ACTUARIAL ANALYSIS OF THE MULTIPLE LIFE ENDOWMENT INSURANCE CONTRACT
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1 ACTARIA ANAYSIS OF THE MTIPE IFE ENDOWMENT INSRANCE CONTRACT Werer Hürl FRSGlobl Swzerld Bederre CH-827 Zürh E-l Abr We eed he ll l of he edowe or o gle lfe o ulle le. The wo le e oerg he o-lfe d he l-urorh u dued horoughl. I re url lue of he rff book re luled uder he lfg uo of deede fuure lfee. I herefore or o eure he of h uo uder he obero h deedee o fulflled rel lfe. I he wo le e he l be eured ug he well-kow Höffdg-Fréhe lower d uer boud. The deedee uo oeree he o-lfe e gle d leel reu d uderee he l-uror e gle d leel reu. The l deo re obed b erfe oe deedee. Soe forul llure he lo o ulle lfe ure or for ore h wo le whh o ou o furher oble deeloe. Ke word edowe lfe ure ulle le o-lfe l-urorh lfee deedee
2 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 2. Iroduo The uroe of h ud o eed he ll l of he edowe or o gle lfe o ulle le. The wo le e oerg he o-lfe d he lurorh u dued horoughl. Soe forul llure he lo o ulle lfe ure or for ore h wo le whh o ou o furher oble deeloe. 2. The Noo of Geerl fe Su The eeo of lfe ure for gle lfe o ulle le bed o he oo of geerl lfe u for whh here re defo of url d flure. Coder grou of g le ged 2... g d le T k T k deoe he rdo fuure lfee of he gle lfe ged k k... g. Bed o hee elee u u wh rdo fuure lfee T u T wll be defed uh h u P T u > d P T u he robbl h he u wll ure o e > q u he robbl h he u wll fl o e >. We wll deelo odel for lfe ure ble uo he flure of he u d lfe ue ble log he u ure. Deoe b D u he e gle reu NSP of lfe ure wh oe u of beef e ble he ed. For erl of he -hl erod of er followg flure of he u where [ ] erod oe e b oeo D u D u oe wre D u D u. Slrl deoe b u d for ouou e ode he NSP of lfe u of oe u er er ble le of frol u he begg of eh of er log he u ure. For erl erod e le of legh [ ] oe e b oeo u u e oe defe u u. d for he lg e of ouou The o or e of geerl lfe u whh wll uffe o ef oleel he edowe ure o wo le re he followg four oe Sgle lfe u A gle lfe ged defe u u h ure whle le. Jo-lfe u A u h e log ll eber of he grou re le d fl uo he fr deh lled o-lfe u d deoed b u.... I fuure lfee derbed b he rdo rble T u T.... T g 2 -urorh u A u h e log le oe eber of he grou le d fl uo he l deh lled l-urorh u d deoed b u.... I h he fuure lfee T u T.... T g g 2 g
3 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 3 Ter er u The er er u whh deoed b u defe lfe u urg for el er d he flg. I h he deer fuure lfee T. Th rulr u ueful whe derbg eorr lfe ure d lfe ue. For ele he geerl lfe u u defe he NSP of gle lfe edowe ure of he e A. I defe lo he NSP of eorr lfe ue of he e [ ] [ ] ure wh beef e le [ ] d reu e le [ ]. I h frework he e leel reu NP of gle lfe edowe d oe u of u ured deered b he quoe NP A [ ] [ ] Probble of Surl d Flure To derbe he edowe ure o wo le oe requre robble for he o-lfe u u d he l-urorh u u. e X Y be he ge--deh of he le d T X T Y he orreodg fuure lfee. Ge he o url fuo S P X > Y > of he oule X Y he T T T d url robble of he fuure lfee T T T re obed follow S S P T > P X > Y > X > Y > S P T > P X > Y > X > Y > S S S Though geerl he rdo rble how o-rl deedee ruure oo re o ue for rg uroe h he le d re deede. I h lfed uo he robble of url deed o he lfe ble of he gle le ol d re ge b 3.3 Rerk 3.. I ree er You e l.22 he de horough l of he ore geerl uo of rl deedee uggeed b Bower e l.986 whh lfe uh ulle lfe lulo. I rulr e of rred oule he how h h de hold uder he uo h he orl re of he wfe or he hubd hould o deed o wheher he he urg oue or
![I h frework he e leel reu NP of gle lfe edowe d oe u of u ured deered b he quoe NP A [ ] [ ]. 2. 3.](/docs-images/45/3291602/images/page_3.jpg "Probble of Surl d Flure To derbe he edowe ure o wo le oe requre robble for he o-lfe u u d he l-urorh u u. e X Y be he ge--deh of he le d T X T Y he orreodg fuure lfee.")
4 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 4 o or o he urg oue ge. Th geerll ued re. Iure oe do o lf ordg o wheher oe h urg oue or o or o oue ge. I be how h he rl deedee uo lo hold for er el url fuo Coo hok url odel S S S2 R S S R where 2 re url drbuo ee Deu e l.26 for lo Fréhe oul odel C u θ u θ u θ [ ] 4. Preu e [ ] [ ] be he beef d reu e le of he edowe ure o wo le. Ne gle reu of lfe ure A NSP of -er edowe ure for gle lfe D NSP of -er er ure ble flure of gle lfe E NSP of -er ure edowe ble url of gle lfe A NSP of -er edowe ure for o-lfe D NSP of -er er ure ble flure of o-lfe E NSP of -er ure edowe ble url of o-lfe A D NSP of -er edowe ure for l-uror NSP of -er er ure ble flure of l-uror E NSP of -er ure edowe ble url of l-uror The edowe NSP he u of he er ure d ure edowe NSP A A A D E D E D E 4.3 Deoe b / he dou for o he ehl ere re. The NSP re deered b he forul D E µ d >
![Preu e [ ] [ ] be he beef d reu e le of he edowe ure o wo le.](/docs-images/45/3291602/images/page_4.jpg "Ne gle reu of lfe ure A NSP of -er edowe ure for gle lfe D NSP of -er er ure ble flure of gle lfe E NSP of -er ure edowe ble url of gle lfe A NSP of -er edowe ure for o-lfe D NSP of -er er ure ble")
5 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 5 D > d µ µ 4.6 E 4.7 D D D D 4.8 E E E E 4.9 Ne gle reu of lfe ue NSP of -er lfe u for gle lfe NSP of -er lfe u for o-lfe NSP of -er lfe u for l-uror The NSP re deered b he followg forul > d 4. > d Ne leel reu of edowe ure NP erl NP of -er edowe ure for gle lfe NP erl NP of -er edowe ure for o-lfe NP erl NP of -er edowe ure for l-uror The e leel reu re NPR for u of u ured d he NP for u ured SI d geerl lfe u u re deered b he followg forul
6 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 6 NPR NP A u u u u NPR u SI eel reu of edowe ure Coder he followg leel reu P of edowe ure wh u ured SI for wo-lfe u P P P P of -er edowe ure for gle lfe P of -er edowe ure for o-lfe P of -er edowe ure for l-uror eel reu lude he oer for ll kd of eee ure o he. Slrl o Gerber986 Ch. four e of eee re odered Aquo o Eee reled o he le of ew lfe ure or re gle o whh re d he begg of he or he re of α er u of u ured. Preu roorol oerg o Thee re rble reurrg eee he re of β er u of leel reu. Co oerg o Thee re reurrg oerg eee of fed o lue β f. Oerg o roorol o he ure beef Thee re rble reurrg oerg eee he re of γ er u of u ured. The leel reu uffe o fe eeed lue he ure beef d o. The leel reu re PR er u of u ured of -er edowe ure wh geerl lfe u u fe he equo PR β α 4.5 SI f u u A u β PR u γ u I follow h he leel reu for geerl lfe u u ge b P u NP u α γ SI β f u β 4.6
7 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 7 5. Mhel Reere Aurl Reere d Preu Reere For lfe u u wh rdo fuure lfee T Tu oe oder he followg rdo rble oed o u [ T u ] K he uber of oleed fuure er led b he u u lo lled ure-fuure-lfee S u T u K u he frol oro of er he u le he er of flure S T u [ S u ] u K u S u he frol oro S u rouded u o he e -h of er ] he oe of beef e e he u fl For grou of g le ged 2 g wh u... u d e > le u deoe he u obed fro u wh eh lfe ged k k... g. For ele f u he o-lfe u o wo le he u. The rdo roee lo or e > of -er edowe ure wh lfe u u d u ured SI he rdo rble defed b T u u SI NP u & T u where & d d & deoe -er u er of oe u er er ble le of frol. I he lg e of u he begg of eh e le of legh ] ouou e oe defe d e & where δ l δ he fore of ere. Oboul oe h l &. To defe url reere roerl eer o oder he oble e u ke oer fuure e. I he e of gle lfe ged or e oe obere h wh ree o he orl X 2 e > whh re defed b rk he lfe be wo dffere e { } X X T T > le e > 2 ded e > Geerlzg o oule of le ged d oe obere h wh ree o X 234 e > he orl rk he oule be four dffere e { } X X X X T > T T > T T T T T > boh d re le e > 2 le d ded e > > 3 ded d le e > 4 boh d re ded e >
![oro of er he u le he er of flure S T u [ S u ] u K u S u he frol oro S u rouded u o he e -h of er ] he oe of beef e e he u fl For grou of g le ged 2 g wh u.](/docs-images/45/3291602/images/page_7.jpg ".. u d e > le u deoe he u obed fro u wh eh lfe ged k k... g. For ele f u he o-lfe u o wo le he u.")
8 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 8 2 dffere e { } g Oboul grou of g le be X g fuure e >. I geerl he hel reere e > of -er u e X e > defed o be edowe ure wh lfe u he odol eeo of he roee lo ge u e > whh deoed d luled follow For gle lfe [ X ] { 2... u E u X e V he hel reere e X 2 he d he hel reere e X ge b drog uul he de V A SI NP 5.4 Slrl for o-lfe he hel reere he e X 23 4 h d he hel reere e X ge b V SI NP A I or o h for l-uror u 5.5 ol he hel reere he e X 4 he d he hel reere he e X 2 3 re ge b ee e.g. Bower e l for he el e V A V A 2 V A 3 E[ T > T > ] SI NP E[ T > T ] SI NP E[ T T > ] SI NP Bede he hel reere whh deed o he e of u oe oder he e reu reere e > whh defed o be he odol eeo of he roee lo ge url o e > e.g. Bower e l.986 Ch V g 2 [ > ] V u P X T u > u E u T u 5.9 For gle lfe reeel o-lfe he e reu reere 5.9 ode wh he hel reere 5.4 reeel 5.5. For l-uror u u he e reu reere robbl weghed u of he hel reere 5.6 o 5.8 deered follow e.g. Bower e l.986 Ch
9 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 9 V V V2 V3 5. Correodg o he e deede hel reere we oder e deede deferred quo o whh for lfe u u e X e > re deoed d defed b { 2... u SI V u VE α Bede he e deede e reu reere we oder he e deede eee u e > deoed d defed b reere whh for lfe u VE u SI V u α. 5.2 The e deede url reere re defed o be he u of he hel reere d he deferred quo o d re deoed b u V u VE u { 2... VA The e deede reu reere defed o be he u of he e reu reere d he eee reere d deoed b u V u VE u. VA 5.4 Rerk 5.2. The oe of e deede reere for he l-uror u h bee rodued b Frer978 ee lo The Aur978 d Mrgu22. The hoe bewee e deede d e deede reere deed uo lo reogo he ble hee reogo or o of u hge. Wh e deede reere he ure o der he or f hd o kowledge of deh log le oe ured ure. 6. Preu Cooe I he followg ol he o rel e > dued. Correodg o he dul uo of e deede d e deede reere wo e of reu deooo re odered. 6.. Se deede Preu Cooe Que re ereed url fuo of he u u d he e dree e k k... wheeer rele. X he
10 MTIPE IFE ENDOWMENT INSRANCE CONTRACT Rk Preu { 2... u q SI V u RP u 2 6. Sg Preu u V u V u { 2... SP The u of he rk reu d g reu oe wh he e leel reu or ue Eee Preu u SP u NP u { 2... RP EP Rk Cooe Eee Preu u P u NP u 6.4 u RP u { 2... REP α Sg Cooe Eee Preu Oerg Co Chrge u EP u REP u { 2... SEP VE u { 2... u VE u SEP u OC Aquo Co Chrge u EP u OC u { 2... AC Se deede Preu Cooe Que re ereed url fuo of he u k k... wheeer rele. u he dree e Rk Preu Sg Preu RP SP u q SI V u 6.9 u u V u V u 6.
11 MTIPE IFE ENDOWMENT INSRANCE CONTRACT The u of he rk reu d g reu oe wh he e leel reu Eee Preu RP EP u SP u NP u 6. Rk Cooe Eee Preu REP Sg Cooe Eee Preu Oerg Co Chrge OC SEP Aquo Co Chrge u P u NP u 6.2 u RP u α 6.3 u EP u RP u 6.4 VE u u VE u SEP u 6.5 AC u EP u OC u Reduo of Clulo The reedg forul how h ll url lue reled o he ulle lfe edowe ure deed olel o he url fuo A u d u. I f uder he oulr ufor drbuo of deh uo he forul be furher redued o he lulo of D u d E u ol. Followg Gerber oe how h D u > u D u D δ 7. where equl o he -hl ol ehl ere re oerble e er e > whh defed b 7.2 d he ehl fore of ere drd oo δ whh defed he lg lue of d equl o
12 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 2 d δ l l l. 7.3 d Slrl d equl o he -hl ol ehl ere re de defed e > b d. 7.4 Se oe reoer he lg e he ehl fore of d ere d l d l δ. 7.5 Fro 7. follow h A u A u D u. 7.6 Furherore followg Gerber d ug 7.6 oe how he reloh u A u D u A u. 7.7 d d Clerl he relo 7.6 d 7.7 olee he dered reduo of lulo. 8. I of Ideedee Auo o Preu Clulo Se url lue of he rff book he bee luled uder he lfg uo of deede fuure lfee T d T or o eure he of h uo uder he obero h deedee o fulflled rel lfe. I he wo le e he l be eured ug he Fréhe lower d uer boud rodued Höffdg94 d Fréhe95 d fr led o lfe ure b Crrère d Ch986. Coder he Fréhe l of ll bre drbuo wh fed P T P T. The Fréhe uer boud he rg d q q drbuo F P T T q q he drbuo P T T q q d he Fréhe lower boud drbuo F P T T d below b F. A o wh fed rg ored fro boe F F F. 8. The Fréhe boud geere four dffere fuure lfee for he o-lfe d l-uror u. Ther url drbuo re deoed d deered b
13 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 3 > 8.2 P T > 8.3 P T > P T 8.4 >. 8.5 P T For oro uroe he url drbuo of he fuure lfee for he o-lfe d l-uror u uder he deedee uo re deoed d deered b > P T 8.6 > P T The defed url drbuo f he equle whh l h he orreodg rdo fuure lfee re ordered he oh doe ee uh T T T T T T. 8.9 The reduo of lulo obed Seo 3.6 how h uffe o lze he l of he deedee uo o reu lulo for he oe-er e. For geerl lfe u u he NSP d he NP of he ul-lfe -er edowe ure h uo re deered b he url fuo NSP u d u NP u d 8. u where he ul-lfe -er lfe u luled fro he forul u k k. 8. Ierg he dffere lfe drbuo o 8. d ug he oh equle oe ob he followg boudg equle bewee he dffere o-lfe d l-uror -er lfe ue k u
14 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 4 whh ordg o 8. l he followg equle bewee he NSP d he NP NSP NP NSP NP NSP NSP NP NP NSP NSP NP NP To eure he of he deedee uo url o ue h he fuure lfee T d T re oel qudr deede d follow le Fréhe drbuo θ q q θ q q [ ] θ F θ 8.8 θ whh fe he equl F F F. g 8. oe ob he followg deo bewee deedee uo d Fréhe uo 8.8 for he NSP d he NP NSP NP θ u NSP u d θ u u θ u NP u 8.9 u u θ u u. 8.2 g 8.2 d 8.3 ler h 8.9 d 8.2 re o-ege for he o-lfe u u d o-oe for he l-urorh u u. Therefore he deedee uo oeree he o-lfe NSP d NP d uderee he l-uror NSP d NP. The l deo re obed for erfe oe deedee θ. Noe h he deedee reer θ be erreed Ser grde orrelo oeffe e.g. Hürl24 Seo 3.. The Tble 3. below llure he l deo uerll. We ue h he rgl fuure lfee follow he drbuo of Goerz825 ee lo Wllee d Koelr2 whh defed b b b e e e. 8.2 The reer of he Goerz drbuo re e equl o 85 b whh re loe o hoe led he ud b Mlek d Poer2 Tble 4.
15 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 5 Tble 3. Ml deo for NSP d NP for he wo-le edowe ure ere le fele er l deo er ll SP SP P P 2% % % % % % % %
16 MTIPE IFE ENDOWMENT INSRANCE CONTRACT 6 Referee Bower N.. Gerber H.. Hk J.C. Joe D.A. d C.J. Neb 986. Aurl Mhe. Soe of Aure I. Crrère J.F. d.k. Ch 986. The boud of bre drbuo h l he lue of l-uror ue. Tro of he Soe of Aure Deu M. Frog E. d B. eko 26. Shf ere re d oo hok odel for ouled le. Belg Aurl Bulle 6-4. Frer W. M Seod o De Jo fe Ch Vlue d Reere. The Aur Arl 4. Fréhe M. 95. Sur le bleu de orrélo do le rge o doée. Ale de l eré de o Se. A Gerber H ebeerherughek. Srger-Verlg. Goerz B O he ure of he fuo eree of he w of Hu Morl d o ew ode of deerg he Vlue of fe Cogee. Phl. Tr. Rol So W. Nol odo. Höffdg W. 94. Mbre Korreloheore. Shrfe de Mh. Iu ud de Iu für Agewde Mhek der er Berl Hürl W. 24. Mulre Fréhe oul d odol lue--rk. Ierol Jourl of Mhe d Mhel See Mrgu P. 22. Geerlzed Frer l re uder urorh lfe ure ole. Norh Aer Aurl Jourl Mlek M. d S.E. Poer 2. The oo luo of he gureed u deh beef rble ue d uul fud. The Jourl of Rk d Iure The Aur 978. Seod o De. eer. The Aur Jue 3. Wllee W.J. d H. Koelr 2. Kowledge elo of Goerz lw of orl. Sd Aurl Jourl You H. Shek A. d E. Her 22. A re-eo of he o orl fuo. Norh Aer Aurl Jourl
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W h a t is m e tro e th e rn e t
110 tv c h a n n e ls to 10 0 0 0 0 u s e rs U lf V in n e ra s C is c o S y s te m s 2 0 0 2, C is c o S y s te m s, In c. A ll rig h ts re s e rv e d. 1 W h a t is m e tro e th e rn e t O b je c tiv
U S B Pay m e n t P r o c e s s i n g TM
U S B Pay m e n t P r o c e s s i n g T h a t s S m a r t P r o c e s s i n g TM USB was simple to enroll in. They had competitive rates and all the fees were listed clearly with no surprises. Everyone