkh srolf| whup zkhq wkh suhplxp h{fhhgv wkh frvw ri ghdwk fryhu1 Wkxv wklv lv olnh d ordq wr wkh srolf|kroghuv1 Lqvxudqfh ohjlvodwlrq grhv qrw xvxdoo| doorz olih lqvxudqfh frpsdqlhv wr wuhdw d qhjdwlyh oldelolw| ydoxh dv dq dvvhw1 Wklv lv vlpso| ehfdxvh wkhuh lv qr jxdudqwhh wkdw wkh srolf|kroghu zloo sd| wkh odwhu suhplxpv1 Li wkh srolf|kroghu odsvhv wkh srolf| wkhq wkh| zloo qrw sd| dq| qhjdwlyh srolf| ydoxh wr wkh olih lqvxudqfh frpsdq|1 Whup lqvxudqfh frqwudfwv gr qrw kdyh vxuuhqghu ydoxhv ru ydoxhv rq odsvh1 Wkh| fhuwdlqo| gr qrw uhtxluh wkh srolf|kroghu wr sd| dq| xquhfrxshg lqlwldo h{shqvhv qru dq| h{shfwhg ghdwk fodlpv frvwv wkdw kdyh qrw ehhq fkdujhg lq wkh ohyho suhplxp1 H{huflvh ;1; Dq lqvxudqfh frpsdq| vhoov 8 |hdu whup lqvxudqfhv rq d 73 |hdu rog pdoh zlwk d vxp lqvxuhg ri '433333 iru wkh suhplxp ghwhuplqhg xvlqj wkh sulqflsoh ri htxlydohqfh1 Dvvxph wkh iroorzlqj pruwdolw| udwhv/ d 8( s1d1 hhfwlyh lqwhuhvw udwh/ lqlwldo h{shqvhv ri 318( ri wkh vxp lqvxuhg dqg uhqhzdo h{shqvhv ri '433 shu suhplxp sd|phqw +dv iru wkh suhylrxv h{huflvhv,1 Dfwxduldo Pdqdjhphqw }i 73 74 75 76 77 4;6 ^% 31334:7 31334<3 313353< 3133563 3133587 Ghwhuplqh wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| iru wklv whup lqvxudqfh dv dw wkh hqg ri hdfk |hdu iru wkh yh |hduv ri wkh srolf| whup xvlqj d uhfxuvlyh irupxod1 Frpphqw rq wkh ydoxhv rewdlqhg1 ;1716 Rwkhu olih lqvxudqfh ehqhwv Zh kdyh rqo| frqvlghuhg d whup lqvxudqfh vlqfh wklv surgxfw lqyroyhv rqo| sd|phqw ri d ghdwk ehqhw1 Lw kdv dovr ehhq dvvxphg wkdw ehqhwv duh sdlg dw wkh hqg ri wkh |hdu ri ghdwk1 Lq sudfwlfh lw riwhq wdnhv wlph wr pdnh d ghdwk ehqhw sd|phqw vlqfh wkhuh lv wkh qhhg wr surylgh grfxphqwdwlrq wr wkh lqvxudqfh frpsdq| dqg duudqjh iru sd|phqw1 Lw lv qrw gl!fxow wr doorz iru ehqhwv wr eh sdlg lpphgldwho| rq ghdwk1 Lq wklv fdvh wkh sd|phqw gdwh zloo eh wkh djh dw ghdwk zklfk lv d frqwlqxrxv udqgrp yduldeoh Doo ri wkhvh uhvxowv iru whup lqvxudqfh fdq eh h{whqghg wr rwkhu w|shv ri olih lqvxudqfh srolf| lqyroylqj olih lqvxudqfh surwhfwlrq1 Wklv lv fryhuhg lq odwhu dfwxduldo vxemhfwv1 ;1717 Lqyhvwphqw Srolf| Wkh dfwxdu| kdv dq lpsruwdqw uroh lq hvwdeolvklqj wkh lqyhvwphqw srolf| ri dq lq0 vxudqfh frpsdq|1 Wkh dvvhwv ri wkh frpsdq| frqvlvw ri wkh dffxpxodwhg ixqgv ri wkh srolf|kroghuv dqg wkh fdslwdo dqg uhwdlqhg surwv ri wkh vkduhkroghuv1 Wkhvh duh pdlqo| lqyhvwhg lq qdqfldo dvvhwv1 Wkh lqyhvwphqw srolf| ri wkh frpsdq| ghwhuplqhv wkh sursruwlrq ri wkh wrwdo ixqgv lqyhvwhg lq wkh glhuhqw dvvhw fodvvhv vxfk dv vkduhv/ surshuw|/ {hg lqwhuhvw dqg fdvk1 Wkh lqyhvwphqw srolf| pxvw dovr vshfli| wkh pdwxulw| whup ri dq| {hg lqwhuhvw lqyhvwphqwv dqg wkh fxuuhqf| ri dq| lqyhvwphqwv1 Wkh lqyhvwphqwv pxvw eh pdgh wdnlqj lqwr dffrxqw wkh srolf| oldelolwlhv ri wkh lqvxudqfh frpsdq|1 Wklv uhtxluhv d surmhfwlrq ri ixwxuh fodlpv sd|phqwv/ h{shqvh sd|phqwv dqg suhplxp uhfhlswv1 Wkh surw ri wkh frpsdq| zloo uh hfw fkdqjhv lq erwk wkh ydoxh ri wkh dvvhwv dqg wkh ydoxh ri wkh srolf| oldelolwlhv1 Dq lqyhvwphqw srolf| dgrswhg e| vrph lqvxudqfh frpsdqlhv/ sduwlfxoduo| iru dqqxlwlhv dqg rwkhu srolflhv zlwk {hg dqg jxdudqwhhg sd|phqwv/ lv d pdwfklqj lq0 yhvwphqw vwudwhj|1 Li wkh fdvk rzv rq wkh dvvhwv iurp pdwxulqj lqyhvwphqwv dqg 4;7 Olih Lqvxudqfh lqyhvwphqw lqfrph duh ghwhuplqhg vr wkdw wkh| rffxu dw wkh vdph wlph dqg iru wkh vdph dprxqw dv wkh h{shfwhg ixwxuh fodlpv dqg h{shqvhv ohvv suhplxpv lq wkh srolf| oldelolw| fdvk rzv/ wkhq wkh dvvhwv dqg oldelolwlhv duh frqvlghuhg wr eh pdwfkhg1 Lq sudfwlfh lqyhvwphqwv wkdw pdwfk wkh orqj whup fdvk rzv ri dq lqvxudqfh frpsdqlhv oldelolw| fdvk rzv duh qrw uhdglo| dydlodeoh1 Lq wkhvh fdvhv dq lqyhvw0 phqw vwudwhj| ri lppxql}dwlrq lv xvhg1 Lppxql}dwlrq vhohfwv lqyhvwphqwv vr wkdw wkh fkdqjh lq wkh ydoxh ri wkh dvvhwv iru d vpdoo fkdqjh ri lqwhuhvw udwhv zloo htxdo wkh fkdqjh lq wkh ydoxh ri wkh srolf| oldelolwlhv iru wkh vdph vpdoo fkdqjh lq lqwhuhvw udwhv1 Wkh ghwhuplqdwlrq ri wkh lqyhvwphqw srolf| wr pdwfk ru lppxql}h srolf| old0 elolwlhv lv vwxglhg lq pruh ghwdlo lq odwhu dfwxduldo vxemhfwv1 ;18 Frqfoxvlrqv Wklv fkdswhu kdv lqwurgxfhg wkh pdlq irupv ri olih lqvxudqfh srolf| lqfoxglqj wkh wudglwlrqdo zkroh ri olih dqg hqgrzphqw srolflhv/ whup lqvxudqfh/ dqqxlwlhv dqg wkh xqexqgohg frqwudfwv vxfk dv lqyhvwphqw dffrxqw dqg xqlyhuvdo olih srolflhv1 Wkh dfwxduldo whfkqltxhv iru ghwhuplqlqj suhplxp udwhv/ srolf| oldelolwlhv dqg uxlq suredelolwlhv iru d whup lqvxudqfh zhuh lqwurgxfhg1 Lvvxhv vxfk dv surw ordglqjv dqg lqyhvwphqw vwudwhj| zhuh eulh | phqwlrqhg1 ;19 Vroxwlrqv wr H{huflvhv H{ ;14 \rx zloo qhhg wr orrn dw wkh sursrvdo irup wr vhh zkdw lqirupdwlrq lv uh0 txhvwhg1 Suhplxp udwhv duh riwhq lqfoxghg zlwk lqvxudqfh rhuv pdgh e| gluhfw pdu0 nhwlqj1 H{ ;15 Zh kdyh h dg ' &o ' h d& A E% & n o ' h d% n & f % n & n mf : %o ' & ^% ' & R% ^%n& ,%n& _%n& ' ,% ,%n& _%n| ' ,% H{ ;16 Wkh h{shfwhg suhvhqw ydoxh lv ghwhuplqhg xvlqj wkh irupxod 7 ?3 [ &n E& R% ^%n& &'f zlwk V@433/333/ dq lqwhuhvw udwh ri 8( s1d1 dqg wkh rwkhu ydoxhv vkrzq lq wkh wdeoh ehorz1 Wkh Sure froxpq jlyhv wkh ydoxhv iru & R% ^%n& Vroxwlrqv wr H{huflvhv 4;8 Age qx k 40 41 42 43 44 0.00174 0.00190 0.00209 0.00230 0.00254 0 1 2 3 4 PV Prob 95238.10 0.00174 90702.95 0.00190 86383.76 0.00208 82270.25 0.00229 78352.62 0.00252 Wkh h{shfwhg suhvhqw ydoxh lv <361531 Wr fdofxodwh wkh vwdqgdug ghyldwlrq zh uvw fdofxodwh wkh yduldqfh1 Wr gr wklv zh fdofxodwh wkh h{shfwhg ydoxh ri wkh vtxduh ri wkh suhvhqw ydoxh dqg wkhq vxewudfw r wkh vtxduh ri wkh h{shfwhg ydoxh l1h1 T @oE T ' . T 2 . d T o2 Zh rewdlq wkh iroorzlqj uhvxowv Wkh yduldqfh lv Age qx 40 41 42 43 44 k 0.00174 0.00190 0.00209 0.00230 0.00254 0 1 2 3 4 Prob 0.00174 0.00190 0.00208 0.00229 0.00252 PV^2 9070294785 8227024748 7462153966 6768393620 6139132535 Iljxuh 4 ..c H.2c e.. Ebf2f2 ' ..c fDSc .2 dqg wkh vwdqgdug ghyldwlrq lv s ..c fDSc .2 ' Hc ..H Qrwh wkdw vrph fduh lv uhtxluhg lq fdofxodwlqj wkh yduldqfh vlqfh wkh udqgrp yduldeoh wdnhv wkh ydoxhv }hur diwhu wkh h{slu| ri wkh whup dqg lw zrxog eh qhfhvvdu| wr doorz iru wkhvh }hur ydoxhv li wkh vwdqgdug irupxod iru yduldqfh zdv xvhg1 H{ ;17 Wkh h{shfwhg suhvhqw ydoxh ri suhplxpv ri '4 s1d1 sd|deoh lq dgydqfh lv ghwhuplqhg xvlqj wkh irupxod ?3 [ &'f & E& R% 4;9 Olih Lqvxudqfh Wkh iroorzlqj wdeoh jlyhv wkh ydoxhv iru wkh irupxod Age qx 40 41 42 43 44 k 0.00174 0.00190 0.00209 0.00230 0.00254 0 1 2 3 4 PV 1.000000 0.952381 0.907029 0.863838 0.822702 kp20 1.000000 0.998260 0.996363 0.994281 0.991994 Wkh h{shfwhg suhvhqw ydoxh lv 7185<7:1 H{ ;18 Wkh surjudp qhhgv wr lqsxw ydoxhv ri 7/ / % dqg ? iurp d oh ru wkh nh|erdug D ohqdph iru wkh pruwodlw| udwhv vkrxog eh lqsxw dqg wkh ^% ydoxhv uhdg iurp wklv oh ru wdeoh1 Wkh uhfxuvlyh dssurdfk vwduwv zlwk wkh h{shfwhg ydoxh htxdo wr }hur iru wkh roghvw djh dqg wkhq vwhsv edfnzdugv rqh djh dw d wlph wr wkh suhvhqw gdwh1 D gr orrs fdofxodwhv wkh h{shfwhg suhvhqw ydoxh ri fodlpv sd|phqwv e| glvfrxqwlqj e| rqh shulrg dqg pxowlso|lqj e| wkh suredelolw| ri vxuylydo1 H{ ;19 Wkh sulqflsoh ri htxlydohqfh htxdwhv wkh h{shfwhg suhvhqw ydoxh ri suhplxpv wr wkh h{shfwhg suhvhqw ydoxh ri fodlpv dqg h{shqvhv1 Zh kdyh eD2be. ' bf2f n Dff n ff eD2be. Vr wkdw bf2f n Dff n ff eD2be. ' efb.b ' H{ ;1: Wkh suhplxp lqfrph lv ffff 22f ' 2c 2ffc fff Wkh uhtxluhg suredelolw| lv wkh suredelolw| wkdw wrwdo fodlpv A h{fhhgv wkh suhplxp lqfrph soxv fdslwdo1 Zh uhtxluh h dA : 2c 2ffc fff n o fff Li zh glylgh e| wkh vxp lqvxuhg ri hdfk srolf| zh uhtxluh wkh suredelolw| wkdw 2c 2ffc fff n h : fff ffc fff Vroxwlrqv wr H{huflvhv 4;: zkhuh lv wkh qxpehu ri fodlpv zklfk kdv d Elqrpldo+43333/ 31334:7, glvwulexwlrq1 Zh qhhg wr ghwhuplqh wkh qxpehu ri fodlpv n wkdw zloo pdnh wkh suredelolw| 40ELQRPGLVW+n/43333/31334:7/WUXH, ohvv wkdq 313341 Xvlqj H{fho zh txlfno| qg wkdw 40ELQRPGLVW+64/43333/31334:7/WUXH,@313343: dqg 40ELQRPGLVW+65/43333/31334:7/WUXH,@313338791 Wkxv n@65 dqg wkh dprxqw ri fdslwdo uhtxluhg lv jlyhq e| 2c 2ffc fff n ' 2 ffc fff ru F@4/333/3331 H{ ;1; Xvlqj wkh uhfxuuhqfh irupxod zh jhw wkh iroorzlqj ydoxhv iru wkh h{shfwhg ydoxh ri wkh srolf| oldelolw|1 Age 40 41 42 43 44 qx 0.00174 0.00190 0.00209 0.00230 0.00254 Expected value of policy liability 0.00 -374.37 -258.30 -155.25 -67.89 0.00 Qrwh wkdw zh kdyh fff2De fffff n E fff2De f n ff efb.b fD ' S.Hb fff2f fffff n E fff2f S.Hb n ff efb.b fD ' DD2D fff2fb fffff n E fff2fb DD2D n ff efb.b fD ' 2DHf fffbf fffff n E fffbf 2DHf n ff efb.b fD ' .e. 4;; Olih Lqvxudqfh fff.e fffff n E fff.e .e. n Dff n ff efb.b fD ' f Qrz qrwh wkdw wkh ydoxhv duh qhjdwlyh1 Li zh uhpryh wkh lqlwldo h{shqvhv wkhq zh jhw d suhplxp ri '5<<1731 Wkh h{shfwhg ydoxh ri wkh srolf| oldelolw| ehfrphv1 Age 40 41 42 43 44 qx 0.00174 0.00190 0.00209 0.00230 0.00254 Expected value of policy liability 0.00 35.44 56.69 60.03 42.50 0.00 Qrz wkh ydoxhv duh srvlwlyh1 Wklv lv zkdw zh zrxog qrupdoo| h{shfw vlqfh wkh ulvn lq lqfuhdvlqj iru d ohyho suhplxp1 Hduo| lq wkh srolf| whup wkh suhplxp lv kljkhu wkdq wkh ghdwk ulvn1 Wkh srolf| oldelolw| phdvxuhv wkh h{fhvv ri wkh ghdwk ulvn frvw ryhu wkh ixwxuh suhplxpv dqg odwhu lq wkh srolf| wkh suhplxp lv orzhu wkdq wkh ghdwk ulvn frvw1 Wkxv wkh srolf| oldelolw| lv srvlwlyh hduo| lq wkh whup1 Wkh hhfw 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'i T >+ n |j 7% 2 +'% ru/ vlqfh dw djh % wkh ydoxh ri 7% lv nqrzq +dqg lv qrw udqgrp,/ zh kdyh & %+'%n|3 [ 2 >+ n |j . d7%n| o ' 7% i T 2 +'% H{dpsoh 43143 Dvvxph wkdw wkh frqwlqxrxv frpsrxqglqj vdodu| jurzwk udwh iru dq| |hdu kdv d qrupdo glvwulexwlrq zlwk h{shfwhg ydoxh 3136 dqg vwdqgdug ghyldwlrq 3134 dqg hdfk |hduv*v jurzwk udwh lv lqghshqghqw1 Ghwhuplqh wkh h{shfwhg vdodu| dw djh 98 iru d 53 |hdu rog lqglylgxdo zlwk d fxuuhqw vdodu| ri '68/3331 Vroxwlrq 43143 Wkh h{shfwhg vdodu| dw djh 98 zloo eh 2 . d7SD o ' Dc fff i T eD ff n eD Eff 2 D22D ' Dc fffe ' Dc fff HSS ' Dc e H{huflvh 4316 Ghwhuplqh wkh h{shfwhg ixwxuh vdodu| dw djh 98 iru d 53 |hdu rog rq d fxuuhqw vdodu| ri '68/333 dvvxplqj wkh frqwlqxrxv frpsrxqglqj vdodu| jurzwk udwh iru dq| |hdu lv lqghshqghqw dqg kdv d qrupdo glvwulexwlrq zlwk h{shfwhg ydoxh 3137 dqg vwdqgdug ghyldwlrq 31341 d orj0qrupdo glvwulexwlrq iru wkh vdodu| jurzwk udwh/ wkh yduldqfh k Dvvxplqj l 7%n| ri *? 7% lqfuhdvhv olqhduo| zlwk |1 Iru odujh ydoxhv ri | wklv fdq phdq wkdw wkh yduldqfh ehfrphv wrr odujh1 Lw lv srvvleoh wr kdyh wkh yduldelolw| lqfuhdvh dw d vorzhu udwh dqg hyhqwxdoo| ehfrph frqvwdqw e| doorzlqj wkh vdodu| jurzwk udwh wr ghshqg rq suhylrxv vdodu| jurzwk udwhv1 Iru h{dpsoh/ d orzhu h{shfwhg jurzwk udwh pljkw eh dvvxphg iru |hdu | n zkhqhyhu wkh dfwxdo vdodu| jurzwk udwh lv deryh wkdw h{shfwhg lq |hdu | dqg ylfh yhuvd1 Glvwulexwlrqv zlwk vxfk ghshqghqfh iurp rqh |hdu wr wkh qh{w duh vwxglhg lq odwhu dfwxduldo vxemhfwv1 431715 Wkh Vhuylfh Wdeoh 0 Vxuylydo dqg Rwkhu Suredelolwlhv Wkh ydoxh ri ixwxuh ehqhwv ri vxshudqqxdwlrq dqg glvdelolw| ixqgv ghshqg rq wkh fkdqfh wkdw dq lqglylgxdo zloo uhdfk uhwluhphqw/ glh/ ehfrph vlfn ru zlwkgudz iurp wkh ixqg1 Wr ghwhuplqh wkh h{shfwhg suhvhqw ydoxh ri ehqhwv zh uhtxluh wkh suredelolwlhv wkdw wkhvh hyhqwv rffxu1 Dfwxduldo Prghoolqj 564 Iru vxshudqqxdwlrq sodqv/ dfwxdulhv xvh dq h{whqvlrq ri wkh olih wdeoh wr lqfoxgh wkh suredelolwlhv ri ghdwk dqg glvdelolw| dv zhoo dv wkh suredelolwlhv wkdw d phpehu zloo zlwkgudz iurp wkh ixqg dw hdfk djh1 Wkh wdeoh wkdw vkrzv wkhvh suredelolwlhv e| djh lv fdoohg d vhuylfh wdeoh1 Wkh iroorzlqj wdeoh lv dq h{dpsoh ri d wdeoh ri wkhvh suredelolwlhv1 Lw vkrzv wkh suredelolw| wkdw d phpehu ri wkh ixqg zloo ohdyh wkh ixqg dw dq| djh iurp hlwkhu zlwkgudzdo +uhvljqdwlrq, ru iurp ghdwk dqg glvdelolw|1 Wkh vxp ri wkhvh wzr sured0 elolwlhv lv wkh suredelolw| wkdw wkh| ohdyh wkh ixqg iru hdfk djh1 Wkh suredelolw| wkdw d phpehu zloo uhfhlyh d uhwluhphqw ehqhw lv wkh suredelolw| wkdw wkh| gr qrw ohdyh wkh ixqg e| uhvljqdwlrq ru ghdwk dqg glvdelolw| ehiruh uhwluhphqw djh +98,1 Age Resignation 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0.16650 0.16020 0.15390 0.14760 0.14130 0.13500 0.12870 0.12240 0.11610 0.10980 0.10350 0.09720 0.09090 0.08460 0.07830 0.07200 0.06795 0.06390 0.05985 0.05580 0.05175 0.04770 0.04365 0.03960 0.03555 0.03150 0.02745 0.02340 0.01935 0.01530 0.01125 0.00900 0.00675 0.00450 0.00225 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Death and Disability 0.00090 0.00089 0.00083 0.00075 0.00067 0.00063 0.00063 0.00064 0.00066 0.00068 0.00072 0.00077 0.00082 0.00087 0.00093 0.00101 0.00110 0.00121 0.00132 0.00146 0.00163 0.00183 0.00205 0.00231 0.00262 0.00297 0.00338 0.00385 0.00439 0.00501 0.00572 0.00655 0.00751 0.00860 0.00983 0.01123 0.01282 0.01462 0.01665 0.01895 0.02154 0.02443 0.02784 0.03185 0.03659 0.04215 565 Uhwluhphqw/ Vrfldo Vhfxulw| dqg Khdowk Fduh Ilqdqflqj Wkh vhuylfh wdeoh vkrzv wkh suredelolwlhv wkdw d olih djhg % zloo uhvljq iurp d vxshudqqxdwlrq ixqg ru glh ru ehfrph glvdeohg iru djhv 53 wr 98 dqg zloo vkrz wkh suredelolw| wkdw d phpehu zloo uhpdlq lq wkh ixqg iru hdfk djh1 Wkhvh suredelolwlhv fdq eh xvhg wr ghwhuplqh wkh rwkhu ixqfwlrqv vkrzq lq d olih wdeoh1 H{dpsoh 43144 Xvlqj wkh vhuylfh wdeoh jlyhq/ ghwhuplqh wkh suredelolw| wkdw d olih djhg 58 zloo uhvljq lq wkh iroorzlqj |hdu dqg wkh suredelolw| wkdw d olih djhg 88 zloo glh ru ehfrph glvdeohg lq wkh iroorzlqj |hdu1 Vroxwlrq 43144 Wkh suredelolw| wkdw d olih djhg 58 zloo uhvljq lv 31468 dqg wkh sure0 delolw| wkdw d olih djhg 88 zloo glh ru ehfrph glvdeohg zloo eh 31344561 H{dpsoh 43145 Xvlqj wkh vhuylfh wdeoh jlyhq/ ghwhuplqh wkh suredelolw| wkdw d php0 ehu djhg 73 zloo vwloo eh d phpehu dw djh 781 Vroxwlrq 43145 Wkh suredelolw| wkdw d phpehu djhg 73 zloo vwloo eh d phpehu dw djh 74 zloo eh ffD.D fffS ' fbeSS2 Wkh suredelolw| wkdw d phpehu djhg 74 zloo vwloo eh d phpehu dw djh 75 zloo eh ffe..f fffH ' fbDfe. Vlploduo|/ iru djh 75 wkh suredelolw| wkdw wkh phpehu zloo vwloo eh d phpehu dw djh 76 lv fbDef dqg iru djh 76 wkh suredelolw| wkdw wkh| zloo eh d phpehu dw wkh hqg ri dqrwkhu |hdu zloo eh fbDHfb1 Dw djh 77 wkh suredelolw| zloo eh fbSH Wkh uhtxluhg suredelolw| lv fbeSS2 fbDfe. fbDef fbDHfb fbSH ' f.b2 H{dpsoh 43146 Ghwhuplqh wkh suredelolw| wkdw d 93 |hdu rog zloo uhdfk djh 98 dqg uhfhlyh d uhwluhphqw ehqhw xvlqj wkh vhuylfh wdeoh jlyhq1 Vroxwlrq 43146 Wkh uhtxluhg suredelolw| lv E ff2De E ff2ee E ff2.He E ffHD E ffSDb ' fHSDDD H{huflvh 4317 Dvvxplqj wkdw 43333 phpehuv mrlq d ixqg dw djh 53/ xvh wkh wdeoh ri suredelolwlhv jlyhq wr frqvwuxfw d vhuylfh wdeoh vkrzlqj wkh h{shfwhg qxpehu ri phpehuv ri wkh ixqg dw hdfk djh iurp 54 wkurxjk 98 dqg wkh h{shfwhg qxpehu ri |hduv ri phpehuvkls ri wkh ixqg iru d qhz phpehu hqwhulqj wkh ixqg dw hdfk ri wkhvh djhv1 +\rx zloo qhhg wr xvh d vsuhdgvkhhw,1 H{huflvh 4318 Xvh wkh vhuylfh wdeoh suredelolwlhv jlyhq wr ghwhuplqh wkh suredelolw| wkdw d qhz phpehu ri wkh ixqg hqwhulqj dw djh 68 zloo ohdyh wkh ixqg ehfdxvh ri ghdwk dw djh 731 Dfwxduldo Prghoolqj 566 431716 Ghqhg ehqhwv H{shfwhg suhvhqw ydoxh ri ehqhwv Wkh h{shfwhg suhvhqw ydoxh ri wkh ehqhwv iru d ghqhg ehqhw vxshudqqxdwlrq ixqg duh ghwhuplqhg xvlqj wkh ehqhw uxohv iru wkh ghqhg ehqhw sd|deoh rq ghdwk dqg glvdelolw|/ zlwkgudzdo dqg uhwluhphqw1 Wkh vdodu| dw hdfk djh ri ohdylqj wkh ixqg lv iruhfdvw xvlqj d suredelolw| glvwulexwlrq/ vxfk dv wkh orj0qrupdo glvwulexwlrq1 Wklv lv xvhg wr ghwhuplqh wkh glvwulexwlrq ri wkh ehqhwv1 Wkh suhvhqw ydoxh ri wkhvh ehqhwv lv wkh surmhfwhg ydoxh glvfrxqwhg iru lqwhuhvw wr wkh fxuuhqw gdwh1 Wkh h{shfwhg ydoxh ri wkh ixwxuh ehqhwv fdq wkhq eh fdofxodwhg xvlqj wkh vhuylfh wdeoh suredelolwlhv wr ghwhuplqh wkh suredelolw| wkdw wkh ehqhwv zloo eh sdlg dw hdfk djh1 Dv d vlpsoh h{dpsoh/ frqvlghu d phpehu djhg % ri d ixqg rq d fxuuhqw vdodu| ri 7% wkdw sd|v d oxps vxp uhwluhphqw ehqhw ri & ? 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