Faculy of Acuaal Scnc and nsuanc Acuaal Rsach Pa No. 93 Sulus Analyss fo aabl Annus wh a GMDB Oon Svn Haban Gablla Pscoo Januay SBN 978--9575-5-6 Cass Busnss School 6 Bunhll Row London CY 8TZ Tl +44 74 847 www.cass.cy.ac.u
Any onons ssd n hs a a y/ou own and no ncssaly hos of y/ou loy o anyon ls /w hav dscussd h wh. You us no coy hs a o quo whou y/ou sson.
Sulus Analyss fo aabl Annus wh a GMDB oon Haban S. Pscoo G. Absac n hs a w analyz h nsuanc sulus fo a aabl Annuy conac wh a Guaand Mnu Dah Bnf GMDB oon. nally w dv h fs wo ons of h dsbuon of h sulus; and subsqunly w dvlo h whol dsbuon usng a sochasc odl whch nvolvs an ngad analyss of fnancal and oaly s fo a ofolo of annus wh GMDB bddd oons. W off a odl accodng whch h u can b odfd as h focass of oaly obabls ns a and fund voluon. Moov h sudy nabls us o dn h u ha lads o a qud obably of nsolvncy and so can b usd fo an valuaon of h adquacy of solvncy. Nucal als llusa h suls. JL classfcaon: G Kywods: Guaand Mnu Dah Bnf oon fnancal s oaly s sulus aabl Annuy.. noducon Th aabl Annuy a has ncasd consdably n h as dcad whn bullsh fnancal as and low ns as hav d nvsos o loo fo hgh uns. aabl Annus a vy aacv o consus bcaus hy ovd acaon n h soc a. Thy a un-lnd annuy conacs usually wh a sngl u ayn u-fon whch s nvsd n on of sval funds and hy noally a dsgnd wh so bddd guaans. As As a ssnally a nw oduc class n h U.K. an ndusy sandad dfnon dos no y ss hy a sla o un lnd n savng vhcl such as un lnd annus howv h avalably of guaans dsngush h Ldl al. 8. On of hs faus s h Guaand Mnu Dah Bnf an ncasng-s u oon wh a sochasc auy da. n h basc fo of oduc whn h nsud ds h bnfcay obans a dah bnf whch s qual o h au of h nvsd u and h accoun valu lnd o h Pofsso of Acuaal Scnc and Duy Dan of Cass Busnss School Cy Unvsy 6 Bunhll Row London CY 8TZ UK. nn addss: s.haban@cy.ac.u Dano d Maaca Sasca Facolà d conoa Unvsà dgl Sud d Naol Fdco va Chna Colsso Mon S.Anglo 86 Naol- aly. nn addss: Gablla.Pscoo.@cy.ac.u
fund. Ths guaan s ad fo by h A hold n h fo of a ual f ha s dducd fo h accoun valu lnd o h undlyng ass. Th uos of hs a s o sudy h nsuanc sulus ov fo a ofolo of aabl Annus wh GMDB oons. Th a hocal foundaons fo hs wo: on h on hand w a no accoun h acuaal lau concnng h valuaon of h aabl Annuy and GMDB oon Bau Klng and Russ 6; Colan Yuyng and Paon 6; Mlvsy M. and Posn Mlvsy M. and Salsbuy Mlvsy M.A. and Poslow ; on h oh hand w loo a h acuaal sach lau on nsuanc sulus and nsolvncy obably Coola al. 3 Dahl 4 Hodas al. 5 Lysno and Pa 7 Macau and Galladz 999 Pa 996 and Pa 994. Th abovnond as dal wh h sochascally dscound valu of fuu cash flows n sc of lf nsuanc and lf annuy conacs. Th nnovav conbuon of ou wo s o aly hs hodology o a nw oduc l a aabl Annuy wh a GMDB oon ndng h odls aang n h lau n od o sudy a oduc wh a ayns lnd o a fund accoun. n h ann of Lysno and Pa 7 w ado a dfnon of sulus as h dffnc bwn h oscv gan and oscv loss: f w f a valuaon da h accuulad valu o of h nsuanc cash flows ha occud bwn s and sns h oscv gan and h sn valu a of h cash flows ha occu af s h oscv loss. W odfy h odl oosd by Lysno and Pa 7 n od o cau h uncany of a dah bnf lnd o a fund accoun. Fuh w do no aoa h u obably funcon of sulus by s lng dsbuon as n Lysno and Pa whch as no accoun h nvsn s bu as h cash flows as gvn and qual o h cd valu. nsad n od o lo h longvy s w sula h ac of boh h fnancal and oaly facos on h oscv gans and oscv losss. W ado h sa fnancal assuons as n h Blac and Schols fawo. Th oaly hyohss s basd on h sochasc odl suggsd by Co and Ln 5 and dvlod by Balloa soso and Haban 6. Th a s oganzd as follows: n scon w dscb h odl; n scon 3 w dfn h sulus as h dffnc bwn h oscv gan and oscv loss and dv h fs wo ons of s dsbuon; n scon 4 w dvlo h fnancal odl. Nucal suls a shown n scon 5 und a dnsc aoach. n scon 6 w dvlo h sulaons and consuc h sulus dsbuon followng a sochasc aoach and n acula w dnfy h coonns lang scvly o ns fund and oaly ss. Concludng as a offd n scon 7.. Th odl n hs wo w consd a ofolo of dncal aabl Annus wh a GMDB oon whch a ssud o a gou of olcyholds who a agd wh h sa s chaacscs and whos suvval obably dsbuon a ndndn and dncal; h fnal ag s n. Th oduc s coosd of an annuy wh annual ayn R and a GMDB oon; h s a sngl u ad a and nvsd n a fund. L b h valu of h accoun a whch s lnd o
a un fund. Followng h sandad assuons n h lau w odl h voluon of h accoun valu as: d μ η d + σ dw wh W s a sandad Bownan oon und h al obably sac η s h df a δ s h chag ad fo h GMDB oon. Th s nual ocss fo s: d η d + σ dw Q wh s h s f a and W Q asu Q. s a Bownan oon und a nw Gsanov ansfod Th ayoff of h GMDB oon a h s: g [ ] wh s h sochasc of dah and g s h guaand a. Th u s calculad accodng o h quvalnc ncl: G Ma o n 3 P 4 Ran + D wh a n s h acuaal valu of an annuy s h chncal a usd o c h annuy and D s h valu of h GMDB oon a s Haban and Pscoo 8. And A ovds dals on how o dscb h GMDB ayoff and calcula s cd valu. As l un lnd conacs can b sucud n dffn ways: boh of h consun lvng and dah bnfs o us on of h can b lnd o a fund accoun. n ou cas only h dah bnf s nvsd n a fund and so h u can b dally dcoosd no a slng a and a un a.: ' '' P P + P 5 wh P s h slng a lang o h annuy and P s h un a lang o h GMDB oon and whch s nvsd n a fund. L b a valuaon da a whch w sa h sulus lnd o hs conac. L RC b h n cash flow a fo ; s calld oscv cash nflow a. s gvn by: 3
4 { } { } { } [ ] { } { } { } { } { } { } G R P G R P G R P RC δ α δ α δ α 6 wh α f olcyhold s alv a ohws ; δ f olcyhold s ds n ya ohws α s h nub of ol fo h nal gou of olcyhold who suvv o and δ s h nub of dahs n ya. L b h sz of h ofolo a ; fo < n- w hav: { } { } q BN BN + + α δ α α W consd snc w sudy all cash flows as vwd fo. W hav fo <: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] q Cov q Cov q Cov q q Cov Cov q q a a q δ α α δ α δ δ δ α α δ α δ α Calculaon of h cash flow ons s saghfowad. Und h asonabl assuon of ndndnc bwn G and δ o α w hav: [ ] { } [ ] { } [ ] [ ] { } G R P RC δ α 7
g wh [ G ] [ Ma ] o can b calculad by sulang h voluon of h fund valu and h cosondng GMDB valu. Moov w can calcula h vaanc of h oscv cash flow: a [ RC ] R a[ ] { } + a[ G δ ] { } + RCov[ α G δ ] { } and h covaanc of h oscv cash flows: α 8 Cov [ RC RC ] [ α α ] + Cov[ G δ G δ ] + RCov[ α G δ ] + RCov[ α G δ ] R Cov 9 Now w f ou anon on h od af. L PC b h n cash flow lus h valu of h shas nvsd n h fund ha occus uns af fo n- wh n s h fnal ag undlyng h lf abl; hs s calld h oscv cash ouflow a. s gvn by: PC R Rα + [ Rα ] + { } + G+ δ + { } α { } + + + + { } G δ { } + G+ δ + { } W can dv foula fo h ons of h cash flow n h sa ann as bfo. a [ PC ] R[ + ] { } + [ G+ ] [ δ + ] { } α [ PC ] R a[ + ] { } + a[ G+ δ + ] { } + RCov[ α + G+ δ + ] { } α Cov [ PC PC ] R Cov[ α α ] + Cov[ G δ G δ ] + RCov [ α G δ ] + RCov[ α G δ ] + + + + + + + + + + + + + 3 N w noduc wo ando vaabls h oscv gan and h oscv loss whch wll b usd o dfn h sulus. 3. Roscv gan oscv loss and sulus Th oscv gan a s h dffnc bwn h accuulad valu o of as us collcd and bnfs ad. can b ssd n s of RC as follows: RG RC 4 wh s dnos h foc of ns accuulaon funcon bwn s s and f s and h foc of ns acualzaon funcon f s n- ; s gvn by: 5
s+ λ f s < s f s + λ f s > s and λ s h foc of ns n od -]. s asonabl o assu ndndnc bwn h fund valu and ns a. Thus w oban: [ RG ] [ RC ] [ ] 5 a [ RG ] [ RG ] [ RG ] [ ] + [ RC ] [ ] [ RC ] [ ] RC + { Cov[ RC RC ] + [ RC ] [ RC ]} [ ] [ RC ] [ ] 6 Th oscv loss a s h dffnc bwn h dscound valus o of fuu bnfs o b ad and us o b collcd alhough n hs cas h a no fuu us snc h conac has a sngl u a. Th oscv loss can b ssd n s of PC as follows: PL n + PC 7 Th ons of PL can b calculad n a sla way o h ons of RG. A hs on of h analyss w dfn h n sochasc sulus as h dffnc bwn h oscv gan and h oscv loss: S RG PL n 8 wh s h gnc cash flow ouflow o nflow a. Thans o ou vous suls w can calcula h cd valu and vaanc of sulus olcy: n [ S ] [ RG / ] [ PL / ] [ ] [ ] / 9 6
a [ S / ] n a n a n n [ ] + Cov n And B w dvlo h abov foula. 4. Fnancal hyohss n accodanc wh h Blac & Schols fawo w odl h voluon of h un fund as n : d μ η d + σ dw Snc W s a sandad Bownan oon follows ha: a Cov [ ] { μ η } [ ] { μ η + σ } g g [ G ] [ Ma ] Ma [ ] [ G ] a[ ] [ G G ] Cov[ ] Moov w odl h foc of ns by a condonal auogssv ocss AR gvn h foc of ns a zo. Ths odl s consdd by Bllhous and Pan 98 and Macau and Galladz 999. L δ h foc of ns n h od - ]: Wh { } [ δ ] γε δ δ ϕ δ + ε s a squnc of ndndn and dncally dsbud sandad noal vaabls and δ s h long an of h ocss. W assu φ < o nsu h ocss s saonay n covaanc. Th on of h accuulaon funcon a dvd n Cans and Pa 997. W assu h ndndnc bwn h fund and h focs of ns. 5.Nucal Rsuls: h fs wo ons of h Suls n hs scon w aly h odl and show nucal suls fo a ofolo of dncal aabl Annus wh a GMDB oon. W consd a gou of olcyholds agd 5 wh h sa s chaacscs whos suvval obably dsbuons a ndndn and dncally. Th oaly abl usd n ou calculaon s h SM basd on h alan al oulaon wh h au ag n. W s R and.4; und hs hyohss h u calculad 7
accodng o h quvalnc ncl s qual o 7; wh h slng a 6 and h un a '' P D s qual o. W s ' P a5 s qual o Δ.6 δ.5 Φ.8 Γ. σ.3 µ-η.6 G.4 W hav cad ou sulaons. W valua h fs ons of h sulus a dffn das and show h suls n Fgus : Fgu : cd alu and vaanc of h Sulus olcy W no ha as h valuaon da ncass h sandad dvaon of h sulus ncass. n od o undsand hs w hav o consd ha h sandad dvaon of h sulus s affcd by h uncany abou h cash flows followng h u and by h vaanc of h ns a. Whn ncass w hav o accuula a ga nub of oscv cash flows fo a long and dscoun a sall nub of oscv cash flows fo a sho od. Consqunly h vaanc of h caalzd cash gans ncass and ha of h dscound losss dcass. Nucal nvsgaon shows ha h fs ffc vals ov h scond on. 6. Dsbuon Funcon of Sulus: a sochasc aoach n h vous scon w hav sudd h fs wo ons of h sochasc sulus fo a hoognous ofolo of aabl Annuy conacs wh GMDB oons. Alhough h analyss of ons s usful s only h fs s owads long h ando bhavou of h sulus. W no ha h sandad dvaon as a s asu s naoa whn dalng wh asyc dsbuons and s ncssay o sudy h whol obably funcon of sulus. 8
Lysno and Pa 7 suggs a cusv hod o consuc hs dsbuon; h coly of h oduc w a consdng as ncssay a sulaon aoach. On of h obcvs of hs sudy s o assss h obably of nsolvncy.. h obably ha wll fall blow zo. n od o achv hs uos w sula h voluon of sulus und a oaly and fnancal sochasc odl. Unl h aoach of Lysno and Pa 7 w do no aoa h u obably funcon of sulus by s lng dsbuon whch as no accoun h nvsn s bu as cash flows as gvn and qual o h cd valu. nsad n od o a accoun also h longvy s w sula h ac of boh fnancal and oaly facos on oscv gans and oscv losss. Th fnancal assuons a h sa as dscbd vously. Also w nd a oaly assuon n od o avod undsaon o ovsaon of h sulus. n hs sc w consd h sochasc odl suggsd by Co and Ln 5 and dvlod by Balloa soso and Haban 6 and whch s dscbd blow. Ou calculaon s basd on h acuaal abl usd bfo; howv w sa h cd valu of h nub of suvvos a ag + [ l+ ] n a sochasc fawo. s ossbl o ov ha l+ s aoaly dsbud as a noal ando vaabl wh an qual o l and vaanc qual o l -. Howv h cn acuaal lau hghlghs h fac ha h cal daa show ubaons n suvval obabls du o ando shocs. Accodngly w sula h suvval obabls adusd fo shocs as follows: ε ' + + wh ε s h shoc n h cd obably a. Balloa soso and Haban 6 assu ha ε follows a ba dsbuon wh aa a and b and h sgn of h shocs dnds on h ando nub sulad fo h unfo dsbuon U. n acula w s: ε f < c - ε f c wh c s a aa whch dnds on h caon of fuu oaly nd. Th oanc of assgnng a ando sgn o ε s ha n hs way h odl caus no only h long od vaaons n oaly as bu also h sho od flucuaons du o conal ccusancs. W cay ou sulaons und dffn fnancal and oaly hyohss. Th suls a shown n h scons concnng ns a s fund s and oaly s. a. ns Ra Rs: Nucal Rsuls W consuc h dsbuon of h sulus olcy a dffn valuaon das und h hyohss of h vous scon. Rsuls a shown n Fgu and suazd n Tabl : 9
Fgu : Bolo of sulus olcy a dffn valuaon das S S S S3 Mn. : 8.36 Mn. : 4.94 s Qu.:.65 s Qu.: 4.637 Mdan : 4.698 Mdan : 8.586 Man : 4.767 Man : 9.8 3d Qu.: 6.88 3d Qu.: 3.63 Ma. :.88 Ma. : 5. Mn. : 7.6 s Qu.:.7 Mdan :.833 Man :.7 3d Qu.: 3.83 Ma. : 7.87 S4 S5 S6 Mn. : 83.98 s Qu.: 6.6 Mdan : 5.89 Man : 58.5 3d Qu.: 8.9 Ma. :53.9 Mn. : 36.4 s Qu.: 4.65 Mdan : 8.9 Man : 3.8 3d Qu.: 44.9 Ma. :89.38 Mn. : 88.8 s Qu.: 47.6 Mdan : 9.5 Man : 7.7 3d Qu.: 5.8 Ma. :57.44 Mn. : 9.94 s Qu.: 8.67 Mdan : 5.59 Man : 7.9 3d Qu.: 4.47 Ma. :37. Tabl : Suay ndcs of h sulus dsbuon olcy a dffn valuaon das W wan o vfy wha hans und a dffn scnao fo h foc of ns. n acula w nvsga h ffc of a ducon n h long an of h foc of ns. W coa h dsbuon of h Sulus olcy a h valuaon da und h followng scnaos: Scnao Scnao δ.6.4 δ.5.5 φ.8.8 Γ..
Th dsbuons of sulus a shown and coad n h n fgu and abl: Fgu 3: Th Sulus olcy a und dffn scnaos fo h focs of ns As h long a of un of h asss n whch h nsu nvss u dcass h cuulav dsbuon of h sulus ovs o h lf and consqunly h obably of nsolvncy ncass. Scnao Scnao PobS/ 5.48% 64.35% Quanl %.586 7.447 5%.938 4.948 %.694 3.947 9% 4.654.737 95% 5.3.366 99% 5.988 3.4636 Tabl : Pobably of nsolvncy and sgnfcan cnl of h sulus dsbuon Ths coason hghlghs h oanc of a coc nvsn sagy n od o avod h nsolvncy. n hs cas h nsu has o nvs h collcd us no asss wh a long an of h a of un qual o.6 n od o hav a osv sulus snc h fs ya and no as oh ony o shaholds. W no ha f h nsu nvss h us no asss ha n an yld a un qual o h guaand a on GMDB h obably of nsolvncy a s 64.35%.
b. Fund Rs: Nucal Rsuls n hs scon w sudy h ffc of shfs n h dsbuons of fund valu. As w wsh o oduc a snsvy analyss w f h hyohss concnng h ns a dsbuon accodng h aas usd n scon 5 and chang hos concnng h fund. n acula w coa h sulus dsbuons und h followng fou scnaos: Scnao µ-η.6.6.6.5 σ.3.4.3.3 g.4.4.5.4 Th suls a suazd n h n abl: Scnao Scnao Scnao Scnao Mn. : 9.94 Mn. : 4.943 Mn. :.354 Mn. : 4.547 s Qu. : 8.67 Mdan : 5.59 Man : 7.9 3d Qu. : 4.47 s Qu.: 8.3 Mdan : 5.434 Man : 7. 3d Qu.: 4.5 s Qu.: 7.6 Mdan : 5.9 Man : 6.64 3d Qu.: 3.849 s Qu.: 9.485 Mdan : 6.9 Man : 8.53 3d Qu.: 5.76 Ma. :37. Ma.: 36.487 Ma. : 36.568 Ma. : 38.754 Tabl 3 : Suay ndcs of h sulus dsbuon olcy und dffn scnaos As cd as h volaly of h fund ncass h vaanc of h sulus ncass and as h guaand a ncass h an of h sulus dsbuon dcass. Moov as h df of h fund ocss dcass h dsbuon of h sulus ovs on h gh as shown n Fgu 4 bcaus h aouns of dah bnfs ad dcas. Fgu 4: h dsbuon of sulus olcy und wo dffn hyohss fo h fund ocss.
c. Moaly Rs: Nucal Rsuls n hs scon w sudy h ffc of shfs n h aas of sochasc oaly odl. n h sa ann as h vous scons w a o oduc a snsvy analyss and so w f h hyohss concnng h ns a and h fund voluon as n scon 5 and chang h oaly abl. n acula w us h oaly odl dscbd and s a.5 and b4.5. w valua h sulus olcy a 3. W consd h cass fo h valu of c: c.5. n h fs cas h wll b ovns n lf cancy a vy da; n oh wods all of h shocs a cd o b osv. Convsly n h scond cas fuh ovns of an alady hgh cancy of lf a ossbl and all shocs a cd o b ngav. Und h hyohss c h ouflows lnd o annus ncas and und h hyohss c hy dcas. n Fgu 6 w show h cdf of h nub of dahs ha occu n ach ya und h hyohss c c5 and c. Und h hyohss c h df of h nub of dahs s anslad o h lf and has a fa lf al and a lss fa gh al han h oh wo cuvs. On h conay f c h df of h nub of dahs s anslad o h gh and has a lss fa lf al and a fa gh al han h oh wo cuvs Fgu 5: Pdf of h nub of dahs und h hyohss c c5 and c. As c ncass h ouflows lnd o h annuy ncas; oov h ayns lad o h GMDB oon ncas oo bcaus hy a olld ov and hy a lnd o a fund valu ha ncass wh. Consqunly h cuulav dsbuon of h sulus ovs on h lf and h obably of nsolvncy ncass. Ths suls a llusad n Fgu 6 and Tabl 4. Fgu 5 shows an unusual fau: ach cuv has wo ods. Th odal fau can b found also n a cn Blgan als abl s Pacco al. 8. 3
Fgu 6: h dsubuon of h Sulus olcy und dffn oaly hyohss c c.5 c PobS/ 36% 57% 85% Quanl %.4539 5.57 8.8 5%.53.77 3.356 % 5.478.8347.379 9% 36.749 34.9 3.7354 95% 43.347 4.648 37.356 99% 57.683 54.8743 5.649 Tabl 4: Pobably of nsolvncy and sgnfcan cnl of h sulus dsbuon und dffn oaly hyohss. 7. Conclusons Th sulus s an oan ndcao of an nsuanc coany s fnancal oson and h ss a consdabl acuaal lau on h oc s fo al Coola al. 3 Hodas al. 5 Lysno and Pa 7 Macau and Galladz 999 and Pa 996. Th conbuon of hs a has bn o analyz h bhavou of h nsuanc sulus fo a ofolo of aabl Annus wh GMDB oons. n od o achv hs uos w hav sulad h voluon of h sulus und a oaly and fnancal sochasc odl. Th suls a snd on h bass of a slfd odl; hus h cash flows as fo h bnf and u ayn sas and a assud o b dndn only on h oaly nc of h ofolo so ha 4
nss and las oons a no consdd; and fuh ando lfs and as of un a assud o b ndndn. Ths a laons of ou fawo and n subsqun wo h odl wll b ovd by noducng a o col and alsc odl sucu and assuons. Ds h abov laons w blv ha h a s usful n nhancng an nsu s undsandng of h sochasc bhavou undlyng a aabl Annuy oduc wh a GMDB oon and ha ovds h fs sudy of h sulus n sc of hs cnly dvlod nsuanc oduc. ndd u o hs h lau has offd only cng odls fo GMDB bu has no sudd h voluon of cash flows. W d hs consdaon s oan n h scv of h lqudy and nsolvncy s anagn. W hav consdd boh fnancal and oaly s and hav oulnd a cohnsv dscon of h nacon of dffn s facos on h GMDB valu.as gnal ul f a dah bnf s addd o an annuy h s a so of oaly naual hdgng ffc.. h ac of longvy s s ducd bcaus h annuy s ad fo a long od bu h acual valu of h dah bnf dcass. Th GMDB oons can sn an con o hs ul; n scon 6.3 w hav shown ha as h sad lf nds h ouflows lnd o h annuy ncas and a h sa h ayns lad o h GMDB oon ncas oo bcaus hy a olld ov and hy a lnd o a fund valu ha ncass wh. Thfo und h hyohss of a gowng fund h ffc of naual hdgng s nullfd. Hnc s no suffcn o sudy h ac of ach s faco on h GMDB valu bu s ncssay o an h nacon. n h a nucal als show a sgnfcan ac of h ns fund and oaly ss on h sulus dsbuon nsouch as h nsolvncy obably ncass consdably n any cass. Wh gad o hs on an advanag of h odl usd s ha allows an an assssn of h nsu s solvncy houghou h duaon of conac. Consqunly a chang o h dsgn of h oduc can b ad and n acula h u can b odfd accodng o h focass of oaly obabls ns a and fund voluon. Moov h odl nabls us o dn h u ha lads o a qud obably of nsolvncy and so can b usd fo an valuaon of h adquacy of solvncy whch s conssn wh cn gulaoy changs. Rfncs Balloa L. soso G. Haban S. 6. Modllng h Fa alu of Annus Conacs: Th ac of ns Ra Rs and Moaly Rs. Acuaal Rsach Pa No.76 Cass Busnss School. Balloa L. Haban S. 6 Th fa valuaon obl of guaand annuy oons: Th sochasc oaly nvonn cas nsuanc: Mahacs and conocs 38 95-4 Bau D. Klng A. Russ J. 6. A Unvsal Pcng Fawo fo Guaand Mnu Bnfs n aabl Annus. Wong Pa ULM Unvsy 5
Bllhous D.R. Pan H.H. 98. Sochasc odlng of ns as wh alcaons o lf conngncs. Jounal of Rs and nsuanc 484 68-637 Cans A.J.G. Pa G. 997. Sochasc nson fund odlng. nsuanc: Mahacs and conocs 43-79 Colan T.F. L Y. Paon M. 6. Hdgng guaans n vaabl annus und boh quy and ns a ss. nsuanc: Mahacs and conocs 38 5-8 Coola M. D Lonzo. Sbllo M. 3. Sochasc analyss n lf offc anagn: alcaons o lag annuy ofolos. Ald Sochasc Modls n Busnss and ndusy 9 3-4. Co S.H. Ln Y. 5. Scuzaon of oaly s n lf annus. Th Jounal of Rs and nsuanc 7 7-5 Dahl M. 4. Sochasc oaly n lf nsuanc: a svs and oaly- lnd nsuanc conacs. nsuanc: Mahacs and conocs 35 3-36 Haban S. Pscoo G. 8. Moaly Rs and h aluaon of Annus wh Guaand Mnu Dah Bnf Oons: Alcaon o h alan Poulaon. Acuaal Rsach Pa No. 87 Cass Busnss School. Hodas T. Dawcz G. Goovas M. 5. Aoaons fo lf annuy conacs n a sochasc fnancal nvonn. nsuanc: Mahacs and conocs 37 39-69. Ldl M.C. Coy D.P. Fnlsn G.S. Rch A.J. Su K. Wlson D.C. 8. aabl annus. Psnd o h nsu of Acuas. Lysno N. Pa G. 7. Sochasc Analyss of Lf nsuanc Sulus. Psnd o h 7 AFR Colloquu. Macau. Galladz P. 999. On lf nsuanc svs n a sochasc oaly and ns as nvonn. nsuanc: Mahacs and conocs 5 6-8. Mlvsy M. Posn S... Th Tanc Oon: aluaon of h Guaand Mnu Dah Bnf n aabl Annus and Muual Funds. Th Jounal of Rs and nsuanc 68 9-6. Mlvsy M. Salsbuy T.S. Th Ral Oon o Las and h aluaon of Dah- Pocd nvsns. Wong Pa Yo Unvsy and h Flds nsu Toono. Mlvsy M.A. Poslow S.D. Moaly dvavs and h oon o annus. nsuanc: Mahacs and conocs 9 99-38 Pa G. 994. Lng dsbuon of h sn valu of a ofolo. ASTN Bulln 4 47-6. 6
Pa G. 996. A ofolo of ndown olcs and s lng dsbuon. ASTN Bulln 6 5-33. Pacco. Dnu M. Haban S. Olv A. 8. Modllng Longvy Dynacs fo Pnsons and Annuy Busnss. Ofod Unvsy Pss Ofod. 7
APPNDX A L T b h fuu lf ando vaabl ssd n connuous F b s cdf and f b s df; hfo fo an ndvdual agd h obably of dah bfo s F P T - - ς + s ds wh ζ dnos h foc of oaly. L b h accoun valu a lnd o fund valu. Followng sandad assuons n h lau w odl h voluon of h accoun as: wh W s a sandad Bownan oon µ s h df a η s h chag ad fo h GMDB oon. Th s nual ocss fo s: d μ η d + σ dw d η d + σ dw Q wh s h s f a and W Q asu Q. Th soluon of h SD s: s a Bownan oon und a nw Gsanov ansfod η σ / + σ Q W. Now w dscb h GMDB ayoff. A h ando da of dah h bnfcay wll cv D g g g a a + wh g s h guaand a. Th valu of h GMDB oon a s h su of h fund valu and a u oon whos s c s h nal valu wh an undlyng ass dscound by h guaand gowh a g. Snc h auy s sochasc and and a ndndn h sn valu of GMDB s gvn by h caons und and : D Q { { D } f w fd h da w hav a an uoan Oon whos acual valu can b calculad wh Blac and Schols foula. Thfo h vous foula can b nd as a dcooson of h acual valu of GMDB n h acual valu of a connuous squnc of uoan u oon. Subsung h sson of D 8
9 { } { } D g g Q + a W can obsv ha { } Q η snc w hav assud ha s a goc Bownan oon wh df qual o -δ and so s cd valu s: { } Q η Consqunly: { } { } { } { } D g g Q g g Q + + η a a W obsv ha fo a fd da T { } [ ] [ ] T T T T T T gt T g Q T BS d N d N η η η η σ η + + + ~ a ~ wh σ σ η T d / ~ + ; T d d σ ; g ~ ; N. s h cuulav obably funcon fo a ando vaabl noally dsbud. f w consd boh h caons: [ ]d BS f D δ ω σ η + ~ n h dsc cas w hav: [ ] BS q D δ ω σ η + +. ~ fo a olcyhold agd a ncon of h conac. Th valu of GMDB s a wghd avag of h valus of ω- uoan u oons wh h wghs a h osond obably of dah n.. h obably of suvval unl and dah bwn and +.
APPNDX B Th vaanc of h cash flows boh oscv o oscv s gvn by h followng foula: wh a a[ G δ ] [ G ] [ δ ] [ G ] [ ] R a[ α ] + a[ G δ ] + RCov[ α G δ ] [ δ ] ; Cov[ α G δ ] [ G ] Cov[ α δ ] n fac Cov [ α G δ ] [ α G δ ] [ α ] [ G δ ] [ G ] [ α δ ] [ α ] [ G ] [ δ ] [ ]{ [ α δ ] [ α ] [ δ ]} [ G ] Cov[ α δ ] G ; Th covaanc of h cash flows s: wh R Cov Cov [ ] [ α α ] + Cov[ G δ G δ ] + RCov[ α G δ ] + RCov[ α G δ ] Cov[ Gδ G δ ] [ Gδ G δ ] [ Gδ ] [ G δ ] [ G G ] [ δ δ ] [ G ] [ δ ] [ G ] [ δ ] [ G ] [ G ] [ δ δ ] [ G ] [ δ ] [ G ] [ δ ] [ G ] [ G ]{ [ δ δ ] [ δ ] [ δ ]} [ G ] G ] Cov[ δ δ ] Cov[ α G δ ] [ G ] Cov[ α δ ] Cov[ α G δ ] [ G ] Cov[ α δ ] Fnally h vaanc of h sulus can b calculad:
[ ] [ ] + n n n n Cov a a S a / wh [ ] a [ ] [ ] { } [ ] [ ] [ ] [ ] [ ] Cov [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] Cov + +
FACULTY OF ACTUARAL SCNC AND NSURANC Acuaal Rsach Pas snc Ro Nub Da Publcaon Tl Auho 6. Januay 5 Moaly Rducon Facos ncooang Coho ffcs. SBN 96584 7 6. Fbuay 5 Th Managn of D-Cuulaon Rss n a Dfnd Conbuon nvonn. SBN 965 85 5. 6. May 5 Th ASB nsuanc Poc fo Lf nsuanc Conacs: a on Rsvng Mhods and Solvncy Rquns. SBN -965 86 3. 63. Sb 5 Asyoc and Nucal Analyss of h Oal nvsn Sagy fo an nsu. SBN -965-88-X 64. Ocob 5. Modllng h Jon Dsbuon of Cong Rss Suvval Ts usng Coula Funcons. SBN -965-89-8 65. Novb 5. css of Loss Rnsuanc Und Jon Suvval Oaly. SBN-965-9- Ahu. Rnshaw Svn Haban Russll J. Gad Svn Haban lna gna Laua Balloa Goga soso Svn Haban Paul s Svn Haban lad Kashv Dna S Dova Svn Haban lad K. Kashv Dna S. Dova 66. Novb 5. L-Ca Gos Rs-Nual. An Alcaon o h alan Annuy Ma. SBN -965-9-X 67. Novb 5 L-Ca Moaly Focasng: Alcaon o h alan Poulaon. SBN -965-93-6 68. Fbuay 6 Th Pobaonay Pod as a Scnng Dvc: Cov Mas. SBN -965-95- 69. Fbuay 6 Tys of Dndnc and T-dndn Assocaon bwn Two Lfs n Sngl Paa Coula Modls. SBN -965-96- 7. Al 6 Modllng Sochasc Bvaa Moaly SBN -965-97-9 7. Fbuay 6 Oal Sags fo Pcng Gnal nsuanc. SBN 965-98-7 7. Fbuay 6 Dynac Pcng of Gnal nsuanc n a Cov Ma. SBN-965-99-5 73. Fbuay 6 Pcng Gnal nsuanc wh Consans. SBN -9575--8 74. May 6 nvsgang h Ma Ponal fo Cusosd Long T Ca nsuanc Poducs. SBN -9575--6 nco Bffs Mchl Dnu Svn Haban Maa Russolllo Jaa Suw Man Kalsson Jaa Suw lsa Lucano Jaa Suw lna gna. Paul s Sv Haban n Savoull Paul s Paul s Man Kalsson Ls Mayhw Bn Rcayzn
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