Actuarial Research Paper No. 193

Transcription

1 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 Cass Busnss School 6 Bunhll Row London CY 8TZ Tl

2 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.

3 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: [email protected] Dano d Maaca Sasca Facolà d conoa Unvsà dgl Sud d Naol Fdco va Chna Colsso Mon S.Anglo 86 Naol- aly. nn addss: [email protected]

4 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

5 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

6 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

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 ] 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

8 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

9 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

10 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

11 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

12 Fgu : Bolo of sulus olcy a dffn valuaon das S S S S3 Mn. : 8.36 Mn. : 4.94 s Qu.:.65 s Qu.: Mdan : Mdan : Man : Man : 9.8 3d Qu.: d 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. : s Qu.: 6.6 Mdan : 5.89 Man : d 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..

13 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 % % % % % % 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%.

14 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 σ g Th suls a suazd n h n abl: Scnao Scnao Scnao Scnao Mn. : 9.94 Mn. : Mn. :.354 Mn. : s Qu. : 8.67 Mdan : 5.59 Man : 7.9 3d Qu. : 4.47 s Qu.: 8.3 Mdan : Man : 7. 3d Qu.: 4.5 s Qu.: 7.6 Mdan : 5.9 Man : d Qu.: s Qu.: Mdan : 6.9 Man : d Qu.: 5.76 Ma. :37. Ma.: Ma. : Ma. : 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.

15 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

16 Fgu 6: h dsubuon of h Sulus olcy und dffn oaly hyohss c c.5 c PobS/ 36% 57% 85% Quanl % % % % % % 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

17 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 Bau D. Klng A. Russ J. 6. A Unvsal Pcng Fawo fo Guaand Mnu Bnfs n aabl Annus. Wong Pa ULM Unvsy 5

18 Bllhous D.R. Pan H.H. 98. Sochasc odlng of ns as wh alcaons o lf conngncs. Jounal of Rs and nsuanc Cans A.J.G. Pa G Sochasc nson fund odlng. nsuanc: Mahacs and conocs Colan T.F. L Y. Paon M. 6. Hdgng guaans n vaabl annus und boh quy and ns a ss. nsuanc: Mahacs and conocs 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 Co S.H. Ln Y. 5. Scuzaon of oaly s n lf annus. Th Jounal of Rs and nsuanc Dahl M. 4. Sochasc oaly n lf nsuanc: a svs and oaly- lnd nsuanc conacs. nsuanc: Mahacs and conocs 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 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 On lf nsuanc svs n a sochasc oaly and ns as nvonn. nsuanc: Mahacs and conocs 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 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 Pa G Lng dsbuon of h sn valu of a ofolo. ASTN Bulln

19 Pa G A ofolo of ndown olcs and s lng dsbuon. ASTN Bulln Pacco. Dnu M. Haban S. Olv A. 8. Modllng Longvy Dynacs fo Pnsons and Annuy Busnss. Ofod Unvsy Pss Ofod. 7

20 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

21 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 +.

22 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:

23 [ ] [ ] + n n n n Cov a a S a / wh [ ] a [ ] [ ] { } [ ] [ ] [ ] [ ] [ ] Cov [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] Cov + +

24 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 Fbuay 5 Th Managn of D-Cuulaon Rss n a Dfnd Conbuon nvonn. SBN May 5 Th ASB nsuanc Poc fo Lf nsuanc Conacs: a on Rsvng Mhods and Solvncy Rquns. SBN Sb 5 Asyoc and Nucal Analyss of h Oal nvsn Sagy fo an nsu. SBN X 64. Ocob 5. Modllng h Jon Dsbuon of Cong Rss Suvval Ts usng Coula Funcons. SBN Novb 5. css of Loss Rnsuanc Und Jon Suvval Oaly. SBN 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 X 67. Novb 5 L-Ca Moaly Focasng: Alcaon o h alan Poulaon. SBN Fbuay 6 Th Pobaonay Pod as a Scnng Dvc: Cov Mas. SBN Fbuay 6 Tys of Dndnc and T-dndn Assocaon bwn Two Lfs n Sngl Paa Coula Modls. SBN Al 6 Modllng Sochasc Bvaa Moaly SBN Fbuay 6 Oal Sags fo Pcng Gnal nsuanc. SBN Fbuay 6 Dynac Pcng of Gnal nsuanc n a Cov Ma. SBN Fbuay 6 Pcng Gnal nsuanc wh Consans. SBN May 6 nvsgang h Ma Ponal fo Cusosd Long T Ca nsuanc Poducs. SBN 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

25 Ro Nub Da Publcaon Tl Auho 75. Dcb 6 Pcng and Caal Rquns fo Wh Pof Conacs: Modllng Consdaons. SBN Dcb 6 Modllng h Fa alu of Annus Conacs: Th ac of ns Ra Rs and Moaly Rs. SBN Dcb 6 Usng Quung Thoy o Analys Colon Ts n Accdn and gncy Dans n h Lgh of h Govnn 4-hou Tag. SBN Al 7 n Scnss and n Halh? Dynacs of Halh and Cohabaon n h Und Kngdo. SBN Al 7 GD Sln saon of Mulvaa Achdan Coulas. SBN May 7 An Analyss of Dsably-lnd Annus. SBN May 7 On Sulaon-basd Aoachs o Rs Masun n Moaly wh Scfc Rfnc o Posson l-ca Modllng. SBN July 7 Hgh Dnsonal Modllng and Sulaon wh Asyc Noal Mus. SBN Augus 7 noal Dynac Ass Allocaon fo Dfnd Conbuon Pnson Schs. SBN Ocob 7 To sl o no o sl: Caal allocaon wh conv s asus. SBN Al 8 On So Mu Dsbuon and Th Bhavou. SBN Ocob 8 Oal Fundng and nvsn Sags n Dfnd Conbuon Pnson Plans und sn-zn Uly. SBN May 8 Moaly Rs and h aluaon of Annus wh Guaand Mnu Dah Bnf Oons: Alcaon o h alan Poulaon. SBN Januay 9 Th Ma Ponal fo Pvaly Fnancd Long T Ca Poducs n h UK. SBN Jun 9 Whh Huan Suvval and Longvy o h Sha of hngs o Co. SBN Ocob 9 lc: A Collcon of R Funcons fo Fng a Class of L Ca Moaly Modls usng av fng Algohs* SBN Ocob 9 Dcooson of Dsas and Dsably Lf cancs n ngland SBN Laua Balloa Laua Balloa Goga soso Svn Haban Ls Mayhw Davd Sh Man Kalsson Ls Mayhw Bn Rcayzn Dna Dova lad Kashv Sdon Pnv Bn Rcayzn Ahu Rnshaw Svn Haban Andas Tsanaas Andw Sh Davd Bla Douglas Wgh Yung Zhang Andas Tsanaas lad Kashv Ja Hoon Jho Davd Bla Douglas Wgh Yung Zhang Svn Haban Gablla Pscoo Lsl Mayhw Lsl Mayhw Davd Sh Zolan Bu Svn Haban Donca Rasulo Lsl Mayhw Bn Rcayzn

26 Ro Nub Da Publcaon Tl Auho 9. Ocob 9 loaon of a Novl Boosa Tchnqu fo sang h Dsbuon of Ousandng Clas Rsvs n Gnal nsuanc. SBN Januay Sulus Analyss fo aabl Annus wh a GMDB Oon. SBN Rob Cowll Svn Haban Gablla Pscoo Sascal Rsach Pas. Dcb 995. So Rsuls on h Dvavs of Ma Funcons. SBN Mach 996 Cohn Ca fo Oal nal Dsgn. SBN Mach 996 Mau noy Salng and Oal Baysan nal Dsgn. SBN May 996 A No on D-oal Dsgns fo a Logsc Rgsson Modl. SBN Augus 996 Fs-od Oal Dsgns fo Non Lna Modls. SBN Sb 996 A Busnss Pocss Aoach o Mannanc: Masun Dcson and Conol. SBN Sb 996. Mons and Gnang Funcons fo h Absoon Dsbuon and s Ngav Bnoal Analogu. SBN P. Sbasan A.P. Dawd P. Sbasan P. Sbasan H.P. Wynn P. Sbasan R. S P. Sbasan R. S Man J. Nwby Man J. Nwby 8. Novb 996. Mu Rducon va Pdcv Scos. SBN Rob G. Cowll. 9. Mach 997. Robus Paa Lanng n Baysan Nwos wh Mssng Daa. SBN P.Sbasan M. Raon. Mach 997. Gudlns fo Cocv Rlacn Basd on Low Sochasc Sucu Assuons. SBN Mach 997 Aoaons fo h Absoon Dsbuon and s Ngav Bnoal Analogu. SBN 965 M.J. Nwby F.P.A. Cooln Man J. Nwby. Jun 997 Th Us of ognous Knowldg o Lan Baysan Nwos fo ncol Daabass. SBN M. Raon P. Sbasan 3. Jun 997 Lanng Baysan Nwos fo ncol Daabass. SBN 965 M. Raon P.Sbasan 4. Jun 997 Rs Basd Oal Dsgns. SBN P.Sbasan H.P. Wynn 5. Jun 997. Salng whou Rlacn n Juncon Ts. SBN Rob G. Cowll 6. July 997 Oal Ovhaul nvals wh fc nscon and Ra. SBN Ocob 997 Baysan nal Dsgn and Shannon nfoaon. SBN Rchad A. Dagg Man J. Nwby P. Sbasan. H.P. Wynn

27 Ro Nub Da Publcaon Tl Auho 8. Novb 997. A Chaacsaon of Phas Ty Dsbuons. SBN Dcb 997 A Coason of Modls fo Pobably of Dcon POD Cuvs. SBN Fbuay 999. Paa Lanng fo ncol Daa Usng Mau noy : Pncls. SBN Lnda C. Wolsnhol Wolsnhol L.C Rob G. Cowll. Novb 999 Paa Lanng fo ncol Daa Usng Mau noy : Alcaon o Baysan Nwos. SBN Mach FNX : Fonsc dnfcaon by Nwo Syss. SBN 965 6X 3. Mach. Wn Lanng Baysan Nwos fo Daa usng Condonal ndndnc Tss s quvalan o a Scong Mc SBN Augus 4 Auoac Cou Add Goc Dsgn of F- Kno Rgsson Slns. SBN Dcb 4 dnfcaon and Saaon of DNA Mus Usng Pa Aa nfoaon. SBN Novb 5. Th Qus fo a Dono : Pobably Basd Mhods Off Hl. SBN Rob G. Cowll Rob G.Cowll Rob G Cowll lad K Kashv Dna S.Dova Svn Haban Rchad J. all R.G.Cowll S.L.Lauzn J Moa P.F.Mosad T. gland. R.G. Cowll. Bosns Ø. Baan 7. Fbuay 6 dnfcaon and Saaon of DNA Mus Usng Pa Aa nfoaon. Udad son of Rsach Ro Nub 5. SBN R.G.Cowll S.L.Lauzn J Moa 8. Ocob 6 Gocally Dsgnd aabl Kno Rgsson Slns : Asyocs and nfnc. SBN Ocob 6 Gocally Dsgnd aabl Kno Rgsson Slns : aaon Dnshng Oaly of Knos. SBN Novb 8 Schdulng Rnan Jobs on Paalll Machns wh a Ro Sv. SBN lad K Kashv Dna S.Dova Svn Haban Rchad J. all lad K Kashv Dna S.Dova Svn Haban Rchad J. all Konsann Chahlvch Cla Glass Pas can b downloadd fo h://

28 Faculy of Acuaal Scnc and nsuanc Cass Busnss School Coygh Faculy of Acuaal Scnc and nsuanc Cass Busnss School 6 Bunhll Row London CY 8TZ. SBN